
The Mathematical Processes

"The mathematical processes cannot be separated from the knowledge and skills that students acquire throughout the year. Students must problem solve, communicate, reason, reflect, and so on, as they develop the knowledge, the understanding of concepts, and the skills required in all the strands in every grade."
The Ontario Curriculum, Grades 18: Mathematics, 2005
The mathematical processes are:

The Kindergarten Program

"Mathematics in the Kindergarten program builds on children’s desire to make sense of their world, and helps them develop and demonstrate their mathematical understanding. Young children use mathematics intuitively and develop their understanding of mathematics through their individual approaches to learning, as well as through their prior experience of their linguistic, family, cultural, and community backgrounds. It is therefore important for children’s existing conceptual understanding of mathematics to be valued and for children to be introduced to mathematical concepts in an appropriate manner and at an appropriate time in their development."
The Kindergarten Program, 2016
As children progress through the Kindergarten program, they:
demonstrate an understanding of numbers, using concrete materials to explore and investigate counting, quantity, and number relationships
measure, using nonstandard units of the same size, and compare objects, materials, and spaces in terms of their length, mass, capacity, area, and temperature, and explore ways of measuring the passage of time, through inquiry and playbased learning
describe, sort, classify, build, and compare twodimensional shapes and threedimensional figures, and describe the location and movement of objects through investigation
recognize, explore, describe, and compare patterns, and extend, translate, and create them, using the core of a pattern and predicting what comes next
collect, organize, display, and interpret data to solve problems and to communicate information, and explore the concept of probability in everyday contexts
apply the mathematical processes to support the development of mathematical thinking, to demonstrate understanding, and to communicate thinking and learning in mathematics, while engaged in playbased learning and in other contexts

Grade 1

*For more on what your child is learning in Grade 1 mathematics, read "Key Learning for Grade 1" from mathies.ca.
Number Sense and Numeration
represent and order whole numbers to 50
establish conservation1 of number
represent money amounts to 20¢
compose and decompose numbers to 20
count by 1’s, 2’s, 5’s, and 10’s
add and subtract numbers to 20
Measurement
measure using nonstandard units2
tell time to the nearest halfhour
developing a sense of area
compare objects using measurable attribute3
compare objects using nonstandard units
investigate relationships when measuring the length of an object
Geometry and Spatial Sense
sort and classify4 twodimensional shapes and three dimensional figures by attributes;
recognize symmetry
relate shapes to other shapes, to designs, and to figures
describe location using positional language (e.g., over, under, inside, outside, etc.)
Patterning and Algebra
Data Management and Probability
organize objects into categories using one attribute
collect and organize categorical data6
read and display data using concrete graphs7 and pictographs
describe the likelihood that an event will occur
1 Conservation. The property by which something remains the same, despite changes such as physical arrangement. For example, with conservation of number, whether three objects are close together or far apart, the quantity remains the same. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 122)
2 Nonstandard units. Common objects used as measurement units; for example, paper clips, cubes, and hand spans. Nonstandard units are used in the early development of measurement concepts. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.128)
3 Attribute. A quantitative or qualitative characteristic of a shape, an object, or an occurrence; for example, colour, size, thickness, or number of sides. An attribute may or may not be a property. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 120)
4 Classify. Make decisions about how to sort or categorize things. Classifying objects and numbers in different ways helps students recognize attributes and properties of objects and numbers, and develop flexible thinking. (A Guide to Effective Instruction in Mathematics, Grades 1 to 3 – Geometry and Spatial Sense, 2016, p. 165)
5 Concrete materials. Objects that students handle and use in constructing or demonstrating their understanding of mathematical concepts and skills. Some examples of concrete materials are base ten blocks, connecting cubes, construction kits, number cubes, games, geoboards, geometric solids, measuring tapes, Miras, pattern blocks, spinners, and tiles. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 122)
6 Categorical data. Data that can be sorted by type or quality, rather than by measured or counted values. Eye colour and favourite food are examples of categorical data. (A Guide to Effective Instruction in Mathematics, Grades 1 to 3 – Geometry and Spatial Sense, 2016, p. 121)
7 Concrete graph. A graph in which real objects are used to represent pieces of information; for example, coloured candy directly placed on a template of a bar graph. (A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 3 – Data Management and Probability, p. 142, 2007)

Grade 2

*For more on what your child is learning in Grade 2 mathematics, read "Key Learning for Grade 2" from mathies.ca.
Number Sense and Numeration
represent and order numbers to 100
represent money amounts to 100¢
compose and decompose twodigit numbers
investigate fractions of a whole
count by 1’s, 2’s, 5’s, 10’s, and 25’s
add and subtract twodigit numbers
relate equalsized groups to multiplication
relate sharing equally to division
Measurement
measure length using centimetres and metres
choose benchmarks (personal referents) for the centimetre and the metre
measure perimeter, area, mass, and capacity using nonstandard units1
compare the mass and capacity of objects using nonstandard units
tell time to the nearest quarterhour
describe and establish temperature change
relate days to weeks and months to years
Geometry and Spatial Sense
distinguish between attributes2 that are geometric properties3 and attributes that are not geometric properties
classify twodimensional shapes by geometric properties (number of sides and vertices)
classify threedimensional figures by geometric properties (number and shape of faces)
locate a line of symmetry4
compose and decompose shapes
describe relative locations of objects
represent objects on a map
Patterning and Algebra
identify and describe repeating patterns
identify and describe growing and shrinking patterns
develop the concept of equality using the addition and subtraction of numbers to 18 and the equal sign
investigate the commutative property5 in addition
investigate the property of zero6 in addition and subtraction
Data Management and Probability
organize objects into categories using two attributes
collect and organize categorical7 and discrete8 data
read and display data using line plots9 and simple bar graphs10
describe probability, in simple games and experiments, as the likelihood that an event will occur
1 Nonstandard units. Common objects used as measurement units; for example, paper clips, cubes, and hand spans. Nonstandard units are used in the early development of measurement concepts. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.128)
2 Attribute. A quantitative or qualitative characteristic of a shape, an object, or an occurrence; for example, colour, size, thickness, or number of sides. An attribute may or may not be a property. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 120)
3 Property (geometric). An attribute that remains the same for a class of objects or shapes. A property of any parallelogram, for example, is that its opposite sides are congruent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.168)
4 Line of symmetry. A line that divides a shape into two parts that can be matched by folding the shape in half along this line. (A Guide to Effective Instruction in Mathematics, Grades 1 to 3 – Geometry and Spatial Sense, 2016, p. 125)
5 Commutative property. A property of addition and multiplication that allows the numbers to be added or multiplied in any order, without affecting the sum or product of the operation. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 121)
6 Properties of zero. The properties of zero in addition and subtraction (i.e., when you add zero to a number, the number does not change; when you subtract zero from a number, the number does not change). (The Ontario Curriculum Grades 18, Mathematics, 2005, p.50)
7 Categorical data. Data that can be sorted by type or quality, rather than by measured or counted values. Eye colour and favourite food are examples of categorical data. (A Guide to Effective Instruction in Mathematics, Grades 1 to 3 – Geometry and Spatial Sense, 2016, p. 121)
8 Discrete data. Data that can include only certain numerical values (often whole numbers) within the range of the data. Discrete data usually represent things that can be counted; for example, the number of times a word is used or the number of students absent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.123)
9 Line plot. A graph that shows a mark (usually an “X”) above a value on the number line for each entry in the data set. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.128)
10 Bar graph. A graph consisting of horizontal or vertical bars that represent the frequency of an event or outcome. There are gaps between the bars to reflect the categorical or discrete nature of the data. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 125)

Grade 3

*For more on what your child is learning in Grade 3 mathematics, read "Key Learning for Grade 3" from mathies.ca.
Number Sense and Numeration
represent and order numbers to 1000
represent money amounts to $10
compose and decompose threedigit numbers
investigate fractions of a set
count by 1’s, 2’s, 5’s, 10’s, 25’s, and 100’s
add and subtract threedigit numbers in a variety of ways
relate onedigit multiplication to reallife situations
relate division, by onedigit divisors to real life situations
Measurement:
measure distance using kilometres
tell time to the nearest 5 minutes
identify temperature benchmarks1
measure perimeter using standard units (e.g, cm, m, km)
measure mass in kilograms
measure capacity in litres
measure area using grid paper
compare the length, mass, and capacity of objects using standard units (e.g, cm, m, km, l, kg)
relate minutes to hours, hours to days, days to weeks, and weeks to years
Geometry and Spatial Sense:
use a reference tool to identify right angles and to compare angles with a right angle
classify twodimensional shapes by geometric properties2 (number of sides and angles)
classify threedimensional figures by geometric properties (number of faces, edges, and vertices)
relate different types of quadrilaterals3
name prisms and pyramids
identify congruent4 shapes
describe movement on a grid map
recognize transformations5
Patterning and Algebra:
create and extend growing and shrinking patterns
represent geometric patterns with a number sequence, a number line, and a bar graph
determine the missing numbers in equations involving addition and subtraction of one and twodigit numbers
investigate the properties of zero6 and one in multiplication
Data Management and Probability:
organize objects into categories using two or more attributes7
collect and organize categorical8 and discrete data9
read and display data using vertical and horizontal bar graphs10
understand mode11
predict the frequency of an outcome
relate fair games to equally likely events
1 Benchmarks. A number or measurement that is internalized and used as a reference to help judge other numbers or measurements. For example, the width of the tip of the little finger is a common benchmark for one centimetre. Also called referent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 121)
2 Geometric Properties. An attribute that remains the same for a class of objects or shapes. A property of any parallelogram, for example, is that its opposite sides are congruent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.168)
3 Quadrilaterals. A polygon with four sides (The Ontario Curriculum Grades 18, Mathematics, 2005, p.131)
4 Congruent. Having the same size and shape. For example, in two congruent triangles, the three corresponding pairs of sides and the three corresponding pairs of angles are equal. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 122)
5 Transformations. 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). (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 134)
6 Properties of Zero. The properties of zero in addition and subtraction (i.e., when you add zero to a number, the number does not change; when you subtract zero from a number, the number does not change). (The Ontario Curriculum Grades 18, Mathematics, 2005, p.50)
7 Attributes. A quantitative or qualitative characteristic of a shape, an object, or an occurrence; for example, colour, size, thickness, or number of sides. An attribute may or may not be a property. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 120)
8 Categorical Data. Data that can be sorted by type or quality, rather than by measured or counted values. Eye colour and favourite food are examples of categorical data. (A Guide to Effective Instruction in Mathematics, Grades 1 to 3 – Geometry and Spatial Sense, 2016, p. 121)
9 Discrete Data. Data that can include only certain numerical values (often whole numbers) within the range of the data. Discrete data usually represent things that can be counted; for example, the number of times a word is used or the number of students absent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.123)
10 Bar Graph. A graph consisting of horizontal or vertical bars that represent the frequency of an event or outcome. There are gaps between the bars to reflect the categorical or discrete nature of the data. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 125)
11 Mode. The value 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. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 128)

Grade 4

Number Sense and Numeration:
 represent and order numbers to 10 000
represent money amounts to $100
develop the concept of place value to tenths
represent and compare fractions using fractional notation (e.g ½ is bigger than ⅓)
add and subtract threedigit numbers in a variety of ways
multiply and divide twodigit whole numbers by onedigit whole numbers
relate halves, fifths, and tenths to decimals
Measurement:
measure length using millimetres
measure time intervals to the nearest minute
determine elapsed time
measure mass in grams and capacity in millilitres
measure volume using concrete materials
determine area and perimeter relationships for rectangles
compare the mass and capacity of objects using standard units (e.g: g, kg, l, ml)
relating years to decades and decades to centuries
Geometry and Spatial Sense:
identify geometric properties1 of parallelograms2
classify twodimensional shapes by geometric properties (number of sides, angles, and symmetry)
identify a straight angle, a right angle, and half a right angle
classify prisms and pyramids by geometric properties
construct threedimensional figures in a variety of ways
describe location using a grid system
perform and describe reflections3
Patterning and Algebra:
relate the term and the term number in a numeric sequence
generate patterns that involve addition, subtraction, multiplication, and reflections
determine the missing numbers in equations involving multiplication of one and twodigit numbers
use the commutative4 and distributive properties5 to facilitate computation
Data Management and Probability:
collect and organize discrete data6
read and display data using stemandleaf plots7 and double bar graphs8
understand median9
compare two related sets of data
predict the frequency of an outcome
investigate how the number of repetitions of a probability experiment affects the conclusion drawn
1 Geometric properties. An attribute that remains the same for a class of objects or shapes. A property of any parallelogram, for example, is that its opposite sides are congruent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.168)
2 Parallelogram. A quadrilateral whose opposite sides are parallel. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 130)
3 Reflection. A transformation that flips a shape over an axis to form a congruent shape. A reflection image is the mirror image that results from a reflection. Also called flip.(The Ontario Curriculum Grades 18, Mathematics, 2005, p. 132)
4 Commutative property. A property of addition and multiplication that allows the numbers to be added or multiplied in any order, without affecting the sum or product of the operation. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 121)
5 Distributive property. The property that allows a number in a multiplication expression to be decomposed into two or more numbers; for example, 51 x 12 = 51 x 10 + 51 x 2. More formally, the distributive property holds that, for three numbers, a, b, and c, a x (b + c) = (a x b) + (a x c) and a x (b – c) = (a x b) – (a x c); for example, 2 x (4 + 1) = 2 x 4 + 2 x 1 and 2 x (4 – 1) = 2 x 4 – 2 x 1. Multiplication is said to be distributed over addition and subtraction. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.123)
6 Discrete data. Data that can include only certain numerical values (often whole numbers) within the range of the data. Discrete data usually represent things that can be counted; for example, the number of times a word is used or the number of students absent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.123)
7 Stemandleaf plot. An organization of data into categories based on place values. The plot allows easy identification of the greatest, least, and median values in a set of data. The following stemandleaf plot represents these test results: 72, 64, 68, 82, 75, 74, 68, 70, 92, 84, 77, 59, 77, 70, 85.(The Ontario Curriculum Grades 18, Mathematics, 2005, p.126)
8 Double bar graph. A graph that combines two bar graphs to compare two aspects of the data in related contexts; for example, comparing the populations of males and females in a school in different years. Also called comparative bar graph. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.125)
9 Median. The middle value in a set of values 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 average of the two middle numbers. For example, 11 is the median of 5, 10, 12, and 28. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.128)

Grade 5

Number Sense and Numeration:
 represent and order numbers to 100 000
represent money amounts to $1000
develop the concept of place value to hundredths
compare and order fractional amounts with like denominators
add and subtract decimal amounts to hundredths
multiply twodigit whole numbers by twodigit whole numbers
divide threedigit whole numbers by onedigit whole numbers
relate simple fractions to decimals
Measurement:
measure time intervals to the nearest second
determine elapsed time
measure temperature
convert from metres to centimetres and from kilometres to metres
relate the 12hour clock to the 24hour clock
develop and apply area and perimeter relationships for a rectangle
relate capacity and volume
develop and apply the volume relationship for a right rectangular prism
Geometry and Spatial Sense:
distinguish among polygons1 and among prisms
identify acute, right, obtuse, and straight angles
measure angles to 90° with a protractor
construct triangles
construct nets of prisms and pyramids
locate objects using the cardinal directions
perform and describe translations2
Patterning and Algebra:
represent a pattern using a table of values
predict terms in a pattern
determine the missing numbers in equations involving addition, subtraction, multiplication, or division and one or twodigit numbers
investigate variables3 as unknown quantities
demonstrate equality using multiplication or division in equations with unknown quantities on both sides
Data Management and Probability:
collect and organize discrete4 and continuous data5
display data using brokenline graphs6
sample data from a population
understand mean7
compare two related sets of data
represent probability using fractions
1 Polygon. A closed shape formed by three or more line segments; for example, triangle, quadrilateral, pentagon, octagon. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.130)
2 Translation. A transformation that moves every point on a shape the same distance, in the same direction, to form a congruent shape. A translation image is the result of a translation. Also called slide. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 134)
3 Variable. A letter or symbol used to represent an unknown quantity, a changing value, or an unspecified number (e.g., a x b = b x a). (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 134)
4 Discrete data. Data that can include only certain numerical values (often whole numbers) within the range of the data. Discrete data usually represent things that can be counted; for example, the number of times a word is used or the number of students absent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.123)
5 Continuous data. Data that can include any numerical value that is represented on a number line and that falls within the range of the data, including decimals and fractions. Continuous data usually represent measurements, such as time, height, and mass. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.122)
6 Brokenline graph. A graph formed by line segments that join points representing the data. The horizontal axis represents discrete quantities such as months or years, whereas the vertical axis can represent continuous quantities. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 125)
7 Mean. One measure of central tendency. The mean of a set of numbers is found by dividing the sum of the numbers by the number of numbers in the set. For example, the mean of 10, 20, and 60 is (10 + 20 + 60) ÷ 3 = 30. A change in the data produces a change in the mean, similar to the way in which changing the load on a lever affects the position of the fulcrum if balance is maintained. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.127)

Grade 6

Number Sense and Numeration:
 represent and order numbers to 1 000 000
develop the concept of place value to thousandths
compare and order fractional amounts with unlike denominators
estimate 10%, 25%, 50%, and 75% of a quantity
add and subtract decimal amounts to thousandths
multiply and divide fourdigit whole numbers by twodigit whole number
multiply and divide decimals to tenths by whole numbers and twodigit by twodigit whole numbers
divide threedigit whole numbers by onedigit whole numbers
apply order of operations1 in expressions without brackets
relate simple fractions, decimals, and percents
Measurement:
measure quantities using metric units
convert from larger to smaller metric units, including square metres to square centimetres
develop and apply area relationships for a parallelogram and a triangle
develop and apply the volume relationships for a triangular prism
determine and apply surface area relationships for rectangular and triangular prisms
relate square metres and square centimetres
Geometry and Spatial Sense:
classify quadrilaterals2 by geometric properties3
sort polygons by lines of symmetry4 and by rotational symmetry5
measure angles to 180° with a protractor
construct polygons6
represent figures using views and isometric sketches
perform and describe rotations
plot points in the first quadrant
Patterning and Algebra:
represent patterns using ordered pairs7 and graphs
describe pattern rules in words
calculate any term when given the term number
investigate variables as changing quantities
solve equations using concrete materials8 and guess and check
Data Management and Probability:
collect and organize discrete9 and continuous data10
display data using continuous line graphs
select appropriate graphical representations
use continuous line graphs and mean to compare sets of data
find theoretical probabilities11
predict the frequency of an outcome based on the theoretical probability
1 Order of operations. A convention or rule used to simplify expressions. The acronym BEDMAS is often used to describe the order: – brackets – exponents – division or multiplication, whichever comes first – addition or subtraction, whichever comes first (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 129)
2 Quadrilateral. A polygon with four sides (The Ontario Curriculum Grades 18, Mathematics, 2005, p.131)
3 Property (geometric). An attribute that remains the same for a class of objects or shapes. A property of any parallelogram, for example, is that its opposite sides are congruent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.168)
4 Line of symmetry. A line that divides a shape into two parts that can be matched by folding the shape in half along this line. (A Guide to Effective Instruction in Mathematics, Grades 1 to 3 – Geometry and Spatial Sense, 2016, p. 125)
5 Rotational symmetry. A geometric property of a shape whose position coincides with its original position after a rotation of less than 360º about its centre. (A Guide to Effective Instruction in Mathematics, Grades 1 to 3 – Geometry and Spatial Sense, 2016, p. 132)
6 Polygon. A closed shape formed by three or more line segments; for example, triangle, quadrilateral, pentagon, octagon. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.130)
7 Ordered pair. Two numbers, in order, that are used to describe the location of a point on a plane, relative to a point of origin (0,0); 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. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 129)
8 Concrete materials. Objects that students handle and use in constructing or demonstrating their understanding of mathematical concepts and skills. Some examples of concrete materials are base ten blocks, connecting cubes, construction kits, number cubes, games, geoboards, geometric solids, measuring tapes, Miras, pattern blocks, spinners, and tiles. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 122)
9 Discrete data. Data that can include only certain numerical values (often whole numbers) within the range of the data. Discrete data usually represent things that can be counted; for example, the number of times a word is used or the number of students absent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.123)
10 Continuous data. Data that can include any numerical value that is represented on a number line and that falls within the range of the data, including decimals and fractions. Continuous data usually represent measurements, such as time, height, and mass. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.122)
11 Theoretical probability. A mathematical calculation of the chances that an event will happen in theory; if all outcomes are equally likely, it is calculated as the number of favourable outcomes divided by the total number of possible outcomes. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 133)

Grade 7

Number Sense and Numeration
 represent and order decimals (to hundredths), fractions, and integers
represent squares and square roots
divide whole numbers by simple fractions and decimals
add and subtract simple fractions and integers
multiply and divide decimal numbers to thousandths by onedigit whole numbers
apply order of operations1 in expressions with brackets
relate fractions, decimals, and percents
solve problems involving wholenumber percents and unit rates
Measurement
convert between metric units, including converting between square centimetres and square metres
develop the area relationship for a trapezoid
develop and apply the formula for the volume of a prism
determine and apply surfacearea relationships for prisms
relate millilitres and cubic centimetres
Geometry and Spatial Sense
construct parallel, perpendicular, and intersecting lines
sort and classify triangles and quadrilaterals2 by geometric properties3
construct angle bisectors and perpendicular bisectors
investigate relationships among congruent4 shapes
relate enlarging and reducing to similar5 shapes
compare similar and congruent shapes
performing and describing dilatations
tile6 a plane
plot points in all four quadrants
Patterning and Algebra
represent linear growing patterns
represent patterns algebraically
model reallife relationships involving constant rates graphically and algebraically
translate phrases, using algebraic expressions
find the term in a pattern algebraically when given any term number
solve linear equations7 using concrete materials8 or inspection and guess and check
Data Management and Probability
collect and organizing categorical9, discrete10, and continuous11 data
display data in relative frequency12 tables and circle graphs
identify bias in data
relate changes in data to changes in central tendency13
make inferences based on data
investigate realworld applications of probability
determine the theoretical probability14 of two independent events
1 Order of operations. A convention or rule used to simplify expressions. The acronym BEDMAS is often used to describe the order: – brackets – exponents – division or multiplication, whichever comes first – addition or subtraction, whichever comes first (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 129)
2 Quadrilateral. A polygon with four sides (The Ontario Curriculum Grades 18, Mathematics, 2005, p.131)
3 Property (geometric). An attribute that remains the same for a class of objects or shapes. A property of any parallelogram, for example, is that its opposite sides are congruent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.168)
4 Congruent. Having the same size and shape. For example, in two congruent triangles, the three corresponding pairs of sides and the three corresponding pairs of angles are equal. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.122)
5 Similar. Having the same shape but not always the same size. If one shape is similar to another shape, there exists a dilatation that will transform the first shape into the second shape. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.132)
6 Tiling. The process of using repeated shapes, which may or may not be congruent, to cover a region completely. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.134)
7 Linear equation. An algebraic representation of a linear relationship. The relationship involves one or more firstdegree variable terms; for example, y = 2x – 1; 2x + 3y = 5; y = 3. The graph of a linear equation is a straight line. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.127)
8 Concrete materials. Objects that students handle and use in constructing or demonstrating their understanding of mathematical concepts and skills. Some examples of concrete materials are base ten blocks, connecting cubes, construction kits, number cubes, games, geoboards, geometric solids, measuring tapes, Miras, pattern blocks, spinners, and tiles. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 122)
9 Categorical data. Data that can be sorted by type or quality, rather than by measured or counted values. Eye colour and favourite food are examples of categorical data. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.121)
10 Discrete data. Data that can include only certain numerical values (often whole numbers) within the range of the data. Discrete data usually represent things that can be counted; for example, the number of times a word is used or the number of students absent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.123)
11 Continuous data. Data that can include any numerical value that is represented on a number line and that falls within the range of the data, including decimals and fractions. Continuous data usually represent measurements, such as time, height, and mass. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.122)
12 Relative frequency.The frequency of a particular outcome or event expressed as a percent of the total number of outcomes. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.132)
13 Measure of central tendency. A value that summarizes a whole set of data; for example, the mean, the median, or the mode. A measure of central tendency represents the approximate centre of a set of data. Also called central measure. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.128)
14 Theoretical probability. A mathematical calculation of the chances that an event will happen in theory; if all outcomes are equally likely, it is calculated as the number of favourable outcomes divided by the total number of possible outcomes. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 133)

Grade 8

Number Sense and Numeration
 represent and order rational numbers1
represent numbers using exponential notation (e.g, 24)
solve multistep problems2 involving whole numbers and decimals
multiply and divide fractions and integers (e.g., 10)
multiply and divide decimals by powers of ten (e.g., 230 000 ÷ 106)
apply order of operations3 in expressions with brackets and exponents
solving problems involving percents to one decimal place and percents greater than 100
solving problems involving rates and proportions
Measurement
convert between cubic centimetres and cubic metres and between millilitres and cubic centimetres
develop circumference4 and area relationships for a circle
develop and apply the formula for the volume of a cylinder
determine and apply surfacearea relationships for cylinders
Geometry and Spatial Sense
sort quadrilaterals5 by geometric properties6 involving diagonals
construct circles
investigate relationships among similar7 shapes
determine and apply angle relationships for parallel and intersecting lines;
relating the numbers of faces, edges, and vertices of a polyhedron8
determine and apply the Pythagorean relationship9 geometrically
plotting the image of a point on the coordinate plane10 after applying a transformation
Patterning and Algebra
represent the general term in a linear sequence, using one or more algebraic expressions
translate statements, using algebraic equations
find the term number in a pattern algebraically when given any term
solve linear equations11 involving one variable terms with integer solutions using a “balance” model
Data Management and Probability
collect categorical12, discrete13, and continuous14 data
organize data into intervals
display data using histograms15 and scatter plots16
use measures of central tendency17 to compare sets of data
compare two attributes18 using data management tools; comparing experimental19 and theoretical probabilities20
calculate the probability of complementary events21
1 Rational number. A number that can be expressed as a fraction in which the denominator is not 0. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 131)
2 Multistep problem. A problem that is solved by making at least two calculations The Ontario Curriculum Grades 18, Mathematics, 2005, p. 128)
3 Order of operations. A convention or rule used to simplify expressions. The acronym BEDMAS is often used to describe the order: – brackets – exponents – division or multiplication, whichever comes first – addition or subtraction, whichever comes first (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 129)
4 Perimeter. The length of the boundary of a shape, or the distance around a shape. For example, [...] the perimeter of a circle is its circumference. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.130)
5 Quadrilateral. A polygon with four sides (The Ontario Curriculum Grades 18, Mathematics, 2005, p.131)
6 Property (geometric). An attribute that remains the same for a class of objects or shapes. A property of any parallelogram, for example, is that its opposite sides are congruent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.168)
7 Similar. Having the same shape but not always the same size. If one shape is similar to another shape, there exists a dilatation that will transform the first shape into the second shape. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.132)
8 Polyhedron. A threedimensional figure that has polygons as faces. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.130)
9 Pythagorean relationship. The relationship that, for a right triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two sides. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.131)
10 Coordinate plane. A plane that contains an xaxis (horizontal) and a yaxis (vertical), which are used to describe the location of a point. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.122)
11 Linear equation. An algebraic representation of a linear relationship. The relationship involves one or more firstdegree variable terms; for example, y = 2x – 1; 2x + 3y = 5; y = 3. The graph of a linear equation is a straight line. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.127)
12 Categorical data. Data that can be sorted by type or quality, rather than by measured or counted values. Eye colour and favourite food are examples of categorical data. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.121)
13 Discrete data. Data that can include only certain numerical values (often whole numbers) within the range of the data. Discrete data usually represent things that can be counted; for example, the number of times a word is used or the number of students absent. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.123)
14 Continuous data. Data that can include any numerical value that is represented on a number line and that falls within the range of the data, including decimals and fractions. Continuous data usually represent measurements, such as time, height, and mass. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.122)
15 Histogram. A type of bar graph in which each bar represents a range of values, and the data are continuous. No spaces are left between the bars, to reflect the continuous nature of the data. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.125)
16 Scatter plot. A graph designed to show a relationship between corresponding numbers from two sets of data measurements associated with a single object or event; for example, a graph of data about marks and the corresponding amount of study time. Drawing a scatter plot involves plotting ordered pairs on a coordinate grid. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.126)
17 Measure of central tendency. A value that summarizes a whole set of data; for example, the mean, the median, or the mode. A measure of central tendency represents the approximate centre of a set of data. Also called central measure. (The Ontario Curriculum Grades 18, Mathematics, 2005, p.128)
18 Attribute. A quantitative or qualitative characteristic of a shape, an object, or an occurrence; for example, colour, size, thickness, or number of sides. An attribute may or may not be a property. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 120)
19 Experimental probability. The likelihood of an event occurring, determined from experimental results rather than from theoretical reasoning. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 124)
20 Theoretical probability. A mathematical calculation of the chances that an event will happen in theory; if all outcomes are equally likely, it is calculated as the number of favourable outcomes divided by the total number of possible outcomes. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 133)
21 Complementary events. Two events that have no outcome(s) in common but that account for all possible outcomes of an experiment. For example, rolling an even number and rolling an odd number using a number cube are complementary events. The sum of the probabilities of complementary events is 1. (The Ontario Curriculum Grades 18, Mathematics, 2005, p. 121)

