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Lesson 4; More Balanced Equations

Author:
Arpit Kesharwani
Topic:
Equations

Warm Up

Equation 1x−3=2−4xWhich of these have the same solution as Equation 1?  Be prepared to explain your reasoning.Equation A2x−6=4−8xEquation Bx−5=-4xEquation C2(1−2x)=x−3Equation D-3=2−5x

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Activity Summary

Select students previously identified to share how they determined whether each equation had the same solution as Equation 1 in the sequence listed in the Activity Narrative. Point out that the question did not ask students what the solution was, only whether each equation had the same solution.To help students make connections between the different methods their classmates used to solve the warm-up, ask:
  • “Which method of answering the question was 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 x. A disadvantage would be that I never discovered what the value for x is that makes the equations true.)
  • “How is this method (manipulating the equation to look like Equation 1) 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.)
By showing that two equations are related by a move (or series of moves), we know they must have the same solution.If time allows, have students create another equation with the same solution as Equation 1 and trade with a partner. They should then explain the step(s) necessary to make it look like Equation 1 to each other.

Here is an equation, and then all the steps Clare wrote to solve it: 14x−2x+312x+33(4x+1)4x+11-8=3(5x+9)=3(5x+9)=3(5x+9)=5x+9=x+9=xHere is the same equation, and the steps Lin wrote to solve it:14x−2x+312x+312x+312x-3xx=3(5x+9)=3(5x+9)=15x+27=15x+24=24=-8

  1. Are both of their solutions correct? Explain your reasoning.
  2. Describe some ways the steps they took are alike and different.
  3. Mai and Noah also solved the equation, but some of their steps have errors. Find the incorrect step in each solution and explain why it is incorrect.Mai:14x−2x+312x+37x+37x+37xx=3(5x+9)=3(5x+9)=3(9)=27=24=247Noah:14x−2x+312x+327x+327xx=3(5x+9)=15x+27=27=24=2427

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Before students work on solving complex equations on their own, in this activity they examine the work (both good and bad) of others. The purpose of this activity is to build student fluency solving equations by examining the solutions of others for both appropriate and inappropriate strategies (MP3).Encourage students to use precise language when discussing the different steps made by the four students in the problem (MP6). For example, if a student says Clare distributed to move from 12x+3=3(5x+9) to 3(4x+1)=3(5x+9), ask them to be more specific about how Clare used the distributive property to help the whole class follow along. (Clare used the distributive property to re-write 12x+3 as 3(4x+1).)LaunchArrange students in groups of 2. Give 4–5 minutes of quiet work time and ask students to pause after the first two problems for a partner discussion. Give 2–3 minutes for partners to work together on the final problem followed by a whole-class discussion. Refer to MLR 3 (Critique, Correct, and Clarify) to guide students in using language to describe the wrong steps.

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