few_shots: Example:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Your response:
Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001097

7
options: Interpret the structure of expressions.

Use polynomial identities to solve problems.

Create equations that describe numbers or relationships

Rewrite rational expressions.

Understand solving equations as a process of reasoning and explain the reasoning.

Use the four operations with whole numbers to solve problems.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Work with radicals and integer exponents.

Understand and apply properties of operations and the relationship between addition and subtraction.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Write expressions in equivalent forms to solve problems.

Write and interpret numerical expressions.

Gain familiarity with factors and multiples.

Use properties of operations to generate equivalent expressions.

Reason about and solve one-variable equations and inequalities.

Understand the relationship between zeros and factors of polynomials.

Represent and solve problems involving addition and subtraction.

Solve systems of equations.

Perform arithmetic operations on polynomials.

Represent and analyze quantitative relationships between dependent and independent variables.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Represent and solve problems involving multiplication and division.

Work with addition and subtraction equations.

Generate and analyze patterns.

Add and subtract within 20.

Understand properties of multiplication and the relationship between multiplication and division.

Represent and solve equations and inequalities graphically.

Solve equations and inequalities in one variable.

Multiply and divide within 100.

Analyze patterns and relationships.

Work with equal groups of objects to gain foundations for multiplication.

Understand the connections between proportional relationships, lines, and linear equations.
problem_activity: Narrative
The purpose of this activity is for students to write equivalent expressions in order to find the volume of a figure composed of two right rectangular prisms. Students decompose the figure in two different ways, and write matching expressions to find the volume. For extra support, provide students with colored pencils to shade the two parts of the prism before finding the side lengths they need to calculate the volume.
Monitor and select a student with each of the following strategies to share in the synthesis:
decomposed the figure into a prism with the side lengths 4 ft by 4 ft by 3 ft and a prism with the side lengths 10 ft by 4 ft by 3 ft and wrote this expression (or one written in a different order) to represent the volume:
\((4 \times 4 \times 3) + (10 \times 4 \times 3)\)
decomposed the figure into a prism with the side lengths 4 ft by 8 ft by 3 ft and a prism with the side lengths 6 ft by 4 ft by 3 ft and wrote this expression (or one written in a different order) to represent the volume:
\((4 \times 8 \times 3) + (6 \times 4 \times 3)\)
Action and Expression: Internalize Executive Functions.
Invite students to plan a strategy, including the tools they will use, for finding the volume of the figures. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Organization, Memory, Attention
Launch
Groups of 2
Activity
10 minutes: individual work time
5 minutes: partner discussion
As students work, consider asking, “Why did you choose to decompose the prism that way?”
Student Facing
Find the volume of the figure by decomposing the figure 2 different ways. Show your thinking. Organize it so it can be followed by others.
[IMG]
[IMG]
For each way you decomposed the figure, write an expression that represents the volume.
Mai used this expression to find the volume of the figure:
\((10 \times 8 \times 3) - (6 \times 4 \times 3)\)
.
Use the diagram to interpret Mai's expression. Show your thinking. Organize it so it can be followed by others.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Ask the two selected students to display their work side by side for all to see.
“How are the diagrams the same? How are they different?”
“How do the expressions relate to the diagrams?”
Display:
\((10 \times 8 \times 3) - (6 \times 4 \times 3)\)
“How does this expression represent the volume of the prism?” (The larger rectangular prism has the side lengths
\(10 \times 8 \times 3\)
cubic feet. We can subtract a rectangular prism with the side lengths
\(6 \times 4 \times 3\)
cubic feet.)
“What is the value of
\((10 \times 8 \times 3) - (6 \times 4 \times 3)\)
?” (168)


correct_option_index: 12