• 8
Top Mathematicians
• Patterns and Relations
• Shape and Space
• 8.SS.1
Develop and apply the Pythagorean theorem to solve problems.
Students who have achieved this outcome should be able to:
A. Model and explain the Pythagorean theorem concretely, pictorially or using technology.
B. Explain, using examples, that the Pythagorean theorem applies only to right triangles.
C. Determine whether or not a given triangle is a right triangle by applying the Pythagorean theorem.
D. Determine the measure of the third side of a right triangle, given the measures of the other two sides, to solve a given problem.
E. Solve a given problem that involves Pythagorean triples, e.g., 3, 4, 5 or 5, 12, 13.
• 8.SS.2
Draw and construct nets for 3-D objects.
Students who have achieved this outcome should be able to:
A. Match a given net to the 3-D object it represents.
B. Construct a 3-D object from a given net.
C. 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.
D. Predict 3-D objects that can be created from a given net and verify the prediction.
• 8.SS.3
Determine the surface area of:
right rectangular prisms;
right triangular prisms;
right cylinders
to solve problems.

Students who have achieved this outcome should be able to:
A. Explain, using examples, the relationship between the area of 2-D shapes and the surface area of a given 3-D object.
B. Identify all the faces of a given prism, including right rectangular and right triangular prisms.
C. Describe and apply strategies for determining the surface area of a given right rectangular or right triangular prism.
D. Describe and apply strategies for determining the surface area of a given right cylinder.
E. Solve a given problem involving surface area.
• 8.SS.4
Develop and apply formulas for determining the volume of right prisms and right cylinders.
Students who have achieved this outcome should be able to:
A. Determine the volume of a given right prism, given the area of the base.
B. Generalize and apply a rule for determining the volume of right cylinders.
C. Explain the connection between the area of the base of a given right 3-D object and the formula for the volume of that object.
D. Demonstrate that the orientation of a given 3-D object does not affect its volume.
E. Apply a formula to solve a given problem involving the volume or a right cylinder or a right prism.
• 8.SS.5
Draw and interpret top, front and side views of 3-D objects composed of right rectangular prisms.
Students who have achieved this outcome should be able to:
A. Draw and label the top, front and side views for a given 3-D object on isometric dot paper.
B. Compare different views of a given 3-D object to the object.
C. Predict the top, front and side views that will result from a described rotation (limited to multiples of 90 degrees) and verify predictions.
D. Draw and label the top, front and side views that result from a given rotation (limited to multiples of 90 degrees).
E. Build a 3-D block object, given the top, front and side views, with or without the use of technology.
F. Sketch and label the top, front and side views of a 3-D object in the environment with or without the use of technology.
• 8.SS.6
Demonstrate an understanding of tessellation by:
explaining the properties of shapes that make tessellating possible;
creating tessellations;
identifying tessellations in the environment.

Students who have achieved this outcome should be able to:
A. 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.
B. Identify, in a given set of irregular polygons, those shapes and combinations of shapes that will tessellate, and use angle measurements to justify choices.
C. Identify a translation, reflection or rotation in a given tessellation.
D. Identify a combination of transformations in a given tessellation.
E. Create a tessellation using one or more 2-D shapes, and describe the tessellation in terms of transformations and conservation of area.
F. 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.
G. Identify and describe tessellations in the environment.
• Statistics & Probability
• 8.SP.1
Critique ways in which data is presented.
Students who have achieved this outcome should be able to:
A. 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.
B. 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.
C. Justify the choice of a graphical representation for a given situation and its corresponding data set.
D. 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.
E. Explain how a given formatting choice could misinterpret the data.
F. Identify conclusions that are inconsistent with a given data set or graph and explain the misinterpretation.
• 8.SP.2
Solve problems involving the probability of independent events.
Students who have achieved this outcome should be able to:
A. Determine the probability of two given independent events and verify the probability using a different strategy.
B. Generalize and apply a rule for determining the probability of independent events.
C. Solve a given problem that involves determining the probability of independent events.
• Number
• 8.N.1
Demonstrate an understanding of perfect square and square root, concretely, pictorially and symbolically (limited to whole numbers).
Students who have achieved this outcome should be able to:
A. Represent a given perfect square as a square region using materials, such as grid paper or square shapes.
B. Determine the factors of a given perfect square, and explain why one of the factors is the square root and the others are not.
C. 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.
D. Determine the square root of a given perfect square and record it symbolically.
E. Determine the square of a given number.
• 8.N.2
Determine the approximate square root of numbers that are not perfect squares (limited to whole numbers).
Students who have achieved this outcome should be able to:
A. Estimate the square root of a given number that is not a perfect square using the roots of perfect squares as benchmarks.
B. Approximate the square root of a given number that is not a perfect square using technology, e.g., calculator, computer.
C. Explain why the square root of a number shown on a calculator may be an approximation.
D. Identify a number with a square root that is between two given numbers.
• 8.N.3
Demonstrate an understanding of percents greater than or equal to 0%.
Students who have achieved this outcome should be able to:
A. Provide a context where a percent may be more than 100% or between 0% and 1%.
B. Represent a given fractional percent using grid paper.
C. Represent a given percent greater than 100 using grid paper.
D. Determine the percent represented by a given shaded region on a grid, and record it in decimal, fractional and percent form.
E. Express a given percent in decimal or fractional form.
F. Express a given decimal in percent or fractional form.
G. Express a given fraction in decimal or percent form.
H. Solve a given problem involving percents.
I. Solve a given problem involving combined percents.
J. 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.
• 8.N.4
Demonstrate an understanding of ratio and rate.
Students who have achieved this outcome should be able to:
A. Express a two-term ratio from a given context in the forms 3 : 5 or 3 to 5.
B. Express a three-term ratio from a given context in the forms 4 : 7 : 3 or 4 to 7 to 3.
C. Express a part to part ratio as a part to whole fraction, e.g., frozen juice to water; 1 can of 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.
D. Identify and describe ratios and rates from real-life examples, and record them symbolically.
E. Express a given rate using words or symbols, e.g., 20 L per 100 km or 20 L/100 km.
F. Express a given ratio as a percent and explain why a rate cannot be represented as a percent.
• 8.N.5
Solve problems that involve rates, ratios and proportional reasoning.
Students who have achieved this outcome should be able to:
A. Explain the meaning of a
b within a given context.
B. Provide a context in which a/b represents a:
fraction;
rate;
ratio;
quotient;
probability.
C. Solve a given problem involving rate, ratio or percent.
• 8.N.6
Demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially and symbolically.
Students who have achieved this outcome should be able to:
A. Identify the operation required to solve a given problem involving positive fractions.
B. Provide a context that requires the multiplying of two given positive fractions.
C. Provide a context that requires the dividing of two given positive fractions.
D. Estimate the product of two given positive proper fractions to determine if the product will be closer to 0, 1/2 or 1.
E. Estimate the quotient of two given positive fractions and compare the estimate to whole number benchmarks.
F. Express a given positive mixed number as an improper fraction and a given positive improper fraction as a mixed number.
G. Model multiplication of a positive fraction by a whole number concretely or pictorially and record the process.
H. Model multiplication of a positive fraction by a positive fraction concretely or pictorially using an area model and record the process.
I. Model division of a positive proper fraction by a whole number concretely or pictorially and record the process.
J. Model division of a positive proper fraction by a positive proper fraction pictorially and record the process.
K. Generalize and apply rules for multiplying and dividing positive fractions, including mixed numbers.
L. Solve a given problem involving positive fractions, taking into consideration order of operations (limited to problems with positive solutions).
• 8.N.7
Demonstrate an understanding of multiplication and division of integers, concretely, pictorially and symbolically.
Students who have achieved this outcome should be able to:
A. Identify the operation required to solve a given problem involving integers.
B. Provide a context that requires multiplying two integers.
C. Provide a context that requires dividing two integers.
D. Model the process of multiplying two integers using concrete materials or pictorial representations and record the process.
E. Model the process of dividing an integer by an integer using concrete materials or pictorial representations and record the process.
F. Solve a given problem involving the division of integers (2-digit by 1-digit) without the use of technology.
G. Solve a given problem involving the division of integers (2-digit by 2-digit) with the use of technology.
H. Generalize and apply a rule for determining the sign of the product and quotient of integers.
I. Solve a given problem involving integers taking into consideration order of operations.