Math Games

Critique ways in which data is presented in circle graphs, line graphs, bar graphs and pictographs.
• Compare information provided for the same data set by a given set of graphs, including circle graphs, line graphs, bar graphs and pictographs, to determine the strengths and limitations of each graph.
• Identify the advantages and disadvantages of different graphs, including circle graphs, line graphs, bar graphs and pictographs, in representing a given set of data.
• Justify the choice of a graphical representation for a given situation and its corresponding data set.
• Explain how the format of a given graph, such as the size of the intervals, the width of the bars and the visual representation, may lead to misinterpretation of the data.
• Explain how a given formatting choice could misrepresent the data.
• Identify conclusions that are inconsistent with a given data set or graph, and explain the misinterpretation.

Solve problems involving the probability of independent events.
• Determine the probability of two given independent events; and verify the probability, using a different strategy.
• Generalize and apply a rule for determining the probability of independent events.
• Solve a given problem that involves determining the probability of independent events.

Graph and analyze two-variable linear relations.
• Determine the missing value in an ordered pair for a given equation.
• Create a table of values by substituting values for a variable in the equation of a given linear relation.
• Construct a graph from the equation of a given linear relation (limited to discrete data).
• Describe the relationship between the variables of a given graph.

Model and solve problems concretely, pictorially and symbolically, using linear equations of the form:
• ax = b where a, b and c are integers.
• x/a = b, a ≠ 0 where a, b and c are integers.
• ax + b = c where a, b and c are integers.
• x/a + b = c, a ≠ 0 where a, b and c are integers.
• a(x + b) = c where a, b and c are integers.
• Model a given problem with a linear equation; and solve the equation, using concrete models, e.g., counters, integer tiles.
• Verify the solution to a given linear equation, using a variety of methods, including concrete materials, diagrams and substitution.
• Draw a visual representation of the steps used to solve a given linear equation, and record each step symbolically.
• Solve a given linear equation symbolically.
• Identify and correct an error in a given incorrect solution of a linear equation.
• Apply the distributive property to solve a given linear equation; e.g., 2(x + 3) = 5 is equivalent to 2x + 6 = 5.
• Solve a given problem, using a linear equation, and record the process.

Demonstrate an understanding of perfect squares and square roots, concretely, pictorially and symbolically (limited to whole numbers).
• Represent a given perfect square as a square region, using materials such as grid paper or square shapes.
• Determine the factors of a given perfect square, and explain why one of the factors is the square root and the others are not.
• Determine whether or not a given number is a perfect square, using materials and strategies such as square shapes, grid paper or prime factorization, and explain the reasoning.
• Determine the square root of a given perfect square, and record it symbolically.
• Determine the square of a given number.

Determine the approximate square root of numbers that are not perfect squares (limited to whole numbers).
• Estimate the square root of a given number that is not a perfect square, using the roots of perfect squares as benchmarks.
• Approximate the square root of a given number that is not a perfect square, using technology; e.g., a calculator, a computer.
• Explain why the square root of a number shown on a calculator may be an approximation.
• Identify a number with a square root that is between two given numbers.

Demonstrate an understanding of percents greater than or equal to 0%, including greater than 100%.
• Provide a context where a percent may be more than 100% or between 0% and 1%.
• Represent a given fractional percent, using grid paper.
• Represent a given percent greater than 100%, using grid paper.
• Determine the percent represented by a given shaded region on a grid, and record it in decimal, fraction and percent form.
• Express a given percent in decimal or fraction form.
• Express a given decimal in percent or fraction form.
• Express a given fraction in decimal or percent form.
• Solve a given problem involving percents.
• Solve a given problem involving combined percents, e.g., addition of percents, such as GST + PST.
• Solve a given problem that involves finding the percent of a percent; e.g., "A population increased by 10% one year and by 15% the next year. Explain why there was not a 25% increase in population over the two years."

Demonstrate an understanding of ratio and rate.
• Express a two-term ratio from a given context in the forms 3:5 or 3 to 5.
• Express a three-term ratio from a given context in the forms 4:7:3 or 4 to 7 to 3.
• Express a part to part ratio as a part to whole fraction; e.g., frozen juice to water: 1 can concentrate to 4 cans of water can be represented as 1/5, which is the ratio of concentrate to solution, or 4/5, which is the ratio of water to solution.
• Identify and describe ratios and rates (including unit rates) from real-life examples, and record them symbolically.
• Express a given rate, using words or symbols; e.g., 20 L per 100 km or 20 L/100 km.
• Express a given ratio as a percent, and explain why a rate cannot be represented as a percent.

Solve problems that involve rates, ratios and proportional reasoning.
• Explain the meaning of ab within a given context.
• Provide a context in which a/b represents a:
- fraction.
- rate.
- ratio.
- quotient.
- probability.
• Solve a given problem involving rate, ratio or percent.

Demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially and symbolically.
• Identify the operation required to solve a given problem involving positive fractions.
• Provide a context that requires the multiplying of two given positive fractions.
• Provide a context that requires the dividing of two given positive fractions.
• Estimate the product of two given positive proper fractions to determine if the product will be closer to 0,1/2 or 1.
• Estimate the quotient of two given positive fractions, and compare the estimate to whole number benchmarks.
• Express a given positive mixed number as an improper fraction and a given positive improper fraction as a mixed number.
• Model multiplication of a positive fraction by a whole number concretely or pictorially, and record the process.
• Model multiplication of a positive fraction by a positive fraction concretely or pictorially, using an area model, and record the process.
• Model division of a positive proper fraction by a whole number concretely or pictorially, and record the process.
• Model division of a whole number by a positive proper fraction concretely or pictorially, using an area model, and record the process.
• Model division of a positive proper fraction by a positive proper fraction pictorially, and record the process.
• Generalize and apply rules for multiplying and dividing positive fractions, including mixed numbers.
• Solve a given problem involving positive fractions, taking into consideration order of operations (limited to problems with positive solutions).
• Apply a personal strategy to solve, symbolically, a given division problem involving improper fractions.
• Refine personal strategies to increase their efficiency.

Demonstrate an understanding of multiplication and division of integers, concretely, pictorially and symbolically.
• Identify the operation required to solve a given problem involving integers.
• Provide a context that requires multiplying two integers.
• Provide a context that requires dividing two integers.
• Model the process of multiplying two integers, using concrete materials or pictorial representations, and record the process.
• Model the process of dividing an integer by an integer, using concrete materials or pictorial representations, and record the process.
• Generalize and apply a rule for determining the sign of the product and quotient of integers.
• Solve a given problem involving the division of integers (2-digit by 1-digit) without the use of technology.
• Solve a given problem involving the division of integers (2-digit by 2-digit) with the use of technology.
• Solve a given problem involving integers, taking into consideration order of operations.

Develop and apply the Pythagorean theorem to solve problems.
• Model and explain the Pythagorean theorem concretely, pictorially or using technology.
• Explain, using examples, that the Pythagorean theorem applies only to right triangles.
• Determine whether or not a given triangle is a right triangle by applying the Pythagorean theorem.
• Determine the measure of the third side of a right triangle, given the measures of the other two sides, to solve a given problem.
• Solve a given problem that involves Pythagorean triples; e.g., 3, 4, 5 or 5, 12, 13.

Draw and construct nets for 3-D objects.
• Match a given net to the 3-D object it represents.
• Construct a 3-D object from a given net.
• Draw nets for a given right cylinder, right rectangular prism and right triangular prism, and verify by constructing the 3-D objects from the nets.
• Predict 3-D objects that can be created from a given net, and verify the prediction.

Determine the surface area of:
• right rectangular prisms to solve problems.
• right triangular prisms to solve problems.
• right cylinders to solve problems.
• Explain, using examples, the relationship between the area of 2-D shapes and the surface area of a given 3-D object.
• Identify all the faces of a given prism, including right rectangular and right triangular prisms.
• Identify all the faces of a given right cylinder.
• Describe and apply strategies for determining the surface area of a given right rectangular or right triangular prism.
• Describe and apply strategies for determining the surface area of a given right cylinder.
• Solve a given problem involving surface area.

Develop and apply formulas for determining the volume of right rectangular prisms, right triangular prisms and right cylinders.
• Determine the volume of a given right prism, given the area of the base.
• Generalize and apply a rule for determining the volume of right cylinders.
• Explain the connection between the area of the base of a given right 3-D object and the formula for the volume of the object.
• Demonstrate that the orientation of a given 3-D object does not affect its volume.
• Apply a formula to solve a given problem involving the volume of a right cylinder or a right prism.

Draw and interpret top, front and side views of 3-D objects composed of right rectangular prisms.
• Draw and label the top, front and side views for a given 3-D object on isometric dot paper.
• Compare different views of a given 3-D object to the object.
• Predict the top, front and side views that will result from a described rotation (limited to multiples of 90º), and verify predictions.
• Draw and label the top, front and side views that result from a given rotation (limited to multiples of 90º).
• Build a 3-D block object given the top, front and side views, with or without the use of technology.
• Sketch and label the top, front and side views of a 3-D object in the environment, with or without the use of technology.

Demonstrate an understanding of the congruence of polygons.
• Determine the coordinates of the vertices of an image following a given combination of transformations of the original figure.
• Draw the original figure and determine the coordinates of its vertices, given the coordinates of the image's vertices and a description of the transformation (translation, rotation, reflection).