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Standards for Mathematical Practice: Sample Problems
8.G.9. Know the formulas 1. Make sense of problems and
for the volumes of cones,
persevere in solving them.
cylinders, and spheres
and use them to solve
Reason abstractly and
real-world and
quantitatively.
mathematical problems.
Construct viable arguments and
critique the reasoning of others.
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.
Model with mathematics.
Use appropriate tools
strategically.
Attend to precision.
Look for and make use of
structure.
7.EE.1. Apply properties
of operations as
strategies to add,
subtract, factor, and
expand linear
expressions with
rational coefficients.
Look for and express regularity in
repeated reasoning.
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
w  w  2w  2w . Write the expression in two other ways.
Solution: 6w OR 2( w)  2(2w) .
Standards for Mathematical Practice: Sample Problems
6RP3. Use ratio and rate 3. Construct viable arguments
reasoning to solve realand critique the reasoning of
world and mathematical
others.
problems, e.g., by
reasoning about tables of
equivalent ratios, tape
diagrams, double
number line diagrams,
or equations.
1.
2.
3.
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
The board is 5 feet long and 4 feet wide.
The area of the board is 20 ft 2 .
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, 2,880 in 2 ×1 ft 2 144 in 2 =2,880÷144 ft 2 =20 ft 2 .
Standards for Mathematical Practice: Sample Problems
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 wholenumber measurements,
find missing values in
the tables, and plot the
pairs of values on the
coordinate plane. Use
tables to compare ratios.
7.G.3. Describe the twodimensional figures that
result from slicing threedimensional figures, as
in plane sections of right
rectangular prisms and
right rectangular
pyramids.
4. Models with Mathematics
Reason abstractly and quantitatively.
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?
Model with mathematics
Black
4
White
3
Use appropriate tools strategically.
Look for and make use of structure.
5. Use appropriate tools
strategically.
Reason abstractly and
quantitatively.
Model with mathematics.
Look for and make use of
structure.
4
0
3
0
2
0
1
5
6
0
4
5
?
6
0
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.
Standards for Mathematical Practice: Sample Problems
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.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).
6. Attend to precision.
Reason abstractly and
quantitatively.
7. Look for and make use of
structure.
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.
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.)
Standards for Mathematical Practice: Sample Problems
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.