options: Work with radicals and integer exponents.

Solve systems of equations.

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

Understand the relationship between zeros and factors of polynomials.

Generate and analyze patterns.

Represent and solve problems involving addition and subtraction.

Use polynomial identities to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

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

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Reason about and solve one-variable equations and inequalities.

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

Use properties of operations to generate equivalent expressions.

Write and interpret numerical expressions.

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

Perform arithmetic operations on polynomials.

Multiply and divide within 100.

Gain familiarity with factors and multiples.

Work with addition and subtraction equations.

Analyze patterns and relationships.

Use the four operations with whole numbers to solve problems.

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

Create equations that describe numbers or relationships

Represent and solve equations and inequalities graphically.

Solve equations and inequalities in one variable.

Add and subtract within 20.

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

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

Write expressions in equivalent forms to solve problems.

Represent and solve problems involving multiplication and division.

Interpret the structure of expressions.

Rewrite rational expressions.
problem_activity: Narrative
The purpose of this activity is for students to find the volume of a figure in different ways. The given figure can be decomposed in two ways into rectangular prisms by making different cuts. However, it can also be found using a single, larger rectangular prism by removing a smaller rectangular prism. This provides an opportunity to express its volume as a difference of volumes of rectangular prisms. Students may notice this feature, and it is highlighted in the activity synthesis.
When students decide whether or not they have the same expressions, they need to reason carefully about what “the same” means. They consider if the order of the factors is different, is it the same expression and if the order of the addends is different, is it the same expression. Students use what they know about volume, geometric figures, and the properties of operations to justify the equivalence of the expressions and critique their peers' reasoning (MP2, MP3, MP7).
MLR8 Discussion Supports.
Display sentence frames to support partner discussion: “First, I _____ because . . . .” , “_____ and _____ are the same because . . .”, “_____ and _____ are different because . . .”
Advances: Conversing, Representing
Launch
Groups of 2
“You are going to look for different expressions to calculate the volume of a figure.”
Activity
5 minutes: partner work time
Monitor for students who draw a vertical line to show where they decompose the figure to share during the synthesis.
Student Facing
[IMG]
Write an expression to represent the volume of the figure in unit cubes.
Compare expressions with your partner.
How are they the same?
How are they different?
If they are the same, try to find another way to represent the volume.
Student Response
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Advancing Student Thinking
If students did not see the figure as a cube missing a rectangular prism, ask them to build the figure and the missing piece that can be added to make it a cube.
Activity Synthesis
Invite students to share different expressions for the volume of the figure.
Display expression:
\((3 \times 5 \times 5) + (2 \times 2 \times 5)\)
“How does the expression represent the volume of the figure?” (
\(3 \times 5 \times 5\)
represents the prism that is 3 by 5 by 5 cubes tall and the
\(2 \times 2 \times 5\)
represents the prism that is 2 by 2 by 5 cubes tall.)
If no student wrote the expression
\((5 \times 5 \times 5) – (2 \times 3 \times 5)\)
, display this expression.
“How does the expression show the volume of the figure in cubic units?” (There is a 5 by 5 by 5 cube and a piece has been taken away. The piece that is taken away measures 2 cubes by 3 cubes by 5 cubes.)


correct_option_index: 15