Lesson 4

Reasoning about Equations and Tape Diagrams (Part 1)

4.1: Algebra Talk: Seeing Structure (10 minutes)

Warm-up

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.

Launch

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.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Find a solution to each equation without writing anything down.

\(x + 1 = 5\)

\(2(x+1) = 10\)

\(3(x+1) = 15\)

\(500 = 100(x+1)\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

4.2: Situations and Diagrams (15 minutes)

Activity

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.

Launch

Ensure students understand that the work of this task is to draw a tape diagram to represent each situation. There is no requirement to write an equation or solve a problem yet.

Arrange students in groups of 2. Give 5–10 minutes to work individually or with their partner, followed by a whole-class discussion.

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to 3 out of the 5 situations. Require students to choose the last situation as it is different from the others.
Supports accessibility for: Organization; Attention

Student Facing

Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.

  1. Diego has 7 packs of markers. Each pack has \(x\) markers in it. After Lin gives him 9 more markers, he has a total of 30 markers.
  2. Elena is cutting a 30-foot piece of ribbon for a craft project. She cuts off 7 feet, and then cuts the remaining piece into 9 equal lengths of \(x\) feet each.
  3. A construction manager weighs a bundle of 9 identical bricks and a 7-pound concrete block. The bundle weighs 30 pounds. 
  4. A skating rink charges a group rate of \$9 plus a fee to rent each pair of skates. A family rents 7 pairs of skates and pays a total of \$30.
  5. Andre bakes 9 pans of brownies. He donates 7 pans to the school bake sale and keeps the rest to divide equally among his class of 30 students. 

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Ensure students understand that the work of this task is to draw a tape diagram to represent each situation. There is no requirement to write an equation or solve a problem yet.

Arrange students in groups of 2. Give 5–10 minutes to work individually or with their partner, followed by a whole-class discussion.

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to 3 out of the 5 situations. Require students to choose the last situation as it is different from the others.
Supports accessibility for: Organization; Attention 

Student Facing

Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.

  1. Diego has 7 packs of markers. Each pack has \(x\) markers in it. After Lin gives him 9 more markers, he has a total of 30 markers.

  2. Elena is cutting a 30-foot piece of ribbon for a craft project. She cuts off 7 feet, and then cuts the remaining piece into 9 equal lengths of \(x\) feet each.
  3. A construction manager weighs a bundle of 9 identical bricks and a 7-pound concrete block. The bundle weighs 30 pounds. 



     
  4. A skating rink charges a group rate of \$9 plus a fee to rent each pair of skates. A family rents 7 pairs of skates and pays a total of \$30.



     
  5. Andre bakes 9 pans of brownies. He donates 7 pans to the school bake sale and keeps the rest to divide equally among his class of 30 students. 



     

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.)
Conversing, Representing: MLR7 Compare and Connect. Use this routine to support student understanding of the connections between tape diagrams and the situations they represent. Because all situations in this activity share the same quantities of 7, 9, and 30, students can compare how each situation affects how quantities appear in a tape diagram. Invite students to compare their diagrams with their partner. Display the following sentence frames: “One thing our diagrams have in common is ….” and “One thing that is different about our diagrams is …”. This routine will help students identify, explain, and verbally respond to correspondences between context and mathematical representations.
Design Principle(s): Optimize output (for comparison); Cultivate conversation

4.3: Situations, Diagrams, and Equations (10 minutes)

Activity

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.

Launch

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

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as “To find the solution, first I ____ because...”, “I made this match because I noticed...”, “Why did you...?”, or “I agree/disagree because…”
Supports accessibility for: Language; Social-emotional skills

Student Facing

Each situation in the previous activity is represented by one of the equations.

  • \(7x+9=30\)
  • \(30=9x+7\)
  • \(30x+7=9\)
  1. Match each situation to an equation.
  2. Find the solution to each equation. Use your diagrams to help you reason.
  3. What does each solution tell you about its situation?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

While in New York City, is it a better deal for a group of friends to take a taxi or the subway to get from the Empire State Building to the Metropolitan Museum of Art? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

Speaking, Representing: MLR3 Clarify, Critique, Correct. Present an incorrect match of an equation with a situation and/or diagram (e.g., \(30 = 9x + 7\) with Situation 1). Invite students to clarify and then critique the error (e.g., there are 7 packs of an unknown amount of markers in Situation 1; not 9 as illustrated in the equation). Press for details, if needed. For example, students can reference “markers” in Situation 1, not “x”, as the unknown value represent by the variable \(x\). This will help students reflect on the quantity of the unknown values that would be a reasonable solution and improve on their reasoning when matching an equation to a situation.
Design Principle(s): Optimize output (for justification)

Lesson Synthesis

Lesson Synthesis

Display one of the situations from the lesson and its corresponding equation. Ask students to explain:

  • “What does each number and letter in the equation represent in the situation?”
  • “What is the reason for each operation (multiplication or addition) used in the equation?”
  • “What is the solution to the equation? What does it mean to be a solution to an equation? What does the solution represent in the situation?”

4.4: Cool-down - Finding Solutions (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Many situations can be represented by equations. Writing an equation to represent a situation can help us express how quantities in the situation are related to each other, and can help us reason about unknown quantities whose value we want to know. Here are three situations:

  1. An architect is drafting plans for a new supermarket. There will be a space 144 inches long for rows of nested shopping carts. The first cart is 34 inches long and each nested cart adds another 10 inches. The architect wants to know how many shopping carts will fit in each row. 

  2. A bakery buys a large bag of sugar that has 34 cups. They use 10 cups to make some cookies. Then they use the rest of the bag to make 144 giant muffins. Their customers want to know how much sugar is in each muffin.

  3. Kiran is trying to save \$144 to buy a new guitar. He has \$34 and is going to save \$10 a week from money he earns mowing lawns. He wants to know how many weeks it will take him to have enough money to buy the guitar.

We see the same three numbers in the situations: 10, 34, and 144. How could we represent each situation with an equation?

In the first situation, there is one shopping cart with length 34 and then an unknown number of carts with length 10. Similarly, Kiran has 34 dollars saved and then will save 10 each week for an unknown number of weeks. Both situations have one part of 34 and then equal parts of size 10 that all add together to 144. Their equation is \(34+10x=144\).

Since it takes 11 groups of 10 to get from 34 to 144, the value of \(x\) in these two situations is \((144-34)\div{10}\) or 11. There will be 11 nested shopping carts in each row, and it will take Kiran 11 weeks to raise the money for the guitar.

In the bakery situation, there is one part of 10 and then 144 equal parts of unknown size that all add together to 34. The equation is \(10+144x=34\). Since 24 is needed to get from 10 to 34, the value of \(x\) is \((34-10)\div{144}\) or \(\frac16\). There is \(\frac16\) cup of sugar in each giant muffin.