# Grades 7 & 8 Mathematics

This course covers the topics typically taught in Canadian Grade 7 and 8 Mathematics curricula and, in some instances, extends ideas beyond grade level. Letters are included beside the unit names to help group the units into similar themes.

For more information about the structure and general use of this courseware, see the *Course Information* unit.

*The courseware is currently 90% complete. New units will be added as they are developed over the coming months.*

### Modules

## Representing and Comparing Numbers (N)

Part A (Lessons 1–7)

Topics include representing and comparing positive rational numbers (integers, fractions, and decimals), finding multiples and factors of positive integers, and determining the least common multiple (LCM) and the greatest common factor (GCF) of a pair of positive integers.

Part B (Lessons 8–12)

Topics include representing negative fractions and negative decimals, comparing the values of any two rational numbers, exponential notation, and using factor trees and prime factorizations to find the LCM or the GCF of a pair of positive integers.

This lesson examines three different number systems: whole numbers, integers, and rational numbers. Connections between different number systems are highlighted to set the groundwork for comparisons and operations.

Mathematicians often use the number line to solve problems. In this lesson, we review the number line, focusing on plotting fractions.

In math, symbols are important for communication. In this lesson, we review the “greater than” and “less than” symbols. In addition, we present two techniques used to compare fractions.

Rational numbers can be written as fractions or decimals. In this lesson, we discuss the connections between fractional representations and decimal representations, specifically, when it comes to plotting numbers on the number line.

In this lesson, we review how to generate a list of multiples of an integer. Using our lists, we identify common multiples of two integers, paying particular attention to the least common multiple (LCM).

Factors, like multiples, have to do with multiplication. In this lesson, we solve problems by identifying factors of positive integers.

Expanding on the factors lesson, we compare the factors of two positive integers to find common factors; specifically, we are often interested in identifying the greatest common factor (GCF). We conclude by solving word problems that require us to apply factors to different contexts.

Fractional quantities can be positive or negative. Similar to negative integers, negative fractions lie to the left of zero on the number line. In this lesson, we plot negative fractions on the number line to help us understand and compare the values of these numbers.

Rational numbers can be written as fractions or decimals. In this lesson, we compare negative decimal numbers by plotting them on the number line. We then compare negative fractions with negative decimals. The decimal equivalents of common fractions are determined and strategies for converting a fraction into a decimal are shown. Finally, we learn how to compare any two rational numbers.

In this lesson, we learn to represent repeated multiplication using exponential notation. Exponential notation is then used to represent whole numbers in expanded form using powers of ten. Square numbers and cube numbers are investigated.

In this lesson, we review prime and composite numbers. We learn how to represent a composite number as a product of its prime factors using a factor tree.

Prime factorizations can be used to determine the greatest common factor (GCF) and the least common multiple (LCM) of a pair of positive integers. We explore how this can be done, and we use these strategies to solve word problems.

## Operations (N)

Part A (Lessons 1–11)

Topics include adding and subtracting rational numbers, multiplying and dividing a whole number by a positive rational number, and evaluating expressions using the order of operations.

Part B (Lessons 12–19)

Topics include multiplying, and dividing integers, fractions and decimal numbers, approximating square roots of positive integers, and evaluating expressions that include exponents using the order of operations.

We begin our discussion of addition by studying how number lines can be used to show addition. In this lesson, we focus on the addition of integers, specifically how positive and negative numbers can be added using a number line.

We can add integers without using a calculator or a number line. In this lesson, we extend our previous discussion on integer addition and examine strategies for performing integer addition mentally.

This lesson explores equivalent fractions in preparation for when we must add and subtract fractions. In the process of finding equivalent fractions, you will be given the opportunity to practise finding common multiples, using improper fractions and mixed numbers, plotting on the number line, and comparing rational numbers.

In this lesson, we build upon our understanding of addition to include rational numbers. To do this, we revisit the number line and incorporate our strategies for plotting rational numbers so that we can find their sum.

This lesson introduces strategies for the addition of fractions without the use of a number line. We use number lines as motivation for finding a common denominator, then we move to adding fractions without the use of visual aids.

We begin our discussion of subtraction by focusing on the integers. In this lesson, we review the operation of subtraction, show subtraction on the number line, and learn how to subtract integers both with and without a number line.

Continuing our discussion on subtraction, in this lesson we explore strategies for subtracting fractions. Our goal is to use equivalent fractions to solve subtraction problems without the use of a calculator or the number line.

This lesson explores strategies for multiplying whole numbers by fractions and decimals. We solve examples and highlight rules for performing calculations without using a calculator.

Multiplication is the operation that is used to scale or resize a quantity. In this lesson, we explore scale factors and discuss why we must start thinking about multiplication in terms of scaling.

In this lesson, we learn how to solve calculations that involve the division of whole numbers by fractions and decimals. Through examples, we highlight rules for performing these calculations without a calculator.

The order of operations is reviewed and used to perform calculations involving integers, fractions, and decimals. Additionally, we explore the importance of brackets; when they are needed and when they can be removed from an expression. We conclude by using the distributive property to simplify calculations.

In this lesson, we learn how to multiply integers mentally. Specifically, we look at how the sign of each integer in a product impacts the sign of the product.

Division is the opposite operation of multiplication, and so the strategies we learn for dividing integers will be similar to those we used when multiplying integers. In this lesson, we examine how the signs of the dividend and divisor impact the sign of the quotient.

We begin this lesson by reviewing how to multiply a fraction by a whole number. We then expand our understanding to include the multiplication of any two fractions. Additionally, some focus is given to estimating the values of products.

In this lesson, we review how to divide a whole number by a fraction. We then explore how to adapt this strategy in order to divide a fraction by another fraction, without the use of a calculator.

We begin this lesson by examining the multiplication of decimal numbers by powers of ten, including a discussion of scientific notation. We then learn how to multiply two decimal numbers, first by converting the numbers to fractions, and second by working with the decimal numbers themselves.

In this lesson, we develop strategies to evaluate division expressions that involve whole numbers and decimal numbers. We also extend these strategies to discuss division with two decimal numbers.

This lesson focuses on the relationship between squaring a number and taking the square root of a number. We discuss perfect squares and examine how to approximate the square root of a positive integer that is not a perfect square.

In this lesson, we revisit the order of operations for arithmetic. We solve problems involving integers, fractions, and decimals, paying special attention to exponents.

## Ratios, Rates, and Proportions (N)

Part A (Lessons 1–5)

Topics include writing and interpreting ratios; finding equivalent ratios; converting between fractions, decimals, and percents; converting between units of measurement; and solving problems involving unit rates.

Part B (Lessons 6–10)

Topics include recognizing proportional situations in word problems, tables and graphs; connecting unit relates to proportional relationships and their representations in tables, graphs and equations; and fractional percents and percents greater than 100 percent.

This lesson discusses the meaning of a ratio and goes through how to write and interpret ratios. We conclude by solving problems that require a ratio to be applied to large quantities.

We begin our discussion on equivalent ratios using diagrams and exploring how two ratios can represent the same relationship between two quantities. Then, we develop strategies for finding equivalent ratios numerically. This lesson concludes with solving ratio problems.

In this lesson, we define a percent and explore the relationships among fractions, decimals and percents. We conclude by solving some word problems involving percents.

This lesson explores strategies for converting between different metric units of length, mass, and capacity. We then apply these strategies to convert between units of time and units of area.

In this lesson, we learn about rates which are comparisons of two measurements with different units. We focus on how to write unit rates and how unit rates can be used to solve word problems. Also included are some examples on how to convert a rate into different units.

In this lesson, we explore the notion of proportionality using examples like image enlargement and paint mixing. We explore proportional relationships between two quantities and learn how to recognize when a situation is and is not proportional.

In this lesson, we examine how to recognize a proportional relationship between two quantities when the data is displayed in a table or a graph.

The relationship between proportional quantities is often given in the form of a unit rate. In this lesson, we explore how this unit rate manifests in an equation, a table, or a graph representing the relationship between the two quantities.

In this lesson, we discuss fractional percents and percents greater than 100 percent. Some focus is given to where percentages appear in everyday life and how estimation can be helpful when working with percentages.

Proportional situations can be presented in many ways including: unit rates, tables, graphs, or equations. In this lesson, we practise comparing proportional relationships that are presented in different ways.

## Bisectors and Properties of Shapes (G)

Part A (Lessons 1–6)

Topics include constructions of angle bisectors and perpendicular bisectors, and the various properties of triangles, quadrilaterals, and more general polygons. In particular, different polygons are classified based on their side lengths and angle measurements.

Part B (Lessons 7–10)

Topics include quadrilateral diagonals, circle terminology and construction, and applications of circles in the real-world.

This lesson introduces the terminology and notation of basic geometric objects, with a focus on written and oral communication.

We review how to classify triangles according to side lengths and angle measurements. We then investigate the side-angle relationship in triangles. This lesson concludes with an application of triangle properties to construct a 60-degree angle using a compass.

A compass and a straightedge can be used to divide an angle in half perfectly without ever taking a measurement. In this lesson, we discuss the properties of angle bisectors and how to use these properties to construct an angle bisector of a given angle using only a compass and a straightedge. We extend our discussion to triangles and explore the relationship of the three angle bisectors in any triangle.

Continuing our discussion on constructions, we look at the properties of perpendicular bisectors and how to use these properties to construct a perpendicular bisector of a given line segment using only a compass and a straightedge. We extend our discussion to triangles and explore the relationship of the three perpendicular bisectors in any triangle.

In this lesson, we look at the properties of six special quadrilaterals. We examine the similarities and differences between each and use a diagram to represent all of the relationships that we discuss.

Expanding on quadrilaterals, in this lesson we discuss the properties of general polygons. In particular, we investigate the sum of the interior angles in a polygon and how polygons are connected to prisms. This lesson concludes with an extension that explores how prisms can be sliced to produce various polygonal faces.

In this lesson, we investigate various properties of the diagonals in quadrilaterals. In particular, we consider when the diagonals bisect each other, are perpendicular to each other, or are equal in length. We then use these properties to help us classify quadrilaterals.

This lesson begins with a discussion of how to describe a circle. Since circles are very different from polygons, we introduce new terminology to use when studying circles. In particular, we define the centre, radius, diameter, and circumference of a circle. We also explore how to use polygons to help us estimate the circumference and the area enclosed by a circle.

In this lesson, we discuss strategies for drawing accurate circles. Specifically, we look at drawing circles when given a centre and a radius, a centre and a point that must lie on the circle, and also given two or more points that must all lie on the circle. We discuss where larger circles appear in the real world and what tools and strategies can be used to create them.

In this lesson, we take the application of circles beyond the wheel and discuss the role of circles in roundabout design, the use of circles in the design of structures, and how circles of different diameters interact in machines that use gears.

## Area, Volume, and Angles (G)

Part A (Lessons 1–5)

Topics include calculating the area of parallelograms, triangles, trapezoids, and composite shapes; calculating the surface area, volume, and capacity of prisms; and representing 3D objects in different ways.

Part B (Lessons 6–10)

Topics include calculating the circumference and area of circles; calculating the volume and surface area of cylinders; and properties of angles formed by intersecting lines including parallel lines and transversals.

This lesson reviews the definition of area and how to calculate the area of a rectangle. We then develop and apply the formulas for finding the areas of parallelograms, triangles, and trapezoids.

Continuing our discussion on area, we explore how to decompose and calculate the area of composite shapes.

In this lesson, we learn how to visualize the surface of a 3D solid using a net. We then calculate the surface area of prisms and solve word problems involving surface area.

In this lesson, we develop and apply the formula for finding the volume of a prism. We relate volume and capacity, and explore how to convert between units of volume.

We wrap up our discussion on prisms and composite solids by learning how to draw them on triangular dot paper. We also learn how to recognize and sketch different 2D views of a 3D object.

In this lesson, we review the circumference and area of circles. We then develop and apply the formulas for calculating the circumference and the area of a circle given the radius (or diameter) of the circle.

We begin our discussion on cylinders by comparing cylinders to prisms. We develop and apply the formula for finding the volume of a cylinder and solve word problems involving the volume or the capacity of a cylinder.

Continuing our discussion on cylinders, in this lesson, we explore the net of a cylinder and use the net to develop a formula for the surface area of a cylinder. We then calculate the surface area of cylinders and solve word problems involving surface area.

In this lesson, we begin our discussion of intersecting lines by exploring the properties of angles formed by two intersecting lines. We define supplementary, complementary, and opposite angles, and use angle relationships to find unknown angles in a diagram.

Continuing our discussion on intersecting lines, in this lesson we explore the angles formed by parallel lines and transversals. We define corresponding, alternate, and co-interior angles, and use angle relationships to solve for unknown angles in a diagram.

## Transformations of Shapes (G)

Part A (Lessons 1–7)

Topics include congruence of polygons, triangle congruence rules, plotting points on the Cartesian plane, the image of a polygon on the Cartesian plane under translations, reflections and/or rotations on the Cartesian plane, and tessellations.

Part B (Lessons 8–11)

Topics include similarity of polygons, triangle similarity rules, dilatations of polygons, and indirect measurements.

In this lesson, we review the definition of congruence and match the sides and angles of two congruent polygons. We also take a look at the perimeter and area of congruent polygons.

Continuing our discussion on congruence, in this lesson, we explore congruence rules for triangles. Our goal is to show two triangles are congruent by matching only three corresponding parts.

This lesson introduces the Cartesian plane. We examine how to construct the Cartesian coordinate system, how to plot points on the Cartesian plane, as well as examine the vertical/horizontal distances between two points on the Cartesian plane.

In this lesson, we begin our discussion on transformations by exploring the translations of polygons. We learn how to draw the image of a polygon under a translation and relate the definition of congruence to translations.

Continuing our discussion on transformations, we now explore reflections of polygons. In this lesson, we learn how to graph the image of a polygon under a reflection on the Cartesian plane and explain how the image is congruent to the original polygon.

In this lesson, we learn how to graph the image of a polygon under a rotation. We also combine all three transformations and graph the image of a polygon under a translation, reflection, and rotation on the Cartesian plane.

This lesson explores the art of tessellations. We define a tessellation and explore what polygons can tessellate the plane. Then, using polygons that we know tessellate the plane, we explore how to create interesting designs that tessellate.

In geometry, the word “similar” is used to indicate when two objects have the same shape, but not necessarily the same size. In this lesson, we learn the precise definition of similar polygons, explore the scale factor between two similar polygons, and learn how to use the scale factor to solve problems.

Every triangle has three angles and three sides, but it turns out that we do not need to know the measures of each to determine the shape of the triangle. In this lesson, we explore the minimum conditions needed to verify that two triangles are similar. We learn the Angle-Angle, Side-Side-Side, and Side-Angle-Side similarity rules and practise constructing similar triangles.

In this lesson, we explore how to draw similar polygons without measuring any angles. This can be done by performing a particular type of transformation: a dilatation.

Indirect measurements allow us to find unknown lengths without actually measuring line segments. In this lesson, we explore how to use the triangle similarity rules to make indirect measurements in different scenarios.

## Representing Patterns (A)

Part A (Lessons 1–6)

Topics include representing sequences using tables, general terms and graphs, describing patterns using variables and expressions, extending sequences, and solving problems involving unknown quantities.

Part B (Lessons 7–11)

Topics include equivalent expressions for the general term of a sequence, describing relationships and patterns using equations, and decreasing and naturally occurring sequences.

We begin our discussion of patterning by examining number and image sequences. In this lesson, we focus on stating the pattern rule which describes how to generate the next term in a sequence.

This lesson explores the relationship between the term number and the term value, that is, the relationship between a term in a sequence and its position in that sequence. We then use the general term to find the value of a term in a sequence given its term number.

We continue to find the general term of sequences, with emphasis on how to use a variable to represent an unknown quantity. This lesson concludes with a discussion on substitution, where we evaluate expressions by substituting a number for a variable in the general term.

In this lesson, we encounter sequences that have a different type of relationship than what we have previously seen. You will continue to practise finding the general term of a sequence, concluding the lesson with some application problems.

In this lesson, we explore how to represent a sequence graphically. With a sequence represented on a graph, we then use the graph to determine the term number that corresponds to a given term in the sequence. Finally, you will practise how to find the general term of a sequence given its graph.

In this lesson, we connect the different sequences that we have studied so far. We continue using tables, graphs, and general terms to study the patterns that sequences represent.

In this lesson, we review how to represent a sequence using a table, a general term, or a graph. Emphasis is put on determining which of these three representations is most appropriate in a particular problem-solving situation.

In this lesson, we analyze different patterns that generate the same sequence of numbers. We generate various expressions to represent the different interpretations of a pattern, and learn how to determine whether two expressions are equivalent.

In this lesson, we learn the difference between an expression and an equation, and explore how each can be used when describing patterns. In particular, we use expressions for the general term of a sequence to form equations to represent relationships in sequences.

In this lesson, we define and explore decreasing sequences. You are challenged to consider how strategies for finding the general terms of increasing sequences can be used to write an equation representing a decreasing sequence. We also examine how some sequences of numbers that arise from physical situations cannot continue forever due to real-world boundaries.

In this lesson, we look beyond the typical sequences discussed in this unit and explore more naturally occurring sequences. The examples focus on popular puzzles and real-life growth and depreciation scenarios. We conclude by discussing, through an example, how apparent patterns can sometimes be deceiving.

## Equations and the Pythagorean Theorem (A)

Part A (Lessons 1–5)

Topics include using variables in expressions and equations, identifying and exploring linear relationships, and solving equations by inspection, trial and error, and using visual models.

Part B (Lessons 6–10)

Topics include solving equations using algebraic techniques, comparing the differences between evaluating an expression and solving an equation, exploring equations with multiple variables, and the Pythagorean Theorem.

In this lesson, we review variables and expressions. We discuss common notation for operations in algebra and practice translating English phrases to mathematical expressions.

In this lesson, we explore linear relationships between two quantities. We learn how to identify a linear relationship represented in a graph, in a table of values, or in an equation.

In this lesson, we use expressions and equations to model and solve real-world problems.

In this lesson, we use graphs and a visual model of weights on a scale to help solve equations. We also practise solving simple equations by inspection.

In this lesson, we practise solving equations by trial and error. These methods are applied to solve word problems and to solve equations that have fractional solutions.

In this lesson, we visualize equations using weights and a balanced scale. We solve equations with one operation using algebraic techniques and learn how to verify a solution of an equation.

We continue to solve equations using algebra by extending our strategies to solve equations with more than one operation.

This lesson explores the forwards and backwards movement through a math machine, and makes connections to the differences between evaluating an expression and solving an equation.

In this lesson, we find solutions to equations with two or more unknown quantities using trial and error and algebra.

In this lesson, we investigate the relationship between the side lengths of a right triangle. We will develop the Pythagorean Theorem and use it to solve for the missing side length of a right triangle.

## Data Collection and Graphs (D)

Part A (Lessons 1–5)

Topics include different types of data; population, sample and census; bias in data collection arising from question wording, accepted answers and choice of sample group; frequency and relative frequency tables and graphs; reading and creating circle graphs; choosing an appropriate graph type for a data set; bias in data representation arising from the chosen graph type, graph structure and shape, and axis labels and scales.

Part B (Lessons 6–9)

Topics include organizing continuous data into stem-and-leaf plots and frequency tables with intervals; as well as creating and reading histograms, and scatter plots.

In this lesson, we discuss different types of data including primary, secondary, categorical, and numerical data. We discuss the terms population, sample, and census and learn the difference between discrete and continuous numerical data.

In this lesson, we explore how data can be influenced by the wording of survey questions, the types of answers accepted in a survey, and the sample group that is being used in the survey to represent the population.

In this lesson, we learn how to organize data into frequency tables, calculate relative frequencies, and create and compare frequency and relative frequency graphs.

In this lesson, we focus on reading and creating circle graphs (or pie charts). We also discuss the appropriate graph types (circle, bar, or line) that can be used to display various data sets.

In this lesson, we explore how choices made while creating a graph can lead to a misrepresentation of the underlying data. In particular, we discuss how the type of graph, the structure and shape of the graph, or the axis labels and scales of the graph can potentially mislead the viewer.

In this lesson, we focus on working with continuous data sets. We explore different ways in which continuous data might be organized and graphed as well as discuss how to display paired data sets.

In this lesson, we study different ways to organize numerical data sets into intervals. We start by organizing data using stem-and-leaf plots and then exploring how frequency tables can be used if we divide the data into intervals. We discuss the advantages and disadvantages of these organization tools and practise choosing appropriate intervals for given data sets.

Standard bar graphs are not always an appropriate way to display a given numerical data set. A histogram is a similar type of graph in which numerical data are first grouped into ranges and then the frequency of each range is plotted using a bar. In this lesson, we discuss the features of a histogram and practise creating histograms from numerical data sets. We discuss what information might be gained or lost by presenting data in a histogram, and explore the effects of interval choice on the shape of the graph.

A scatter plot is a graph consisting of points which are formed using the values of two variable quantities. Scatter plots are used to display a relationship between the two variables in question. In this lesson, we discuss the features of a scatter plot and practise creating scatter plots from paired data sets. We discuss the roles that the two variables play in a scatter plot and explore what information might be revealed when we consider the shape formed by the data points as a whole.

## Data Analysis (D)

Part A (Lessons 1–4)

Topics include determining the mean, median, and mode of data sets; studying the effects of adding data to a data set or removing data from a data set; exploring the effect of outliers on the mean, median, and mode; and practising drawing conclusions and making predictions from data in graphs.

Part B (Lessons 5–8)

**Under Construction**

Topics include interpreting data, histograms, and scatter plots and drawing conclusions from these graphs; describing relationships between the two variables in a scatter plot; estimating rates of change associated with scatter plots; making predictions supported by the data in histograms and scatter plots; and using appropriate measures of central tendency to compare two data sets.

It can be helpful to use a single value to summarize the information in a large data set. Measures of central tendency, like the mean, median, and mode, attempt to summarize data by measuring the middle (or centre) of a data set. In this lesson, we will learn how to determine the mean, median, and mode of different data sets and discuss how they can be used to analyze data.

In this lesson, we discuss the effects of adding data to (or removing data from) a data set. We focus on how this might affect the mean, median, and mode in different ways.

Some data sets contain outliers, which is data that is separated from the rest of the values in the data set. In this lesson, we discuss the effect of outliers on the mean, median, and mode of data sets, and explore different contexts in which one particular measure might be the most appropriate for summarizing the given data.

In this lesson, we practise interpreting the underlying data displayed in different graphs. We discuss the difference between statements that can be verified using the information in a graph and predictions that are supported by the trends in the graph but cannot be verified using the graph alone.