Connecting Equations to Graphs (Part 2)

5 min

Narrative

In this Warm-up, students review how to apply the distributive property to rewrite expressions that involve division, preparing them to do so in the next activity in the lesson.

Launch

Arrange students in groups of 2.

Tell students that solutions should be in a form like ax+bax + b. For example, 3x6,2312x,3x - 6, \frac{2}{3} - \frac{1}{2}x, or 10x+410x + 4 would be valid answers.

Give students a moment of quiet time to work on the first two questions and then time to discuss their responses with their partner before moving on the last two questions.

Student Task

Rewrite each quotient as a sum or a difference. 

  1. 4x102\dfrac {4x-10}{2}
  2. 150x-2\dfrac {1 - 50x}{\text-2}
  3. 5(x+10)25\dfrac {5(x+10)}{25}
  4. -15x+52\dfrac {\text- \frac15 x + 5}{2}

Sample Response

  1. 2x52x - 5
  2. -12+25x\text- \frac12 + 25x
  3. 15x+2\frac15 x + 2
  4. -110x+52\text- \frac {1}{10}x + \frac52
Activity Synthesis (Teacher Notes)

Invite students to share their equivalent expressions and how they reasoned about them. Display their expressions for all to see. Make sure students see that rewriting the expressions as sums or differences involves distributing the division (or applying the distributive property to division). 

If not mentioned in students' explanations, point out that each division could be thought of in terms of multiplication. For example, 4x102\frac {4x-10}{2} is equivalent to 12(4x10)\frac12 (4x-10), because dividing by a number (in this case, 2) gives the same result as multiplying by the reciprocal of that number (in this case, 12\frac12). Applying the distributive property of multiplication to 12(4x10)\frac12 (4x-10) enables us to rewrite this product as a difference.

Anticipated Misconceptions

Expect some students to give 2x102x-10 or 4x54x-5 as an answer to the first question. To illustrate why these are incorrect, take an example like 4+62\frac{4+6}{2}. Explain that we know that 10 divided by 2 is 5, but if we divide only the 4 or only the 6 by 2 we won’t get 5. Alternatively, remind students that fraction bars can be interpreted as division, so each expression can be rewritten as, say, (4x10)÷2(4x-10) \div 2, and we can apply the distributive property.

The signs of the numbers in the second expression might be a source of confusion. Students might be unsure if the expression should be -1225x\text - \frac12 -25x, 12+25x\frac12 + 25x, or another expression. Encourage students to substitute a number into the original expression, then try it in each potential answer. To explain why 12+25x\text- \frac12 + 25x is correct, appeal to the distributive property again.

Standards
Building On
  • 6.EE.3·Apply the properties of operations to generate equivalent expressions. <em>For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.</em>
  • 6.EE.A.3·Apply the properties of operations to generate equivalent expressions. <span>For example, apply the distributive property to the expression <span class="math">\(3 (2 + x)\)</span> to produce the equivalent expression <span class="math">\(6 + 3x\)</span>; apply the distributive property to the expression <span class="math">\(24x + 18y\)</span> to produce the equivalent expression <span class="math">\(6 (4x + 3y)\)</span>; apply properties of operations to <span class="math">\(y + y + y\)</span> to produce the equivalent expression <span class="math">\(3y\)</span>.</span>

15 min

15 min