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

Work with addition and subtraction equations.

Create equations that describe numbers or relationships

Understand the relationship between zeros and factors of polynomials.

Represent and solve problems involving addition and subtraction.

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

Represent and solve problems involving multiplication and division.

Add and subtract within 20.

Represent and solve equations and inequalities graphically.

Perform arithmetic operations on polynomials.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Generate and analyze patterns.

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

Understand the connections between proportional relationships, lines, and linear equations.

Interpret the structure of expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Analyze patterns and relationships.

Write expressions in equivalent forms to solve problems.

Solve equations and inequalities in one variable.

Write and interpret numerical expressions.

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

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

Work with radicals and integer exponents.

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

Use the four operations with whole numbers to solve problems.

Rewrite rational expressions.

Gain familiarity with factors and multiples.

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

Reason about and solve one-variable equations and inequalities.

Use properties of operations to generate equivalent expressions.

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

Solve systems of equations.

Use polynomial identities to solve problems.
problem_activity: Narrative
The goal of this activity is to represent expressions as decompositions of a figure made of two non-overlapping right rectangular prisms. This gives students an opportunity to interpret parentheses in expressions while also checking their understanding of different ways to represent the volume of rectangular prism; namely, length times width times height and area of a base times the corresponding height.
Students work abstractly and quantitatively in this problem (MP2) as they relate abstract expressions to decompositions of figures composed of two rectangular prisms.
MLR7 Compare and Connect.
Invite students to think about details they can include that will help others understand their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams or drawings. Give students time to investigate each others’ work. During the whole-class discussion, ask, “What kinds of details or language helped you understand your partner’s work?”
Advances: Representing, Conversing
Launch
Groups of 2
“Look at the first figure and think about how you might decompose it into 2 prisms.”
1 minute: quiet think time
“Now you will consider how different expressions can represent the volume of figures.”
Activity
5 minutes: independent work time
5 minutes: partner discussion time
Monitor for students who
use the numbers in the expressions to determine how to break up the figure
break up the figure and use this to identify the expressions.
Student Facing
Explain how each expression represents the volume of the figure. Show your thinking. Organize it so it can be followed by others.
\(((2 \times 3) \times4) + ((3 \times 3) \times 2)\)
\((5 \times 6) + (3 \times 4)\)
[IMG]
How does each expression represent the volume of the prism? Explain or show your thinking. Organize it so it can be followed by others.
\((5\times8\times6) + (5\times4\times9)\)
cubic inches
\((5\times4\times3) + (5\times12\times6)\)
cubic inches
[IMG]
Student Response
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Advancing Student Thinking
If students do not describe how the expressions represent the volume of the figure, show them part of the expression, the numbers inside the parentheses for example, and ask them to explain which part of the figure is represented.
Activity Synthesis
Invite students to share how the expressions for the first prism represent its volume.
“Why are there 3 factors in the expression
\((2 \times 3) \times 4)\)
but only 2 factors in the expression
\((5 \times 6)\)
?” (The 2, 3, and 4 are the length, width, and height of the prism in the back. The 5 is the height of the prism on bottom whose base area is 6.)
“In the expression
\((3 \times 3) \times 2\)
, what do the parentheses tell you?” (They tell me to first multiply 3 times 3, which gives me the area of the base, and then I multiply that by 2 to get the volume of the prism.)


correct_option_index: 20