More Balanced Moves

10 min

Teacher Prep
Setup
Give students 2–3 minutes quiet think time, then whole-class discussion.

Narrative

The purpose of this Warm-up is for students to use the structure of equations to recognize when they are equivalent.

Monitor for students who use these different strategies:

  • Solve each equation, then compare the solutions.
  • Solve Equation 1, then substitute it into the other equations.
  • Use valid moves correctly to make equations look alike.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.

Launch

Give students 2–3 minutes of quiet think time, and then facilitate a whole-class discussion.

Select work from students with different strategies, such as those described in the Activity Narrative, to share later.

Student Task

Equation 1

x−3=2−4xx-3=2-4x

Which of these have the same solution as Equation 1?  Be prepared to explain your reasoning.

Equation A

2x−6=4−8x2x-6=4-8x

Equation B

x−5=-4xx-5=\text-4x

Equation C

2(1−2x)=x−32(1-2x)=x-3

Equation D

-3=2−5x\text-3=2-5x

Sample Response

All of the other equations have the same solution as the first equation, x=1x=1.

Sample reasoning:

  • Equation A: If you multiply each side of Equation 1 by 2, the result is Equation A.  So if xx makes Equation 1 true, then it makes Equation A true as well.
  • Equation B: If you subtract 2 from each side of Equation 1, the result is Equation B. So if xx makes Equation 1 true, then it makes Equation B true, too.
  • Equation C: If you switch everything to the left of the equal sign and everything to the right of the equal sign on Equation C, and then rewrite the expression 2(1−2x)2(1-2x) as 2−4x2-4x using the distributive property, the result is Equation 1. So if xx makes Equation 1 true, then it makes Equation C true.
  • Equation D: If you subtract xx from each side of Equation 1, the result is Equation D. So if xx makes Equation 1 true, then it also makes Equation D true.
Activity Synthesis (Teacher Notes)

The goal of this discussion is for students to recognize that there are multiple ways to check that two equations are equivalent. In particular, students should recognize that equations do not need to be solved to determine that they have the same solution.
Display 2–3 strategies from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different strategies. Here are some questions for discussion:

  • “What do the strategies have in common? How are they different?”
  • “Which method of answering the question is the most efficient? After seeing all these ways to answer the question, which would you choose?”
  • “What is an advantage of changing the equation to look like Equation 1? What is a disadvantage?” (An advantage is that I could see quickly whether it would be the same as Equation 1, and I didn't have to keep going to actually figure out the value of xx. A disadvantage would be that I never discovered what the value for xx is that makes the equations true.)
  • “How is writing equivalent equations similar to what we did in previous lessons with the balance hangers?” (In order to keep the hangers balanced, I had to make sure to do the same thing to each side of the hanger. In order to have each equation still be true, I have to make sure to do the same thing to each side of an equation.)

If time allows, have students create another equation with the same solution as Equation 1 and trade with a partner. They should then explain to each other the step(s) necessary to make it look like Equation 1.

Standards
Addressing
  • 8.EE.C·Analyze and solve linear equations and pairs of simultaneous linear equations.
  • 8.EE.C·Analyze and solve linear equations and pairs of simultaneous linear equations.

15 min

10 min