Solving Equations Using Reasoning, Part 2

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Objective

SWBAT solve equations using reasoning to find the number that makes the equation true.

Big Idea

What steps can we take to reason about the value of a variable, when the equation is more difficult?

Think About It

8 minutes

My students really enjoy today's Think about It problem. Students work with a partner to figure out the average weight of each banana in shown on the image.  After 2-3 minutes of thinking and discussion, I ask for hands of students who want to share out their calculation of the weight of each banana.

Typically, the first student explains that each banana must be two pounds, because there 5 bananas which equal the same weight as the 10 pound pineapple. When this explanation is offered I ask, "How did you know that the five bananas and the pineapple weighed the same amount?"  I want a student to explain the role of the balance in understanding the problem.

Then, I call on another student and say, "Talk to me about the apples."  I want everyone to understand that because there are three apples on each side, ignoring them will help us get to the weight of the bananas.

I think it's important here to name for students that this picture isn't realistic, as a real-world scenario.  I want my students always thinking about the reasonableness of a situation.  It is very unlikely that we'd find 2 groups of 3 apples each that have the exact same weight.  An average banana weighs under half a pound, not a whopping 2 pounds!  An average pineapple weighs about 2 pounds.  Talking about giant fruit also infuses a moment of joy into this section.

I frame the lesson by explaining that we'll use the idea of the balance and the apples today to think about equations.  I say, "Equations are two balanced expressions - their values are the same.  We'll be solving more complex equations using reasoning, and we'll need to sometimes look for the apples."  I want the students to remember the apples as we look for constants that are the same in both expressions, and, ignore them.

 

Intro to New Material

15 minutes

After the Think About It problem, I guide students through the four examples that follow on the TAB and INM handout.

For Example 1, I begin by asking a student to read the problem to the class. I hope that they phrase it something like, "3 times some number plus seven is equal to twelve plus seven." Then, I use these questions to help students think about the problem:

  • What is the same on both sides?
  • What do we call that 7? 
  • What are the terms that are different on each side? 

Then, I say, "When I think about this problem, one of the first things that I decide is to ignore the 7's.  I call this eliminating the 7's." After giving the students a few seconds to think about this I ask, "Why can I eliminate the sevens?" After the TAB warm-up a few students are likely to volunteer something like, "they are equal so eliminating them won’t change equality of the equation, nor the value.  They are like the apples."

At this point I like to ask the students to think about the fact that the equal sign allows us to assume that the equation is balanced. I ask, "If the 7’s are equal and the equation is balanced, what has to be true about the 3x and the 12?" At this point it is important to press the students to explain their thinking. Once we agree that 3x and 12 represent the same quantity, we proceed with a solution by reasoning. I confirm with the students that we can eliminate 7 from both sides of the equation. Then, I cross out 7 on both sides or the original equation and I rewrite the equation as 3x = 12. Then, I ask the students to use reasoning to determine the value of x that makes the equation true.

Possible Extension: At this point, I make a decision about whether or not this is a good place to ask the students about the number of solutions an equation like this can have. Is our answer unique? Are there other possible answers?

I use the same question sequence for the remaining questions.  Questions 2 and 3 require the use of the Distributive Property.

Once many of the students have completed Question 3, I ask all of the students look at Question 4. After a few seconds I ask "What do you notice about this problem?"  I use this question often. I want my students to learn that there are many things we can notice in math.  I generally need to be patient at this point and call on a few students before one reports that the constants are different on each side of the equation. Once this point is made I have students watch me on the document camera.  I tell them that I want them to explain the choice I make as I rewrite the problem. I write:

8g + 12 =52

8g + 12 = 40 + 12

Then I ask students to turn-and-talk with a partner about why I made the move that I made. After a few minutes, we come back together as a class, and share out.  When students explain that I broke the 52 down into 40 + 12, I push them to be specific about why I made that choice. I may ask, "Why didn't I break 52 down into 50 + 2 or 26 + 26?"  I hope that they will identify the fact that I have changed the structure of the equation to appear like the earlier problems. It would be a big win if anyone makes the connections to the apples.

This is a key point for today's lesson: the structure of a problem can change and some structures are more useful than others.  In this case, a structure with the same constant on both sides of the equation makes an equation easier to solve. By manipulating or transforming the right side, we make progress.

After discussing Example 4, we fill in the scaffolded notes.  See the visual anchor for the words we use in the blanks. 

Partner Practice

17 minutes

Students move into Partner Practice.  While they are working, I leave the list of steps and Example 4 on the document camera, so that students can refer to them as they work.  As partners work, I am circulating and looking for:

  • Are students determining the correct solution?
  • Are students using the distributive property correctly, when needed?
  • Are students breaking up the constant correctly, when needed?
  • Are students boxing off their solution?
  • Are students using substitution to prove/check their solutions?

If I sense that students need a bit more practice with my help before they move into partner work, I will have us work on problem 3 together.  

Questions I ask as I circulate:

  • How did you find the value of the variable?
  • How do you know that x equals ___?
  • What property did you use here?
  • Why did you make that choice?
  • How do you know your answer is correct?

After 10 minutes of  partner work, we discuss Problem 7 and Problem 8. Before we move into independent practice, students complete the final check for understanding on their own. I pull a popsicle stick and have the lucky student explain his/her thinking.  I then ask for feedback from the group around what they like about the organization of the student's work and what can be improved.

Independent Practice

15 minutes

Students work on Independent Practice on their own.  As I circulate I am looking for and asking the same things as in the partner practice:

  • Are students determining the correct solution?
  • Are students using the distributive property correctly, when needed?
  • Are students breaking up the constant correctly, when needed?
  • Are students boxing off their solution?
  • Are students using substitution to prove/check their solutions?

Questions I ask as I circulate:

  • How did you find the value of the variable?
  • How do you know that x equals ___?
  • What property did you use here?
  • Why did you make that choice?
  • How do you know your answer is correct?

After a few minutes of independent work time, I complete Problems 1a and 1d on the document camera.  I don't say anything to interrupt students as I do this.  Once the exemplars are up, I say "If you'd like to check your work, my answers for 1a and 1d are up."

 

 

Closing and Exit Ticket

5 minutes

After independent work time, I bring the class together to discuss their work.  First, I have a student show work on the document camera for problem 2a.  The student talks through how (s)he reasoned about the solution.

Then, I have students turn and talk with their partners about how they reasoned about problem 2g.  Students may have struggled with the fractions in this problem.  This problem also has an infinite number of solutions.  It offers opportunity for students to have a rich mathematical conversation.  

Students work independently on the Exit Ticket to end the lesson.  Sample student work shows what the exit ticket could look like.