Lesson 4

The purpose of this Algebra Talk is to elicit strategies and understandings students have for solving equations. These understandings help students develop fluency and will be helpful later in this unit when students will need to be able to come up with ways to solve equations of this form. While four equations are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.

Students should understand the meaning of solution to an equation from grade 6 work as well as from work earlier in this unit, but this is a good opportunity to re-emphasize the idea.

In this string of equations, each equation has the same solution. Digging into why this is the case requires noticing and using the structure of the equations (MP7). Noticing and using the structure of an equation is an important part of fluency at solving equations.

Display one equation at a time. Give students 30 seconds of quiet think time for each equation and ask them to give a signal when they have an answer and a strategy. Keep all equations displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the equation in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

An important idea to highlight is the meaning of a solution to an equation; a solution is a value that makes the equation true.

For the second and third equations, some students may first think about applying the distributive property before reasoning about the solution. As students see the third and fourth equations, they are likely to notice commonalities among the equations that can support solving them. One likely observation to highlight is that each equation has the same solution. It is worth asking why each equation has the same solution. A satisfying answer to this question requires attending to the structure of the equations.

The purpose of this activity is to work toward showing students that some situations can be represented by an equation of the form \(px+q=r\) (or equivalent). In this activity, students are simply tasked with drawing a tape diagram to represent each situation. In the following activity, they will work with corresponding equations.

The last question is tough to represent with a tape diagram, because you would have to divide the diagram into 30 equal pieces. This is intentional, and can be used to make the point that we are trying to develop more efficient ways of solving problems than drawing a diagram every time.

For each question, monitor for one student with a correct diagram. Press students to explain what any variables used to label the diagram represent in the situation.

Select one student for each situation to present their correct diagram. Ensure that students explain the meaning of any variables used to label their diagram. Possible questions for discussion:

• “For the situations with no \(x\), how did you decide what quantity to represent with a variable?” (Think about which amount is unknown but has a relationship to one or more other amounts in the story.)
• “Did any situations have the same diagrams? How can you tell from the story that the diagrams would be the same?” (Same number of equal parts, same amount for unequal parts, same amount for the total.)
• “How is the last situation different from the others?” (It’s the only one where 30 is the coefficient of \(x\) rather than the total.)
• “Why was it tough to draw a diagram for the last question?” (You would have to divide the diagram into 30 equal pieces.)

This activity is a continuation of the previous one. Students match each situation from the previous activity with an equation, solve the equation by any method that makes sense to them, and interpret the meaning of the solution. Students are still using any method that makes sense to them to reason about a solution. In later lessons, a hanger diagram representation will be used to justify more efficient methods for solving. For example, when they are using tape diagrams, they could just say “I subtracted the 9 extra markers and then divided the remaining 21 markers by 7.” Later, when working with hanger diagrams, we can press them to say “I can subtract 9 from each side and then divide each side by 7.”

For each equation, monitor for a student using their diagram to reason about the solution and a student using the structure of the equation to reason about the solution.

Keep students in the same groups. 5 minutes to work individually or with a partner, followed by a whole-class discussion.

For each equation, ask one student who reasoned with the diagram and one who reasoned only about the equation to explain their solutions. Display the diagram and the equation side by side, drawing connections between the two representations. If no students bring up one or both of these approaches, demonstrate maneuvers on a diagram side by side with maneuvers on the corresponding equation. For example, “I subtracted the 9 extra markers and then divided the remaining 21 markers by 7,” can be shown on a tape diagram and on a corresponding equation. It is not necessary to invoke the more abstract language of “doing the same thing to each side” of an equation yet.