Math Games

Develop and apply the Pythagorean theorem to solve problems.
• Achievement Indicators
- 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.
• Achievement Indicators
- 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 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 given net and verify the prediction.

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

• Achievement Indicators
- 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.
- 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 prisms and right cylinders.
• Achievement Indicators
- 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.
• Achievement Indicators
- 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 90 degrees) and verify predictions.
- Draw and label the top, front and side views that result from a given rotation (limited to multiples of 90 degrees).
- 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.

• Achievement Indicators
- Identify, in a given set of regular polygons, those shapes and combinations of shapes that will tessellate, and use angle measurements to justify choices, e.g., squares, regular n-gons.
- Identify, in a given 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 given tessellation.
- Identify a combination of transformations in a given 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 given tessellating polygon, e.g., one by M. C. Escher, and describe the resulting tessellation in terms of transformations and conservation of area.
- Identify and describe tessellations in the environment.

Critique ways in which data is presented.
• Achievement Indicators
- Compare the information that is provided for the same data set by a given set of graphs, including circle graphs, line graphs, bar graphs, double 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, double bar graphs and pictographs, in representing a specific 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 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.
• Achievement Indicators
- 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.
• Achievement Indicators
- 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 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.

• Achievement Indicators
- 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; 2 x + 6 = 5; ...
- Solve a given problem using a linear equation and record the process.

Demonstrate an understanding of perfect square and square root, concretely, pictorially and symbolically (limited to whole numbers).
• Achievement Indicators
- 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).
• Achievement Indicators
- 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., 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%.
• Achievement Indicators
- 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, fractional and percent form.
- Express a given percent in decimal or fractional form.
- Express a given decimal in percent or fractional 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 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.
• Achievement Indicators
- 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 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.
• Achievement Indicators
- Explain the meaning of a/b 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.
• Achievement Indicators
- 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 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).

Demonstrate an understanding of multiplication and division of integers, concretely, pictorially and symbolically.
• Achievement Indicators
- 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.
- 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.
- Generalize and apply a rule for determining the sign of the product and quotient of integers.
- Solve a given problem involving integers taking into consideration order of operations.