Standards for Mathematical Practice: Sample Problems
8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. / 1. Make sense of problems and persevere in solving them.Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning. / James wanted to plant pansies in his new planter. He wondered how much potting soil he should buy to fill it. Use the measurements in the diagram below to determine the planter’s volume.
7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. / 2. Reason abstractly and quantitatively.
Attend to precision.
Look for and make use of structure. / A rectangle is twice as long as wide. One way to write an expression to find that represents the perimeter would be . Write the expression in two other ways.
Solution: OR .
6RP3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. / 3. Construct viable arguments and critique the reasoning of others. / Jada has a rectangular board that is 60 inches long and 48 inches wide.
1. How long is the board measured in feet? How wide is the board measured in feet?
2. Find the area of the board in square feet.
3. Jada said,
To convert inches to feet, I should divide by 12.
The board has an area of 48 in × 60 in = 2,880 in 2 .
If I divide the area by 12, I can find out the area in square feet.
So the area of the board is 2,880 ÷ 12 = 240 ft 2 .
What went wrong with Jada's reasoning? Explain.
Solution: You must square the conversion factor too
1. The board is 5 feet long and 4 feet wide.
2. The area of the board is 20 ft 2 .
3. While it is true that you convert inches to feet by dividing by 12, that doesn’t work for converting square inches to square feet. Because a square foot is 12 inches on each side, there are 12 2 = 144 square inches per square foot (see the picture).
Thus, 2,880in2×1ft2144in2=2,880÷144ft2=20ft2.
6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
1. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. / 4. Models with Mathematics
Reason abstractly and quantitatively.
Model with mathematics
Use appropriate tools strategically.
Look for and make use of structure. / Compare the number of black to white circles. If the ratio remains the same, how many black circles will you have if you have 60 white circles?
Black / 4 / 40 / 20 / 60 / ?
White / 3 / 30 / 15 / 45 / 60
7.G.3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. / 5. Use appropriate tools strategically.
Reason abstractly and quantitatively.
Model with mathematics.
Look for and make use of structure. / Using a model of a rectangular prism, describe the shapes that are created when planar cuts are made diagonally, perpendicularly, and parallel to the base.
8.F.1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.) / 6. Attend to precision.
Reason abstractly and quantitatively. / Using Use the rule that takes x as input and gives x2+5x+4 as output to determine a set of ordered pairs and graph the function.
6.NS.4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9+2). / 7. Look for and make use of structure. / What is the greatest common factor (GCF) of 24 and 36? How can you use factor lists or the prime factorizations to find the GCF?
Solution: 22 3 = 12. Students should be able to explain that both 24 and 36 have 2 factors of 2 and one factor of 3, thus 2 x 2 x 3 is the greatest common factor.)
7.G.4. Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference and area of a circle / 8. Look for and express regularity in repeated reasoning.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure. / Measure the circumference and diameter of several circular objects in the room (clock, trash can, door knob, wheel, etc.). Explore the relationship between circumference and diameter by looking for patterns in the ratio of the measures. Write an expression that could be used to find the circumference of a circle with any diameter and check the expression on other circles.