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.
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.
Mathematicians often use the number line to solve problems. In this lesson, we review the number line, focusing on plotting fractions.
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.
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.
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 explores strategies for multiplying whole numbers by fractions and decimals. We solve examples and highlight rules for performing calculations without using a calculator.
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.
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 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.
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.
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.
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.
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.
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.
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 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.
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 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.
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.
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 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 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 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.
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 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 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)
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.
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.
In this lesson, we practise identifying and interpreting information provided in a histogram, and drawing conclusions supported by the histogram. We also explore how the interval size of a histogram can affect the conclusions drawn by someone who is analyzing the data in a histogram.
In this lesson, we practise identifying and interpreting information provided in a scatter plot, and drawing conclusions supported by the scatter plot. We explore how to identify and describe a general relationship that might exist between the two variables in a scatter plot.
Scatter plots are often used to identify and study a relationship between two variables. When the data points in a scatter plot seem to roughly follow the path of a line, we can use our knowledge of linear patterns to study the data and make predictions. In this lesson, we explore drawing lines that approximate the trend observed in a scatter plot, and estimating rates of change associated to scatter plots. We compare rates of change of different scatter plots and use them to make predictions.
In this lesson, we practise using measures of central tendency to compare two data sets, draw conclusions, and discuss factors that might influence which measure of central tendency is most appropriate for a particular comparison. We also explore how to compare data presented in histograms.
Part A (Lessons 1–4)
Topics include random experiments, outcomes, and events; calculating theoretical probabilities of single events; comparing probabilities of different events; independent events; experimental probability; and using probabilities to make predictions.
Part B (Lessons 5–8)
Topics include comparing theoretical probabilities and experimental probabilities; exploring how the number of trials impacts probability estimates; complementary events; setting up and running simulations using probability models; and revisiting independent events.
A random experiment is an experiment where the set of possible outcomes is known but the actual outcome cannot be predicted with any certainty. Probability theory is the study of random experiments including different ways to measure the likelihood that a particular outcome or event will occur. In this lesson, we review the notion of probability and practise calculating theoretical probabilities of different events in various experiments.
Often random experiments include more than one object, for example, an experiment might include tossing a fair coin and rolling a standard die. In this lesson, we explore how to calculate the probability that two independent events occur, for example, the probability that a head is tossed and an even number is rolled. We define and identify independent events and use tables and tree diagrams to systematically list all outcomes of an experiment in order to calculate probabilities of various events.
Theoretical probability is a ratio that describes what we expect to happen in an experiment and experimental probability is a ratio that describes what actually happened during trials of an experiment. In this lesson, we calculate experimental probabilities of different events and explore how these compare to known theoretical probabilities. We also explore situations where experimentation is our only option for studying probabilities.
If you can determine the chances that a particular event will occur in an experiment, then you can use this information to make predictions involving this experiment. In this lesson, we use theoretical and experimental probabilities to make predictions. We discuss how reliable, or unreliable, our predictions might be and explore how we might design experiments in a way that makes our predictions as reliable as possible.
In this lesson, we compare theoretical probabilities to probability estimates found through experimentation, and explore how the number of trials performed in an experiment might impact probability estimates.
For many real-world situations involving probabilities, it can be difficult to collect data directly by running real experiments. In these situations, mathematicians often run simulations that resemble the real situation in terms of probabilities. In this lesson, we will learn how to choose appropriate models for simulations and practise running simulations to obtain probability estimates.
In this lesson, we review how to determine probabilities of independent events using lists, tables, and tree diagrams to display all possible outcomes. We also explore how to count the number of possible outcomes and favourable outcomes without explicitly writing them down. These skills can be helpful for experiments with too many outcomes to list efficiently.