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Ontario Math Standards - 8th Grade
MathScore aligns to the Ontario Math Standards for 8th Grade. The standards appear below along with the MathScore topics that match. If you click on a topic name, you will see sample problems at varying degrees of difficulty that MathScore generated. When students use our program, the difficulty of the problems will automatically adapt based on individual performance, resulting in not only true differentiated instruction, but a challenging game-like experience.
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Number Sense and Numeration
Overall ExpectationsBy the end of Grade 8, students will:
• represent, compare, and order equivalent representations of numbers, including those involving positive exponents; (Compare Mixed Values , Compare Mixed Values 2 , Positive Number Line , Number Line , Compare Integers , Exponent Basics )
• solve problems involving whole numbers, decimal numbers, fractions, and integers, using a variety of computational strategies; (Small Decimal Division , Unit Cost , Fraction Addition , Fraction Subtraction , Fraction Multiplication , Fraction Division , Fraction Word Problems , Fraction Word Problems 2 , Decimal Addition , Decimal Subtraction , Decimal Multiplication , Decimal Division , Integer Addition , Integer Subtraction , Positive Integer Subtraction , Integer Multiplication , Integer Division , Integers In Word Problems )
• solve problems by using proportional reasoning in a variety of meaningful contexts. (Unit Cost , Proportions 2 , Distance, Rate, and Time )
Specific Expectations
Quantity Relationships
By the end of Grade 8, students will:
- express repeated multiplication using exponential notation (e.g., 2 x 2 x 2 x 2 = 24); (Exponent Basics )
- represent whole numbers in expanded form using powers of ten
- represent, compare, and order rational numbers (i.e., positive and negative fractions and decimals to thousandths); (Compare Mixed Values , Compare Mixed Values 2 , Positive Number Line , Number Line , Compare Integers )
- translate between equivalent forms of a number (i.e., decimals, fractions, percents) (e.g., 3/4 = 0.75); (Fractions to Decimals , Decimals To Fractions , Percentages )
- determine common factors and common multiples using the prime factorization of numbers (e.g., the prime factorization of 12 is 2 x 2 x 3; the prime factorization of 18 is 2 x 3 x 3; the greatest common factor of 12 and 18 is 2 x 3 or 6; the least common multiple of 12 and 18 is 2 x 2 x 3 x 3 or 36). (Prime Factoring , Greatest Common Factor , Least Common Multiple )
Operational Sense
By the end of Grade 8, students will:
- solve multi-step problems arising from real-life contexts and involving whole numbers and decimals, using a variety of tools (e.g., graphs, calculators) and strategies (e.g., estimation, algorithms); (Making Change , Unit Cost )
- solve problems involving percents expressed to one decimal place (e.g., 12.5%) and whole-number percents greater than 100 (e.g., 115%) (Sample problem: The total cost of an item with tax included [115%] is $23.00. Use base ten materials to determine the price before tax.); (Percentage Change , Percent of Quantity )
- use estimation when solving problems involving operations with whole numbers, decimals, percents, integers, and fractions, to help judge the reasonableness of a solution;
- represent the multiplication and division of fractions, using a variety of tools and strategies (e.g., use an area model to represent 1/4 muitiplied by 1/3); (Fraction Multiplication , Fraction Division )
- solve problems involving addition, subtraction, multiplication, and division with simple fractions; (Fraction Addition , Fraction Subtraction , Fraction Multiplication , Fraction Division , Fraction Word Problems , Fraction Word Problems 2 )
- represent the multiplication and division of integers, using a variety of tools [e.g., if black counters represent positive amounts and red counters represent negative amounts, you can model 3 x (-2) as three groups of two red counters]; (Integer Multiplication , Integer Division )
- solve problems involving operations with integers, using a variety of tools (e.g., two colour counters, virtual manipulatives, number lines); (Integer Addition , Integer Subtraction , Positive Integer Subtraction , Integer Multiplication , Integer Division , Integer Equivalence , Integers In Word Problems )
- evaluate expressions that involve integers, including expressions that contain brackets and exponents, using order of operations; (Using Parentheses , Order Of Operations , Variable Substitution 2 )
- multiply and divide decimal numbers by various powers of ten (e.g.,"To convert 230 000 cm3 to cubic metres, I calculated in my head 230 000 ÷ 106 to get 0.23 m3.") (Sample problem: Use a calculator to help you generalize a rule for dividing numbers by 1 000 000.); (Multiply By Multiples Of 10 )
- estimate, and verify using a calculator, the positive square roots of whole numbers, and distinguish between whole numbers that have whole-number square roots (i.e., perfect square numbers) and those that do not (Sample problem: Explain why a square with an area of 20 cm2 does not have a whole-number side length.). (Estimating Square Roots , Perfect Squares )
Proportional Relationships
By the end of Grade 8, students will:
- identify and describe real-life situations involving two quantities that are directly proportional (e.g., the number of servings and the quantities in a recipe, mass and volume of a substance, circumference and diameter of a circle); (Mixture Word Problems , Work Word Problems )
- solve problems involving proportions, using concrete materials, drawings, and variables (Sample problem: The ratio of stone to sand in HardFast Concrete is 2 to 3. How much stone is needed if 15 bags of sand are used?); (Unit Cost , Area And Volume Proportions , Proportions 2 , Distance, Rate, and Time )
- solve problems involving percent that arise from real-life contexts (e.g., discount, sales tax, simple interest) (Sample problem: In Ontario, people often pay a provincial sales tax [PST] of 8% and a federal sales tax [GST] of 7% when they make a purchase. Does it matter which tax is calculated first? Explain your reasoning.); (Purchases At Stores , Restaurant Bills , Commissions , Simple Interest , Compound Interest )
- solve problems involving rates (Sample problem: A pack of 24 CDs costs $7.99. A pack of 50 CDs costs $10.45. What is the most economical way to purchase 130 CDs?). (Unit Cost , Distance, Rate, and Time )
Measurement
Overall ExpectationsBy the end of Grade 8, students will:
• research, describe, and report on applications of volume and capacity measurement;
• determine the relationships among units and measurable attributes, including the area of a circle and the volume of a cylinder. (Circle Area , Cylinders )
Specific Expectations
Attributes, Units, and Measurement Sense
By the end of Grade 8, students will:
- research, describe, and report on applications of volume and capacity measurement (e.g., cooking, closet space, aquarium size) (Sample problem: Describe situations where volume and capacity are used in your home.).
Measurement Relationships
By the end of Grade 8, students will:
- solve problems that require conversions involving metric units of area, volume, and capacity (i.e., square centimetres and square metres; cubic centimetres and cubic metres; millilitres and cubic centimetres) (Sample problem: What is the capacity of a cylindrical beaker with a radius of 5 cm and a height of 15 cm?); (Area and Volume Conversions Metric )
- measure the circumference, radius, and diameter of circular objects, using concrete materials (Sample Problem: Use string to measure the circumferences of different circular objects.); (Circle Measurements , Circle Circumference )
- determine, through investigation using a variety of tools (e.g., cans and string, dynamic geometry software) and strategies, the relationships for calculating the circumference and the area of a circle, and generalize to develop the formulas (Sample problem: Use string to measure the circumferences and the diameters of a variety of cylindrical cans, and investigate the ratio of the circumference to the diameter.); (Circle Area , Circle Circumference )
- solve problems involving the estimation and calculation of the circumference and the area of a circle; (Circle Area , Circle Circumference )
- determine, through investigation using a variety of tools and strategies (e.g., generalizing from the volume relationship for right prisms, and verifying using the capacity of thin-walled cylindrical containers), the relationship between the area of the base and height and the volume of a cylinder, and generalize to develop the formula (i.e., Volume = area of base x height); (Cylinders )
- determine, through investigation using concrete materials, the surface area of a cylinder (Sample problem: Use the label and the plastic lid from a cylindrical container to help determine its surface area.); (Cylinders )
- solve problems involving the surface area and the volume of cylinders, using a variety of strategies (Sample problem: Compare the volumes of the two cylinders that can be created by taping the top and bottom, or the other two sides, of a standard sheet of paper.). (Cylinders )
Geometry and Spatial Sense
Overall ExpectationsBy the end of Grade 8, students will:
• demonstrate an understanding of the geometric properties of quadrilaterals and circles and the applications of geometric properties in the real world; (Circle Measurements , Quadrilateral Angles , Parallelogram Area , Circle Area , Circle Circumference , Irregular Shape Areas , Perimeter and Area Word Problems )
• develop geometric relationships involving lines, triangles, and polyhedra, and solve problems involving lines and triangles; (Triangle Angles , Triangle Area , Triangle Area 2 , Triangle Angles 2 , Identifying Angles , Pythagorean Theorem , Solving For Angles , Angle Measurements , Angle Measurements 2 )
• represent transformations using the Cartesian coordinate plane, and make connections between transformations and the real world. (Translations and Reflections )
Specific Expectations
Geometric Properties
By the end of Grade 8, students will:
- sort and classify quadrilaterals by geometric properties, including those based on diagonals, through investigation using a variety of tools (e.g., concrete materials, dynamic geometry software) (Sample problem: Which quadrilaterals have diagonals that bisect each other perpendicularly?); (Quadrilateral Types )
- construct a circle, given its centre and radius, or its centre and a point on the circle, or three points on the circle;
- investigate and describe applications of geometric properties (e.g., properties of triangles, quadrilaterals, and circles) in the real world. (Perimeter and Area Word Problems )
Geometric Relationships
By the end of Grade 8, students will:
- determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials, geoboard), relationships among area, perimeter, corresponding side lengths, and corresponding angles of similar shapes (Sample problem: Construct three similar rectangles, using grid paper or a geoboard, and compare the perimeters and areas of the rectangles.); (Area And Volume Proportions , Proportions 2 )
- determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials, protractor) and strategies (e.g., paper folding), the angle relationships for intersecting lines and for parallel lines and transversals, and the sum of the angles of a triangle; (Triangle Angles , Identifying Angles )
- solve angle-relationship problems involving triangles (e.g., finding interior angles or complementary angles), intersecting lines (e.g., finding supplementary angles or opposite angles), and parallel lines and transversals (e.g., finding alternate angles or corresponding angles); (Triangle Angles , Triangle Angles 2 , Solving For Angles , Angle Measurements , Angle Measurements 2 )
- determine the Pythagorean relationship, through investigation using a variety of tools (e.g., dynamic geometry software; paper and scissors; geoboard) and strategies; (Pythagorean Theorem )
- solve problems involving right triangles geometrically, using the Pythagorean relationship; (Pythagorean Theorem )
- determine, through investigation using concrete materials, the relationship between the numbers of faces, edges, and vertices of a polyhedron (i.e., number of faces + number of vertices = number of edges + 2) (Sample problem: Use Polydrons and/or paper nets to construct the five Platonic solids [i.e., tetrahedron, cube, octahedron, dodecahedron, icosahedron], and compare the sum of the numbers of faces and vertices to the number of edges for each solid.).
Location and Movement
By the end of Grade 8, students will:
- graph the image of a point, or set of points, on the Cartesian coordinate plane after applying a transformation to the original point(s) (i.e., translation; reflection in the x-axis, the y-axis, or the angle bisector of the axes that passes through the first and third quadrants; rotation of 90°, 180°, or 270° about the origin);
- identify, through investigation, real-world movements that are translations, reflections, and rotations. (Translations and Reflections )
Patterning and Algebra
Overall ExpectationsBy the end of Grade 8, students will:
• represent linear growing patterns (where the terms are whole numbers) using graphs, algebraic expressions, and equations; (Determining Slope , Function Tables , Function Tables 2 )
• model linear relationships graphically and algebraically, and solve and verify algebraic equations, using a variety of strategies, including inspection, guess and check, and using a "balance" model. (Linear Equations , Single Variable Equations , Single Variable Equations 2 , Single Variable Equations 3 , Determining Slope , Graphs to Linear Equations , Graphs to Linear Equations 2 , Graphs to Linear Inequalities , Single Variable Inequalities )
Specific Expectations
Patterns and Relationships
By the end of Grade 8, students will:
- represent, through investigation with concrete materials, the general term of a linear pattern, using one or more algebraic expressions (e.g.,"Using toothpicks, I noticed that 1 square needs 4 toothpicks, 2 connected squares need 7 toothpicks, and 3 connected squares need 10 toothpicks. I think that for n connected squares I will need 4 + 3(n - 1) toothpicks, because the number of toothpicks keeps going up by 3 and I started with 4 toothpicks. Or, if I think of starting with 1 toothpick and adding 3 toothpicks at a time, the pattern can be represented as 1 + 3n."); (Function Tables , Function Tables 2 )
- represent linear patterns graphically (i.e., make a table of values that shows the term number and the term, and plot the coordinates on a graph), using a variety of tools (e.g., graph paper, calculators, dynamic statistical software);
- determine a term, given its term number, in a linear pattern that is represented by a graph or an algebraic equation (Sample problem: Given the graph that represents the pattern 1, 3, 5, 7,…, find the 10th term. Given the algebraic equation that represents the pattern, t = 2n - 1, find the 100th term.).
Variables, Expressions, and Equations
By the end of Grade 8, students will:
- describe different ways in which algebra can be used in real-life situations (e.g., the value of $5 bills and toonies placed in a envelope for fund raising can be represented by the equation v = 5f + 2t); (Mixture Word Problems , Work Word Problems , Integer Word Problems )
- model linear relationships using tables of values, graphs, and equations (e.g., the sequence 2, 3, 4, 5, 6,… can be represented by the equation t = n + 1, where n represents the term number and t represents the term), through investigation using a variety of tools (e.g., algebra tiles, pattern blocks, connecting cubes, base ten materials) (Sample problem: Leah put $350 in a bank certificate that pays 4% simple interest each year. Make a table of values to show how much the bank certificate is worth after five years, using base ten materials to help you. Represent the relationship using an equation.); (Linear Equations , Determining Slope , Graphs to Linear Equations , Graphs to Linear Equations 2 , Graphs to Linear Inequalities )
- translate statements describing mathematical relationships into algebraic expressions and equations (e.g., for a collection of triangles, the total number of sides is equal to three times the number of triangles or s = 3n); (Phrases to Algebraic Expressions , Algebraic Sentences 2 , Algebraic Sentences )
- evaluate algebraic expressions with up to three terms, by substituting fractions, decimals, and integers for the variables (e.g., evaluate 3x + 4y = 2z, where x = 1/2, y = 0.6, and z = -1); (Variable Substitution 2 )
- make connections between solving equations and determining the term number in a pattern, using the general term (e.g., for the pattern with the general term 2n + 1, solving the equation 2n + 1 = 17 tells you the term number when the term is 17);
- solve and verify linear equations involving a one-variable term and having solutions that are integers, by using inspection, guess and check, and a "balance" model (Sample problem: What is the value of the variable in the equation 30x - 5 = 10?). (Linear Equations , Single Variable Equations , Single Variable Equations 2 , Single Variable Equations 3 , Single Variable Inequalities )
Data Management and Probability
Overall ExpectationsBy the end of Grade 8, students will:
• collect and organize categorical, discrete, or continuous primary data and secondary data and display the data using charts and graphs, including frequency tables with intervals, histograms, and scatter plots; (Requires outside materials )
• apply a variety of data management tools and strategies to make convincing arguments about data;
• use probability models to make predictions about real-life events. (Probability , Probability 2 , Object Picking Probability )
Specific Expectations
Collection and Organization of Data
By the end of Grade 8, students will:
- collect data by conducting a survey or an experiment to do with themselves, their environment, issues in their school or community, or content from another subject, and record observations or measurements; (Requires outside materials )
- organize into intervals a set of data that is spread over a broad range (e.g., the age of respondents to a survey may range over 80 years and may be organized into ten-year intervals);
- collect and organize categorical, discrete, or continuous primary data and secondary data (e.g., electronic data from websites such as E-Stat or Census At Schools), and display the data in charts, tables, and graphs (including histograms and scatter plots) that have appropriate titles, labels (e.g., appropriate units marked on the axes), and scales (e.g., with appropriate increments) that suit the range and distribution of the data, using a variety of tools (e.g., graph paper, spreadsheets, dynamic statistical software); (Requires outside materials )
- select an appropriate type of graph to represent a set of data, graph the data using technology, and justify the choice of graph (i.e., from types of graphs already studied, including histograms and scatter plots);
- explain the relationship between a census, a representative sample, sample size, and a population (e.g.,"I think that in most cases a larger sample size will be more representative of the entire population.").
Data Relationships
By the end of Grade 8, students will:
- read, interpret, and draw conclusions from primary data (e.g., survey results, measurements, observations) and from secondary data (e.g., election data or temperature data from the newspaper, data from the Internet about lifestyles), presented in charts, tables, and graphs (including frequency tables with intervals, histograms, and scatter plots); (Stem And Leaf Plots , Tally and Pictographs , Bar Graphs , Line Graphs )
- determine, through investigation, the appropriate measure of central tendency (i.e., mean, median, or mode) needed to compare sets of data (e.g., in hockey, compare heights or masses of players on defence with that of forwards);
- demonstrate an understanding of the appropriate uses of bar graphs and histograms by comparing their characteristics (Sample problem: How is a histogram similar to and different from a bar graph? Use examples to support your answer.);
- compare two attributes or characteristics (e.g., height versus arm span), using a scatter plot, and determine whether or not the scatter plot suggests a relationship (Sample problem: Create a scatter plot to compare the lengths of the bases of several similar triangles with their areas.);
- identify and describe trends, based on the rate of change of data from tables and graphs, using informal language (e.g., "The steep line going upward on this graph represents rapid growth. The steep line going downward on this other graph represents rapid decline.");
- make inferences and convincing arguments that are based on the analysis of charts, tables, and graphs (Sample problem: Use data to make a convincing argument that the environment is becoming increasingly polluted.);
- compare two attributes or characteristics, using a variety of data management tools and strategies (i.e., pose a relevant question, then design an experiment or survey, collect and analyse the data, and draw conclusions) (Sample problem: Compare the length and width of different-sized leaves from a maple tree to determine if maple leaves grow proportionally. What generalizations can you make?).
Probability
By the end of Grade 8, students will:
- compare, through investigation, the theoretical probability of an event (i.e., the ratio of the number of ways a favourable outcome can occur compared to the total number of possible outcomes) with experimental probability, and explain why they might differ (Sample problem: Toss a fair coin 10 times, record the results, and explain why you might not get the predicted result of 5 heads and 5 tails.); (Probability , Probability 2 , Object Picking Probability )
- determine, through investigation, the tendency of experimental probability to approach theoretical probability as the number of trials in an experiment increases, using class-generated data and technology-based simulation models (Sample problem: Compare the theoretical probability of getting a 6 when tossing a number cube with the experimental probabilities obtained after tossing a number cube once, 10 times, 100 times, and 1000 times.);
- identify the complementary event for a given event, and calculate the theoretical probability that a given event will not occur (Sample problem: Bingo uses the numbers from 1 to 75. If the numbers are pulled at random, what is the probability that the first number is a multiple of 5? Is not a multiple of 5?). (Probability , Probability 2 )
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