Extract from
1. Number Sense and Numeration
It is important
for students to develop the mathematical competence that comes from understanding
numbers, number systems, and their related operations.
Number is a
complex and multifaceted concept. A well-developed understanding of number
includes a grasp not only of counting and numeral recognition but also of a complex
system of more-and-less relationships, part-whole relationships, the role of
special numbers such as five and ten, connections between numbers and real
quantities and measures in the environment, and much more. Experience suggests
that students do not grasp these relationships automatically. It is the
teacher's job to provide students with a broad range of activities that will
help them develop many of these ideas about number.
Helping
students to understand numbers must always include introducing them to
procedures for accurately performing operations with numbers. Key facts and
processes must be mastered. Students also need to develop "number
sense". Number sense includes:
an appreciation of and ability to make quick
order-of-magnitude approximations with emphasis on quick and accurate estimates
in computation and measurement;
the ability to detect arithmetic errors;
knowledge of place value and the effects of
arithmetic operations;
a grasp of ideas about the role of numbers and about
their multiple relationships;
an appreciation of the need for numbers beyond whole
numbers.
Mental Mathematics and Estimation
By the end of
Grade 6, students should have consolidated their understanding of basic computational
facts and be able to use computational strategies to do mental mathematics.
This ability develops over time, supported by regular practice, as students
identify relationships between numbers and learn when and how to use the
various operations effectively. Techniques of mental mathematics should be
introduced along with concepts of place value and the use of pencil-and-paper
calculations. For example, the mental skill of adding numbers ending in zero,
such as 20 + 40 + 70, can be learned by a student who understands place value
and can add 2 + 4 + 7. Instruction in computational estimation should include
not only applications involving whole numbers and decimals, but also those
involving fractions and percent. Early experiences in estimating with percent
and fractions can help students develop number sense.
Pencil-and-Paper Computation
Students (and
adults) require facility with pencil-and-paper computations. It is important
for pencil-and-paper computational procedures to be introduced through the use
of concrete materials. Students should use these materials until they
understand the concepts well enough to move from the manipulative stage to the
semiconcrete medium of pictures and then to the abstract form of numbers on a
page.
Many methods of
pencil-and-paper calculation have been appropriately replaced by operations of
a calculator or computer. For example, long division, operations with long
lists of large numbers, and the calculation of square root can be done more
efficiently using technology. Teachers should also provide students with a
variety of experiences and investigations involving number.
Fractions and Rationals
Concepts and
operations with fractions should be introduced using concrete materials.
Models, tiles, manipulatives, and diagrams should be used to relate fractions
to decimals, to find equivalent fractions, and to explore operations with
fractions and decimals. Fraction symbols build on the understanding developed
in these ways. Mathematics instruction should help students gain conceptual
understanding as well as use fractions and rational numbers effectively and
accurately.
It is
recommended that, initially, simple denominators such as 2, 3, 4, 5, and 10 be
used. As students gain more experience and skill in working with fractions,
denominators such as 6, 8, and 12 can be included. Later, the focus shifts to
using fractions in ratios, rates, and percent. As well, students will extend
their fraction sense to include skill in operations with fractions. It should
be remembered, however, that the use of fractions in real-life situations often
involves estimating (e.g., "My friend lives half a block from here"),
and this skill should be developed along with accurate calculation.
Calculators
The ability to
use calculators intelligently is an integral part of number sense. It should be
noted that the use of calculators does not do away with the necessity for
students to master the fundamental mathematical operations. Students should use
calculators in their schoolwork, just as adults use calculators for many
purposes in the course of their daily lives. More importantly, students must
learn when it is appropriate to use a calculator and when it is not. They must
learn from experience with calculators when to estimate and when to seek an exact
answer, and how to estimate answers to verify the plausibility of calculator
results. Calculators allow teachers to engage students in meaningful
mathematical investigations, such as solving science problems with large
numbers, before their skill with pencil-and-paper computation is equal to the
task. Proper calculator use stimulates the growth of number sense in students.
Computers
The computer is
an important tool used by mathematicians to perform a wide variety of tasks; the
ability to use computers effectively and appropriately is central to students'
development of mathematical competence.
An important
use of computer software is to engage students in the exploration of concepts.
Computer programs should help students develop number sense and deal with large
amounts of data in an organized way. Spreadsheets should be used by all
students to manage and operate on long lists of numbers. Also, the computer can
serve as an aid to students in clarifying operations rules that will help them
develop concepts used in early algebra.
Number Sense and Numeration: Grade 1
Overall Expectations
By the end of
Grade 1, students will:
understand whole numbers by exploring number
relationships using concrete materials (e.g., demonstrate with blocks that 7 is
one less than 8 or two more than 5);
understand numerals, ordinals, and the corresponding
words, and demonstrate the ability to print them;
understand the concept of order by sequencing events
(e.g., the steps in washing a dog);
compare and order whole numbers using concrete
materials and drawings to develop number meanings (e.g., to show place value,
arrange 32 counters in groups of 3 tens and 2 ones);
represent fractions (halves as part of a whole) using
concrete materials;
understand and explain basic operations (addition and
subtraction) of whole numbers by modelling and discussing a variety of problem
situations (e.g., show that addition involves joining);
develop proficiency in adding one-digit whole
numbers;
solve simple problems involving counting, joining,
and taking one group away from another (e.g., how many buttons are on the
table?), and describe and explain the strategies used;
estimate quantity in everyday life (e.g., guess, then
count how many beans are in the jar);
use a calculator to explore counting and to solve
problems beyond the required pencil-and-paper skills.
Specific Expectations
Students will:
|
Understanding Number
|
Computations
Applications
|
Number Sense and Numeration: Grade 2
Overall Expectations
By the end of
Grade 2, students will:
represent whole numbers using concrete materials,
drawings, numerals, and number words;
compare and order whole numbers using concrete materials,
drawings, numerals, and number words to develop an understanding of place
value;
compare proper fractions using concrete materials;
understand and explain basic operations (addition,
subtraction, multiplication, and division) of whole numbers by modelling and
discussing a variety of problem situations (e.g., show that division is
sharing, show addition and subtraction with money amounts);
develop proficiency in adding and subtracting one-
and two-digit whole numbers;
solve number problems involving addition and
subtraction, and describe and explain the strategies used;
use and describe an estimation strategy (e.g.,
grouping, comparing, rounding to the nearest ten), and check an answer for
reasonableness using a defined procedure;
use a calculator to skip count, explore number
patterns, and solve problems beyond the required pencil-and-paper skills.
Specific Expectations
Students will:
|
Understanding Number
Computations
|
Applications
|
Number Sense and Numeration: Grade 3
Overall Expectations
By the end of
Grade 3, students will:
represent whole numbers using concrete materials,
drawings, numerals, and number words;
compare and order whole numbers using concrete
materials, drawings, and ordinals;
represent common fractions and mixed numbers using
concrete materials;
understand and explain basic operations (addition,
subtraction, multiplication, division) involving whole numbers by modelling and
discussing a variety of problem situations (e.g., show division as sharing,
show multiplication as repeated addition);
develop proficiency in adding and subtracting
three-digit whole numbers;
develop proficiency in multiplying and dividing
one-digit whole numbers;
select and perform computation techniques (addition,
subtraction, multiplication, division) appropriate to specific problems and
determine whether the results are reasonable;
solve problems and describe and explain the variety
of strategies used;
justify in oral or written expression the method
chosen for addition and subtraction, estimation, mental computation, concrete
materials, algorithms, calculators;
use a calculator to solve problems beyond the
required pencil-and-paper skills.
Specific Expectations
Students will:
|
Understanding Number
Computations
|
Applications
|
Number Sense and Numeration: Grade 4
Overall Expectations
By the end of
Grade 4, students will:
represent, and explore the relationships between,
decimals, mixed numbers, and fractions using concrete materials and drawings;
compare and order whole numbers and decimals using
concrete materials and drawings;
compare and order mixed numbers and proper and
improper fractions with like denominators using concrete materials and
drawings;
understand and explain basic operations (addition and
subtraction) of decimals by modelling and discussing a variety of problem
situations;
develop proficiency in adding and subtracting decimal
numbers to tenths;
select and perform computation techniques appropriate
to specific problems involving whole numbers and decimals, and determine
whether the results are reasonable;
solve problems involving whole numbers and decimals,
and describe and explain the variety of strategies used;
justify in oral or written expression the method
chosen for calculations beyond the proficiency expectations for
pencil-and-paper operations: estimation, mental computation, concrete
materials, algorithms (rules for calculations), or calculators.
For
the following operations, students will be proficient at pencil-and-paper
calculations. For computations that are more complex, students may use
calculators and/or estimation.
Addition: 2
four-digit numbers
Subtraction: a four-digit number subtract a three-digit number
Multiplication: a three-digit number by a one-digit number
Division: a three-digit number by a one-digit number
Specific Expectations
Students will:
|
Understanding Number
|
Computations
Applications
|
Number Sense and Numeration: Grade 5
Overall Expectations
By the end of
Grade 5, students will:
represent, and explore relationships between,
decimals, mixed numbers, and fractions using concrete materials and drawings;
compare, order, and represent whole numbers,
decimals, and fractions using concrete materials and drawings;
understand and explain basic operations
(multiplication and division) of decimals by modelling and discussing a variety
of problem situations;
develop proficiency in multiplying by tenths and
hundredths and dividing by tenths;
understand the significance of numbers within the
surrounding environment;
compare and order, and represent the relationship
between, fractions, improper fractions, and mixed numbers using concrete
materials and drawings;
select and perform computation techniques appropriate
to specific problems involving whole numbers, decimals, and equivalent
fractions, and determine whether the results are reasonable;
solve problems involving decimals and fractions, and
describe and explain the variety of strategies used;
justify in oral and written expression the method
chosen for calculations: estimation, mental computation, concrete materials,
algorithms, or calculators.
For
the following operations, students will be proficient at pencil-and-paper
calculations. For computations that are more complex, students may use
calculators and/or estimation.
Addition: 3
four-digit numbers
Subtraction: 2 four-digit numbers
Multiplication: a two-digit number by a two-digit number
Division: a four-digit number by a one-digit number
Specific Expectations
Students will:
|
Understanding Number
|
Computations
Applications
|
Number Sense and Numeration: Grade 6
Overall Expectations
By the end of
Grade 6, students will:
represent, and explore the relationships between,
decimals, percents, rates, and ratios using concrete materials and drawings;
compare, order, and represent decimals, percents,
rates, and ratios using concrete materials and drawings;
develop proficiency in multiplying by tenths,
hundredths, and thousandths, and in dividing by 100;
understand and explain the characteristics of multiples
and factors and of composite and prime numbers;
compare and order, and represent the relationship
between, fractions with unlike denominators using concrete materials and
drawings;
understand the significance of numbers in the greater
world and evaluate the use of numbers in the media;
select and perform computation techniques appropriate
to specific problems involving unlike denominators in fractions and the
multiplication and division of decimals, and determine whether the results are
reasonable;
solve and explain multi-step problems using the
multiplication and division of decimals and percents;
justify and verify the method chosen for calculations
with whole numbers, fractions, decimals, and percents;
use and verify estimation strategies (e.g., rounding)
to determine the reasonableness of solutions to problems and justify the choice
of strategy.
For
the following operations, students will be proficient at pencil-and-paper
calculations. For computations that are more complex, students may use
calculators and/or estimation.
Addition: 4
three-digit numbers
Subtraction: a five-digit number subtract a four-digit number
Multiplication: a three-digit number by a two-digit number
Division: a four-digit number by a two-digit number
Specific Expectations
Students will:
|
Understanding Number
|
Computations
Applications
|
Number Sense and Numeration: Grade 7
Overall Expectations
By the end of
Grade 7, students will:
compare, order, and represent decimals, integers,
multiples, factors, and square roots;
understand and explain operations with fractions
using manipulatives;
demonstrate an understanding of the order of
operations with brackets;
understand and explain that exponents represent
repeated multiplication;
use estimation to justify or assess the
reasonableness of calculations;
solve and explain multi-step problems involving
simple fractions, decimals, and percents;
explain, in writing, the process of problem solving
using appropriate mathematical language;
use a calculator to solve number questions that are
beyond the proficiency expectations for operations using pencil and paper.
Specific Expectations
Students will:
|
Understanding Number
Computations
|
Applications
|
Number Sense and Numeration: Grade 8
Overall Expectations
By the end of
Grade 8, students will:
compare, order, and represent fractions, decimals, integers,
and square roots;
demonstrate proficiency in operations with fractions;
understand and apply the order of operations with
brackets and exponents in evaluating expressions that involve fractions;
understand and apply the order of operations with
brackets for integers;
demonstrate an understanding of the rules applied in
the multiplication and division of integers;
use a calculator to solve number questions that are
beyond the proficiency expectations for operations using pencil and paper;
justify the choice of method for calculations:
estimation, mental computation, concrete materials, pencil and paper,
algorithms (rules for calculations), or calculators;
solve and explain multi-step problems involving
fractions, decimals, integers, percents, and rational numbers;
use mathematical language to explain the process used
and the conclusions reached in problem solving.
Specific Expectations
Students will:
|
Understanding Number
Computations
|
Applications
|
2. Measurement
Measurement
concepts and skills are directly applicable to the world in which students
live.
The topic of measurement
lends itself naturally to the introduction of fractions and decimals. It also
requires students to be actively involved in solving and discussing problems.
Students should be encouraged to compare objects directly by covering them with
various units and counting the units. Concrete experience in solving
measurement problems is the best foundation for using instruments and formulas.
As students develop increasing skills in numeration, they can be challenged to
undertake increasingly complex measurement problems, thereby strengthening
their facility in both areas of mathematics.
Estimation
activities are important to help students become familiar with different
measures and the process of measuring, and to gain an awareness of the size of
units. Often, only an estimate is required in order to make a decision about
the solution to a problem. Students should develop a fund of informal
measurement guidelines, so that they know, for example, that their fingers are
about 1 cm wide, that a can of pop is about 350 mL, and so on. These guidelines
will help them estimate sizes in the world around them.
Measurement: Grade 1
Overall Expectations
By the end of
Grade 1, students will:
demonstrate an understanding of and ability to apply
measurement terms: height, length, width, time (hour, half-hour), money
(pennies, nickels, dimes), temperature;
identify relationships between and among measurement
concepts (e.g., winter time – colder temperatures);
solve problems related to their day-to-day
environment using concrete experiences of measurement and estimation;
compare the areas of shapes using non-standard units;
estimate, measure, and record the capacity of
containers and the mass of familiar objects using non-standard units, and
compare the measures.
Specific Expectations
Students will:
|
Units of Measure
|
Perimeter and Area
Capacity, Volume, and
Mass
|
Measurement: Grade 2
Overall Expectations
By the end of
Grade 2, students will:
demonstrate an understanding of and ability to apply
measurement terms: centimetre, metre, second, minute, hour, day, week, month,
year, coins to $1 value;
identify relationships between and among measurement
concepts (e.g., shorter time, longer length, colder temperatures);
solve problems related to their day-to-day
environment using concrete experiences of measurement and estimation;
estimate, measure, and record the perimeter and the
area of two-dimensional shapes and compare the perimeters and areas;
estimate, measure, and record the capacity of
containers and the mass of familiar objects using non-standard units, and
compare the measures.
Specific Expectations
Students will:
|
Units of Measure
|
Perimeter and Area
Capacity, Volume, and
Mass
|
Measurement: Grade 3
Overall Expectations
By the end of
Grade 3, students will:
demonstrate an understanding of and ability to apply
measurement terms: centimetre, metre, kilometre; millilitre, litre; gram,
kilogram; degree Celsius; week, month, year;
identify relationships between and among measurement
concepts;
solve problems related to their day-to-day environment
using measurement and estimation (e.g., in finding the height of the school
fence);
estimate, measure, and record the perimeter and the
area of two-dimensional shapes, and compare the perimeters and areas;
estimate, measure, and record the capacity of
containers and the mass of familiar objects, and compare the measures.
Specific Expectations
Students will:
|
Units of Measure
|
Perimeter and Area
Capacity, Volume, and
Mass
|
Measurement: Grade 4
Overall Expectations
By the end of
Grade 4, students will:
demonstrate an understanding of and ability to apply
appropriate metric prefixes in measurement and estimation activities;
identify relationships between and among measurement
concepts (e.g., millimetre to kilometre);
solve problems related to their day-to-day
environment using measurement and estimation;
estimate, measure, and record the perimeter and the
area of two-dimensional shapes, and compare the perimeters and areas;
estimate, measure, and record the capacity of
containers and the mass of familiar objects, compare the measures, and model
the volume of three-dimensional figures.
Specific Expectations
Students will:
|
Units of Measure
Perimeter and Area
|
Capacity, Volume, and
Mass
|
Measurement: Grade 5
Overall Expectations
By the end of
Grade 5, students will:
demonstrate an understanding of and ability to apply appropriate
metric prefixes in measurement and estimation activities;
identify relationships between and among measurement
concepts (linear, temporal, monetary);
solve problems related to the calculation of the perimeter
and the area of regular and irregular two-dimensional shapes;
estimate, measure, and record the capacity of
containers, the mass of familiar objects, and the volume of irregular
three-dimensional figures, and compare the measures.
Specific Expectations
Students will:
|
Units of Measure
|
Perimeter and Area
Capacity, Volume, and
Mass
|
Measurement: Grade 6
Overall Expectations
By the end of
Grade 6, students will:
demonstrate an understanding of and ability to apply
appropriate metric prefixes in measurement and estimation activities;
identify relationships between and among measurement
concepts (linear, square, cubic, temporal, monetary);
solve problems related to the calculation and
comparison of the perimeter and the area of regular polygons;
estimate, measure, and record the mass of objects and
the volume of prisms, and compare the measures.
Specific Expectations
Students will:
|
Units of Measure
|
Perimeter and Area
Capacity, Volume, and
Mass
|
Measurement: Grade 7
Overall Expectations
By the end of
Grade 7, students will:
demonstrate a verbal and written understanding of and
ability to apply accurate measurement strategies that relate to their
environment;
identify relationships between and among measurement
concepts (linear, square, cubic, temporal, monetary);
solve problems related to the calculation and
comparison of the perimeter and the area of irregular two-dimensional shapes;
apply volume formulas to problem-solving situations
involving rectangular prisms.
Specific Expectations
Students will:
|
Units of Measure
Perimeter and Area
|
Capacity, Volume, and
Mass
|
Measurement: Grade 8
Overall Expectations
By the end of
Grade 8, students will:
demonstrate a verbal and written understanding of and
ability to apply accurate measurement and estimation strategies that relate to
their environment;
identify relationships between and among measurement
concepts (linear, square, cubic, temporal, monetary);
solve problems related to the calculation of the
radius, diameter, and circumference of a circle;
apply volume and area formulas to problem-solving
situations involving triangular prisms.
Specific Expectations
Students will:
|
Units of Measure
Perimeter,
Circumference, and Area
|
Capacity, Volume, and
Mass
|
3. Geometry and Spatial Sense
Spatial sense
is the intuitive awareness of one's surroundings and the objects in them.
Geometry helps us represent and describe, in an orderly manner, objects and
their interrelationships in space. A strong sense of spatial relationships and
competence in using the concepts and language of geometry can improve students'
understanding of number and measurement.
Spatial sense is
necessary for interpreting, understanding, and appreciating our inherently
geometric world. Insights and intuitions about the characteristics of
two-dimensional shapes and three-dimensional figures, the interrelationships of
shapes, and the effects of changes to shapes are important aspects of spatial
sense.
Students need
to visualize, draw, and compare shapes in various positions in order to develop
their spatial sense. Although students do need to learn the formal language of
geometry, instruction in the correct terminology should not be the only focus
of the program. Students must also explore and understand relationships among
figures. As students' conceptual understanding develops, technical terms will
become meaningful to them, and they will develop the ability to use correct
terminology in presenting their own views and arguments.
Students'
experiences with graphing and interpreting should include using technology to
explore both linear and non-linear relations. All students should have access
to computers and graphing calculators as powerful tools that can help them
expand their understanding of analytic geometry.
Geometry and Spatial Sense: Grade 1
Overall Expectations
By the end of
Grade 1, students will:
describe and classify three-dimensional figures and
two-dimensional shapes using concrete materials and drawings;
build three-dimensional objects and models;
understand basic concepts in transformational
geometry using concrete materials and drawings.
Specific Expectations
Students will:
|
Three- and
Two-Dimensional Geometry
|
Transformational
Geometry
|
Geometry and Spatial Sense: Grade 2
Overall Expectations
By the end of
Grade 2, students will:
investigate the attributes of three-dimensional figures
and two-dimensional shapes using concrete materials and drawings;
build three-dimensional objects and models;
understand key concepts in transformational geometry
using concrete materials and drawings;
describe location and movements on a grid;
use language effectively to describe geometric
concepts, reasoning, and investigations.
Specific Expectations
Students will:
|
Three- and
Two-Dimensional Geometry
|
Transformational
Geometry
Grids and Coordinate
Geometry
|
Geometry and Spatial Sense: Grade 3
Overall Expectations
By the end of
Grade 3, students will:
investigate the attributes of three-dimensional
figures and two-dimensional shapes using concrete materials and drawings;
draw and build three-dimensional objects and models;
explore transformations of geometric figures;
understand key concepts in transformational geometry
using concrete materials and drawings;
describe location and movements on a grid;
use language effectively to describe geometric
concepts, reasoning, and investigations.
Specific Expectations
Students will:
|
Three- and
Two-Dimensional Geometry
|
Transformational
Geometry
Grids and Coordinate
Geometry
|
Geometry and Spatial Sense: Grade 4
Overall Expectations
By the end of
Grade 4, students will:
solve problems using geometric models;
investigate the attributes of three-dimensional
figures and two-dimensional shapes using concrete materials and drawings;
draw and build three-dimensional objects and models;
explore transformations of geometric figures;
understand key concepts in transformational geometry
using concrete materials and drawings;
describe location and movements on a grid;
use language effectively to describe geometric
concepts, reasoning, and investigations, and coordinate systems.
Specific Expectations
Students will:
|
Three- and
Two-Dimensional Geometry
|
Transformational
Geometry
Coordinate Geometry
|
Geometry and Spatial Sense: Grade 5
Overall Expectations
By the end of
Grade 5, students will:
identify, describe, compare, and classify geometric
figures;
draw and build three-dimensional objects and models;
explore transformations of geometric figures;
understand key concepts in transformational geometry
using concrete materials and drawings;
identify congruent and similar figures using
transformations;
use mathematical language effectively to describe
geometric concepts, reasoning, and investigations, and coordinate systems.
Specific Expectations
Students will:
|
Three- and
Two-Dimensional Geometry
|
Transformational
Geometry
Coordinate Geometry
|
Geometry and Spatial Sense: Grade 6
Overall Expectations
By the end of
Grade 6, students will:
identify, describe, compare, and classify geometric
figures;
draw and construct three-dimensional geometric
figures from nets;
identify congruent and similar figures;
explore transformations of geometric figures;
understand, apply, and analyse key concepts in
transformational geometry using concrete materials and drawings;
use mathematical language effectively to describe
geometric concepts, reasoning, and investigations, and coordinate systems.
Specific Expectations
Students will:
|
Three- and
Two-Dimensional Geometry
|
Transformational
Geometry
Coordinate Geometry
|
Geometry and Spatial Sense: Grade 7
Overall Expectations
By the end of
Grade 7, students will:
identify, describe, compare, and classify geometric
figures;
identify, draw, and construct three-dimensional
geometric figures from nets;
identify congruent and similar figures;
explore transformations of geometric figures;
understand, apply, and analyse key concepts in
transformational geometry using concrete materials and drawings;
use mathematical language effectively to describe
geometric concepts, reasoning, and investigations.
Specific Expectations
Students will:
|
Three- and Two-Dimensional
Geometry
|
Transformational
Geometry
|
Geometry and Spatial Sense: Grade 8
Overall Expectations
By the end of
Grade 8, students will:
identify, describe, compare, and classify geometric
figures;
identify, draw, and represent three-dimensional
geometric figures;
identify and investigate the relationships of angles;
construct and solve problems involving lines and
angles;
investigate geometric mathematical theories to solve
problems;
use mathematical language effectively to describe
geometric concepts, reasoning, and investigations.
Specific Expectations
Students will:
|
Three- and
Two-Dimensional Geometry
|
|
4. Patterning and Algebra
One of the
central themes in mathematics is the study of patterns and functions. This study
requires students to recognize, describe, and generalize patterns and to build
mathematical models to predict the behaviour of real-world phenomena that
exhibit observed patterns. Exploring patterns helps students develop both
mathematical competence and an appreciation of the aesthetic qualities of
mathematics.
In Grades 1 to
3, instruction should focus on helping students identify regularities in
events, shapes, designs, and sets of numbers. Students will begin to see that
regularity is the essence of mathematics. Physical materials and pictorial
displays can be used to help students recognize and create patterns and
relationships. Through observing varied relationships of the same pattern,
students can begin to identify its properties. Encouraging students to label
and describe patterns, using letters and other symbols, prepares them to use
variables in the future.
In Grades 4 to
6, the focus of instruction shifts from exploring patterns to exploring
functions. When students use graphs, data tables, expressions, equations, or
verbal descriptions to represent a single relationship, they discover that
different representations yield different interpretations of a situation.
Through such activities, students learn informally that functions are things that
can vary (variables) and that therefore have a changing relationship with other
variables: changes in one variable result in changes in another.
Algebra is the
language through which most of mathematics is communicated. The focus of the
study in Grades 7 and 8 is first on understanding how the language of algebra
can be used to generalize a pattern or a relationship. A second focus is on
using algebra as a problem-solving tool – a means of clarifying concepts at an
abstract level before applying them. Experience with this process often helps
students to develop generalizations and insights that extend their learning
beyond the original application.
Patterning and Algebra: Grade 1
Overall Expectations
By the end of
Grade 1, students will:
explore patterns and pattern rules;
identify relationships between and among patterns.
Specific Expectations
Students will:
|
|
|
Patterning and Algebra: Grade 2
Overall Expectations
By the end of
Grade 2, students will:
identify, extend, and create number, geometric, and
measurement patterns, and patterns in their environment;
explore patterns and pattern rules;
identify relationships between and among patterns.
Specific Expectations
Students will:
|
|
|
Patterning and Algebra: Grade 3
Overall Expectations
By the end of Grade
3, students will:
recognize that patterning results from repetition;
identify, extend, and create linear and non-linear
geometric patterns, number and measurement patterns, and patterns in their
environment;
create charts to display patterns;
identify relationships between and among patterns.
Specific Expectations
Students will:
|
|
|
Patterning and Algebra: Grade 4
Overall Expectations
By the end of
Grade 4, students will:
demonstrate an understanding of mathematical
relationships in patterns using concrete materials, drawings, and symbols;
identify, extend, and create linear and non-linear geometric
patterns, number and measurement patterns, and patterns in their environment;
recognize and discuss patterning rules;
apply patterning strategies to problem-solving
situations.
Specific Expectations
Students will:
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|
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Patterning and Algebra: Grade 5
Overall Expectations
By the end of
Grade 5, students will:
recognize and discuss the mathematical relationships
between and among patterns;
identify, extend, and create patterns in a variety of
contexts;
analyse and discuss patterning rules;
create tables to display patterns;
apply patterning strategies to problem-solving
situations.
Specific Expectations
Students will:
|
|
|
Patterning and Algebra: Grade 6
Overall Expectations
By the end of
Grade 6, students will:
recognize and discuss the mathematical relationships between
and among patterns;
identify, extend, and create patterns in a variety of
contexts;
analyse and discuss patterning rules;
display pattern relationships graphically and
numerically;
apply patterning strategies to problem-solving
situations.
Specific Expectations
Students will:
|
|
|
Patterning and Algebra: Grade 7
Overall Expectations
By the end of
Grade 7, students will:
identify the relationships between whole numbers and
variables;
identify, extend, create, and discuss patterns using
whole numbers and variables;
identify, create, and solve simple algebraic
equations;
apply and discuss patterning strategies in
problem-solving situations.
Specific Expectations
Students will:
|
Modelling
|
Linear Equations
|
Patterning and Algebra: Grade 8
Overall Expectations
By the end of
Grade 8, students will:
identify the relationships between whole numbers and
variables;
identify, create, and discuss patterns in algebraic
terms;
evaluate algebraic expressions;
identify, create, and solve simple algebraic
equations;
apply and defend patterning strategies in
problem-solving situations.
Specific Expectations
Students will:
|
Modelling
|
Linear Equations
|
5. Data Management and Probability
The related
topics of probability and statistics are highly relevant to everyday life.
Graphs and statistics bombard the public in advertising, opinion polls,
reliability estimates, population trends, descriptions of discoveries by
scientists, estimates of health risks, and analyses of students' performance in
schools and schools' performance within school systems, to name just a few.
Students should
actively explore situations by experimenting with and simulating a variety of
probability models. The focus should be on real-world questions – such as the
probable outcome of a sports event or whether it will rain on the day of the school
picnic. Students should talk about their ideas and use the results of their
experiments to model situations or predict events. The topic of probability is
rich in interesting problems that can fascinate students and provide bridges to
other topics, such as ratios, fractions, percents, and decimals.
Data Management and Probability: Grade 1
Overall Expectations
By the end of
Grade 1, students will:
collect, organize, and describe data using concrete
materials and drawings;
interpret displays of data using concrete materials,
and discuss the data;
demonstrate an understanding of probability and
demonstrate the ability to apply probability in familiar day-to-day situations.
Specific Expectations
Students will:
|
Collecting, Organizing,
and Analysing Data
Concluding and
Reporting
|
Probability
|
Data Management and Probability: Grade 2
Overall Expectations
By the end of
Grade 2, students will:
sort and classify objects and data using concrete
materials;
collect and organize data;
create and interpret displays of data, and present
and discuss the information;
demonstrate an understanding of probability and
demonstrate the ability to apply probability in familiar day-to-day situations.
Specific Expectations
Students will:
|
Collecting, Organizing,
and Analysing Data
Concluding and Reporting
|
Probability
|
Data Management and Probability: Grade 3
Overall Expectations
By the end of
Grade 3, students will:
sort, classify, and cross-classify objects and data;
collect and organize data;
interpret displays of data, present the information,
and discuss it using mathematical language;
demonstrate an understanding of probability and
demonstrate the ability to apply probability in familiar day-to-day situations;
relate meaningful experiences about probability.
Specific Expectations
Students will:
|
Collecting and
Organizing Data
Concluding and
Reporting
|
Probability
|
Data Management and Probability: Grade 4
Overall Expectations
By the end of
Grade 4, students will:
collect and organize data and identify their use;
predict the results of data collected;
interpret displays of data and present the
information using mathematical terms;
demonstrate an understanding of probability and use
language appropriate to situations involving probability experiments;
solve simple problems involving the concept of probability.
Specific Expectations
Students will:
|
Collecting and
Organizing Data
Analysing Data
Concluding and
Reporting
|
Probability
|
Data Management and Probability: Grade 5
Overall Expectations
By the end of
Grade 5, students will:
use computer applications to record the results of
data collected;
predict the validity of the results of data
collected;
interpret displays of data and present the information
using mathematical terms;
evaluate and use data from graphic organizers;
demonstrate an understanding of probability concepts
and use mathematical symbols;
pose and solve simple problems involving the concept
of probability.
Specific Expectations
Students will:
|
Collecting and
Organizing Data
Analysing Data
Concluding and
Reporting
|
Probability
|
Data Management and Probability: Grade 6
Overall Expectations
By the end of
Grade 6, students will:
systematically collect, organise, and analyse data;
use computer applications to examine data in a
variety of ways;
construct graphic organizers using computer
applications;
interpret displays of data and present the
information using mathematical terms;
evaluate data and make conclusions from the analysis
of data;
use a knowledge of probability to pose and solve
problems;
examine the concepts of possibility and probability;
compare experimental probability results with
theoretical results.
Specific Expectations
Students will:
|
Collecting and
Organizing Data
Analysing Data
Concluding and
Reporting
|
Probability
|
Data Management and Probability: Grade 7
Overall Expectations
By the end of
Grade 7, students will:
systematically collect, organize, and analyse data;
recognize the different levels of data collection;
use computer applications to examine and interpret
data in a variety of ways;
develop an appreciation for statistical methods as
powerful means of decision making;
construct graphic organizers using computer
applications;
interpret displays of data and present the
information using mathematical terms;
evaluate data and make conclusions from the analysis
of data;
use and apply a knowledge of probability.
Specific Expectations
Students will:
|
Collecting and
Organizing Data
Analysing Data
|
Concluding and
Reporting
Probability
|
Data Management and Probability: Grade 8
Overall Expectations
By the end of
Grade 8, students will:
systematically collect, organize, and analyse primary
data;
use computer applications to examine and interpret
data in a variety of ways;
interpret displays of data and present the
information using mathematical terms;
evaluate data and draw conclusions from the analysis
of data;
identify probability situations and apply a knowledge
of probability;
appreciate the power of using a probability model by
comparing experimental results with theoretical results.
Specific Expectations
Students will:
|
Collecting and
Organizing Data
Analysing Data
|
Concluding and
Reporting
Probability
|
6. Mathematics - Grade 9
Principles of Mathematics, Grade 9, Academic (MPM1D)
This course enables students to develop generalizations of mathematical ideas and methods through the exploration of applications, the effective use of technology, and abstract reasoning. Students will investigate relationships to develop equations of straight lines in analytic geometry, explore relationships between volume and surface area of objects in measurement, and apply extended algebraic skills in problem solving. Students will engage in abstract extensions of core learning that will deepen their mathematical knowledge and enrich their understanding.
Number Sense and Algebra
Overall Expectations
By the end of this course, students will:
solve multi-step
problems requiring numerical answers, using a variety of strategies and tools;
demonstrate
understanding of the three basic exponent rules and apply them to simplify
expressions;
manipulate
first-degree polynomial expressions to solve first-degree equations;
solve problems,
using the strategy of algebraic modelling.
Specific Expectations
Solving Numerical Problems
By the end of this course, students will:
demonstrate
facility with critical numerical skills, including mental mathematics,
estimation, operations with integers (as necessary for working with equations
and analytic geometry), and operations with rational numbers (as necessary in
analytic geometry, measurement, and equation solving);
distinguish between
exact and approximate representations of the same quantity and choose
appropriately between them in given situations (e.g., use the symbol pi instead
of 3.14 in determining the effect on the volume of a sphere of doubling its
diameter; determine the perimeter of a square having an area of 2);
solve multi-step
problems involving applications of percent, ratio, and rate as they arise
throughout the course;
use a scientific
calculator effectively for applications that arise throughout the course;
judge the
reasonableness of answers to problems by considering likely results within the
situation described in the problem;
judge the
reasonableness of answers produced by a calculator, a computer, or pencil and
paper, using mental mathematics and estimation.
Operating with Exponents
By the end of this course, students will:
evaluate
numerical expressions involving natural-number exponents with rational-number
bases;
substitute into
and evaluate algebraic expressions involving exponents, to support other topics
of the course (e.g., measurement, analytic geometry);
determine the
meaning of negative exponents and of zero as an exponent from activities
involving graphing, using technology, and from activities involving patterning;
represent very large
and very small numbers, using scientific notation;
enter and
interpret exponential notation on a scientific calculator, as necessary in
calculations involving very large and very small numbers;
determine, from
the examination of patterns, the exponent rules for multiplying and dividing
monomials and the exponent rule for the power of a power, and apply these rules
in expressions involving one and two variables.
Manipulating Polynomial Expressions and Solving Equations
By the end of this course, students will:
add and subtract
polynomials;
multiply a
polynomial by a monomial, and factor a polynomial by removing a common factor;
expand and
simplify polynomial expressions involving one variable;
solve first-degree
equations, including equations with fractional coefficients, using an algebraic
method;
calculate sides
in right triangles, using the Pythagorean theorem, as required in topics
throughout the course (e.g., measurement);
rearrange
formulas involving variables in the first degree, with and without
substitution, as they arise in topics throughout the course (e.g., analytic
geometry, measurement).
Using Algebraic Modelling to Solve Problems
By the end of this course, students will:
use algebraic
modelling as one of several problem-solving strategies in various topics of the
course (e.g., relations, measurement, direct and partial variation, the
Pythagorean theorem, percent);
compare
algebraic modelling with other strategies used for solving the same problem;
communicate
solutions to problems in appropriate mathematical forms (e.g., written
explanations, formulas, charts, tables, graphs) and justify the reasoning used
in solving the problems.
Relationships
Overall Expectations
By the end of this course, students will:
determine
relationships between two variables by collecting and analysing data;
compare the
graphs and formulas of linear and non-linear relations;
describe the
connections between various representations of relations.
Specific Expectations
Determining Relationships
By the end of this course, students will:
pose problems,
identify variables, and formulate hypotheses associated with relationships (Sample problem: If you look through a
paper tube at a wall, you can see a region of a certain height on the wall. If
you move farther from the wall, the height of that region changes. What is the
relationship between the height of the visible region and your distance from
the wall? Describe the relationship that you think will occur);
demonstrate an
understanding of some principles of sampling and surveying (e.g.,
randomization, representivity, the use of multiple trials) and apply the
principles in designing and carrying out experiments to investigate the
relationships between variables (Sample
problem: What factors might affect the outcome of this experiment?
How could you design the experiment to account for them?);
collect data,
using appropriate equipment and/or technology (e.g., measuring tools, graphing
calculators, scientific probes, the Internet) (Sample problem: Acquire or construct a paper tube and work
with a partner to measure the heights of visible regions at various distances
from a wall);
organize and
analyse data, using appropriate techniques (e.g., making tables and graphs,
calculating measures of central tendency) and technology (e.g., graphing
calculators, statistical software, spreadsheets) (Sample problem: Enter the data into a spreadsheet. Decide
what analysis would be appropriate to examine the relationship between the
variables – a graph, measures of central tendency, ratios);
describe trends
and relationships observed in data, make inferences from data, compare the
inferences with hypotheses about the data, and explain the differences between
the inferences and the hypotheses (Sample
problem: Describe any trend observed in the data. Does a
relationship seem to exist? Of what sort? Is the outcome consistent with your
original hypothesis? Discuss any outlying pieces of data and provide
explanations for them. Suggest a formula relating the height of the visible
region to the distance from the wall. How might you vary this experiment to
examine other relationships?);
communicate the
findings of an experiment clearly and concisely, using appropriate mathematical
forms (e.g., written explanations, formulas, charts, tables, graphs), and
justify the conclusions reached;
solve and/or
pose problems related to an experiment, using the findings of the experiment.
Comparing Linear and Non-linear Relations
By the end of this course, students will:
construct tables
of values, graphs, and formulas to represent linear relations derived from
descriptions of realistic situations (e.g., the cost of holding a banquet in a
rented hall is $25 per person plus $975 for the hall);
construct tables
of values and scatter plots for linearly related data collected from
experiments (e.g., the rebound height of a ball versus the height from which it
was dropped) or from secondary sources (e.g., the number of calories in fast
food versus the number of grams of fat);
determine the
equation of a line of best fit for a scatter plot, using an informal process
(e.g., a process of trial and error on a graphing calculator; calculation of
the equation of the line joining two carefully chosen points on the scatter
plot);
construct tables
of values and graphs to represent non-linear relations derived from
descriptions of realistic situations (Sample
problem: A triangular prism has a height of 20 cm and a square
base. Represent the relationship between the volume of the prism and the side
length of its base, as the side length varies);
construct tables
of values and scatter plots for non-linearly related data collected from
experiments (e.g., the relationship between height and age) or from secondary
sources (e.g., the population of
demonstrate an
understanding that straight lines represent linear relations and curves
represent non-linear relations.
Describing Connections Between Representations of Relations
By the end of this course, students will:
determine values
of a linear relation by using the formula of the relation and by interpolating
or extrapolating from the graph of the relation (e.g., if a student earns $5/h
caring for children, determine how long he or she must work to earn $143);
describe, in
written form, a situation that would explain the events illustrated by a given
graph of a relationship between two variables (e.g., write a story that matches
the events shown in the graph);
identify, by
calculating finite differences in its table of values, whether a relation is
linear or non-linear;
describe the effect
on the graph and the formula of a relation of varying the conditions of a
situation they represent (e.g., if a graph showing partial variation represents
the cost of producing a yearbook, describe how the appearance of the graph
changes if the cost per book is altered; describe how it changes if the fixed
costs are altered).
Analytic Geometry
Overall Expectations
By the end of this course, students will:
determine,
through investigation, the relationships between the form of an equation and
the shape of its graph with respect to linearity and non-linearity;
determine,
through investigation, the properties of the slope and y-intercept of a linear relation;
solve problems,
using the properties of linear relations.
Specific Expectations
Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph
By the end of this course, students will:
determine,
through investigations, the characteristics that distinguish the equation of a
straight line from the equations of non-linear relations (e.g., use graphing
software to obtain the graphs of a variety of linear and non-linear relations
from their equations; classify the relations according to the shapes of their
graphs; focus on the characteristics of the equations of linear relations and
how they differ from the characteristics of the equations of non-linear
relations);
select the
equations of straight lines from a given set of equations of linear and
non-linear relations;
identify the
equation of a line in any of the forms y=mx
+ b, Ax + By + C=0,
x=a, y=b;
rearrange the
equation of a line from the form y=mx
+ b to the form Ax + By
+ C=0, and vice versa.
Investigating the Properties of Slope
By the end of this course, students will:
determine the
slope of a line segment, using various formulas
|
|
|
rise |
|
|
|
|
|
y2 - y1 |
|
|
|
A |
|
|
(e.g., m |
= |
|
, m |
= |
|
, m |
= |
|
, m |
= |
- |
|
); |
|
|
|
run |
|
|
|
|
|
x2 - x1 |
|
|
|
B |
|
identify the slope
of a linear relation as representing a constant rate of change;
calculate the
finite differences in the table of values of a linear relation and relate the
result to the slope of the relation;
identify the
geometric significance of m and
b in the equation y=mx + b through investigation;
identify the
properties of the slopes of line segments (e.g., direction, positive or
negative rate of change, steepness, parallelism, perpendicularity) through
investigations facilitated by graphing technology, where appropriate.
Using the Properties of Linear Relations to Solve Problems
By the end of this course, students will:
plot points on
the xy-plane and use the
terminology and notation of the xy-plane
correctly;
graph lines by hand,
using a variety of techniques (e.g., making a table of values, using
intercepts, using the slope and y-intercept);
graph lines,
using graphing calculators or graphing software;
determine the
equation of a line, given information about the line (e.g., the slope and y-intercept, the slope and a point, two
points, a line parallel to a given line and having the same x-intercept as another given line);
communicate
solutions to multi-step problems in established mathematical form, with clear
reasons given for the steps taken;
describe the
meaning of the slope and y-intercept
for a linear relation arising from a realistic situation, interpolate and
extrapolate from the graph and the equation of the relation, and identify and
explain any restrictions on the variables in the relation;
describe a
situation that would be modelled by a given linear equation;
determine the
point of intersection of two linear relations, by hand for simple examples, and
using graphing calculators or graphing software for more complex examples;
interpret the intersection point in the context of an application.
Measurement and Geometry
Overall Expectations
By the end of this course, students will:
determine the
optimal values of various measurements through investigations facilitated,
where appropriate, by the use of concrete materials, diagrams, and calculators
or computer software;
solve problems
involving the surface area and the volume of three-dimensional objects;
formulate conjectures
and generalizations about geometric relationships involving two-dimensional
figures, through investigations facilitated by dynamic geometry software, where
appropriate.
Specific Expectations
Investigating the Optimal Value of Measurements
By the end of this course, students will:
identify,
through investigation, the effect of varying the dimensions of a rectangular
prism or cylinder on the volume or surface area of the object;
identify,
through investigation, the relationships between the volume and surface area of
a given rectangular prism or cylinder;
explain the
significance of optimal surface area or volume in various applications (e.g.,
packaging; the relationship between surface area and heat loss);
pose and solve a
problem involving the relationship between the perimeter and the area of a
figure when one of the measures is fixed.
Solving Problems Involving Surface Area and Volume
By the end of this course, students will:
solve simple problems,
using the formulas for the surface area and the volume of prisms, pyramids,
cylinders, cones, and spheres;
solve multi-step
problems involving the volume and the surface area of prisms, cylinders,
pyramids, cones, and spheres;
judge the
reasonableness of answers to measurement problems by considering likely results
within the situation described in the problem;
judge the
reasonableness of answers produced by a calculator, a computer, or pencil and
paper, using mental mathematics and estimation.
Investigating Geometric Relationships
By the end of this course, students will:
illustrate and
explain the properties of the interior and the exterior angles of triangles and
quadrilaterals, and of angles related to parallel lines;
determine the properties
of angle bisectors, medians, and altitudes in various types of triangles
through investigation;
determine the
properties of the sides and the diagonals of polygons (e.g., the diagonals in
quadrilaterals, the diagonals of regular pentagons, the figure that results
from joining the midpoints of sides of quadrilaterals) through investigation;
pose questions
about geometric relationships, test them, and communicate the findings, using
appropriate language and mathematical forms (e.g., written explanations,
diagrams, formulas, tables);
confirm a
statement about the relationships between geometric properties by illustrating
the statement with examples, or deny the statement on the basis of a
counter-example (e.g., confirm or deny the following statement: If a
quadrilateral has perpendicular diagonals, then it is a square).
Foundations of Mathematics, Grade 9, Applied (MFM1P)
This course enables students to develop mathematical ideas and methods through the exploration of applications, the effective use of technology, and extended experiences with hands-on activities. Students will investigate relationships of straight lines in analytic geometry, solve problems involving the measurement of 3-dimensional objects and 2-dimensional figures, and apply key numeric and algebraic skills in problem solving. Students will also have opportunities to consolidate core skills and deepen their understanding of key mathematical concepts.
Number Sense and Algebra
Overall Expectations
By the end of this course, students will:
consolidate numerical skills by using them in a
variety of contexts throughout the course;
demonstrate understanding of the three basic exponent
rules and apply them to simplify expressions;
manipulate first-degree polynomial expressions to solve
first-degree equations;
solve problems, using the strategy of algebraic
modelling.
Specific Expectations
Consolidating Numerical Skills
By the end of this course, students will:
determine strategies for mental mathematics and
estimation, and apply these strategies throughout the course;
demonstrate facility in operations with integers, as
necessary to support other topics of the course (e.g., polynomials, equations,
analytic geometry);
demonstrate facility in operations with percent,
ratio and rate, and rational numbers, as necessary to support other topics of
the course (e.g., analytic geometry, measurement);
use a scientific calculator effectively for
applications that arise throughout the course;
judge the reasonableness of answers to problems by
considering likely results within the situation described in the problem;
judge the reasonableness of answers produced by a
calculator, a computer, or pencil and paper, using mental mathematics and
estimation.
Operating with Exponents
By the end of this course, students will:
evaluate numerical expressions involving
natural-number exponents with rational-number bases;
substitute into and evaluate algebraic expressions
involving exponents, to support other topics of the course (e.g., measurement,
analytic geometry);
determine the meaning of negative exponents and of
zero as an exponent from activities involving graphing, using technology, and
from activities involving patterning;
represent very large and very small numbers, using
scientific notation;
enter and interpret exponential notation on a
scientific calculator, as necessary in calculations involving very large and
very small numbers;
determine, from the examination of patterns, the exponent
rules for multiplying and dividing monomials and the exponent rule for the
power of a power, and apply these rules in expressions involving one variable.
Manipulating Polynomial Expressions and Solving Equations
By the end of this course, students will:
add and subtract polynomials, and multiply a
polynomial by a monomial;
expand and simplify polynomial expressions involving
one variable;
solve first-degree equations, excluding equations
with fractional coefficients, using an algebraic method;
calculate sides in right triangles, using the
Pythagorean theorem, as required in topics throughout the course (e.g.,
measurement);
substitute into measurement formulas and solve for
one variable, with and without the help of technology.
Using Algebraic Modelling to Solve Problems
By the end of this course, students will:
use algebraic modelling as one of several
problem-solving strategies in various topics of the course (e.g., relations,
measurement, direct and partial variation, the Pythagorean theorem, percent);
compare algebraic modelling with other strategies
used for solving the same problem;
communicate solutions to problems in appropriate
mathematical forms (e.g., written explanations, formulas, charts, tables,
graphs) and justify the reasoning used in solving the problems.
Relationships
Overall Expectations
By the end of this course, students will:
determine relationships between two variables by
collecting and analysing data;
compare the graphs of linear and non-linear
relations;
describe the connections between various
representations of relations.
Specific Expectations
Determining Relationships
By the end of this course, students will:
pose problems, identify variables, and formulate
hypotheses associated with relationships (Sample
problem: Does the rebound height of a ball depend on the height
from which it was dropped? Make a hypothesis and then design an experiment to
test it);
demonstrate an understanding of some principles of
sampling and surveying (e.g., randomization, representivity, the use of
multiple trials) and apply the principles in designing and carrying out
experiments to investigate the relationships between variables (Sample problem: What factors might affect
the outcome of this experiment? How could you design the experiment to account
for them?);
collect data, using appropriate equipment and/or
technology (e.g., measuring tools, graphing calculators, scientific probes, the
Internet) (Sample problem: Drop
a ball from varying heights, measuring the rebound height each time);
organize and analyse data, using appropriate
techniques (e.g., making tables and graphs, calculating measures of central
tendency) and technology (e.g., graphing calculators, statistical software,
spreadsheets) (Sample problem:
Enter the data into a spreadsheet. Decide what analysis would be appropriate to
examine the relationship between the variables – a graph, measures of central
tendency, ratios);
describe trends and relationships observed in data, make
inferences from data, compare the inferences with hypotheses about the data,
and explain the differences between the inferences and the hypotheses (Sample problem: Describe any trend
observed in the data. Does a relationship seem to exist? Of what sort? Is the
outcome consistent with your original hypothesis? Discuss any outlying pieces
of data and provide explanations for them. Suggest a formula relating the
rebound height to the height from which the ball was dropped. How might you
vary this experiment to examine other relationships?);
communicate the findings of an experiment clearly and
concisely, using appropriate mathematical forms (e.g., written explanations,
formulas, charts, tables, graphs), and justify the conclusions reached;
solve and/or pose problems related to an experiment,
using the findings of the experiment.
Comparing Linear and Non-linear Relations
By the end of this course, students will:
construct tables of values, graphs, and formulas to represent
linear relations derived from descriptions of realistic situations involving
direct and partial variation (e.g., the cost of holding a banquet in a rented
hall is $25 per person plus $975 for the hall);
construct tables of values and scatter plots for
linearly related data involving direct variation collected from experiments
(e.g., the rebound height of a ball versus the height from which it was
dropped);
determine the equation of a line of best fit for a
scatter plot, using an informal process (e.g., a process of trial and error on
a graphing calculator; calculation of the equation of the line joining two
carefully chosen points on the scatter plot);
construct tables of values and graphs to represent
non-linear relations derived from descriptions of realistic situations (e.g.,
represent the relationship between the volume of a cube and its side length, as
the side length varies);
demonstrate an understanding that straight lines represent
linear relations and curves represent non-linear relations.
Describing Connections Between Representations of Relations
By the end of this course, students will:
determine values of a linear relation by using the
formula of the relation and by interpolating or extrapolating from the graph of
the relation (e.g., if a student earns $5/h caring for children, determine how
long he or she must work to earn $143);
describe, in written form, a situation that would
explain the events illustrated by a given graph of a relationship between two
variables (e.g., write a story that matches the events shown in the graph);
identify, by calculating finite differences in its
table of values, whether a relation is linear or non-linear;
describe the effect on the graph and the formula of a
relation of varying the conditions of a situation they represent (e.g., if a
graph showing partial variation represents the cost of producing a yearbook,
describe how the appearance of the graph changes if the cost per book is altered;
describe how it changes if the fixed costs are altered).
Analytic Geometry
Overall Expectations
By the end of this course, students will:
determine, through investigation, the relationships between
the form of an equation and the shape of its graph with respect to linearity
and non-linearity;
determine, through investigation, the properties of
the slope and y-intercept of a linear relation;
graph a line and write the equation of a line from
given information.
Specific Expectations
Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph
By the end of this course, students will:
determine, through investigations, the
characteristics that distinguish the equation of a straight line from the
equations of non-linear relations (e.g., use graphing software to obtain the
graphs of a variety of linear and non-linear relations from their equations;
classify the relations according to the shapes of their graphs; focus on the
characteristics of the equations of linear relations and how they differ from
the characteristics of the equations of non-linear relations);
select the equations of straight lines from a given
set of equations of linear and non-linear relations;
identify y=mx + b
as a standard form for the equation of a straight line, including the special
cases x=a, y=b.
Investigating the Properties of Slope
By the end of this course, students will:
identify practical situations illustrating slope (e.g.,
ramps, slides, staircases) and calculate the slopes of the inclines;
determine the slope of a line segment, using the
formula
|
|
|
rise |
|
|
m |
= |
|
; |
|
|
|
run |
|
identify the geometric significance of m and b in the
equation y=mx + bthrough
investigation;
identify the properties of the slopes of line
segments (i.e., direction, positive or negative rate of change, steepness,
parallelism, perpendicularity) through investigations facilitated by graphing
technology, where appropriate.
Graphing and Writing Equations of Lines
By the end of this course, students will:
plot points on the xy-plane
and use the terminology and notation of the xy-plane
correctly;
graph lines by hand, using a variety of techniques (e.g.,
making a table of values, using intercepts, using the slope and y-intercept);
graph lines, using graphing calculators or graphing
software;
determine the equation of a line, given the slope and
y-intercept, the slope and a
point on the line, and two points on the line;
communicate solutions in established mathematical
form, with clear reasons given for the steps taken.
Measurement and Geometry
Overall Expectations
By the end of this course, students will:
determine the optimal values of various measurements
through investigations facilitated by the use of concrete materials, diagrams,
and calculators or computer software;
solve problems involving the measurement of
two-dimensional figures and three-dimensional objects;
formulate conjectures and generalizations about
geometric relationships involving two-dimensional figures, through
investigations facilitated by dynamic geometry software, where appropriate.
Specific Expectations
Investigating the Optimal Values of Measurements
By the end of this course, students will:
construct a variety of rectangles for a given
perimeter and determine the maximum area for a given perimeter;
construct a variety of square-based prisms for a
given volume and determine the minimum surface area for a square-based prism
with a given volume;
construct a variety of cylinders for a given volume
and determine the minimum surface area for a cylinder with a given volume;
describe applications in which it would be important
to know the maximum area for a given perimeter or the minimum surface area for
a given volume (e.g., building a fence, designing a container).
Solving Problems Involving Measurement
By the end of this course, students will:
solve problems involving the area of composite plane figures
(e.g., combinations of rectangles, triangles, parallelograms, trapezoids, and
circles);
solve simple problems, using the formulas for the
surface area of prisms and cylinders and for the volume of prisms, cylinders,
cones, and spheres;
solve problems involving perimeter, area, surface
area, volume, and capacity in applications;
judge the reasonableness of answers to measurement
problems by considering likely results within the situation described in the
problem;
judge the reasonableness of answers produced by a
calculator, a computer, or pencil and paper, using mental mathematics and
estimation.
Investigating Geometric Relationships
By the end of this course, students will:
illustrate and explain the properties of the interior
and exterior angles of triangles and quadrilaterals, and of angles related to
parallel lines;
determine the properties of angle bisectors, medians,
and altitudes in various types of triangles through investigation;
determine some properties of the sides and the diagonals
of quadrilaterals (e.g., the diagonals of a rectangle bisect each other);
communicate the findings of investigations, using
appropriate language and mathematical forms (e.g., written explanations,
diagrams, formulas, tables).
Mathematics Glossary, Grade 9
algebraic
expression. One or more
variables and possibly numbers and operation symbols. For example, 3x + 6, x,
and 5x are algebraic expressions.
algorithm. A systematic procedure for carrying out a
computation. For example, the addition algorithm is a set of rules for finding
the sum of two or more numbers.
alternate
angles. Two angles on
opposite sides of a transversal when it crosses two lines. The angles are equal
when the lines are parallel. The angles form one of these patterns: 
,
.
analog
clock. A timepiece that
indicates the time through the position of its hands.
attribute. A quantitative or qualitative
characteristic of an object or a shape, for example, colour, size, thickness.
bar graph. See under graph.
bias. An emphasis on characteristics that are
not typical of an entire population.
binomial. An algebraic expression with two terms,
for example, 2x + 4y, 5k – 3n, and 2y2 + 5.
bisector. A line that divides a segment, an angle,
a line, or a figure into two equal halves.
broken-line
graph. See under graph.
calculation
method. Any of a variety
of methods used for solving problems, for example, estimation, mental
calculation, pencil-and-paper computation, the use of technology (including
calculators, computer spreadsheets).
capacity. The greatest amount that a container can
hold; usually measured in litres or millilitres.
Cartesian
coordinate grid. See
coordinate plane.
Cartesian
plane. See coordinate
plane.
census. The counting of an entire population.
circle
graph. See under graph.
clustering. See under estimation
strategies.
coefficient. Part of a term. In a term, the numerical
factor is the numerical coefficient, and the variable factor is the variable
coefficient. For example, in 5y, 5 is the numerical coefficient and y is the
variable coefficient.
comparative
bar graph. See under
graph.
compatible
numbers. Pairs of
numbers whose sum is a power of 10. For example, 30 + 70 = 100 (102).
complementary
angles. Two angles whose
sum is 90º.
composite
number. A number that
has factors other than itself and 1. For example, the number 8 has four
factors: 1, 2, 4, and 8.
computer
spreadsheet. Software
that helps to organize information using rows and columns.
concrete
graph. See under graph.
concrete
materials. Objects that
students handle and use in constructing their own understanding of mathematical
concepts and skills and in illustrating that understanding. Some examples are
base ten blocks, centicubes, construction kits, dice, games, geoboards,
geometric solids, hundreds charts, measuring tapes, Miras, number lines,
pattern blocks, spinners, and tiles. Also called manipulatives.
cone. A three-dimensional figure with a
circular base and a curved surface that tapers proportionately to an apex.
congruent
figures. Geometric
figures that have the same size and shape.
conservation. The property by which something remains
the same despite changes such as physical arrangement.
coordinate
graph. See under graph.
coordinate
plane. A plane that
contains an X-axis (horizontal) and a Y-axis (vertical). Also called Cartesian
coordinate grid or Cartesian plane.
coordinates. An ordered pair used to describe a
location on a grid or plane. For example, the coordinates (3, 5) describe a
location on a grid found by moving 3 units horizontally from the origin
(0, 0) followed by 5 units vertically.
data. Facts or information.
database. An organized and sorted list of facts or
information; usually generated by a computer.
degree. A unit for measuring angles.
dependent
variable. A variable
that changes as a result of a change in the independent variable.
diameter. A line segment that joins two points on
the circumference of a circle and passes through the centre.
displacement. The amount of fluid displaced by an
object placed in it.
distribution. A classification or an arrangement of
statistical information.
double bar
graph. See comparative
bar graph under graph.
equation. A mathematical statement that has
equivalent terms on either side of the equal sign.
equivalent
fractions. Fractions that
represent the same part of a whole or group, for example, 1/3 , 2/6, 3/9, 4/12.
equivalent
ratios. Ratios that
represent the same fractional number or amount, for example, 1:3, 2:6, 3:9.
estimation
strategies. Mental mathematics
strategies used to obtain an approximate answer. Students estimate when an
exact answer is not required and estimate to check the reasonableness of their
mathematics work. Some estimation strategies are:
clustering. A strategy used for estimating the sum of numbers
that cluster around one particular value. For example, the numbers 42, 47, 56,
55 cluster around 50. So estimate 50 + 50 + 50 + 50 = 200.
front-end
loading. The addition of significant
digits (those with the highest place value) with an adjustment of the remaining
values. Also called front loading. The following is an example of
front-end loading:
Step 1 - Add the
first digits in each number.
193 + 428 + 253
Think 100 + 400 + 200 = 700.
Step 2 - Adjust
the estimate to reflect the size of the remaining digits.
93 + 28 + 53 is approximately 175.
Think 700 + 175 = 875.
rounding. A process of replacing a number by an approximate
value of that number. For example, rounding to the nearest tens for 106 is 110.
event. One of several independent
probabilities.
expanded
form. A way of writing
numbers that shows the value of each digit, for example, 432 = 4 x 100 + 3 x 10
+ 2 x 1.
experimental
probability. The chance
of an event occurring based on the results of an experiment.
exponential form. A shorthand method for writing repeated
multiplication. In 53, 3,
which is the exponent, indicates that 5 is to be multiplied by itself three
times. 53 is in exponential form.
expression. A combination of numbers and variables
without an equal sign, for example, 3x + 5.
factors. See under multiplication.
first-hand
data. See primary
data.
flip. See reflection.
formula. A set of ideas, words, symbols, figures,
characters, or principles used to state a general rule. For example, the
formula for the area of a rectangle is A = l x w.
frequency. The number of times an event or item
occurs.
frequency
distribution. A table or
graph that shows how often each score, event, or measurement occurred.
front-end
loading. See under
estimation strategies.
graph. A representation of data in a pictorial
form. Some types of graphs are:
bar graph.
A diagram consisting of horizontal or vertical bars that represent data.
broken-line graph. On a coordinate grid, a display of data formed by line segments that join
points representing data.
circle graph. A graph in which a circle used to represent a whole is divided into
parts that represent parts of the whole.
comparative bar graph. A graph consisting of two or more bar graphs placed
side by side to compare the same thing. Also called double bar graph.
concrete graph. A graph in which real objects are used to represent pieces of
information.
coordinate graph. A grid that has data points named as ordered pairs of numbers, for
example, (4, 3).
histogram.
A type of bar graph in which each bar represents a range of values, and the
data are continuous.
pictograph.
A graph that illustrates data using pictures and symbols.
histogram. See under graph.
improper
fraction. A fraction whose
numerator is greater than its denominator, for example, 12/5.
independent
events. Two or more
events for which the occurrence or non-occurrence of one does not change the
probability of the other.
independent
variable. A variable
that does not depend on another for its value; a variable that the experimenter
purposely changes. Also called cause variable.
inequality. A statement using symbols to show that
one expression is greater than (>), less than (<), or not equal to
another expression.
integer. Any one of the numbers. . . , –4, –3,
–2, –1, 0, 1, 2, 3, 4, . . .
integral
exponent. A power that
has an integer as an exponent.
intersecting
lines. Two lines with
exactly one point in common, the point of intersection.
interval. A space between two points. For example,
0–10 represents the interval from 0 to 10 inclusively.
irrational
number. A number that
cannot be represented as a terminating or repeating decimal, for example, 
.
irregular
polygon. A polygon whose
side and angle measures are not equal.
isometric
dot paper. Dot paper
formed by the vertices of equilateral triangles, used for three-dimensional
drawings. Also called triangular dot paper or triangle dot paper.
isosceles
triangle. A triangle
that has two sides of equal length.
linear
dimension. Dimension
involving the measurement of only one linear attribute, for example, length,
width, height, depth.
linear
relationship. A relationship
that has a straight-line graph.
line of best
fit. A line that can
sometimes be determined on a scatter plot. If a line of best fit can be found,
a relationship exists between the independent and dependent variables.
line of
symmetry. A line that
divides a shape into two parts that can be matched by folding the shape in
half.
manipulatives. See concrete materials.
many-to-one
correspondence. The
matching of elements in two sets in such a way that more than one element in one
set can be matched with one and only one element in another set, for example, 3
pennies to each pocket.
mass. The amount of matter in an object;
usually measured in grams or kilograms.
mathematical
communication. The use
of mathematical language by students to:
respond to and describe the world around them;
communicate their attitudes about and interests in
mathematics;
reflect and shape their understandings of and skills
in mathematics.
Students
communicate by talking, drawing pictures, drawing diagrams, writing journals,
charting, dramatizing, building with concrete materials, and using symbolic
language, (e.g., 2, >, =).
mathematical
concepts. The
fundamental understandings about mathematics that a student develops within
problem-solving contexts.
mathematical
language.
terminology (e.g., factor, pictograph, tetrahedron);
pictures/diagrams (e.g., 2 x 3 matrix, parallelogram,
tree diagram);
symbols, including numbers (e.g., 2, 1/4), operations
(e.g., 3 x 8 = [3 x 4] + [3 x 4]), and relations (e.g., 1/4 <).
mathematical
procedures. The skills,
operations, mechanics, manipulations, and calculations that a student uses to
solve problems.
mean. The average; the sum of a set of numbers
divided by the number of numbers in the set. For example, the average of 10 +
20 + 30 is 60 ÷ 3 = 20.
measure of
central tendency. A
value that can represent a set of data, for example, mean, median, mode. Also
called central measure.
median. The middle number in a set of numbers,
such that half the numbers in the set are less and half are greater when the
numbers are arranged in order. For example, 14 is the median for the set of
numbers 7, 9, 14, 21, 39. If there is an even number of numbers, the median is
the mean of the two middle numbers. For example, 11 is the median of 5, 10, 12,
and 28.
Mira. A transparent mirror used in geometry to
locate reflection lines, reflection images, and lines of symmetry, and to
determine congruency and line symmetry.
mixed
number. A number that is
the sum of a whole number and a fraction, for example, 81/4.
mode. The number that occurs most often in a
set of data. For example, in a set of data with the values 3, 5, 6, 5, 6, 5, 4,
5, the mode is 5.
modelling. A representation of the facts and
factors of, and the inferences to be drawn from, an entity or a situation.
monomial. An algebraic expression with one term,
for example, 2x or 5xy2.
multiple. The product of a given number and a
whole number. For example, 4, 8, 12, . . . are multiples of 4.
multiplication. An operation that combines numbers
called factors to give one number called a product. For example, 4 x 5 = 20;
thus factor x factor = product.
multi-step
problem. A problem whose
solution requires at least two calculations. For example, shoppers who want to
find out how much money they have left after a purchase follow these steps:
Step 1 - Add all items purchased (subtotal).
Step 2 - Multiply the sum of purchases by % of tax.
Step 3 - Add the tax to the sum of purchases (grand total).
Step 4 - Subtract the grand total from the shopper's
original amount of money.
natural
numbers. The counting
numbers 1, 2, 3, 4, . . .
net. A pattern that can be folded to make a
three-dimensional figure.
network. A set of vertices joined by paths.
non-standard
units. Measurement units
used in the early development of measurement concepts, for example, paper
clips, cubes, hand spans, and so on. 
number line. A line that matches a set of numbers and
a set of points one to one.
number
operations. Mathematical
processes or actions that include the addition, subtraction, multiplication,
and division of numbers.
nth term. The last of a series of terms.
obtuse
angle. An angle that
measures more than 90º and less than 180º.
one-to-one
correspondence. The
matching of elements in two sets in such a way that every element in one set
can be matched with one and only one element in another set.
ordered pair. Two numbers in order, for example, (2,
6). On a coordinate plane, the first number is the horizontal coordinate of a
point, and the second is the vertical coordinate of the point.
order of
operations. The rules
used to simplify expressions. Often the acronym BEDMAS is used to describe this
calculation process:
brackets
exponents
division or
multiplication, whichever comes first
addition or
subtraction, whichever comes first
ordinal
number. A number that shows
relative position or place, for example, first, second, third, fourth.
parallel
lines. Lines in the same
plane that do not intersect.
parallelogram. A quadrilateral whose opposite sides are
parallel.
perfect
square. The product of
an integer multiplied by itself. For example, 9 = 3 x 3; thus 9 is a perfect
square.
perpendicular
lines. Two lines that
intersect at a 90º angle.
pictograph. See under graph.
place value. The value given to the place in which a
digit appears in a numeral. In the number 5473, 5 is in the thousands place, 4
is in the hundreds place, 7 is in the tens place, and 3 is in the ones place.
plane shape. A two-dimensional figure.
polygon. A closed figure formed by three or more
line segments. Examples of polygons are triangles, quadrilaterals, pentagons,
octagons.
polyhedron. A three-dimensional object that has
polygons as faces.
polynomial. An algebraic expression. Examples of
polynomials are 6x, 3x – 2, and 4x2 + 5x – 4.
population. The total number of individuals or
items.
power. A number written in exponential form; a
shorter way of writing repeated multiplication. For example, 102 and 26 are powers.
primary
data. Information that
is collected directly or first-hand. Data from a person-on-the-street survey
are primary data. Also called first-hand data or primary-source data.
prime
factorization. An
expression showing a composite number as a product of its prime factors. The
prime factorization for 42 is 2 x 3 x 7.
prime
number. A whole number
greater than 1 that has only two factors, itself and 1. For example, 7 = 1 x 7.
prism. A three-dimensional figure with two
parallel and congruent bases. A prism is named by the shape of its bases, for
example, rectangular prism, triangular prism.
probability. A number that shows how likely it is
that an event will happen.
product. See under multiplication.
proper
fraction. A fraction
whose numerator is smaller than its denominator, for example, 2/3.
proportion. A number sentence showing that two ratios
are equal, for example, 2/3 = 6/9.
Pythagorean
theorem. In a right
triangle, the square of the length of the hypotenuse is equal to the sum of the
squares of the other two sides.
quadrilateral. A polygon with four straight sides.
radius. A line segment whose endpoints are the
centre of a circle and a point on the circle.
range. The difference between the highest and
lowest number in a group of numbers. For example, in a data set of 8, 32, 15,
10, the range is 24, that is, 32 – 8.
rate. A comparison of two numbers with
different units, such as kilometres and hours, for example, 100 km/h.
ratio. A comparison of numbers with the same
units, for example, 3:4 or 3/4.
rational
number. A number that
can be expressed as the quotient of two integers where the divisor is not 0.
reflection. A transformation that turns a figure
over an axis. The figure does not change size or shape, but it does change
position and orientation. A reflection image is the result of a reflection.
Also called flip.
regular
polygon. A closed figure
in which all sides and angles are equal.
rotation. A transformation that turns a figure
about a fixed point. The figure does not change size or shape, but it does
change position and orientation. A rotation image is the result of a rotation.
Also called turn.
rotational
symmetry. A shape that
fits onto itself after a turn less than a full turn has rotational symmetry.
For example, a square has a turn symmetry of order 4 because it resumes its
original orientation after each of 4 turns: 1/4 turn, 1/2 turn, 3/4 turn, and
full turn. Also called turn symmetry.
rounding. See under estimation
strategies.
sample. A small, representative group chosen
from a population and examined in order to make predictions about the
population. Also called sampling.
scale
drawing. A drawing in
which the lengths are a reduction or an enlargement of actual lengths.
scalene
triangle. A triangle
with three sides of different lengths.
scatter
plot. A graph that
attempts to show a relationship between two variables by means of points
plotted on a coordinate grid. Also called scatter diagram.
scientific
notation. A way of
writing a number as the product of a number between 1 and 10 and a power of 10.
In scientific notation, 58 000 000 is written 5.8 x 107.
secondary
data. Information that
is not collected first-hand, for example, data from a government document or a
database. Also called second-hand data or secondary-source data.
second-hand
data. See secondary
data.
sequence. A succession of things that are
connected in some way, for example, the sequence of numbers 1, 1, 2, 3, 5, . .
.
seriation
line. A line used for
the ordering of objects, numbers, or ideas.
shell. A three-dimensional figure whose
interior is completely empty.
SI. The international system of measurement units,
for example, centimetre, kilogram. (From the French Système International.)
similar
figures. Geometric
figures that have the same shape but not always the same size.
simple
interest. The formula
used to calculate the interest on an investment: I = PRT where P is the
principal, R is the rate of interest, and T is the time chosen to invest the
principal.
simulation. A probability experiment to test the
likelihood of an event. For example, tossing a coin is a simulation of whether
the next person you meet is a male or a female.
skeleton. A three-dimensional figure showing only
the edges and vertices of the figure.
slide. See translation.
standard
form. A way of writing a
number in which each digit has a place value according to its position in relation
to the other digits. For example, 7856 is in standard form.
stem-and-leaf
plot. An organization of
data into categories based on place values.
supplementary
angles. Two angles whose
sum is 180º.
surface
area. The sum of the
areas of the faces of a three-dimensional object.
survey. A sampling of information, such as that
made by asking people questions or interviewing them.
symbol. See under mathematical
language.
systematic
counting. A process used
as a check so that no event or outcome is counted twice.
table. An orderly arrangement of facts set out
for easy reference, for example, an arrangement of numerical values in vertical
or horizontal columns.
tally chart. A chart that uses tally marks to count
data and record frequencies.
tangram. An ancient Chinese puzzle made from a
square cut into seven pieces: two large triangles, one medium-sized triangle,
two small triangles, one square, and one parallelogram.
term. Each of the quantities constituting a
ratio, a sum, or an algebraic expression.
tessellation. A tiling pattern in which shapes are
fitted together with no gaps or overlaps.
theoretical
probability. The number
of favourable outcomes divided by the number of possible outcomes.
tiling. The process of using repeated congruent
shapes to cover a region completely.
transformation. A change in a figure that results in a
different position, orientation, or size. The transformations include the translation
(slide), reflection (flip), rotation (turn), and dilatation
(reduction or enlargement).
translation. A transformation that moves a figure to
a new position in the same plane. The figure does not change size, shape, or
orientation; it only changes position. A translation image is the result of a
translation. Also called slide.
trapezoid. A quadrilateral with exactly one pair of
parallel sides.
tree
diagram. A branching
diagram that shows all possible combinations or outcomes.
turn. See rotation.
variable. A letter or symbol used to represent a
number.
Venn
diagram. A diagram consisting
of overlapping circles used to show what two or more sets have in common.
vertex. The common endpoint of the two segments
or lines of an angle.
volume. The amount of space occupied by an
object; measured in cubic units such as cubic centimetres.
achievement
levels. Brief
descriptions of four different degrees of achievement of the provincial
curriculum expectations for any given grade. Level 3, which is the
"provincial standard", identifies a high level of achievement of the
provincial expectations. Parents of students achieving at level 3 in a
particular grade can be confident that their children will be prepared for work
at the next grade. Level 1 identifies achievement that falls much below the
provincial standard. Level 2 identifies achievement that approaches the
standard. Level 4 identifies achievement that surpasses the standard.
expectations. The knowledge and skills that students
are expected to develop and to demonstrate in their class work, on tests, and
in various other activities on which their achievement is assessed. The new
strands. The five major areas of knowledge and
skills into which the curriculum for Mathematics is organized. The strands for
Mathematics are: Number Sense and Numeration, Measurement, Geometry and Spatial
Sense, Patterning and Algebra, and Data Management and Probability.
7. Mathematics - Grade 10
Principles of Mathematics, Grade 10, Academic (MPM2D)
This course enables students to broaden their understanding of relations, extend their skills in multi-step problem solving, and continue to develop their abilities in abstract reasoning. Students will pursue investigations of quadratic functions and their applications; solve and apply linear systems; solve multi-step problems in analytic geometry to verify properties of geometric figures; investigate the trigonometry of right and acute triangles; and develop supporting algebraic skills.
Quadratic Functions
Overall Expectations
By the end of this course, students will:
solve quadratic
equations;
determine,
through investigation, the relationships between the graphs and the equations
of quadratic functions;
determine, through
investigation, the basic properties of quadratic functions;
solve problems
involving quadratic functions.
Specific Expectations
Solving Quadratic Equations
By the end of this course, students will:
expand and
simplify second-degree polynomial expressions;
factor
polynomial expressions involving common factors, differences of squares, and
trinomials;
solve quadratic
equations by factoring and by using graphing calculators or graphing software;
solve quadratic equations,
using the quadratic formula;
interpret real
and non-real roots of quadratic equations geometrically as the x-intercepts of the graph of a quadratic
function.
Investigating the Connection Between the Graphs and the Equations of
Quadratic Functions
By the end of this course, students will:
identify the
effect of simple transformations (i.e., translations, reflections, vertical
stretch factors) on the graph and the equation of y=x2, using graphing
calculators or graphing software;
explain the role
of a, h, and k in the graph of y=a(x – h)2 + k;
express the
equation of a quadratic function in the form y=a(x
– h)2 + k,
given it in the form y=ax2 + bx + c,
using the algebraic method of completing the square in situations involving no
fractions;
sketch, by hand,
the graph of a quadratic function whose equation is given in the form y=ax2 + bx + c,
using a suitable method [e.g., complete the square; locate the x-intercepts if
the equation is factorable; express in the form y=ax(x – s) + t to locate two points and deduce the vertex].
Investigating the Basic Properties of Quadratic Functions
By the end of this course, students will:
collect data that may be represented by quadratic functions,
from secondary sources (e.g., the Internet, Statistics Canada), or from
experiments, using appropriate equipment and technology (e.g., scientific
probes, graphing calculators);
fit the equation of a quadratic function to a scatter
plot, using an informal process (e.g., a process of trial and error on a
graphing calculator), and compare the results with the equation of a curve of
best fit produced by using graphing calculators or graphing software;
describe the nature of change in a quadratic function,
using finite differences in tables of values, and compare the nature of change
in a quadratic function with the nature of change in a linear function;
report the findings of an experiment in a clear and
concise manner, using appropriate mathematical forms (e.g., written
explanations, tables, graphs, formulas, calculations), and justify the
conclusions reached.
Solving Problems Involving Quadratic Functions
By the end of this course, students will:
determine the zeros and the maximum or minimum value
of a quadratic function, using algebraic techniques;
determine the zeros and the maximum or minimum value
of a quadratic function from its graph, using graphing calculators or graphing
software;
solve problems related to an application, given the graph
or the formula of a quadratic function (e.g., given a quadratic function
representing the height of a ball over elapsed time, answer questions such as
the following: What is the maximum height of the ball? After what length of
time will the ball touch the ground? Over what interval is the height of the
ball greater than 3 m?).
Analytic Geometry
Overall Expectations
By the end of this course, students will:
model and solve problems involving the intersection
of two straight lines;
solve problems involving the analytic geometry
concepts of line segments;
verify geometric properties of triangles and
quadrilaterals, using analytic geometry.
Specific Expectations
Using Linear Systems to Solve Problems
By the end of this course, students will:
determine the point of intersection of two linear
relations graphically, with and without the use of graphing calculators or
graphing software, and interpret the intersection point in the context of a
realistic situation;
solve systems of two linear equations in two
variables by the algebraic methods of substitution and elimination;
solve problems represented by linear systems of two
equations in two variables arising from realistic situations, by using an
algebraic method and by interpreting graphs.
Solving Problems Involving the Properties of Line Segments
By the end of this course, students will:
determine formulas for the midpoint and the length of
a line segment and use these formulas to solve problems;
determine the equation for a circle having centre (0,
0) and radius r, by applying
the formula for the length of a line segment; identify the radius of a circle
of centre (0, 0), given its equation; and write the equation, given the radius;
solve multi-step problems, using the concepts of the
slope, the length, and the midpoint of line segments (e.g., determine the
equation of the right bisector of a line segment, the coordinates of whose end
points are given; determine the distance from a given point to a line whose
equation is given; show that the centre of a given circle lies on the right
bisector of a given chord);
communicate the solutions to multi-step problems in
good mathematical form, giving clear reasons for the steps taken to reach the
solutions.
Using Analytic Geometry to Verify Geometric Properties
By the end of this course, students will:
determine characteristics of a triangle whose vertex
coordinates are given (e.g., the perimeter; the classification by side length;
the equations of medians, altitudes, and right bisectors; the location of the
circumcentre and the centroid);
determine characteristics of a quadrilateral whose
vertex coordinates are given (e.g., the perimeter; the classification by side
length; the properties of the diagonals; the classification of a quadrilateral
as a square, a rectangle, or a parallelogram);
verify geometric properties of a triangle or
quadrilateral whose vertex coordinates are given (e.g., the line joining the
midpoints of two sides of a triangle is parallel to the third side; the
diagonals of a rectangle bisect each other).
Trigonometry
Overall Expectations
By the end of this course, students will:
develop the primary trigonometric ratios, using the
properties of similar triangles;
solve trigonometric problems involving right
triangles;
solve trigonometric problems involving acute
triangles.
Specific Expectations
Developing the Primary Trigonometric Ratios
By the end of this course, students will:
determine the properties of similar triangles (e.g.,
the correspondence and equality of angles, the ratio of corresponding sides,
the ratio of areas) through investigation, using dynamic geometry software;
describe and compare the concepts of similarity and
congruence;
solve problems involving similar triangles in
realistic situations (e.g., problems involving shadows, reflections,
surveying);
define the formulas for the sine, the cosine, and the
tangent of angles, using the ratios of sides in right triangles.
Solving Problems Involving the Trigonometry of Right Triangles
By the end of this course, students will:
determine the measures of the sides and angles in
right triangles, using the primary trigonometric ratios;
solve problems involving the measures of sides and
angles in right triangles (e.g., in surveying, navigation);
determine the height of an inaccessible object in the
environment around the school, using the trigonometry of right triangles.
Solving Problems Involving the Trigonometry of Acute Triangles
By the end of this course, students will:
determine, through investigation, the relationships
between the angles and sides in acute triangles (e.g., the largest angle is
opposite the longest side; the ratio of side lengths is equal to the ratio of
the sines of the opposite angles), using dynamic geometry software;
calculate the measures of sides and angles in acute
triangles, using the sine law and cosine law;
describe the conditions under which the sine law or the
cosine law should be used in a problem;
solve problems involving the measures of sides and
angles in acute triangles;
describe the application of trigonometry in science
or industry.
Foundations of Mathematics, Grade 10, Applied (MFM2P)
This course enables students to consolidate their understanding of key mathematical concepts through hands-on activities and to extend their problem-solving experiences in a variety of applications. Students will solve problems involving proportional reasoning and the trigonometry of right triangles; investigate applications of piecewise linear functions; solve and apply systems of linear equations; and solve problems involving quadratic functions. The effective use of technology in learning and in solving problems will be a focus of the course.
Proportional Reasoning
Overall Expectations
By the end of this course, students will:
solve problems derived from a variety of
applications, using proportional reasoning;
solve problems involving similar triangles;
solve problems involving right triangles, using
trigonometry.
Specific Expectations
Using Proportional Reasoning to Solve Problems from Applications
By the end of this course, students will:
solve problems involving percent, ratio, rate, and
proportion (e.g., in topics such as interest calculation, currency conversion,
similar triangles, trigonometry, direct and partial variation related to linear
functions) by a variety of methods and models (e.g., diagrams, concrete
materials, fractions, tables, patterns, graphs, equations);
draw and interpret scale diagrams related to
applications (e.g., technical drawings);
distinguish between consistent and inconsistent
representations of proportionality in a variety of contexts (e.g., explain the
distortion of figures resulting from irregular scales; identify misleading
features in graphs; identify misleading conclusions based on invalid
proportional reasoning).
Solving Problems Involving Similar Triangles
By the end of this course, students will:
determine some properties of similar triangles (e.g.,
the correspondence and equality of angles, the ratio of corresponding sides)
through investigation, using dynamic geometry software;
solve problems involving similar triangles in
realistic situations (e.g., problems involving shadows, reflections, surveying);
define the formulas for the sine, the cosine, and the
tangent of angles, using the ratios of sides in right triangles.
Solving Problems Involving the Trigonometry of Right Triangles
By the end of this course, students will:
calculate the length of a side of a right triangle,
using the Pythagorean theorem;
determine the measures of the sides and angles in
right triangles, using the primary trigonometric ratios;
solve problems involving the measures of sides and
angles in right triangles (e.g., in surveying, navigation);
determine the height of an inaccessible object in the
environment around the school, using the trigonometry of right triangles;
describe applications of trigonometry in various
occupations.
Linear Functions
Overall Expectations
By the end of this course, students will:
apply the properties of piecewise linear functions as
they occur in realistic situations;
solve and interpret systems of two linear equations
as they occur in applications;
manipulate algebraic expressions as they relate to
linear functions.
Specific Expectations
Applying Piecewise Linear Functions
By the end of this course, students will:
explain the characteristics of situations involving piecewise
linear functions (e.g., pay scale variations, gas consumption costs, water
consumption costs, differentiated pricing, motion);
construct tables of values and sketch graphs to
represent given descriptions of realistic situations involving piecewise linear
functions, with and without the use of graphing calculators or graphing
software;
answer questions about piecewise linear functions by
interpolation and extrapolation, and by considering variations on given
conditions.
Interpreting Systems of Linear Equations
By the end of this course, students will:
determine the point of intersection of two linear
relations arising from a realistic situation, using graphing calculators or
graphing software;
interpret the point of intersection of two linear relations
within the context of a realistic situation;
solve systems of two linear equations in two
variables by the algebraic methods of substitution and elimination;
solve problems represented by linear systems of two equations
in two variables arising from realistic situations, by using an algebraic
method and by interpreting graphs.
Manipulating Algebraic Expressions
By the end of this course, students will:
write linear equations by generalizing from tables of
values and by translating written descriptions;
rearrange equations from the form y=mx + b to the form Ax + By
+ C=0, and vice versa;
solve first-degree equations in one variable,
including those with fractional coefficients, using an algebraic method;
isolate a variable in formulas involving first-degree
terms.
Quadratic Functions
Overall Expectations
By the end of this course, students will:
manipulate algebraic expressions as they relate to
quadratic functions;
determine, through investigation, the relationships
between the graphs and the equations of quadratic functions;
solve problems by interpreting graphs of quadratic
functions.
Specific Expectations
Manipulating Algebraic Expressions
By the end of this course, students will:
multiply two binomials and square a binomial;
expand and simplify polynomial expressions involving
the multiplying and squaring of binomials;
describe intervals on quadratic functions, using
appropriate vocabulary (e.g., greater than,
less than, between, from . . . to, less than 3 or greater than 7);
factor polynomials by determining a common factor;
factor trinomials of the form x2
+ bx + c;
factor the difference of squares;
solve quadratic equations by factoring.
Investigating the Connection Between the Graphs and the Equations of
Quadratic Functions
By the end of this course, students will:
construct tables of values, sketch graphs, and write
equations of the form y=ax2 + b to
represent quadratic functions derived from descriptions of realistic situations
(e.g., vary the side length of a cube and observe the effect on the surface
area of the cube);
identify the effect of simple transformations (i.e., translations,
reflections, vertical stretch factors) on the graph and the equation of y=x2, using graphing calculators or graphing software;
explain the role of a, h, and k
in the graph of y=a(x – h)2 + k;
expand and simplify an equation of the form y=a(x – h)2 + k to
obtain the form y=ax2 + bx + c.
Solving Problems Involving Quadratic Functions
By the end of this course, students will:
obtain the graphs of quadratic functions whose
equations are given in the form y=a(x – h)2 + k or
the form y=ax2 + bx + c,
using graphing calculators or graphing software;
determine the zeros and the maximum or minimum value
of a quadratic function from its graph, using graphing calculators or graphing
software;
solve problems involving a given quadratic function
by interpreting its graph (e.g., given a formula representing the height of a
ball over elapsed time, graph the function, using a graphing calculator or
graphing software, and answer questions such as the following: What is the
maximum height of the ball? After what length of time will the ball touch the
ground? Over what interval is the height of the ball greater than 3 m?).
Mathematics Glossary, Grade 10
acute triangle.
A
triangle in which each of the three interior angles measures less than 90º.
algebraic expression.
One
or more variables and possibly numbers and operation symbols. For example, 3x
+ 6, x, 5x, and 21 – 2w are all algebraic expressions.
algebraic modelling.
The
process of representing a relationship by an equation or a formula, or
representing a pattern of numbers by an algebraic expression.
algorithm.
A
specific set of instructions for carrying out a procedure.
altitude.
A
line segment giving the height of a geometric figure. In a triangle, an
altitude is found by drawing the perpendicular from a vertex to the side
opposite. For example:
analytic geometry.
A
geometry that uses the xy-plane to determine equations that represent
lines and curves.
angle bisector.
A
line that divides an angle into two equal parts.
application.
An
area outside of mathematics within which concepts and skills of mathematics may
be used to solve problems.
binomial.
An algebraic
expression containing two terms, for example, 3x + 2.
centroid of a triangle.
The
point of intersection of the three medians of a triangle. Also called balance
point.
chord.
A
line segment joining two points on a curve.
circumcentre of a triangle.
The
centre of the circle that passes through the three vertices of a triangle.
coefficient.
The
factor by which a variable is multiplied. For example, in the term 5x,
the coefficient is 5; in the term ax, the coefficient is a.
congruence.
The
property of being congruent. Two geometric figures are congruent if they are
equal in all respects.
constant rate of change.
A
relationship between two variables illustrates a constant rate of change when
equal intervals of the first variable are associated with equal intervals of
the second variable. For example, if a car travels at 100 km/h, in the first
hour it travels 100 km, in the second hour it travels 100 km, and so on.
curve of best fit.
The
curve that best describes the distribution of points in a scatter plot.
diagonal.
In
a polygon, a line joining two vertices that are not next to each other (i.e.,
not joined by one side).
difference of squares.
A
technique of factoring applied to an expression of the form a2 – b2, which involves the subtraction of two perfect
squares.
direct variation.
A
relationship between two variables in which one variable is a constant multiple
of the other.
dynamic geometry
software.
Computer
software that allows the user to plot points on a coordinate system, measure
line segments and angles, construct two-dimensional shapes, create
two-dimensional representations of three-dimensional objects, and transform
constructed figures by moving parts of them.
To
determine a value for.
exponent.
A
special use of a superscript in mathematics. For example, in 32, the exponent is 2. An exponent is used to denote repeated
multiplication. For example, 54 means 5 x 5 x
5 x 5.
exponential notation.
The
notation used by calculators to display numbers that are too large or too small
to fit onto the screen of the calculator. For example, the number 25 382 000
000 000 000 might appear as “2.5382 16” on a calculator screen. The digits “16”
to the right of the expression indicate the number of places that the decimal point
should be moved to express the number in normal form.
extrapolate.
To
estimate values lying outside the range of given data. For example, to
extrapolate from a graph means to estimate coordinates of points beyond those
that are plotted.
factor.
To
express a number as the product of two or more numbers, or an algebraic
expression as the product of two or more other algebraic expressions. Also, the
individual numbers or algebraic expressions in such a product.
Given a table of values in
which the x-coordinates are evenly spaced, the first differences are
calculated by subtracting consecutive y-coordinates. The second
differences are calculated by subtracting consecutive first differences, and so
on. In a linear function, the first differences are constant; in a quadratic
function, the second differences are constant. For example:
|
x |
y |
First |
Second |
|
1 |
1 |
|
|
|
2 |
4 |
4 - 1=3 |
|
|
3 |
9 |
9 - 4=5 |
5 - 3=2 |
|
4 |
16 |
16 - 9=7 |
7 - 5=2 |
|
5 |
25 |
25 - 16=9 |
9 -7=2 |
first-degree equation.
An
equation in which the variable has the exponent 1. For example,
5(3x – 1) + 6=–20 + 7x + 5.
first-degree inequation.
An
inequality in which the variable has the exponent 1. For example, 6 + 2x
+ 8 
4x
+ 20.
first-degree polynomial.
A
polynomial in which the variable has the exponent 1. For example, 4x +
20.
first differences.
See
finite differences.
A
relation in which for each value of x there is only one value of y.
generalize.
To determine
a general rule or conclusion from examples. Specifically, to determine a
general rule to represent a pattern or relationship between variables.
graphing calculator.
A
hand-held device capable of a wide range of mathematical operations, including
graphing from an equation, constructing a scatter plot, determining the
equation of a curve of best fit for a scatter plot, making statistical
calculations, performing elementary symbolic manipulation. Many graphing
calculators will attach to scientific probes that can be used to gather data
involving physical measurements (e.g., position, temperature, force).
graphing software.
Computer
software that provides features similar to those of a graphing calculator.
To make
a conclusion based on a relationship identified between variables in a set of
data.
integer.
Any
one of the numbers . . . , –4, –3, –2, –1, 0, +1, +2, +3, +4, . . .
The
distance from the origin of the xy-plane to the point at which a line or curve
cuts a given axis (e.g., x-intercept or y-intercept). For
example:
To
estimate values lying between elements of given data. For example, to
interpolate from a graph means to estimate coordinates of points between those
that are plotted.
linear relation.
A
relation between two variables that appears as a straight line when graphed on
a coordinate system. May also be referred to as a linear function.
linear system.
A
pair of equations of straight lines.
line of best fit.
The
straight line that best describes the distribution of points in a scatter plot.
make inferences from
data.
See
infer from data.
manipulate.
To
apply operations, such as addition, multiplication, or factoring, on algebraic
expressions.
A
mathematical description (e.g., a diagram, a graph, a table of values, an
equation, a formula, a physical model, a computer model) of a real situation.
mathematical modelling.
The
process of describing a real situation in a mathematical form. See also
mathematical model.
measure of central
tendency.
A
value that can represent a set of data, for example, the mean, the median, or
the mode.
median.
Geometry. The line drawn from a vertex of a triangle to the
midpoint of the opposite side. Statistics. The middle number in a set,
such that half the numbers in the set are less and half are greater when the
numbers are arranged in order.
method of elimination.
In
solving systems of linear equations, a method in which the coefficients of one
variable are matched through multiplication and then the equations are added or
subtracted to eliminate that variable.
method of substitution.
In
solving systems of linear equations, a method in which one equation is
rearranged and substituted into the other.
model.
See
mathematical model.
monomial.
An
algebraic expression with one term, for example, 5x2.
multiple trials.
A technique
used in experimentation in which the same experiment is done several times and
the results are combined through a measure such as averaging. The use of
multiple trials “smooths out” some of the random occurrences that can affect
the outcome of an individual trial of an experiment.
non-linear relation.
A
relationship between two variables that does not fit a straight line when
graphed.
non-real root of an
equation.
A
solution to an equation that is not an element of the set of real numbers (e.g.,
).
See real root of an equation.
optimal value.
The
maximum or minimum value of a variable.
partial variation.
A
relationship between two variables in which one variable is a multiple of the
other, plus some constant number. For example, the cost of a taxi fare has two
components, a flat fee and a fee per kilometre driven. A formula representing
the situation of a flat fee of $2.00 and a fee rate of $0.50/km would be F=0.50d
+ 2.00, where F is the total fare and d is the number of
kilometres driven.
piecewise linear function.
A
function composed of two or more linear functions having different slopes.
polygon.
See
polynomial expression.
An
algebraic expression of the form a + bx + cx2 + . . . , where a, b, and c are
numbers.
population.
Statistics. The total number of individuals or items under
consideration in a surveying or sampling activity.
primary trigonometric
ratios..
The
basic ratios of trigonometry (i.e., sine, cosine, and tangent).
prism.
A
three-dimensional figure with two parallel, congruent polygonal bases. A prism
is named by the shape of its bases, for example, rectangular prism, triangular
prism.
Reasoning
or problem solving based on the examination of equal ratios.
Pythagorean theorem.
The
conclusion that, in a right triangle, the square of the length of the longest
side is equal to the sum of the squares of the lengths of the two other sides.
quadratic equation.
An
equation that contains at least one term whose exponent is 2, and no term with
an exponent greater than 2, for example, x2 + 7x + 10=0.
quadratic formula.
A formula
for determining the roots of a quadratic equation, ax2 + bx + c=0.
The formula is phrased in terms of the coefficients of the quadratic equation:
|
|
|
|
|
x |
= |
|
|
|
|
2a |
![(root [b-squared minus 4ac])](academicMATHgrades1to12_files/image014.gif){kind=small}
quadratic function.
A
function whose equation is in quadratic form, for example, y=x2 + 7x
+ 10.
quadrilateral.
A
polygon with four sides.
randomization.
A
principle of data analysis that involves selecting a sample in such a way that
each member of the population has an equally likely chance of being selected.
rational number.
A
number that can be expressed as the quotient of two integers where the divisor
is not 0.
realistic situation.
A
description of an event or events drawn from real life or from an experiment
that provides experience with such an event.
A
solution to an equation that is an element of the set of real numbers. The set
of real numbers includes all numbers commonly used in daily life: all
fractions, all decimals, all negative and positive numbers.
region on the xy-plane.
An
area bounded by a curve or curves and/or lines on the xy-plane.
regression.
A
method for determining the equation of a curve (not necessarily a straight
line) that fits the distribution of points on a scatter plot.
relation.
An
identified relationship between variables that may be expressed as a table of
values, a graph, or an equation.
representivity.
A
principle of data analysis that involves selecting a sample that is typical of
the characteristics of the population from which it is drawn.
right triangle.
A triangle
containing one 90º angle.
sample.
A
small group chosen from a population and examined in order to make predictions
about the population.
sampling technique.
A
process for collecting a sample of data.
scatter plot.
A
graph that attempts to show a relationship between two variables by means of
points plotted on a coordinate grid. Also called scatter diagram.
scientific probe.
A
device that may be attached to a graphing calculator or to a computer in order
to gather data involving measurement (e.g., position, temperature, force).
second-degree polynomial.
A
polynomial in which at least one term, the variable, has an exponent 2, and for
no term is the exponent of the variable greater than 2, for example, 4x2 + 20 or x2 + 7x + 10.
second differences.
See
finite differences.
similar triangles.
Triangles
in which corresponding sides are proportional.
simulation.
A
probability experiment to estimate the likelihood of an event. For example,
tossing a coin is a simulation of whether the next person you meet will be male
or female.
slope.
A
measure of the steepness of a line, calculated as the ratio of the rise (vertical
distance travelled) to the run (horizontal distance travelled).
spreadsheet.
Computer
software that allows the entry of formulas for repeated calculation.
substitution.
The
process of replacing a variable by a value. See also method of substitution.
system of equations.
A
system of linear equations comprises two or more equations in two or more
variables. The solution to a system of linear equations in two variables is the
point of intersection of two straight lines.
table of values.
A
table used to record the coordinates of points in a relation. For example:
|
x |
y=3x -1 |
|
-1 |
-4 |
|
0 |
-1 |
|
1 |
2 |
|
2 |
5 |
variable.
A
symbol used to represent an unspecified number. For example, x and y
are variables in the expression x + 2y.
vertex.
A
point at which two sides of a polygon meet.
vertical stretch factor.
A
coefficient in an equation of a relation that causes stretching of the
corresponding graph in the vertical direction only. For example, the graph of y=3x2 would appear to be narrower than the graph of y=x2 because its y-coordinates are three times as great
for the same x-coordinate.
xy-plane.
A
coordinate system based on the intersection of two straight lines called axes,
which are usually perpendicular. The horizontal axis is the x-axis, and
the vertical axis is the y-axis. The point of intersection of the axes
is called the origin.
zeros of a function.
The
values of x for which a function has a value of zero. The zeros of a
function correspond to the x-intercepts of its graph.
8. Mathematics – Grade 11
Functions and Relations, Grade 11, University Preparation (MCR3U)
This course introduces some financial applications of mathematics, extends
students’ experiences with functions, and introduces second-degree relations.
Students will solve problems in personal finance involving applications of
sequences and series; investigate properties and applications of trigonometric
functions; develop facility in operating with polynomials, rational
expressions, and exponential expressions; develop an understanding of inverses
and transformations of functions; and develop facility in using function
notation and in communicating mathematical reasoning. Students will also
investigate loci and the properties and applications of conics.
Prerequisite: Principles of Mathematics, Grade 10, Academic
Financial Applications of Sequences and Series
Overall Expectations
By the end of this course, students will:
solve problems
involving arithmetic and geometric sequences and series;
solve problems
involving compound interest and annuities;
solve problems
involving financial decision making, using spreadsheets or other appropriate
technology.
Specific Expectations
Solving Problems Involving Arithmetic and Geometric Sequences and Series
By the end of this course, students will:
write terms of a sequence, given the formula for the nth term or given a recursion formula;
determine a formula for the nth term of a given
sequence (e.g., the nth term of
the sequence
,
,
,
. . . is 
);
identify sequences as arithmetic or geometric, or
neither;
determine the value of any term in an arithmetic or a
geometric sequence, using the formula for the nth term of the sequence;
determine the sum of the terms of an arithmetic or a
geometric series, using appropriate formulas and techniques.
Solving Problems Involving Compound Interest and Annuities
By the end of this course, students will:
derive the formulas for compound interest and present
value, the amount of an ordinary annuity, and the present value of an ordinary
annuity, using the formulas for the nth
term of a geometric sequence and the sum of the first n terms of a geometric series;
solve problems involving compound interest and
present value;
solve problems involving the amount and the present
value of an ordinary annuity;
demonstrate an understanding of the relationships
between simple interest, arithmetic sequences, and linear growth;
demonstrate an understanding of the relationships between
compound interest, geometric sequences, and exponential growth.
Solving Problems Involving Financial Decision Making
By the end of this course, students will:
analyse the effects of changing the conditions in
long-term savings plans (e.g., altering the frequency of deposits, the amount
of deposit, the interest rate, the compounding period, or a combination of
these) (Sample problem: Compare
the results of making an annual deposit of $1000 to an RRSP, beginning at age
20, with the results of making an annual deposit of $3000, beginning at age
50);
describe the manner in which interest is calculated
on a mortgage (i.e., compounded semi-annually but calculated monthly) and
compare this with the method of interest compounded monthly and calculated monthly;
generate amortization tables for mortgages, using
spreadsheets or other appropriate software;
analyse the effects of changing the conditions of a
mortgage (e.g., the effect on the length of time needed to pay off the mortgage
of changing the payment frequency or the interest rate);
communicate the solutions to problems and the
findings of investigations with clarity and justification.
Trigonometric Functions
Overall Expectations
By the end of this course, students will:
solve problems involving the sine law and the cosine
law in oblique triangles;
demonstrate an understanding of the meaning and
application of radian measure;
determine, through investigation, the relationships
between the graphs and the equations of sinusoidal functions;
solve problems involving models of sinusoidal
functions drawn from a variety of applications.
Specific Expectations
Solving Problems Involving the Sine Law and the Cosine Law in Oblique
Triangles
By the end of this course, students will:
determine the sine, cosine, and tangent of angles
greater than 90°, using a suitable technique (e.g., related angles, the unit
circle), and determine two angles that correspond to a given single
trigonometric function value;
solve problems in two dimensions and three dimensions
involving right triangles and oblique triangles, using the primary
trigonometric ratios, the cosine law, and the sine law (including the ambiguous
case).
Understanding the Meaning and Application of Radian Measure
By the end of this course, students will:
define the term radian
measure;
describe the relationship between radian measure and
degree measure;
represent, in applications, radian measure in exact
form as an expression involving
(e.g.,
,
2)
and in approximate form as a real number (e.g., 1.05);
determine the exact values of the sine, cosine, and
tangent of the special angles
0,
,
,
,
and
their multiples less than or equal to 2;
prove simple identities, using the Pythagorean
identity, sin2x + cos2x = 1,
and the quotient relation,
;
solve linear and quadratic trigonometric equations
(e.g., 6 cos2x – sin x – 4 = 0) on the interval 0
x 2;
demonstrate facility in the use of radian measure in
solving equations and in graphing.
Investigating the Relationships Between the Graphs and the Equations of
Sinusoidal Functions
By the end of this course, students will:
sketch the graphs of y = sin x
and y = cos x, and describe their periodic
properties;
determine, through investigation, using graphing
calculators or graphing software, the effect of simple transformations (e.g.,
translations, reflections, stretches) on the graphs and equations of y = sin x and y =
cos x;
determine the amplitude, period, phase shift, domain,
and range of sinusoidal functions whose equations are given in the form y = a
sin(kx + d) +
c or y = a cos(kx
+ d) + c;
sketch the graphs of simple sinusoidal functions
[e.g., y = a sin x,
y = cos kx, y
= sin(x + d), y
= a cos kx + c];
write the equation of a sinusoidal function, given
its graph and given its properties;
sketch the graph of y = tan x;
identify the period, domain, and range of the function; and explain the
occurrence of asymptotes.
Solving Problems Involving Models of Sinusoidal Functions
By the end of this course, students will:
determine, through investigation, the periodic
properties of various models (e.g., the table of values, the graph, the
equation) of sinusoidal functions drawn from a variety of applications;
explain the relationship between the properties of a
sinusoidal function and the parameters of its equation, within the context of
an application, and over a restricted domain;
predict the effects on the mathematical model of an
application involving sinusoidal functions when the conditions in the
application are varied;
pose and solve problems related to models of
sinusoidal functions drawn from a variety of applications, and communicate the
solutions with clarity and justification, using appropriate mathematical forms.
Tools for Operating and Communicating with Functions
Overall Expectations
By the end of this course, students will:
demonstrate facility in manipulating polynomials,
rational expressions, and exponential expressions;
demonstrate an understanding of inverses and
transformations of functions and facility in the use of function notation;
communicate mathematical reasoning with precision and
clarity throughout the course.
Specific Expectations
Manipulating Polynomials, Rational Expressions, and Exponential Expressions
By the end of this course, students will:
solve first-degree inequalities and represent the
solutions on number lines;
add, subtract, and multiply polynomials;
determine the maximum or minimum value of a quadratic
function whose equation is given in the form y
= ax2 + bx + c,
using the algebraic method of completing the square;
identify the structure of the complex number system
and express complex numbers in the form a +
bi, where i2 = –1 (e.g., 4i, 3 – 2i);
determine the real or complex roots of quadratic
equations, using an appropriate method (e.g., factoring, the quadratic formula,
completing the square), and relate the roots to the x-intercepts of the graph of the corresponding function;
add, subtract, multiply, and divide complex numbers
in rectangular form;
add, subtract, multiply, and divide rational
expressions, and state the restrictions on the variable values;
simplify and evaluate expressions containing integer
and rational exponents, using the laws of exponents;
solve exponential equations (e.g., 4x = 8x + 3, 22x – 2x = 12).
Understanding Inverses and Transformations and Using Function Notation
By the end of this course, students will:
define the term function;
demonstrate facility in the use of function notation
for substituting into and evaluating functions;
determine, through investigation, the properties of the
functions defined by ƒ(x) =
[e.g.,
domain, range, relationship to ƒ(x)
= x2] and ƒ(x) = 
[e.g.,
domain, range, relationship to ƒ(x)
= x.];
explain the relationship between a function and its
inverse (i.e., symmetry of their graphs in the line y = x; the interchange of x
and y in the equation of the
function; the interchanges of the domain and range), using examples drawn from
linear and quadratic functions, and from the functions ƒ(x) =
and
ƒ(x) = ;
represent inverse functions, using function notation,
where appropriate;
represent transformations (e.g., translations,
reflections, stretches) of the functions defined by ƒ(x) = x,
ƒ(x) = x2, ƒ(x) =
,
ƒ(x) = sin x, and ƒ(x) = cos x,
using function notation;
describe, by interpreting function notation, the
relationship between the graph of a function and its image under one or more
transformations;
state the domain and range of transformations of the
functions defined by ƒ(x) = x, ƒ(x)
= x2, ƒ(x) =
,
ƒ(x) = sin x, and ƒ(x) = cos x.
Communicating Mathematical Reasoning
By the end of this course, students will:
explain mathematical processes, methods of solution,
and concepts clearly to others;
present problems and their solutions to a group, and
answer questions about the problems and the solutions;
communicate solutions to problems and to findings of
investigations clearly and concisely, orally and in writing, using an effective
integration of essay and mathematical forms;
demonstrate the correct use of mathematical language,
symbols, visuals (e.g., diagrams, graphs), and conventions;
use graphing technology effectively (e.g., use
appropriate menus and algorithms; set the graph window to display the
appropriate section of a curve).
Investigations of Loci and Conics
Overall Expectations
By the end of this course, students will:
represent loci, using various models (e.g., a verbal
description, a diagram, a dynamic model, an equation);
determine the equation and the key features of a
conic;
solve problems involving applications of the conics.
Specific Expectations
Representing Loci
By the end of this course, students will:
construct a geometric model (e.g., a diagram created by
hand, a diagram created by using dynamic geometry software) to represent a
described locus of points; determine the properties of the geometric model; and
use the properties to interpret the locus (e.g., the locus of points
equidistant from two fixed points is the right bisector of the line segment
joining the two fixed points);
explain the process used in constructing a geometric
model of a described locus;
determine an equation to represent a described locus
[e.g., determine the equation of the locus of points equidistant from (–2, 7)
and (5, 4)];
construct geometric models to represent the locus
definitions of the conics;
determine equations for conics from their locus
definitions, by hand for simple particular cases [e.g., determine the equation
of the locus of points the sum of whose distances from (–3, 0) and (3, 0) is
10].
Determining the Equation and the Key Features of a Conic
By the end of this course, students will:
identify the standard forms for the equations of parabolas,
circles, ellipses, and hyperbolas having centres at (0, 0) and at (h, k);
identify the type of conic, given its equation in the
form ax2 + by2 + 2gx + 2fy + c
= 0;
determine the key features (e.g., the centre or the vertex,
the focus or foci, the asymptotes, the lengths of the axes) of a conic whose
equation is given in the form ax2 + by2 + 2gx + 2fy + c
= 0, by hand in simple cases (e.g., x2 + 9y2 – 6x + 36y – 36 = 0);
sketch the graph of a conic whose equation is given
in the form ax2 + by2 + 2gx + 2fy + c
= 0;
illustrate the conics as intersections of planes with
cones, using concrete materials or technology.
Solving Problems Involving Applications of the Conics
By the end of this course, students will:
describe the importance, within applications, of the
focus of a parabola, an ellipse, or a hyperbola (e.g., all incoming rays
parallel to the axis of a parabolic antenna are reflected through the focus;
the planets move in elliptical orbits with the sun at one of the foci);
pose and solve problems drawn from a variety of
applications involving conics, and communicate the solutions with clarity and
justification (Sample problem: A
parabolic antenna is 320 m wide at a distance of 50 m above its vertex. Determine
the distance above the vertex of the focus of the antenna);
solve problems involving the intersections of lines
and conics.
Functions, Grade 11, University/College Preparation (MCF3M)
This course introduces some financial applications of mathematics and
extends students’ experiences with functions. Students will solve problems in
personal finance involving applications of sequences and series; investigate
properties and applications of trigonometric functions; develop facility in
operating with polynomials, rational expressions, and exponential expressions;
develop an understanding of inverses and transformations of functions; and
develop facility in using function notation and in communicating mathematical
reasoning.
Prerequisite: Principles of Mathematics, Grade 10, Academic
Financial Applications of Sequences and Series
Overall Expectations
By the end of this course, students will:
solve problems
involving arithmetic and geometric sequences and series;
solve problems involving
compound interest and annuities;
solve problems
involving financial decision making, using spreadsheets or other appropriate
technology.
Specific Expectations
Solving Problems Involving Arithmetic and Geometric Sequences and Series
By the end of this course, students will:
write terms of a sequence, given the formula for the nth term;
determine a formula for the nth term of a given sequence (e.g., the nth term of the sequence
,
,
,
. . . is );
identify sequences as arithmetic or geometric, or
neither;
determine the value of any term in an arithmetic or a
geometric sequence, using the formula for the nth term of the sequence;
determine the sum of the terms of an arithmetic or a
geometric series, using appropriate formulas and techniques.
Solving Problems Involving Compound Interest and Annuities
By the end of this course, students will:
derive the formulas for compound interest and present
value, the amount of an ordinary annuity, and the present value of an ordinary
annuity, using the formulas for the nth
term of a geometric sequence and the sum of the first n terms of a geometric series;
solve problems involving compound interest and
present value;
solve problems involving the amount and the present
value of an ordinary annuity;
demonstrate an understanding of the relationships
between simple interest, arithmetic sequences, and linear growth;
demonstrate an understanding of the relationships
between compound interest, geometric sequences, and exponential growth.
Solving Problems Involving Financial Decision Making
By the end of this course, students will:
analyse the effects of changing the conditions in
long-term savings plans (e.g., altering the frequency of deposits, the amount
of deposit, the interest rate, the compounding period, or a combination of
these) (Sample problem: Compare
the results of making an annual deposit of $1000 to an RRSP, beginning at age
20, with the results of making an annual deposit of $3000, beginning at age
50);
describe the manner in which interest is calculated
on a mortgage (i.e., compounded semi-annually but calculated monthly) and
compare this with the method of interest compounded monthly and calculated
monthly;
generate amortization tables for mortgages, using
spreadsheets or other appropriate software;
analyse the effects of changing the conditions of a
mortgage (e.g., the effect on the length of time needed to pay off the mortgage
of changing the payment frequency or the interest rate);
communicate the solutions to problems and the
findings of investigations with clarity and justification.
Trigonometric Functions
Overall Expectations
By the end of this course, students will:
solve problems involving the sine law and the cosine
law in oblique triangles;
demonstrate an understanding of the meaning and
application of radian measure;
determine, through investigation, the relationships between
the graphs and the equations of sinusoidal functions;
solve problems involving models of sinusoidal
functions drawn from a variety of applications.
Specific Expectations
Solving Problems Involving the Sine Law and the Cosine Law in Oblique Triangles
By the end of this course, students will:
determine the sine, cosine, and tangent of angles
greater than 90°, using a suitable technique (e.g., related angles, the unit
circle), and determine two angles that correspond to a given single
trigonometric function value;
solve problems in two dimensions and three dimensions
involving right triangles and oblique triangles, using the primary
trigonometric ratios, the cosine law, and the sine law (including the ambiguous
case).
Understanding the Meaning and Application of Radian Measure
By the end of this course, students will:
define the term radian
measure;
describe the relationship between radian measure and
degree measure;
represent, in applications, radian measure in exact form
as an expression involving
(e.g.,
,
2)
and in approximate form as a real number (e.g., 1.05);
determine the exact values of the sine, cosine, and
tangent of the special angles 0,
,
,
,
and
their multiples less than or equal to 2;
prove simple identities, using the Pythagorean
identity, sin2x + cos2x = 1,
and the quotient relation,
;
solve linear and quadratic trigonometric equations
(e.g., 6 cos2x – sin x – 4 = 0) on the interval 0
x 2;
demonstrate facility in the use of radian measure in
solving equations and in graphing.
Investigating the Relationships Between the Graphs and the Equations of
Sinusoidal Functions
By the end of this course, students will:
sketch the graphs of y = sin x
and y = cos x, and describe their periodic
properties;
determine, through investigation, using graphing
calculators or graphing software, the effect of simple transformations (e.g.,
translations, reflections, stretches) on the graphs and equations of y = sin x and y =
cos x;
determine the amplitude, period, phase shift, domain,
and range of sinusoidal functions whose equations are given in the form y = a
sin(kx + d) + c
or y = a cos(kx
+ d) + c;
sketch the graphs of simple sinusoidal functions
[e.g., y = a sin x, y
= cos kx, y = sin(x + d), y = a
cos kx + c];
write the equation of a sinusoidal function, given
its graph and given its properties;
sketch the graph of
y = tan x; identify
the period, domain, and range of the function; and explain the occurrence of
asymptotes.
Solving Problems Involving Models of Sinusoidal Functions
By the end of this course, students will:
determine, through investigation, the periodic
properties of various models (e.g., the table of values, the graph, the
equation) of sinusoidal functions drawn from a variety of applications;
explain the relationship between the properties of a
sinusoidal function and the parameters of its equation, within the context of
an application, and over a restricted domain;
predict the effects on the mathematical model of an
application involving sinusoidal functions when the conditions in the
application are varied;
pose and solve problems related to models of
sinusoidal functions drawn from a variety of applications, and communicate the
solutions with clarity and justification, using appropriate mathematical forms.
Tools for Operating and Communicating with Functions
Overall Expectations
By the end of this course, students will:
demonstrate facility in manipulating polynomials,
rational expressions, and exponential expressions;
demonstrate an understanding of inverses and
transformations of functions and facility in the use of function notation;
communicate mathematical reasoning with precision and
clarity throughout the course.
Specific Expectations
Manipulating Polynomials, Rational Expressions, and Exponential Expressions
By the end of this course, students will:
solve first-degree inequalities and represent the
solutions on number lines;
add, subtract, and multiply polynomials;
determine the maximum or minimum value of a quadratic
function whose equation is given in the form y
= ax2 + bx + c,
using the algebraic method of completing the square;
identify the structure of the complex number system
and express complex numbers in the form a +
bi, where i2 = –1 (e.g., 4i, 3 – 2i);
determine the real or complex roots of quadratic
equations, using an appropriate method (e.g., factoring, the quadratic formula,
completing the square), and relate the roots to the x-intercepts of the graph of the corresponding function;
add, subtract, multiply, and divide rational
expressions, and state the restrictions on the variable values;
simplify and evaluate expressions containing integer
and rational exponents, using the laws of exponents;
solve exponential equations (e.g., 4x = 8x + 3, 22x
– 2x = 12).
Understanding Inverses and Transformations and Using Function Notation
By the end of this course, students will:
define the term function;
demonstrate facility in the use of function notation
for substituting into and evaluating functions;
determine, through investigation, the properties of
the functions defined by ƒ(x) =
[e.g.,
domain, range, relationship to ƒ(x)
= x2] and ƒ(x) = [e.g.,
domain, range, relationship to ƒ(x)
= x];
explain the relationship between a function and its
inverse (i.e., symmetry of their graphs in the line y = x; the interchange of x
and y in the equation of the
function; the interchanges of the domain and range), using examples drawn from
linear and quadratic functions, and from the functions ƒ(x) =
and
ƒ(x) = ;
represent inverse functions, using function notation,
where appropriate;
represent transformations (e.g., translations,
reflections, stretches) of the functions defined by ƒ(x) = x,
ƒ(x) = x2, ƒ(x) =
,
ƒ(x) = sin x, and ƒ(x) = cos x,
using function notation;
describe, by interpreting function notation, the
relationship between the graph of a function and its image under one or more
transformations;
state the domain and range of transformations of the
functions defined by ƒ(x) = x, ƒ(x)
= x2, ƒ(x) =
,
ƒ(x) = sin x, and ƒ(x) = cos x.
Communicating Mathematical Reasoning
By the end of this course, students will:
explain mathematical processes, methods of solution,
and concepts clearly to others;
present problems and their solutions to a group, and
answer questions about the problems and the solutions;
communicate solutions to problems and to findings of investigations
clearly and concisely, orally and in writing, using an effective integration of
essay and mathematical forms;
demonstrate the correct use of mathematical language,
symbols, visuals (e.g., diagrams, graphs), and conventions;
use graphing technology effectively (e.g., use
appropriate menus and algorithms; set the graph window to display the
appropriate section of a curve).
Mathematics of Personal Finance, Grade 11, College Preparation (MBF3C)
This course enables students to broaden their understanding of exponential growth and of important areas of personal finance. Students will investigate properties of exponential functions and develop skills in manipulating exponential expressions; solve problems and investigate financial applications involving compound interest and annuities; and apply mathematics in making informed decisions about transportation, accommodation, and career choices.
Prerequisite: Foundations of Mathematics, Grade 10, Applied
Models of Exponential Growth
Overall Expectations
By the end of this course, students will:
demonstrate an
understanding of the nature of exponential growth;
describe the
mathematical properties of exponential functions;
manipulate
expressions related to exponential functions.
Specific Expectations
Understanding the Nature of Exponential Growth
By the end of this course, students will:
describe the significance of exponential growth or
decay within the context of applications represented by various mathematical
models (e.g., tables of values, graphs, equations);
compare the effects of exponential growth within a
context (e.g., interest earned, population size) with the effects of linear or
quadratic growth within the same context;
pose and solve problems related to models of
exponential functions drawn from a variety of applications, and communicate the
solutions with clarity and justification.
Describing the Mathematical Properties of Exponential Functions
By the end of this course, students will:
sketch the graphs of simple exponential functions,
given their equations [e.g., those with equations y = 2x, y = 10x, y =
],
without using technology;
compare the rates of change of different types of
functions (e.g., those with equations y
= 2x, y = x2, y = 2x);
identify, through investigations, using graphing
calculators or graphing software, the key properties of exponential functions
with equations of the form y = ax (a > 0, a
1)
and their graphs (e.g., the domain is the set of the real numbers; the range is
the set of the positive real numbers; the function either increases or
decreases throughout its domain; the graph has the x-axis as an asymptote and has y-intercept = 1).
Manipulating Expressions
By the end of this course, students will:
demonstrate the quick recall or calculation of simple
powers of natural numbers (e.g., 28, 63, 54, 202), without
using technology;
evaluate simple numerical expressions involving
rational exponents, without using technology;
evaluate numerical expressions involving negative and
decimal exponents, using scientific calculators;
simplify algebraic expressions involving integral exponents,
using the laws of exponents;
solve exponential equations involving common bases
(e.g., 2x = 32, 45x – 1 = 4x + 11, 35x
+ 8 = 27x).
Applications of Compound Interest and Annuities
Overall Expectations
By the end of this course, students will:
solve problems involving arithmetic and geometric
sequences and series;
solve problems involving compound interest and
annuities;
demonstrate an understanding of the effect on
investment and borrowing of compounding interest.
Specific Expectations
Solving Problems Involving Arithmetic and Geometric Sequences and Series
By the end of this course, students will:
determine terms that follow three or more given terms
in a sequence;
determine whether a sequence is arithmetic or
geometric, or neither;
solve problems related to the formulas for the nth term and the sum of n terms of arithmetic and geometric
sequences and series.
Solving Problems Involving Compound Interest and Annuities
By the end of this course, students will:
solve problems involving the calculation of any
variable in the simple-interest formula (I
= Prt), using scientific calculators;
solve problems involving the calculation of the
amount (A) and the principal (P) in the compound-interest formula A = P(1 + i)n, using scientific calculators;
solve problems involving the calculation of the
interest rate per period (i)
and the number of periods (n)
in the compound-interest formula A = P(1
+ i)n, using a spreadsheet;
solve problems involving the calculation of the amount
and the regular payment in the formula for the amount of an ordinary annuity,
using scientific calculators;
solve problems involving the calculation of the
present value and the regular payment in the formula for the present value of
an ordinary annuity, using scientific calculators;
demonstrate an understanding of the relationships
between simple interest, arithmetic sequences, and linear growth;
demonstrate an understanding of the relationships
between compound interest, geometric sequences, and exponential growth.
Understanding the Effect of Compounding
By the end of this course, students will:
determine, through investigation, the characteristics
of various savings alternatives available from a financial institution (e.g.,
savings accounts, GICs);
determine the effect of compound interest on deposits
made into savings accounts (e.g., determine the doubling period of a single
deposit; demonstrate the effect of saving a small amount on a regular basis;
compare the effects of different compounding periods);
determine, through investigation, the properties of a
variety of investment alternatives (e.g., stocks, bonds, mutual funds, real
estate), and compare the alternatives from the point of view of risk versus
return;
demonstrate, through calculation, the advantages of
early deposits to long-term savings plans (e.g., compare the results of making
an annual deposit of $1000 to an RRSP, beginning at age 20, with the results of
making an annual deposit of $3000, beginning at age 50);
identify the common terminology and features
associated with mortgages;
describe the manner in which interest is usually
calculated on a mortgage (i.e., compounded semi-annually but calculated
monthly) and compare this with the method of interest compounded monthly and
calculated monthly;
generate an amortization table for a mortgage, using
a spreadsheet or other appropriate software;
calculate the total amount of interest paid over the
life of a mortgage, using a spreadsheet or other appropriate software, and
compare the amount with the original principal of the mortgage or value of the
property;
compare the effects of various payment periods,
payment amounts, and interest rates on the length of time needed to pay off a
mortgage;
demonstrate, through calculations, using technology,
the effect on interest paid of retiring a loan before it is due;
determine, through investigation, the features of
various credit and debit cards;
demonstrate, using technology, the effects of delayed
payment on a credit card balance, on the basis of current credit card rates and
regulations;
calculate the cost of borrowing to purchase a costly
item (e.g., a car, a stereo);
design an effective financial plan to facilitate the
achievement of a long-term goal (e.g., attending college, purchasing a car,
moving into an apartment, purchasing a house, establishing a small business).
Personal Financial Decisions
Overall Expectations
By the end of this course, students will:
demonstrate an understanding of the costs involved in
owning and operating a vehicle;
determine, through investigation, the relative costs
of renting an apartment and buying a house;
design effective personal and household budgets for
individuals and families described in case studies;
demonstrate the ability to make informed decisions
involving life situations;
apply decision making in the investigation of career
opportunities.
Specific Expectations
Owning and Operating a Vehicle
By the end of this course, students will:
identify the procedures, costs, advantages, and
disadvantages involved in buying a new vehicle and a used vehicle;
compare the costs involved in buying versus leasing
the same vehicle;
calculate the fixed and variable costs involved in owning
and operating a vehicle (e.g., the licence fee, insurance, maintenance);
determine, through investigation, the cost of
purchasing or leasing a chosen new vehicle or purchasing a chosen used vehicle,
including financing.
Renting or Buying Accommodation
By the end of this course, students will:
collect, organize, and analyse data involving the
costs of various kinds of accommodation in the community;
compare the costs of maintaining an apartment with
the costs of maintaining a house;
compare the advantages and disadvantages of renting
accommodation with the advantages and disadvantages of buying accommodation;
summarize the findings of investigations in effective
presentations, blending written and visual forms.
Designing Budgets
By the end of this course, students will:
describe and estimate the living costs involved for
different family groupings (e.g., a family of four, including two young
children; a single young person; a single parent with one child);
design a budget suitable for a family described in a
given case study, reflecting the current costs of common items (e.g., interest
rates, utility rates, rents), using technology (e.g., spreadsheets, budgeting
software, the Internet);
explain and justify budgets, using appropriate
mathematical forms (e.g., written explanations, charts, tables, graphs,
calculations);
determine the effect on an overall budget of changing
one component, using a spreadsheet or budgeting software.
Making Informed Decisions
By the end of this course, students will:
describe a decision involving a choice between
alternatives (e.g., Which program should I study at college? What car should I
buy? Should I stay at home or rent an apartment?);
collect relevant information related to the
alternatives to be considered in making a decision;
summarize the advantages and disadvantages of the
alternatives to a decision, using lists and organization charts;
compare alternatives by rating and ranking information
and by applying mathematical calculations and analysis, as appropriate (e.g.,
calculating loan payments or interest rates; constructing graphs or tables),
using technology;
explain the process used in making a decision and
justify the conclusions reached;
identify the advantages and disadvantages to the
purchaser of various types of selling (e.g., retail store, catalogue,
telemarketing, multilevel marketing, Internet) and techniques of selling (the
use of loss leaders, the use of incentives such as coupons or Air Miles);
compare the value of the Canadian dollar with the
values of foreign currencies over a period of time and identify possible
effects on purchasing and travel decisions.
Investigating Career Opportunities
By the end of this course, students will:
identify the advantages and disadvantages of a
variety of occupations of personal interest;
compare the expected income for a variety of
occupations with the costs of the education or training required;
analyse employment trends to identify some
occupations that are in high demand, and identify the skills required and the
education paths recommended in order to qualify for these occupations.
Mathematics for Everyday Life, Grade 11, Workplace Preparation (MEL3E)
This course enables students to broaden their understanding of mathematics
as it is applied in important areas of day-to-day living. Students will solve
problems associated with earning money, paying taxes, and making purchases;
apply calculations of simple and compound interest in saving, investing, and
borrowing; and calculate the costs of transportation and travel in a variety of
situations.
Prerequisite: Mathematics, Grade 9, Academic or Applied
Earning, Paying Taxes, and Purchasing
Overall Expectations
By the end of this course, students will:
solve problems involving different types of
remuneration;
describe various forms of taxation;
solve problems involving the purchasing of items.
Specific Expectations
Earning Money
By the end of this course, students will:
solve problems involving various ways that an
employee can be paid (e.g., salary, hourly rate, overtime, commission), using
calculators or appropriate software;
explain the differences between gross pay and net pay,
and describe possible payroll deductions (e.g., for a pension plan, a savings
plan, employment insurance, union dues);
calculate gross pay and net pay for given situations;
describe the effects on personal spending habits of
the frequency of pay period (e.g., weekly, biweekly, monthly);
determine the remuneration for chosen occupations,
including salary and benefits, and evaluate it in terms of purchasing power and
living standards.
Describing Forms of Taxation
By the end of this course, students will:
solve problems involving the estimation and
calculation of provincial and federal sales taxes;
identify the information and documents required for
filing a personal income tax return, and explain why they are required;
identify agencies in the community that will complete
or help to complete a personal income tax return;
identify other forms of taxation (e.g., taxes
included in the prices of gasoline and tobacco).
Purchasing Items
By the end of this course, students will:
provide the correct change for an amount offered
(e.g., provide the correct change for a charge of $13.87 when the amount
offered is $20.00; provide the correct change for a charge of $13.87 when the
amount offered is $15.12);
select compatible amounts to offer for a given charge
to reduce the number of coins received in the change (e.g., what payments might
a person offer for a charge of $46.36 in order to reduce the number of coins
received in the change?);
estimate and calculate the unit prices of comparable
items to determine the best buy;
estimate and calculate discounts, sale prices, and
after-tax costs;
identify various incentives to make purchases (e.g.,
Air Miles, coupons, stamps, interest-free loans), and explain their
characteristics;
estimate and calculate the price in Canadian funds of
items bought in or ordered from another country;
make a decision regarding the purchase of a costly
item by identifying and ranking criteria for the comparison of possible
choices;
identify, calculate, and compare the interest costs
involved in making purchases under various plans (e.g., instalment, layaway,
credit card, credit line), using technology (e.g., spreadsheets,
money-management software).
Saving, Investing, and Borrowing
Overall Expectations
By the end of this course, students will:
calculate simple and compound interest;
solve problems involving savings and investment
alternatives;
solve problems involving different ways of borrowing.
Specific Expectations
Calculating Simple and Compound Interest
By the end of this course, students will:
calculate interest earned and total amount in
applications involving simple interest;
describe the differences between simple interest and
compound interest;
calculate compound interest by using the
simple-interest formula and a given spreadsheet template;
solve problems involving the amount (A) resulting from compound-interest
calculations, using the formula A = P(1
+ i)n;
identify, through the use of technology, the effects
of different compounding periods on the amount of a loan or an investment;
construct graphs to represent the growth in the value
of an investment over time, using spreadsheets or graphing technology.
Understanding Saving and Investing
By the end of this course, students will:
explain the features of various savings alternatives
commonly available from financial institutions (e.g., savings accounts, GICs,
mutual funds);
identify the types of transactions available through
automated teller machines and online banking, and the fee(s) related to each
type;
interpret and check the accuracy of transaction codes
and entries on a monthly banking or financial statement or in a passbook;
determine the effect of compound interest on deposits
made into savings accounts, using a given spreadsheet template for repeated
calculations (e.g., the effect of saving a small amount on a regular basis);
identify the characteristics of different types of
investments (e.g., mutual funds, bonds, stocks);
demonstrate, through the use of technology, the
advantages of early deposits to long-term savings plans (e.g., compare the
results of making a deposit of $1000 to an RRSP, beginning at age 20, with the results
of making a deposit of $3000, beginning at age 50);
monitor the value of investments (e.g., mutual funds,
stocks) over a period of time, using technology (e.g., a spreadsheet, the
Internet);
demonstrate an understanding of risk tolerance and
how it changes during different life stages.
Understanding Borrowing
By the end of this course, students will:
describe the features of various credit cards and
debit cards;
demonstrate, through the use of technology, the
effects of delayed payment on a credit card balance, using current rates;
describe the features and conditions of various
short-term loans (e.g., car loans, loans to consolidate debt, lines of credit);
generate an amortization table for a personal loan
whose features are described, using a given spreadsheet template;
calculate the total amount of interest paid over the
life of a personal loan, using a given spreadsheet template, and compare this
amount with the original principal of the loan;
compare the effects of various payment periods on the
length of time needed to pay off loans, using a given spreadsheet template;
explain the advantages and disadvantages of
borrowing.
Transportation and Travel
Overall Expectations
By the end of this course, students will:
demonstrate an understanding of the costs involved in
owning and operating an automobile;
demonstrate an understanding of the costs involved in
travelling by automobile;
compare the costs of making a trip by automobile, by
train, by bus, or by airplane.
Specific Expectations
Understanding the Costs of Owning and Operating a Vehicle
By the end of this course, students will:
describe the procedures and costs involved in
obtaining a driver’s licence;
compare the procedures, costs, advantages, and
disadvantages involved in buying a new versus a used vehicle;
compare the costs involved in buying versus leasing
the same new vehicle;
identify the factors and costs involved in insuring a
vehicle;
identify the costs of failing to operate a vehicle
responsibly (e.g., fines, legal costs);
calculate the fixed and variable costs involved in
owning and operating a vehicle;
complete a project involving the purchase or lease of
a new vehicle or the purchase of a used vehicle, including the costs of
insurance;
compare the costs of owning or leasing and
maintaining a vehicle with the costs of using public transportation.
Understanding the Costs of Travelling by Automobile
By the end of this course, students will:
plan a travel route, by considering a variety of
factors (e.g., the estimated distances involved, the purpose of the trip, the
time of year, probable road conditions, personal interest);
estimate the costs involved in a trip by automobile
(e.g., gasoline, accommodation, food, entertainment), using real data acquired
from authentic sources (e.g., automobile association travel books, travel
guides, the Internet);
explain the cost estimate for a trip by automobile in
a clear, detailed presentation.
Comparing Travel Costs
By the end of this course, students will:
identify sources of information for routes,
schedules, and fares for travel by airplane, train, or bus;
interpret airline, train, or bus schedules;
compare the costs of travelling to a given
destination by airplane, train, or bus;
describe the advantages and disadvantages of
travelling to a given destination by airplane, train, and bus.
Mathematics Glossary, Grade 11
algebraic
expression. One or more
variables and possibly numbers and operation symbols. For example, 3x + 6, x,
and 5x are algebraic expressions.
algorithm. A systematic procedure for carrying out a
computation. For example, the addition algorithm is a set of rules for finding
the sum of two or more numbers.
alternate
angles. Two angles on
opposite sides of a transversal when it crosses two lines. The angles are equal
when the lines are parallel. The angles form one of these patterns: ,
.
analog
clock. A timepiece that
indicates the time through the position of its hands.
attribute. A quantitative or qualitative
characteristic of an object or a shape, for example, colour, size, thickness.
bar graph. See under graph.
bias. An emphasis on characteristics that are
not typical of an entire population.
binomial. An algebraic expression with two terms,
for example, 2x + 4y, 5k – 3n, and 2y2 + 5.
bisector. A line that divides a segment, an angle,
a line, or a figure into two equal halves.
broken-line
graph. See under graph.
calculation
method. Any of a variety
of methods used for solving problems, for example, estimation, mental
calculation, pencil-and-paper computation, the use of technology (including
calculators, computer spreadsheets).
capacity. The greatest amount that a container can
hold; usually measured in litres or millilitres.
Cartesian
coordinate grid. See
coordinate plane.
Cartesian
plane. See coordinate
plane.
census. The counting of an entire population.
circle
graph. See under graph.
clustering. See under estimation
strategies.
coefficient. Part of a term. In a term, the numerical
factor is the numerical coefficient, and the variable factor is the variable
coefficient. For example, in 5y, 5 is the numerical coefficient and y is the variable
coefficient.
comparative
bar graph. See under
graph.
compatible
numbers. Pairs of
numbers whose sum is a power of 10. For example, 30 + 70 = 100 (102).
complementary
angles. Two angles whose
sum is 90º.
composite
number. A number that
has factors other than itself and 1. For example, the number 8 has four
factors: 1, 2, 4, and 8.
computer
spreadsheet. Software
that helps to organize information using rows and columns.
concrete
graph. See under graph.
concrete
materials. Objects that
students handle and use in constructing their own understanding of mathematical
concepts and skills and in illustrating that understanding. Some examples are
base ten blocks, centicubes, construction kits, dice, games, geoboards,
geometric solids, hundreds charts, measuring tapes, Miras, number lines,
pattern blocks, spinners, and tiles. Also called manipulatives.
cone. A three-dimensional figure with a
circular base and a curved surface that tapers proportionately to an apex.
congruent
figures. Geometric
figures that have the same size and shape.
conservation. The property by which something remains
the same despite changes such as physical arrangement.
coordinate
graph. See under graph.
coordinate
plane. A plane that
contains an X-axis (horizontal) and a Y-axis (vertical). Also called Cartesian
coordinate grid or Cartesian plane.
coordinates. An ordered pair used to describe a
location on a grid or plane. For example, the coordinates (3, 5) describe a
location on a grid found by moving 3 units horizontally from the origin
(0, 0) followed by 5 units vertically.
data. Facts or information.
database. An organized and sorted list of facts or
information; usually generated by a computer.
degree. A unit for measuring angles.
dependent
variable. A variable
that changes as a result of a change in the independent variable.
diameter. A line segment that joins two points on
the circumference of a circle and passes through the centre.
displacement. The amount of fluid displaced by an
object placed in it.
distribution. A classification or an arrangement of
statistical information.
double bar
graph. See comparative
bar graph under graph.
equation. A mathematical statement that has
equivalent terms on either side of the equal sign.
equivalent
fractions. Fractions that
represent the same part of a whole or group, for example, 1/3 , 2/6, 3/9, 4/12.
equivalent
ratios. Ratios that
represent the same fractional number or amount, for example, 1:3, 2:6, 3:9.
estimation
strategies. Mental
mathematics strategies used to obtain an approximate answer. Students estimate
when an exact answer is not required and estimate to check the reasonableness
of their mathematics work. Some estimation strategies are:
clustering. A strategy used for estimating the sum of numbers
that cluster around one particular value. For example, the numbers 42, 47, 56,
55 cluster around 50. So estimate 50 + 50 + 50 + 50 = 200.
front-end
loading. The addition of significant
digits (those with the highest place value) with an adjustment of the remaining
values. Also called front loading. The following is an example of
front-end loading:
Step 1 - Add the
first digits in each number.
193 + 428 + 253
Think 100 + 400 + 200 = 700.
Step 2 - Adjust
the estimate to reflect the size of the remaining digits.
93 + 28 + 53 is approximately 175.
Think 700 + 175 = 875.
rounding. A process of replacing a number by an approximate
value of that number. For example, rounding to the nearest tens for 106 is 110.
event. One of several independent
probabilities.
expanded form. A way of writing numbers that shows the
value of each digit, for example, 432 = 4 x 100 + 3 x 10 + 2 x 1.
experimental
probability. The chance
of an event occurring based on the results of an experiment.
exponential
form. A shorthand method
for writing repeated multiplication. In 53, 3, which is the exponent, indicates that 5 is to be
multiplied by itself three times. 53 is in exponential form.
expression. A combination of numbers and variables
without an equal sign, for example, 3x + 5.
factors. See under multiplication.
first-hand
data. See primary
data.
flip. See reflection.
formula. A set of ideas, words, symbols, figures,
characters, or principles used to state a general rule. For example, the
formula for the area of a rectangle is A = l x w.
frequency. The number of times an event or item
occurs.
frequency
distribution. A table or
graph that shows how often each score, event, or measurement occurred.
front-end
loading. See under
estimation strategies.
graph. A representation of data in a pictorial
form. Some types of graphs are:
bar graph.
A diagram consisting of horizontal or vertical bars that represent data.
broken-line graph. On a coordinate grid, a display of data formed by line segments that
join points representing data.
circle graph. A graph in which a circle used to represent a whole is divided into
parts that represent parts of the whole.
comparative bar graph. A graph consisting of two or more bar graphs placed
side by side to compare the same thing. Also called double bar graph.
concrete graph. A graph in which real objects are used to represent pieces of
information.
coordinate graph. A grid that has data points named as ordered pairs of numbers, for
example, (4, 3).
histogram.
A type of bar graph in which each bar represents a range of values, and the
data are continuous.
pictograph.
A graph that illustrates data using pictures and symbols.
histogram. See under graph.
improper
fraction. A fraction
whose numerator is greater than its denominator, for example, 12/5.
independent
events. Two or more
events for which the occurrence or non-occurrence of one does not change the
probability of the other.
independent
variable. A variable
that does not depend on another for its value; a variable that the experimenter
purposely changes. Also called cause variable.
inequality. A statement using symbols to show that
one expression is greater than (>), less than (<), or not equal to
another expression.
integer. Any one of the numbers. . . , –4, –3,
–2, –1, 0, 1, 2, 3, 4, . . .
integral exponent. A power that has an integer as an
exponent.
intersecting
lines. Two lines with
exactly one point in common, the point of intersection.
interval. A space between two points. For example,
0–10 represents the interval from 0 to 10 inclusively.
irrational
number. A number that
cannot be represented as a terminating or repeating decimal, for example, .
irregular
polygon. A polygon whose
side and angle measures are not equal.
isometric
dot paper. Dot paper
formed by the vertices of equilateral triangles, used for three-dimensional
drawings. Also called triangular dot paper or triangle dot paper.
isosceles
triangle. A triangle
that has two sides of equal length.
linear
dimension. Dimension
involving the measurement of only one linear attribute, for example, length,
width, height, depth.
linear
relationship. A
relationship that has a straight-line graph.
line of best
fit. A line that can
sometimes be determined on a scatter plot. If a line of best fit can be found,
a relationship exists between the independent and dependent variables.
line of
symmetry. A line that
divides a shape into two parts that can be matched by folding the shape in
half.
manipulatives. See concrete materials.
many-to-one
correspondence. The
matching of elements in two sets in such a way that more than one element in
one set can be matched with one and only one element in another set, for
example, 3 pennies to each pocket.
mass. The amount of matter in an object;
usually measured in grams or kilograms.
mathematical
communication. The use
of mathematical language by students to:
respond to and describe the world around them;
communicate their attitudes about and interests in mathematics;
reflect and shape their understandings of and skills
in mathematics.
Students
communicate by talking, drawing pictures, drawing diagrams, writing journals,
charting, dramatizing, building with concrete materials, and using symbolic
language, (e.g., 2, >, =).
mathematical
concepts. The
fundamental understandings about mathematics that a student develops within
problem-solving contexts.
mathematical
language.
terminology (e.g., factor, pictograph, tetrahedron);
pictures/diagrams (e.g., 2 x 3 matrix, parallelogram,
tree diagram);
symbols, including numbers (e.g., 2, 1/4), operations
(e.g., 3 x 8 = [3 x 4] + [3 x 4]), and relations (e.g., 1/4 <).
mathematical
procedures. The skills, operations,
mechanics, manipulations, and calculations that a student uses to solve
problems.
mean. The average; the sum of a set of numbers
divided by the number of numbers in the set. For example, the average of 10 +
20 + 30 is 60 ÷ 3 = 20.
measure of
central tendency. A
value that can represent a set of data, for example, mean, median, mode. Also
called central measure.
median. The middle number in a set of numbers,
such that half the numbers in the set are less and half are greater when the
numbers are arranged in order. For example, 14 is the median for the set of
numbers 7, 9, 14, 21, 39. If there is an even number of numbers, the median is
the mean of the two middle numbers. For example, 11 is the median of 5, 10, 12,
and 28.
Mira. A transparent mirror used in geometry to
locate reflection lines, reflection images, and lines of symmetry, and to
determine congruency and line symmetry.
mixed
number. A number that is
the sum of a whole number and a fraction, for example, 81/4.
mode. The number that occurs most often in a
set of data. For example, in a set of data with the values 3, 5, 6, 5, 6, 5, 4,
5, the mode is 5.
modelling. A representation of the facts and
factors of, and the inferences to be drawn from, an entity or a situation.
monomial. An algebraic expression with one term,
for example, 2x or 5xy2.
multiple. The product of a given number and a
whole number. For example, 4, 8, 12, . . . are multiples of 4.
multiplication. An operation that combines numbers
called factors to give one number called a product. For example, 4 x 5 = 20;
thus factor x factor = product.
multi-step
problem. A problem whose
solution requires at least two calculations. For example, shoppers who want to
find out how much money they have left after a purchase follow these steps:
Step 1 - Add all items purchased (subtotal).
Step 2 - Multiply the sum of purchases by % of tax.
Step 3 - Add the tax to the sum of purchases (grand
total).
Step 4 - Subtract the grand total from the shopper's
original amount of money.
natural numbers. The counting numbers 1, 2, 3, 4, . . .
net. A pattern that can be folded to make a
three-dimensional figure.
network. A set of vertices joined by paths.
non-standard
units. Measurement units
used in the early development of measurement concepts, for example, paper
clips, cubes, hand spans, and so on.
number line. A line that matches a set of numbers and
a set of points one to one.
number operations. Mathematical processes or actions that
include the addition, subtraction, multiplication, and division of numbers.
nth term. The last of a series of terms.
obtuse
angle. An angle that
measures more than 90º and less than 180º.
one-to-one
correspondence. The
matching of elements in two sets in such a way that every element in one set
can be matched with one and only one element in another set.
ordered
pair. Two numbers in
order, for example, (2, 6). On a coordinate plane, the first number is the horizontal
coordinate of a point, and the second is the vertical coordinate of the point.
order of
operations. The rules
used to simplify expressions. Often the acronym BEDMAS is used to describe this
calculation process:
brackets
exponents
division or
multiplication, whichever comes first
addition or
subtraction, whichever comes first
ordinal
number. A number that
shows relative position or place, for example, first, second, third, fourth.
parallel
lines. Lines in the same
plane that do not intersect.
parallelogram. A quadrilateral whose opposite sides are
parallel.
perfect
square. The product of
an integer multiplied by itself. For example, 9 = 3 x 3; thus 9 is a perfect
square.
perpendicular
lines. Two lines that
intersect at a 90º angle.
pictograph. See under graph.
place value. The value given to the place in which a
digit appears in a numeral. In the number 5473, 5 is in the thousands place, 4
is in the hundreds place, 7 is in the tens place, and 3 is in the ones place.
plane shape. A two-dimensional figure.
polygon. A closed figure formed by three or more
line segments. Examples of polygons are triangles, quadrilaterals, pentagons,
octagons.
polyhedron. A three-dimensional object that has
polygons as faces.
polynomial. An algebraic expression. Examples of
polynomials are 6x, 3x – 2, and 4x2 + 5x – 4.
population. The total number of individuals or
items.
power. A number written in exponential form; a
shorter way of writing repeated multiplication. For example, 102 and 26 are powers.
primary
data. Information that
is collected directly or first-hand. Data from a person-on-the-street survey
are primary data. Also called first-hand data or primary-source data.
prime
factorization. An
expression showing a composite number as a product of its prime factors. The
prime factorization for 42 is 2 x 3 x 7.
prime
number. A whole number
greater than 1 that has only two factors, itself and 1. For example, 7 = 1 x 7.
prism. A three-dimensional figure with two
parallel and congruent bases. A prism is named by the shape of its bases, for
example, rectangular prism, triangular prism.
probability. A number that shows how likely it is
that an event will happen.
product. See under multiplication.
proper
fraction. A fraction
whose numerator is smaller than its denominator, for example, 2/3.
proportion. A number sentence showing that two
ratios are equal, for example, 2/3 = 6/9.
Pythagorean
theorem. In a right
triangle, the square of the length of the hypotenuse is equal to the sum of the
squares of the other two sides.
quadrilateral. A polygon with four straight sides.
radius. A line segment whose endpoints are the
centre of a circle and a point on the circle.
range. The difference between the highest and
lowest number in a group of numbers. For example, in a data set of 8, 32, 15,
10, the range is 24, that is, 32 – 8.
rate. A comparison of two numbers with
different units, such as kilometres and hours, for example, 100 km/h.
ratio. A comparison of numbers with the same
units, for example, 3:4 or 3/4.
rational
number. A number that
can be expressed as the quotient of two integers where the divisor is not 0.
reflection. A transformation that turns a figure
over an axis. The figure does not change size or shape, but it does change
position and orientation. A reflection image is the result of a reflection.
Also called flip.
regular
polygon. A closed figure
in which all sides and angles are equal.
rotation. A transformation that turns a figure
about a fixed point. The figure does not change size or shape, but it does
change position and orientation. A rotation image is the result of a rotation.
Also called turn.
rotational
symmetry. A shape that
fits onto itself after a turn less than a full turn has rotational symmetry.
For example, a square has a turn symmetry of order 4 because it resumes its
original orientation after each of 4 turns: 1/4 turn, 1/2 turn, 3/4 turn, and
full turn. Also called turn symmetry.
rounding. See under estimation
strategies.
sample. A small, representative group chosen
from a population and examined in order to make predictions about the
population. Also called sampling.
scale
drawing. A drawing in
which the lengths are a reduction or an enlargement of actual lengths.
scalene
triangle. A triangle
with three sides of different lengths.
scatter
plot. A graph that
attempts to show a relationship between two variables by means of points
plotted on a coordinate grid. Also called scatter diagram.
scientific
notation. A way of
writing a number as the product of a number between 1 and 10 and a power of 10.
In scientific notation, 58 000 000 is written 5.8 x 107.
secondary
data. Information that
is not collected first-hand, for example, data from a government document or a
database. Also called second-hand data or secondary-source data.
second-hand
data. See secondary
data.
sequence. A succession of things that are
connected in some way, for example, the sequence of numbers 1, 1, 2, 3, 5, . .
.
seriation
line. A line used for
the ordering of objects, numbers, or ideas.
shell. A three-dimensional figure whose interior
is completely empty.
SI. The international system of measurement
units, for example, centimetre, kilogram. (From the French Système
International.)
similar
figures. Geometric
figures that have the same shape but not always the same size.
simple interest. The formula used to calculate the
interest on an investment: I = PRT where P is the principal, R is the rate of
interest, and T is the time chosen to invest the principal.
simulation. A probability experiment to test the
likelihood of an event. For example, tossing a coin is a simulation of whether
the next person you meet is a male or a female.
skeleton. A three-dimensional figure showing only
the edges and vertices of the figure.
slide. See translation.
standard
form. A way of writing a
number in which each digit has a place value according to its position in
relation to the other digits. For example, 7856 is in standard form.
stem-and-leaf
plot. An organization of
data into categories based on place values.
supplementary
angles. Two angles whose
sum is 180º.
surface
area. The sum of the
areas of the faces of a three-dimensional object.
survey. A sampling of information, such as that
made by asking people questions or interviewing them.
symbol. See under mathematical
language.
systematic
counting. A process used
as a check so that no event or outcome is counted twice.
table. An orderly arrangement of facts set out
for easy reference, for example, an arrangement of numerical values in vertical
or horizontal columns.
tally chart. A chart that uses tally marks to count
data and record frequencies.
tangram. An ancient Chinese puzzle made from a
square cut into seven pieces: two large triangles, one medium-sized triangle,
two small triangles, one square, and one parallelogram.
term. Each of the quantities constituting a
ratio, a sum, or an algebraic expression.
tessellation. A tiling pattern in which shapes are
fitted together with no gaps or overlaps.
theoretical
probability. The number
of favourable outcomes divided by the number of possible outcomes.
tiling. The process of using repeated congruent
shapes to cover a region completely.
transformation. A change in a figure that results in a
different position, orientation, or size. The transformations include the translation
(slide), reflection (flip), rotation (turn), and dilatation
(reduction or enlargement).
translation. A transformation that moves a figure to
a new position in the same plane. The figure does not change size, shape, or
orientation; it only changes position. A translation image is the result of a
translation. Also called slide.
trapezoid. A quadrilateral with exactly one pair of
parallel sides.
tree
diagram. A branching
diagram that shows all possible combinations or outcomes.
turn. See rotation.
variable. A letter or symbol used to represent a
number.
Venn
diagram. A diagram
consisting of overlapping circles used to show what two or more sets have in
common.
vertex. The common endpoint of the two segments
or lines of an angle.
volume. The amount of space occupied by an object;
measured in cubic units such as cubic centimetres.
achievement
levels. Brief
descriptions of four different degrees of achievement of the provincial
curriculum expectations for any given grade. Level 3, which is the
"provincial standard", identifies a high level of achievement of the
provincial expectations. Parents of students achieving at level 3 in a
particular grade can be confident that their children will be prepared for work
at the next grade. Level 1 identifies achievement that falls much below the
provincial standard. Level 2 identifies achievement that approaches the
standard. Level 4 identifies achievement that surpasses the standard.
expectations. The knowledge and skills that students
are expected to develop and to demonstrate in their class work, on tests, and
in various other activities on which their achievement is assessed. The new
strands. The five major areas of knowledge and
skills into which the curriculum for Mathematics is organized. The strands for
Mathematics are: Number Sense and Numeration, Measurement, Geometry and Spatial
Sense, Patterning and Algebra, and Data Management and Probability.
Advanced Functions and Introductory Calculus, Grade 12, University Preparation (MCB4U)
This course builds on students’ experience with functions and introduces the
basic concepts and skills of calculus. Students will investigate and apply the
properties of polynomial, exponential, and logarithmic functions; broaden their
understanding of the mathematics associated with rates of change; and develop
facility with the concepts and skills of differential calculus as applied to
polynomial, rational, exponential, and logarithmic functions. Students will apply
these skills to problem solving in a range of applications.
Prerequisite: Functions and Relations, Grade 11, University Preparation, or Functions, Grade 11, University/College Preparation
Advanced Functions
Overall Expectations
By the end of this course, students will:
determine,
through investigation, the characteristics of the graphs of polynomial
functions of various degrees;
demonstrate
facility in the algebraic manipulation of polynomials;
demonstrate an understanding
of the nature of exponential growth and decay;
define and apply
logarithmic functions;
demonstrate an
understanding of the operation of the composition of functions.
Specific Expectations
Investigating the Graphs of Polynomial Functions
By the end of this course, students will:
determine,
through investigation, using graphing calculators or graphing software, various
properties of the graphs of polynomial functions (e.g., determine the effect of
the degree of a polynomial function on the shape of its graph; the effect of
varying the coefficients in the polynomial function; the type and the number of
x-intercepts; the behaviour
near the x-intercepts; the end
behaviours; the existence of symmetry);
describe the
nature of change in polynomial functions of degree greater than two, using
finite differences in tables of values;
compare the
nature of change observed in polynomial functions of higher degree with that
observed in linear and quadratic functions;
sketch the graph
of a polynomial function whose equation is given in factored form;
determine an
equation to represent a given graph of a polynomial function, using methods
appropriate to the situation (e.g., using the zeros of the function; using a
trial-and-error process on a graphing calculator or graphing software; using
finite differences).
Manipulating Algebraic Expressions
By the end of this course, students will:
demonstrate an
understanding of the remainder theorem and the factor theorem;
factor
polynomial expressions of degree greater than two, using the factor theorem;
determine, by
factoring, the real or complex roots of polynomial equations of degree greater
than two;
determine the
real roots of non-factorable polynomial equations by interpreting the graphs of
the corresponding functions, using graphing calculators or graphing software;
write the
equation of a family of polynomial functions, given the real or complex zeros
[e.g., a polynominal function having non-repeated zeros 5, –3, and –2 will be
defined by the equation ƒ(x) = k(x
– 5)(x + 3)(x + 2), for k 
];
describe
intervals and distances, using absolute-value notation;
solve factorable
polynomial inequalities;
solve
non-factorable polynomial inequalities by graphing the corresponding functions,
using graphing calculators or graphing software and identifying intervals above
and below the x-axis;
solve problems
involving the abstract extensions of algorithms (e.g., a problem involving the
nature of the roots of polynomial equations: If h and k are
the roots of the equation 3x2 + 28x – 20 = 0, find
the equation whose roots are h + k
and hk; a problem involving the
factor theorem: For what values of k
does the function ƒ(x) = x3 + 6x2 + kx – 4 give the same remainder when
divided by either x – 1 or x + 2?).
Understanding the Nature of Exponential Growth and Decay
By the end of this course, students will:
identify,
through investigations, using graphing calculators or graphing software, the key
properties of exponential functions of the form ax (a > 0, a 1) and their graphs (e.g., the
domain is the set of the real numbers; the range is the set of the positive
real numbers; the function either increases or decreases throughout its domain;
the graph has the x-axis as an
asymptote and has y-intercept =
1);
describe the
graphical implications of changes in the parameters a, b, and c in the equation y = cax + b;
compare the
rates of change of the graphs of exponential and non-exponential functions
(e.g., those with equations y =
2x, y = x2, y = 
and
y = 2x);
describe the
significance of exponential growth or decay within the context of applications
represented by various mathematical models (e.g., tables of values, graphs);
pose and solve
problems related to models of exponential functions drawn from a variety of
applications, and communicate the solutions with clarity and justification.
Defining and Applying Logarithmic Functions
By the end of this course, students will:
define the
logarithmic function logax (a > 1) as the inverse of the
exponential function ax, and compare the properties of the two functions;
express
logarithmic equations in exponential form, and vice versa;
simplify and
evaluate expressions containing logarithms;
solve
exponential and logarithmic equations, using the laws of logarithms;
solve simple
problems involving logarithmic scales (e.g., the Richter scale, the pH scale,
the decibel scale).
Understanding the Composition of Functions
By the end of this course, student will:
identify
composition as an operation in which two functions are applied in succession;
demonstrate an
understanding that the composition of two functions exists only when the range
of the first function overlaps the domain of the second;
determine the
composition of two functions expressed in function notation;
decompose a
given composite function into its constituent parts;
describe the
effect of the composition of inverse functions [i.e., ƒ(f–1(x)) = x].
Underlying Concepts of Calculus
Overall Expectations
By the end of this course, students will:
determine and
interpret the rates of change of functions drawn from the natural and social
sciences;
demonstrate an
understanding of the graphical definition of the derivative of a function;
demonstrate an
understanding of the relationship between the derivative of a function and the
key features of its graph.
Specific Expectations
Understanding Rates of Change
By the end of this course, students will:
pose problems
and formulate hypotheses regarding rates of change within applications drawn
from the natural and social sciences;
calculate and
interpret average rates of change from various models (e.g, equations, tables
of values, graphs) of functions drawn from the natural and social sciences;
estimate and
interpret instantaneous rates of change from various models (e.g, equations,
tables of values, graphs) of functions drawn from the natural and social
sciences;
explain the
difference between average and instantaneous rates of change within
applications and in general;
make inferences
from models of applications and compare the inferences with the original
hypotheses regarding rates of change.
Understanding the Graphical Definition of the Derivative
By the end of this course, students will:
demonstrate an
understanding that the slope of a secant on a curve represents the average rate
of change of the function over an interval, and that the slope of the tangent
to a curve at a point represents the instantaneous rate of change of the
function at that point;
demonstrate an
understanding that the slope of the tangent to a curve at a point is the
limiting value of the slopes of a sequence of secants;
demonstrate an
understanding that the instantaneous rate of change of a function at a point is
the limiting value of a sequence of average rates of change;
demonstrate an
understanding that the derivative of a function at a point is the instantaneous
rate of change or the slope of the tangent to the graph of the function at that
point.
Connecting Derivatives and Graphs
By the end of this course, students will:
describe the key
features of a given graph of a function, including intervals of increase and
decrease, critical points, points of inflection, and intervals of concavity;
identify the
nature of the rate of change of a given function, and the rate of change of the
rate of change, as they relate to the key features of the graph of that
function;
sketch, by hand,
the graph of the derivative of a given graph.
Derivatives and Applications
Overall Expectations
By the end of this course, students will:
demonstrate an
understanding of the first-principles definition of the derivative;
determine the
derivatives of given functions, using manipulative procedures;
determine the
derivatives of exponential and logarithmic functions;
solve a variety
of problems, using the techniques of differential calculus;
sketch the
graphs of polynomial, rational, and exponential functions;
analyse
functions, using differential calculus.
Specific Expectations
Understanding the First-Principles Definition of the Derivative
By the end of this course, students will:
determine the
limit of a polynomial, a rational, or an exponential function;
demonstrate an
understanding that limits can give information about some behaviours of graphs
of functions [e.g.,
predicts a hole
at (5, 10)];
identify examples
of discontinuous functions and the types of discontinuities they illustrate;
determine the
derivatives of polynomial and simple rational functions from first principles,
using the definitions of the derivative function,
identify
examples of functions that are not differentiable.
Determining Derivatives
By the end of this course, students will:
justify the
constant, power, sum-and- difference, product, quotient, and chain rules for
determining derivatives;
determine the derivatives
of polynomial and rational functions, using the constant, power,
sum-and-difference, product, quotient, and chain rules for determining
derivatives;
determine second
derivatives;
determine
derivatives, using implicit differentiation in simple cases (e.g., 4x2 + 9y2 = 36).
Determining the Derivatives of Exponential and Logarithmic Functions
By the end of this course, students will:
identify e as
|
|
|
|
|
lim |
|
|
|
|
|
|
and approximate
the limit, using informal methods;
define ln x as the inverse function of ex;
determine the
derivatives of the exponential functions ax and ex and the logarithmic functions logax and ln x;
determine the
derivatives of combinations of the basic polynomial, rational, exponential, and
logarithmic functions, using the rules for sums, differences, products,
quotients, and compositions of functions.
Using Differential Calculus to Solve Problems
By the end of this course, students will:
determine the
equation of the tangent to the graph of a polynomial, a rational, an
exponential, or a logarithmic function, or of a conic;
solve problems
of rates of change drawn from a variety of applications (including distance,
velocity, and acceleration) involving polynomial, rational, exponential, or
logarithmic functions;
solve optimization
problems involving polynomial and rational functions;
solve
related-rates problems involving polynomial and rational functions.
Sketching the Graphs of Polynomial, Rational, and Exponential Functions
By the end of this course, students will:
determine, from
the equation of a rational function, the intercepts and the positions of the
vertical and the horizontal or oblique asymptotes to the graph of the function;
determine, from
the equation of a polynomial, a rational, or an exponential function, the key
features of the graph of the function (i.e., intervals of increase and
decrease, critical points, points of inflection, and intervals of concavity),
using the techniques of differential calculus, and sketch the graph by hand;
determine, from
the equation of a simple combination of polynomial, rational, or exponential
functions (e.g., ƒ(x) = 
),
the key features of the graph of the combination of functions, using the
techniques of differential calculus, and sketch the graph by hand;
sketch the
graphs of the first and second derivative functions, given the graph of the
original function;
sketch the graph
of a function, given the graph of its derivative function.
Using Calculus Techniques to Analyse Models of Functions
By the end of this course, students will:
determine the
key features of a mathematical model of an application drawn from the natural
or social sciences, using the techniques of differential calculus;
compare the key
features of a mathematical model with the features of the application it
represents;
predict future
behaviour within an application by extrapolating from a mathematical model of a
function;
pose questions
related to an application and answer them by analysing mathematical models,
using the techniques of differential calculus;
communicate
findings clearly and concisely, using an effective integration of essay and
mathematical forms.
Geometry and Discrete Mathematics, Grade 12, University Preparation (MGA4U)
This course enables students to broaden mathematical knowledge and skills
related to abstract mathematical topics and to the solving of complex problems.
Students will solve problems involving geometric and Cartesian vectors, and
intersections of lines and planes in three-space. They will also develop an
understanding of proof, using deductive, algebraic, vector, and indirect
methods. Students will solve problems involving counting techniques and prove
results using mathematical induction.
Prerequisite: Functions and Relations, Grade 11, University Preparation
Geometry
Overall Expectations
By the end of this course, students will:
perform
operations with geometric and Cartesian vectors;
determine
intersections of lines and planes in three-space.
Specific Expectations
Operating with Geometric and Cartesian Vectors
By the end of this course, students will:
represent
vectors as directed line segments;
perform the
operations of addition, subtraction, and scalar multiplication on geometric
vectors;
determine the
components of a geometric vector and the projection of a geometric vector;
model and solve
problems involving velocity and force;
determine and
interpret the dot product and cross product of geometric vectors;
represent
Cartesian vectors in two-space and in three-space as ordered pairs or ordered
triples;
perform the
operations of addition, subtraction, scalar multiplication, dot product, and
cross product on Cartesian vectors.
Determining Intersections of Lines and Planes in Three-Space
By the end of this course, students will:
determine the
vector and parametric equations of lines in two-space and the vector, parametric,
and symmetric equations of lines in three-space;
determine the
intersections of lines in three-space;
determine the
vector, parametric, and scalar equations of planes;
determine the
intersection of a line and a plane in three-space;
solve systems of
linear equations involving up to three unknowns, using row reduction of
matrices, with and without the aid of technology;
interpret row
reduction of matrices as the creation of a new linear system equivalent to the
original;
determine the
intersection of two or three planes by setting up and solving a system of
linear equations in three unknowns;
interpret a
system of two linear equations in two unknowns and a system of three linear
equations in three unknowns geometrically, and relate the geometrical
properties to the type of solution set the system of equations possesses;
solve problems
involving the intersections of lines and planes, and present the solutions with
clarity and justification.
Proof and Problem Solving
Overall Expectations
By the end of this course, students will:
prove properties
of plane figures by deductive, algebraic, and vector methods;
solve problems,
using a variety of strategies;
complete
significant problem-solving tasks independently.
Specific Expectations
Proving Properties of Plane Figures by Deductive, Algebraic, and Vector
Methods
By the end of this course, students will:
demonstrate an
understanding of the principles of deductive proof (e.g., the role of axioms;
the use of “if . . . then” statements; the use of “if and only if” statements
and the necessity to prove them in both directions; the fact that the converse
of a proposition differs from the proposition) and of the relationship of
deductive proof to inductive reasoning;
prove some
properties of plane figures (e.g., circles, parallel lines, congruent
triangles, right triangles), using deduction;
prove some
properties of plane figures (e.g., the midpoints of the sides of a
quadrilateral are the vertices of a parallelogram; the line segment joining the
midpoints of two sides of a triangle is parallel to the third side)
algebraically, using analytic geometry;
prove some
properties of plane figures, using vector methods;
prove some properties
of plane figures, using indirect methods;
demonstrate an
understanding of the relationship between formal proof and the illustration of
properties that is carried out by using dynamic geometry software.
Using a Variety of Strategies to Solve Problems
By the end of this course, students will:
solve problems
by effectively combining a variety of problem-solving strategies (e.g.,
brainstorming, considering cases, choosing algebraic/geometric/vector or
direct/indirect approaches, working backwards, visualizing by using concrete
materials or diagrams or software, iterating, varying parameters, creating a
model, introducing a coordinate system);
generate
multiple solutions to the same problem;
use technology
effectively in making and testing conjectures;
solve complex
problems and present the solutions with clarity and justification.
Completing Significant Problem-Solving Tasks Independently
By the end of this course, students will:
solve problems of
significance, working independently, as individuals and in small groups;
solve problems
requiring effort over extended periods of time;
demonstrate
significant learning and the effective use of skills in tasks such as solving
challenging problems, researching problems, applying mathematics, creating
proofs, using technology effectively, and presenting course topics or
extensions of course topics.
Discrete Mathematics
Overall Expectations
By the end of this course, students will:
solve problems,
using counting techniques;
prove results,
using mathematical induction.
Specific Expectations
Using Counting Techniques
By the end of this course, students will:
solve problems,
using the additive and multiplicative counting principles;
express the answers
to permutation and combination problems, using standard combinatorial symbols
[e.g., 
,
P(n, r)];
evaluate
expressions involving factorial notation, using appropriate methods (e.g.,
evaluate mentally, by hand, by using a calculator);
solve problems
involving permutations and combinations, including problems that require the
consideration of cases;
explain
solutions to counting problems with clarity and precision;
describe the
connections between Pascal’s triangle, values of ,
and values for the binomial coefficients;
solve problems,
using the binomial theorem to determine terms in the expansion of a binomial.
Using Mathematical Induction to Prove Results
By the end of this course, students will:
demonstrate an
understanding of the principle of mathematical induction;
use sigma
notation to represent a series or the sum of a series;
prove the
formulas for the sums of series, using mathematical induction;
prove the
binomial theorem, using mathematical induction;
prove relationships
between the coefficients in Pascal’s triangle, by mathematical induction and
directly.
Mathematics of Data Management, Grade 12, University Preparation (MDM4U)
This course broadens students’ understanding of mathematics as it relates to
managing information. Students will apply methods for organizing large amounts
of information; apply counting techniques, probability, and statistics in
modelling and solving problems; and carry out a culminating project that
integrates the expectations of the course and encourages perseverance and
independence. Students planning to pursue university programs in business, the
social sciences, or the humanities will find this course of particular
interest.
Prerequisite: Functions and Relations, Grade 11, University Preparation, or Functions, Grade 11, University/College Preparation
Organization of Data for Analysis
Overall Expectations
By the end of this course, students will:
organize data to
facilitate manipulation and retrieval;
solve problems involving
complex relationships, with the aid of diagrams;
model situations
and solve problems involving large amounts of information, using matrices.
Specific Expectations
Organizing Data
By the end of this course, students will:
locate data to
answer questions of significance or personal interest, by searching
well-organized databases;
use the Internet
effectively as a source for databases;
create database
or spreadsheet templates that facilitate the manipulation and retrieval of data
from large bodies of information that have a variety of characteristics (e.g.,
a compact disc collection classified by artist, by date, by type of music).
Using Diagrams to Solve Problems
By the end of this course, students will:
represent simple
iterative processes (e.g., the water cycle, a person’s daily routine, the
creation of a fractal design), using diagrams that involve branches and loops;
represent
complex tasks (e.g., searching a list by using algorithms; classifying
organisms; calculating dependent or independent outcomes in probability) or
issues (e.g., the origin of global warming), using diagrams (e.g., tree
diagrams, network diagrams, cause-and-effect diagrams, time lines);
solve network
problems (e.g., scheduling problems, optimum-path problems, critical-path
problems), using introductory graph theory.
Using Matrices to Model and Solve Problems
By the end of this course, students will:
represent
numerical data, using matrices, and demonstrate an understanding of terminology
and notation related to matrices;
demonstrate
proficiency in matrix operations, including addition, scalar multiplication,
matrix multiplication, the calculation of row sums, and the calculation of
column sums, as necessary to solve problems, with and without the aid of
technology;
solve problems
drawn from a variety of applications (e.g., inventory control, production
costs, codes), using matrix methods.
Counting and Probability
Overall Expectations
By the end of this course, students will:
solve counting
problems and clearly communicate the results;
determine and
interpret theoretical probabilities, using combinatorial techniques;
design and carry
out simulations to estimate probabilities.
Specific Expectations
Solving Counting Problems
By the end of this course, students will:
use Venn
diagrams as a tool for organizing information in counting problems;
solve
introductory counting problems involving the additive and multiplicative
counting principles;
express the
answers to permutation and combination problems, using standard combinatorial
symbols, [e.g., ,
P(n, r)];
evaluate
expressions involving factorial notation, using appropriate methods (e.g.,
evaluating mentally, by hand, by using a calculator);
solve problems,
using techniques for counting permutations where some objects may be alike;
solve problems,
using techniques for counting combinations;
identify
patterns in Pascal’s triangle and relate the terms of Pascal’s triangle to
values of ,
to the expansion of a binomial, and to the solution of related problems (Sample problem: A girl’s school is 5
blocks west and 3 blocks south of her home. Assuming that she leaves home and
walks only west or south, how many different routes can she take to school?);
communicate
clearly, coherently, and precisely the solutions to counting problems.
Determining and Interpreting Theoretical Probabilities
By the end of this course, students will:
solve
probability problems involving combinations of simple events, using counting
techniques [i.e., P(A or B),
P(A and B),
and P(
A)];
identify
examples of discrete random variables (e.g., the sums that are possible when two
dice are rolled);
construct a
discrete probability distribution function by calculating the probabilities of
a discrete random variable;
calculate
expected values and interpret them within applications (e.g., lottery prizes,
tests of the fairness of games, estimates of wildlife populations) as averages
over a large number of trials;
determine
probabilities, using the binomial distribution (Sample problem: A light-bulb manufacturer estimates that
0.5% of the bulbs manufactured are defective. If a client places an order for
100 bulbs, what is the probability that at least one bulb is defective?);
interpret
probability statements, including statements about odds, from a variety of
sources.
Simulating and Predicting
By the end of this course, students will:
identify the
advantages of using simulations in contexts;
design and carry
out simulations to estimate probabilities in situations for which the
calculation of the theoretical probabilities is difficult or impossible (Sample problem: A set of 6 baseball cards
can be collected from cereal boxes. If the different cards are evenly
distributed throughout the boxes, carry out a simulation to determine the
probability of collecting one complete set in a purchase of 14 boxes);
assess the validity
of some simulation results by comparing them with the theoretical
probabilities, using the probability concepts developed in the course (Sample problem: A light-bulb manufacturer
estimates that 0.5% of the bulbs manufactured are defective. Carry out a
simulation to estimate the probability that at least one bulb is defective in
an order of 100 bulbs).
Statistics
Overall Expectations
By the end of this course, students will:
demonstrate an
understanding of standard techniques for collecting data;
analyse data
involving one variable, using a variety of techniques;
solve problems
involving the normal distribution;
describe the
relationship between two variables by interpreting the correlation coefficient;
evaluate the validity
of statistics drawn from a variety of sources.
Specific Expectations
Collecting Data
By the end of this course, students will:
demonstrate an
understanding of the purpose and the use of a variety of sampling techniques
(e.g., a simple random sample, a systematic sample, a stratified sample);
describe
different types of bias that may arise in surveys (e.g., response bias,
measurement bias, non-response bias, sampling bias);
illustrate
sampling bias and variability by comparing the characteristics of a known
population with the characteristics of samples taken repeatedly from that
population, using different sampling techniques;
organize and
summarize data from secondary sources (e.g., the Internet, computer databases),
using technology (e.g., spreadsheets, graphing calculators).
Analysing Data Involving One Variable
By the end of this course, students will:
compute, using
technology, measures of one-variable statistics (i.e., the mean, median, mode,
range, interquartile range, variance, and standard deviation), and demonstrate
an understanding of the appropriate use of each measure;
interpret
one-variable statistics to describe characteristics of a data set;
describe the
position of individual observations within a data set, using z-scores and percentiles.
Solving Problems Involving the Normal Distribution
By the end of this course, students will:
identify
situations that give rise to common distributions (e.g., bimodal, U-shaped,
exponential, skewed, normal);
demonstrate an
understanding of the properties of the normal distribution (e.g., the mean,
median, and mode are equal; the curve is symmetric about the mean; 68% of the
population are within one standard deviation of the mean) and use these
properties to solve problems;
make probability
statements about normal distributions, on the basis of information drawn from a
variety of applications.
Describing the Relationship Between Two Variables
By the end of this course, students will:
define the
correlation coefficient as a measure of the fit of a scatter graph to a linear
model;
calculate the
correlation coefficient for a set of data, using graphing calculators or
statistical software;
demonstrate an
understanding of the distinction between cause-effect relationships and the
mathematical correlation between variables;
describe
possible misuses of regression (e.g., use with too small a sample, use without
considering the effect of outliers, inappropriate extrapolation).
Evaluating Validity
By the end of this course, students will:
explain examples
of the use and misuse of statistics in the media;
assess the
validity of conclusions made on the basis of statistical studies, by analysing
possible sources of bias in the studies (e.g., sampling bias) and by
calculating and interpreting additional statistics, where possible (e.g.,
measures of central tendency, the standard deviation);
explain the
meaning and the use in the media of indices based on surveys (e.g., the
consumer price index, the cost of living index).
Integration of the Techniques of Data Management
Overall Expectations
By the end of this course, students will:
carry out a
culminating project on a topic or issue of significance that requires the
integration and application of the expectations of the course;
present a
project to an audience and critique the projects of others.
Specific Expectations
Carrying Out a Culminating Project
By the end of this course, students will:
pose a significant
problem whose solution would require the organization and analysis of a large
amount of data;
select and apply
the tools of the course (e.g., methods for organizing data, methods for
calculating and interpreting measures of probability and statistics, methods
for data collection) to design and carry out a study of the problem;
compile a clear,
well-organized, and fully justified report of the investigation and its
findings.
Presenting and Critiquing Projects
By the end of this course, students will:
create a summary
of a project to present within a restricted length of time, using
communications technology effectively;
answer questions
about a project, fully justifying mathematical reasoning;
critique the mathematical
work of others in a constructive fashion.
College and Apprenticeship Mathematics, Grade 12, College Preparation (MAP4C)
This course equips students with the mathematical knowledge and skills they
will need in many college programs. Students will use statistical methods to
analyse problems; solve problems involving the application of principles of
geometry and measurement to the design and construction of physical models;
solve problems involving trigonometry in triangles; and consolidate their
skills in analysing and interpreting mathematical models.
Prerequisite: Mathematics of Personal Finance, Grade 11, College Preparation, or Functions, Grade 11, University/College Preparation (or Functions and Relations, Grade 11, University Preparation)
Applications of Statistics
Overall Expectations
By the end of this course, students will:
collect,
analyse, and evaluate data involving one variable;
collect,
analyse, and evaluate data involving two variables;
analyse
significant problems or issues, using statistics;
evaluate the
validity of the use of statistics in the media.
Specific Expectations
Collecting, Analysing, and Evaluating Data Involving One Variable
By the end of this course, students will:
determine
appropriate methods for collecting, storing, and retrieving, from primary or
secondary sources, data involving one variable;
design
questionnaires for gathering data through surveys, giving consideration to
possible sources of bias;
demonstrate an
understanding of the distinction between the terms population and sample;
choose from and
apply a variety of sampling techniques (e.g., random, stratified);
represent data
in appropriate graphical forms (e.g., histograms, bar graphs), using
technology;
identify and describe
properties of common distributions of data (e.g., normal, bimodal, exponential,
skewed);
calculate the
mean, median, mode, range, variance, and standard deviation of a data set,
using standard statistical notation and technology;
describe the significance
of results drawn from analysed data (e.g., the shape of the distribution, the
mean, the standard deviation);
make and justify
statements about a population on the basis of sample data.
Collecting, Analysing, and Evaluating Data Involving Two Variables
By the end of this course, students will:
determine
appropriate methods for collecting, storing, and retrieving, from primary or
secondary sources, data involving two variables;
construct a
scatter plot to represent data, using technology;
determine an
equation of a line of best fit, using the regression capabilities of graphing
technology;
calculate and
interpret the correlation coefficient, using appropriate technology;
describe
possible misuses of regression (e.g., use with too small a sample, use without
considering the effect of outliers, inappropriate extrapolation);
describe the
relationship between two variables suggested by a scatter plot (e.g., no
relationship, a positive correlation, a negative correlation);
make and justify
statements about a population on the basis of sample data.
Analysing Problems
By the end of this course, students will:
collect,
organize, and analyse data to address problems or issues, and calculate
relevant statistical measures;
formulate a summary
conclusion to a problem or an issue, by synthesizing interpretations of
individual statistical measures;
formulate
extending questions related to the conclusion reached in the investigation of a
problem or an issue;
communicate the
process used and the conclusions reached in the investigation of a problem or
an issue, using appropriate mathematical forms (e.g., oral and written
explanations, tables, graphs, formulas).
Evaluating Validity
By the end of this course, students will:
explain the use
and misuse in the media of graphs and commonly used statistical terms (e.g., percentile), and expressions (e.g., 19 times out of 20);
assess the
validity of conclusions made on the basis of statistical studies, by analysing
possible sources of bias in the studies (e.g., sampling bias);
explain the
meaning, and the use in the media, of indices based on surveys (e.g., the
consumer price index).
Applications of Geometry, Measurement, and Trigonometry
Overall Expectations
By the end of this course, students will:
demonstrate an
understanding of the relationship between three-dimensional objects and their
two-dimensional representations;
solve problems
involving measurement;
solve problems
involving trigonometry in triangles.
Specific Expectations
Understanding Two-Dimensional and Three-Dimensional Shapes
By the end of this course, students will:
identify,
through observation and measurement, the uses of geometric shapes and the
reasons for those uses, in a variety of applications (e.g., product design,
architecture, fashion);
represent
three-dimensional objects in a variety of ways (e.g., front, side, and top
views; perspective drawings; scale models), using concrete materials and design
or drawing software;
create nets,
plans, and patterns from physical models related to a variety of applications
(e.g., fashion design, interior decorating, building construction), using
design or drawing software;
design and
construct physical models of things (e.g., structures, equipment, furniture),
satisfying given constraints and using concrete materials, design software, or
drawing software.
Solving Problems Involving Measurement
By the end of this course, students will:
solve problems
related to the perimeter and area of plane figures, and the surface area and
volume of prisms, pyramids, cylinders, spheres, and cones, including problems
involving combinations of these objects;
demonstrate
accuracy and precision in working with metric measures;
demonstrate an
understanding of the use of the imperial system in a variety of applications
(e.g., bolt and screw sizes; tool sizes; quantities of soil, water, or cement);
demonstrate a
working knowledge of the measurement of length and area in the imperial system,
in relation to applications (e.g., design, construction);
perform required
conversions between the imperial system and the metric system, as necessary
within projects and applications;
use calculators effectively
in solving problems involving measurement, and judge the reasonableness of the
answers produced.
Solving Problems Involving Trigonometry in Triangles
By the end of this course, students will:
solve problems
involving trigonometry in right triangles;
demonstrate an
understanding of the signs of the sine, cosine, and tangent of obtuse angles;
determine side
lengths and angle measures in oblique triangles, using the cosine law and the
sine law, and solve related problems;
identify
applications of trigonometry in occupations and in postsecondary programs
related to the occupations.
Analysis of Mathematical Models
Overall Expectations
By the end of this course, students will:
interpret and
analyse given graphical models;
interpret and analyse
given formulaic models;
interpret and
analyse data given in a variety of forms.
Specific Expectations
Interpreting and Analysing Given Graphical Models
By the end of this course, students will:
interpret a
given linear, quadratic, or exponential graph to answer questions, using
language and units appropriate to the context from which the graph was drawn;
interpret the
rate of change and initial conditions (i.e., the slope and y-intercept) of a linear model given
within a context;
make and justify
a decision or prediction and discuss trends based on a given graph;
describe the
effect on a given graph of new information about the circumstances represented
by the graph (e.g., describe the effect of a significant change in population on
a graph representing the size of the population over time);
communicate the
results of an analysis orally, in a written report, and graphically.
Interpreting and Analysing Given Formulaic Models
By the end of this course, students will:
evaluate any
variable in a given formula drawn from an application by substituting into the
formula and using the appropriate order of operations on a scientific
calculator;
construct (e.g.,
combine or modify) formulas to solve multi-step problems in particular situations
(e.g., determine the amount of paint required to paint two coats on a large
cylindrical water tank);
rearrange a
formula to isolate any variable in it (e.g., to determine the values of a
variable in a formula, using a spreadsheet);
judge the reasonableness
of answers to problems;
demonstrate
mastery of key algebraic skills, including the ability to solve linear
equations, to solve systems of linear equations, to graph a linear function
from its equation, and to determine the slope and intercepts of a linear
function from its equation;
factor
expressions of the form ax2 + bx + c;
solve quadratic
equations by factoring.
Interpreting and Analysing Data Given in a Variety of Forms
By the end of this course, students will:
retrieve information
from various sources (e.g., graphs, charts, spreadsheets, schedules);
identify options
that meet certain criteria, using more than one chart, spreadsheet, or schedule
(e.g., the schedules of connecting flights; the spreadsheets of mortgage- payment
plans);
make informed
decisions, using data provided in chart, spreadsheet, or schedule format and
taking into account personal needs and preferences;
enter data or a
formula into a graphing calculator and retrieve other forms of the model (e.g.,
enter data and retrieve a scatter graph or a table of values; enter a formula
and retrieve a table of values or the graph of a function).
Mathematics for College Technology, Grade 12, College Preparation (MCT4C)
This course equips students with the mathematical knowledge and skills
needed for entry into college technology programs. Students will investigate
and apply properties of polynomial, exponential, and logarithmic functions;
solve problems involving inverse proportionality; and explore the properties of
reciprocal functions. They will also analyse models of a variety of functions,
solve problems involving piecewise-defined functions, solve linear-quadratic
systems, and consolidate key manipulation and communication skills.
Prerequisite: Functions, Grade 11, University/College Preparation (or Functions and Relations, Grade 11, University Preparation)
Polynomial Functions and Inverse Proportionality
Overall Expectations
By the end of this course, students will:
determine,
through investigation, the characteristics of the graphs of polynomial
functions of various degrees;
demonstrate
facility in the algebraic manipulation of polynomials;
demonstrate an
understanding of inverse proportionality;
determine, through
investigation, the key properties of reciprocal functions.
Specific Expectations
Investigating the Graphs of Polynomial Functions
By the end of this course, students will:
determine,
through investigation, using graphing calculators or graphing software, various
properties of the graphs of polynomial functions (e.g., determine the effect of
the degree of a polynomial function on the shape of its graph; the effect of
varying the coefficients in the polynomial function; the type and the number of
x-intercepts; the behaviour
near the x-intercepts; the end
behaviours; the existence of symmetry);
describe the
nature of change in polynomial functions of degree greater than two, using
finite differences in tables of values;
compare the
nature of change observed in polynomial functions of higher degree with that
observed in linear and quadratic functions;
sketch the graph
of a polynomial function whose equation is given in factored form;
determine an equation
to represent a given graph of a polynomial function, using methods appropriate
to the situation (e.g., using the zeros of the function; using a
trial-and-error process on a graphing calculator or graphing software; using
finite differences).
Manipulating Algebraic Expressions
By the end of this course, students will:
demonstrate an
understanding of the remainder theorem and the factor theorem;
factor
polynomial expressions of degree greater than two, using the factor theorem;
determine, by
factoring, the real or complex roots of polynomial equations of degree greater
than two;
determine the
real roots of non-factorable polynomial equations by interpreting the graphs of
the corresponding functions, using graphing calculators or graphing software;
write the
equation of a family of polynomial functions, given the real or complex zeros
[e.g., a polynomial function having non-repeated zeros 5, –3, and –2 will be
defined by the equation ƒ(x) = k(x
– 5) (x + 3)(x + 2), for k ];
describe intervals
and distances, using absolute-value notation;
solve factorable
polynomial inequalities;
solve
non-factorable polynomial inequalities by graphing the corresponding functions,
using graphing calculators or graphing software and identifying intervals above
and below the x-axis.
Understanding Inverse Proportionality
By the end of this course, students will:
construct tables
of values, graphs, and formulas to represent functions of inverse
proportionality derived from descriptions of realistic situations (e.g., the
time taken to complete a job varies inversely as the number of workers; the
intensity of light radiating equally in all directions from a source varies
inversely as the square of the distance between the source and the observer);
solve problems
involving relationships of inverse proportionality.
Determining the Key Properties of Reciprocal Functions
By the end of this course, students will:
sketch the graph
of the reciprocal of a given linear or quadratic function by considering the implications
of the key features of the original function as predicted from its equation
(e.g., such features as the domain, the range, the intervals where the function
is positive or negative, the intervals where the function is increasing or
decreasing, the zeros of the function);
describe the
behaviour of a graph near a vertical asymptote;
identify the
horizontal asymptote of the graph of a reciprocal function by examining the
patterns in the values of the given function.
Exponential and Logarithmic Functions
Overall Expectations
By the end of this course, students will:
demonstrate an
understanding of the nature of exponential growth and decay;
define and apply
logarithmic functions.
Specific Expectations
Understanding the Nature of Exponential Growth and Decay
By the end of this course, students will:
identify,
through investigations, using graphing calculators or graphing software, the
key properties of exponential functions of the form ax (a > 0, a 1)
and their graphs (e.g., the domain is the set of the real numbers; the range is
the set of the positive real numbers; the function either increases or
decreases throughout its domain; the graph has the x-axis as an asymptote and has y-intercept = 1);
describe the
graphical implications of changes in the parameters a, b, and c in the equation y = cax + b;
compare the
rates of change of the graphs of exponential and non-exponential functions
(e.g., those with equations y =
2x, y = x2, y = ,
and y = 2x);
describe the
significance of exponential growth or decay within the context of applications
represented by various mathematical models (e.g., tables of values, graphs,
equations);
pose and solve
problems related to models of exponential functions drawn from a variety of
applications, and communicate the solutions with clarity and justification.
Defining and Applying Logarithmic Functions
By the end of this course, students will:
define the
logarithmic function logax (a > 1) as the inverse of the
exponential function ax, and compare the properties of the two functions;
express
logarithmic equations in exponential form, and vice versa;
simplify and
evaluate expressions containing logarithms, using the laws of logarithms;
solve simple
problems involving logarithmic scales (e.g., the Richter scale, the pH scale,
the decibel scale).
Applications and Consolidation
Overall Expectations
By the end of this course, students will:
analyse models
of linear, quadratic, polynomial, exponential, or trigonometric functions drawn
from a variety of applications;
analyse and
interpret models of piecewise-defined functions drawn from a variety of
applications;
solve
linear-quadratic systems and interpret their solutions within the contexts of
applications;
demonstrate
facility in carrying out and applying key manipulation skills.
Specific Expectations
Analysing Models of Functions
By the end of this course, students will:
determine the
key features of a mathematical model (e.g., an equation, a table of values, a
graph) of a function drawn from an application;
compare the key
features of a mathematical model with the features of the application it
represents;
predict future
behaviour within an application by extrapolating from a given model of a
function;
pose questions
related to an application and use a given function model to answer them.
Analysing and Interpreting Models of Piecewise-Defined Functions
By the end of this course, students will:
demonstrate an
understanding that some naturally occurring functions cannot be represented by
a single formula (e.g., the temperature at a particular location as a function
of time);
graph a
piecewise-defined function, by hand and by using graphing calculators or
graphing software;
analyse and
interpret a given mathematical model of a piecewise-defined function, and
relate the key features of the model to the characteristics of the application
it represents;
make predictions
and answer questions about an application represented by a graph or formula of
a piecewise-defined function;
determine the
effects on the graph and formula of a piecewise-defined function of changing
the conditions in the situation that the function represents.
Solving Linear-Quadratic Systems
By the end of this course, students will:
determine the
key properties of a linear function or a quadratic function, given the equation
of the function, and interpret the properties within the context of an
application;
solve
linear-quadratic systems arising from the intersections of the graphs of linear
and quadratic functions;
interpret the
solution(s) to a linear- quadratic system within the context of an application.
Consolidating Key Skills
By the end of this course, students will:
perform
numerical computations effectively, using mental mathematics and estimation;
solve problems
involving ratio, rate, and percent drawn from a variety of applications;
solve problems involving
trigonometric ratios in right triangles and the sine and cosine laws in oblique
triangles;
demonstrate
facility in using manipulation skills related to solving linear, quadratic, and
polynomial equations, simplifying rational expressions, and operating with
exponents.
Mathematics for Everyday Life, Grade 12, Workplace Preparation (MEL4E)
This course enables students to broaden their understanding of mathematics
as it is applied in important areas of day-to-day living. Students will use
statistics in investigating questions of interest and apply principles of
probability in familiar situations. They will also investigate accommodation
costs and create household budgets; solve problems involving estimation and
measurement; and apply concepts of geometry in the creation of designs.
Prerequisite: Mathematics for Everyday Life, Grade 11, Workplace Preparation
Statistics and Probability
Overall Expectations
By the end of this course, students will:
construct and interpret graphs;
formulate questions, and collect and organize data
related to the questions;
apply principles of probability to familiar
situations;
interpret statements about statistics and probability
arising from familiar situations and the media.
Specific Expectations
Constructing and Interpreting Graphs
By the end of this course, students will:
represent given data in a variety of graphical forms,
using spreadsheets or other suitable graphing technology;
select an effective graphical form for a given set of
data and explain reasons for the choice;
interpret graphs by identifying trends and describing
the meaning of the trends within the context of the data.
Collecting and Organizing Data
By the end of this course, students will:
identify issues or questions of interest and collect
related data, using an appropriate sampling technique;
construct tables and graphs to represent collected
data, using spreadsheets or other suitable graphing technology;
draw appropriate conclusions about questions or
issues on the basis of the interpretation of graphs;
explain conclusions clearly.
Applying Principles of Probability
By the end of this course, students will:
express probabilities of simple events as the number
of favourable outcomes divided by the total number of outcomes;
express probabilities as fractions, decimals, and
percents, and interpret probabilities expressed in each of these forms;
describe the results obtained in carrying out probability
experiments related to familiar situations involving chance (e.g., rolling
dice, spinning spinners, flipping coins);
compare predicted and experimental results for
familiar situations involving chance, using technology to extend the number of experimental
trials (e.g., using a random number generator on a spreadsheet or on a graphing
calculator);
simulate familiar situations involving chance and
explain the choice of simulation (e.g., simulate the gender of children in a
family by the repeated flipping of a coin and explain why coin flipping was
used).
Interpreting Statements About Statistics and Probability
By the end of this course, students will:
interpret information about probabilities to assist
in making informed decisions in a variety of situations (e.g., evaluating risk
versus reward in the purchase of lottery tickets);
interpret and assess statistical and probabilistic
information used in the media and in common conversation (e.g., vague
statements such as “four out of five dentists recommend”; statements about
odds; scales on graphs).
Everyday Financing
Overall Expectations
By the end of this course, students will:
determine the costs involved in renting an apartment;
determine the costs involved in buying a house;
design household budgets for given circumstances.
Specific Expectations
Determining the Costs of Renting an Apartment
By the end of this course, students will:
determine, through investigation, the costs of
apartment rentals in the surrounding community;
describe the alternatives available (e.g., leasing,
renting month to month) and the procedures involved (e.g., paying a deposit) in
renting an apartment;
describe the rights and responsibilities of an
apartment tenant and an apartment landlord;
calculate the monthly costs involved in maintaining
an apartment.
Determining the Costs of Buying a House
By the end of this course, students will:
determine, through investigation, patterns in the
cost of housing in the surrounding community (e.g., what kind of house can be
purchased for $75 000? $140 000? $250 000? $400 000?);
describe the procedures and costs involved in
purchasing a house;
identify the costs involved in maintaining a house;
calculate the monthly costs involved in maintaining a
given house.
Designing Budgets
By the end of this course, students will:
identify typical components (e.g., accommodation,
food, savings) and their dispersion in a household budget;
determine the type of housing affordable in the surrounding
community by a person with a given income and family responsibilities;
design an appropriate monthly budget for a person
living in the surrounding community who has a given income, family
responsibilities, and long-term savings goals;
present a budget in a clear fashion, using
appropriate mathematical forms (e.g., written or oral explanations, charts,
tables, graphs, calculations);
investigate the effect on an overall budget of
changing one component, using a given spreadsheet template or budgeting
software.
Applications of Measurement and Geometry
Overall Expectations
By the end of this course, students will:
use measurement and strategies of estimation in a
variety of applications;
solve problems involving measurement and design;
apply transformation geometry in creating effective
designs.
Specific Expectations
Measuring and Estimating
By the end of this course, students will:
demonstrate a working knowledge of the metric system;
measure lengths accurately, using the metric system
and the imperial system;
estimate distances in metric units and in imperial
units by applying personal referents (e.g., the width of a finger is
approximately 1 cm; the length of a piece of standard loose-leaf paper is about
1 foot);
estimate capacities in metric units by applying
personal referents (e.g., a can of pop is about 350 mL);
estimate, with reasonable accuracy, large numbers
that are illustrated visually (e.g., books on a wall in a library, pictures of
crowds, populations of high-rise buildings), and explain the strategies used.
Solving Problems Involving Measurement and Design
By the end of this course, students will:
demonstrate an understanding of the Pythagorean
theorem, by constructing on a floor a rectangular region having accurate
right-angled corners;
determine the perimeter and area of regular and
irregular figures from given diagrams;
estimate, with reasonable accuracy, perimeters and
areas of large regions (e.g., a playing field), and explain the strategies
used;
demonstrate an understanding of the effect on the
area of familiar objects (e.g., a photograph, a television screen, a road map)
of multiplying each dimension by the same factor;
make a two-dimensional scale drawing of a room, using
design or drawing software effectively;
create a three-dimensional drawing of the interior of
a room, using design or drawing software effectively;
construct, with reasonable accuracy, a scale model of
an environment of personal interest (e.g., a building, a garden, a bridge);
estimate and calculate the surface area and volume of
objects and containers in the surrounding environment that approximate the
shape of rectangular prisms and cylinders;
investigate the making of a household improvement
(e.g, landscaping a property, decorating a room), design the improvement, and
estimate and calculate the cost, using technology (e.g., spreadsheets, design
or drawing software).
Applying Transformation Geometry
By the end of this course, students will:
describe the use of translations, reflections,
rotations, and dilatations as they relate to symmetry and design in logos, with
the aid of technology (e.g., dynamic geometry software, design or drawing
software);
analyse the geometric aspects of interesting and
appealing applications (e.g, logos found in advertising, designs found in
fabric or wallpaper);
create a personal logo, using the mathematics of symmetry,
translations, reflections, rotations, or dilatations, with the aid of
technology (e.g., dynamic geometry software, design or drawing software);
determine, through investigations, using concrete
materials and technology, the characteristics of shapes that will tile the
plane;
create designs involving tiling patterns (e.g.,
Escher-type designs, wallpaper or fabric designs), using technology (e.g.,
dynamic geometry software, design or drawing software).