Expressions for Volume

10 min

Narrative

The purpose of this True or False? is for students to demonstrate the strategies and understandings they have for determining the equivalence of numerical expressions. These understandings help students deepen their understanding of the properties of operations and are helpful as students interpret expressions for volume. In this activity, students have an opportunity to notice and make use of structure (MP7) when they use the properties of operations to determine equivalence, without having to calculate.

This is the first time students experience the True or False? routine in grade 5. Students are familiar with this routine from a previous grade, however, they may benefit from a brief review of the steps involved.

Launch

  • Display one statement.
  • “Give me a signal when you know whether the statement is true and can explain how you know.”
  • 1 minute: quiet think time
Teacher Instructions
  • Share and record answers and strategies.
  • Repeat with each statement.

Student Task

Decide if each statement is true or false. Be prepared to explain your reasoning.

  • (4×2)×5=4×(2×5)(4\times2)\times5=4\times(2\times5)
  • (2×5)×4=2×20(2\times5)\times4=2\times20
  • 5×4×2=10×405\times4\times2=10\times40

Sample Response

  • True: The same numbers are being multiplied but grouped differently.
  • True: Since 2×(5×4)=2×202\times(5\times4)=2\times20, (2×5)×4=2×20(2\times5)\times4=2\times20; both sides are equal to 40.
  • False:  5×25\times2 is 10, but one side of the equation is multiplying that 10 by 4 and the other side is multiplying it by 40.
Activity Synthesis (Teacher Notes)
  • “How can you justify your answer without evaluating both sides?” (I could see in the first equation that all of the factors are the same, so it is true.)
  • Consider asking:
    • “Who can restate ___’s reasoning in a different way?”
    • “Does anyone want to add on to _____’s reasoning?”
    • “Can we make any generalizations, based on the statements?”
Standards
Addressing
  • 5.OA.1·Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
  • 5.OA.A.1·Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Building Toward
  • 5.MD.5.b·Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
  • 5.MD.C.5.b·Apply the formulas <span class="math">\(V = l \times w \times h\)</span> and <span class="math">\(V = b \times h\)</span> for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

15 min

10 min