Math Games

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

Model and solve problems using linear equations of the form:
• ax = b
• x/a = b, a ≠ 0
• ax + b = c
• x/a + b = c, a ≠ 0
• a(x + b) = c
concretely, pictorially, and symbolically, where a, b, and c, are integers.

• Model a problem with a linear equation, and solve the equation using concrete models.
• Verify the solution to a linear equation using a variety of methods, including concrete materials, diagrams, and substitution.
• Draw a visual representation of the steps used to solve a linear equation, and record each step symbolically.
• Solve a linear equation symbolically.
• Identify and correct errors in an incorrect solution of a linear equation.
• Solve a linear equation by applying the distributive property [e.g., 2(x + 3) = 5; 2x + 6 = 5; ...].
• Solve a 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 perfect square as a square region using materials, such as grid paper or square shapes.
• Determine the factors of a perfect square, and explain why one of the factors is the square root and the others are not.
• Determine whether or not a 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 perfect square, and record it symbolically.
• Determine the square of a number

Determine the approximate square root of numbers that are not perfect squares (limited to whole numbers).
• Estimate the square root of a number that is not a perfect square using the roots of perfect squares as benchmarks.
• Approximate the square root of a number that is not a perfect square using technology (e.g., calculator, 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%.
• Provide a context where a percent may be more than 100% or between 0% and 1%.
• Represent a fractional percent using grid paper.
• Represent a percent greater than 100% using grid paper.
• Determine the percent represented by a shaded region on a grid, and record it in decimal, fractional, or percent form.
• Express a percent in decimal or fractional form.
• Express a decimal in percent or fractional form.
• Express a fraction in decimal or percent form.
• Solve a problem involving percents.
• Solve a problem involving combined percents (e.g., addition of percents, such as GST + PST).
• Solve a problem that involves finding the percent of a percent (e.g., A population increased by 10% one year and then increased 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 context in the forms 3:5 or 3 to 5.
• Express a three-term ratio from a context in the forms 4:7:3 or 4 to 7 to 3.
• Express a part-to-part ratio as a part to whole ratio (e.g., Given the ratio of frozen juice to water is 1 can to 4 cans, this can be written as 1/4 or 1:4 or 1 to 4, [part-to-part ratio]. Related part-to-whole ratios are 1/5 or 1:5 or 1 to 5, which is the ratio of juice to solution, or 4/5, or 4:5 or 4 to 5, which is the ratio of water to solution).
• Identify and describe ratios and rates from real-life examples, and record them symbolically.
• Express a rate using words or symbols (e.g., 20 L per 100 km or 20 L/100 km).
• Express a 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 a/b within a context.
• Provide a context in which a/b represents a
- fraction
- rate
- ratio
- quotient
- probability
• Solve a 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(s) required to solve a problem involving positive fractions.
• Provide a context involving the multiplying of two positive fractions.
• Provide a context involving the dividing of two positive fractions.
• Express a positive mixed number as an improper fraction and a 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, and record the process.
• Model division of a positive fraction by a whole number, concretely or pictorially, and record the process.
• Generalize and apply rules for multiplying and dividing positive fractions, including mixed numbers.
• Solve a problem involving positive fractions, taking into consideration order of operations(limited to problems with positive solutions).

Demonstrate an understanding of multiplication and division of integers, concretely, pictorially, and symbolically.
• Identify the operation(s) required to solve a 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 or quotient of integers.
• Solve a problem involving integers, taking into consideration order of operations.

Solve problems involving positive rational numbers.
• Identify the operation(s) required to solve a problem involving positive rational numbers.
• Determine the reasonableness of an answer to a problem involving positive rational numbers.
• Estimate the solution and solve a problem involving positive rational numbers.
• Identify and correct errors in the solution to a problem involving positive rational numbers.

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

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

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

Draw and construct nets for 3-D objects.
• Match a net to the 3-D object it represents.
• Construct a 3-D object from a net.
• Draw nets for a right circular 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 net and verify the prediction.

Determine the surface area of
• right rectangular prisms
• right triangular prisms
• right cylinders
to solve problems.

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

Develop and apply formulas for determining the volume of right prisms and right cylinders.
• Determine the volume of a right prism, given the area of the base.
• Generalize and apply a rule for determining the volume of right cylinders.
• Explain the relationship between the area of the base of a right 3-D object and the formula for the volume of the object.
• Demonstrate that the orientation of a 3-D object does not affect its volume.
• Apply a formula to solve a 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 3-D object on isometric dot paper.
• Compare different views of a 3-D object to the object.
• Predict the top, front, and side views that will result from a described rotation (limited to multiples of 90°) and verify predictions.
• Draw and label the top, front, and side views that result from a rotation (limited to multiples of 90°).
• 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 tessellation by
• explaining the properties of shapes that make tessellating possible
• creating tessellations
• identifying tessellations in the environment

• Identify in a set of regular polygons those shapes and combinations of shapes that will tessellate, and use angle measurements to justify choices.
• Identify in a set of irregular polygons those shapes and combinations of shapes that will tessellate, and use angle measurements to justify choices.
• Identify a translation, reflection, or rotation in a tessellation.
• Identify a combination of transformations in a tessellation.
• Create a tessellation using one or more 2-D shapes, and describe the tessellation in terms of transformations and conservation of area.
• Create a new tessellating shape (polygon or non-polygon) by transforming a portion of a tessellating polygon, and describe the resulting tessellation in terms of transformations and conservation of area.
• Identify and describe tessellations in the environment.