Reasoning about Equations with Tape Diagrams

10 min

Teacher Prep
Setup
Groups of 2. 3 minutes of quiet work time, followed by sharing with a partner and whole-class discussion.

Narrative

In this activity, students remember what the distributive property is all about before they will be expected to use it in the process of solving equations of the form p(x+q)=rp(x+q)=r later in this unit. If this activity indicates that students remember little of the distributive property from grade 6, heavier interventions may be needed.

Look for students who:

  • Rule out expressions by testing values.
  • Use the term distributive property.

In order to explain how they know each selected expression is equivalent, students need to attend to precision in the language used (MP6).

Launch

Ask students to think of anything they know about equivalent expressions. Ask if they can:

  • Explain why 2x2x and 2+x2+x are not equivalent. (These expressions are equal when xx is 2, but not equal for other values of xx. Multiplying 2 by a number usually gives a different result than adding that number to 2.)
  • Explain why 3+x3+x and x+3x+3 are equivalent. (These expressions are equal no matter the value of xx. Also, addition is commutative.)
  • Describe ways to decide whether expressions are equivalent. (Test some values, draw diagrams for different values, analyze them for properties of the operations involved.)

Arrange students in groups of 2. Give 3 minutes of quiet work time and then invite students to share their responses with their partner, followed by a whole-class discussion.

Student Task

Select all the expressions that are equivalent to 7(23n)7(2-3n). Explain how you know each expression you select is equivalent. 

  1. 910n9-10n
  2. 143n14-3n
  3. 1421n14-21n
  4. (23n)7(2-3n) \boldcdot 7
  5. 72(-3n)7 \boldcdot 2 \boldcdot (\text- 3n)

Sample Response

1421n14-21n is equivalent because of the distributive property. (23n)7(2-3n) \boldcdot 7 is equivalent because multiplication is commutative.

Activity Synthesis (Teacher Notes)

The purpose of this discussion is to revisit ways of telling whether two expressions are equivalent, and to recall the distributive property.

Select a student who tested values to explain how they know two expressions are not equivalent. For example, 910n9-10n is not equivalent to 7(23n)7(2-3n), because if we use 0 in place of nn, 91009-10\boldcdot 0 is 9 but 7(23n)7(2-3\boldcdot n) is 14. If no one brings this up, demonstrate an example.

Select a student who used the term “distributive property” to explain why 7(23n)7(2-3n) is equivalent to 1421n14-21n and ask them to explain what they mean by that term. In general, an expression of the form a(b+c)a(b+c) is equivalent to ab+acab+ac.

Standards
Building On
  • 6.EE.4·Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). <em>For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.</em>
  • 6.EE.A.4·Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). <span>For example, the expressions <span class="math">\(y + y + y\)</span> and <span class="math">\(3y\)</span> are equivalent because they name the same number regardless of which number <span class="math">\(y\)</span> stands for.</span>

10 min

10 min