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# 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.

## Units

### 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.

### Ratios, Rates, and Proportions (N)

Part A (Lessons 1–6)

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

Part B (Lessons 7–11)

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.

### 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.

### 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.

### 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.

### 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.

### Equations and the Pythagorean Theorem (A)

Part A (Lessons 1–7)

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

Part B (Lessons 8-15)

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

### 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.

### 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.

### Probability (D)

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.