Lesson 4

The purpose of this warm-up is to elicit the idea that discrete diagrams can be inefficient for representing larger numbers, which will be useful when students interpret and use more abstract tape diagrams later in the lesson. While students may notice and wonder many things about the diagram, ideas and questions for how the student could better represent the comparison are the important discussion points.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “Does this diagram show that Clare read 8 times as many pages as Noah?” 
• “Why do you think the student who drew this diagram scribbled parts out?” (Maybe they ran out of space. There wasn’t enough room to draw all the pages the same size and keep them together in groups.)
• “What might be challenging about drawing each page when there are lots of objects?” (It takes a lot of time and space. You might lose count.)
• “What other ways could you make a diagram to show larger amounts?” (Use numbers instead of drawing each part.)
• “Today we are going to look at diagrams that show comparisons with larger numbers and think about when we might use them to represent and solve our own problems.”

The purpose of this activity is for students to interpret and solve multiplicative comparison problems. Students interpret tape diagrams that label each box with a value, which is different from the discrete diagrams from previous lessons. They also write equations to represent the situations and explain how the equations connect to the tape diagrams (MP2). The problems in this activity have unknown smaller quantities, larger quantities, or multipliers. Questions are provided to support students in representing the unknown in tape diagrams and identifying the unknown in given situations.

• Groups of 2
• Display and read the first problem.
• 30 seconds: quiet think time
• 1 minute: partner discussion
• “Is 15 pages too high or low of an estimate for Andre?” (Too low, because the diagram shows he should have read more than Mai.)
• “If Andre read 9 times as much as Mai, what number should replace the question mark in the diagram? Why?” (135 Because the 9 rectangles represent 9 times as many as 15. There should be 9 fifteens.) 
• 30 seconds: quiet think time
• 1 minute: partner discussion

• 3 minutes: independent work time
• 8 minutes: partner work time
• Monitor for students who:
  • explain the connection between their equations and the diagram.
  • show understanding that each rectangle represents the same amount.

If students write equations without unknowns, consider asking, “What information do you know from looking at the diagram? What information is missing? What do you need to find?” Allow students to analyze the diagram and then ask, “Before you find the missing value, what equation could you write to represent this situation? Use a symbol to represent the missing value.”

• Select 2–3 students to share their responses and reasoning. 
• “How do the diagrams help us figure out what we know about the situation and what we need to find out?” (Each diagram shows two quantities being compared, and the ‘times as many’ amount that compares them. The quantity that is missing is the part we need to find out.)
• “How do the diagrams show times as many?” (‘Times as many’ is represented by the number of times each rectangle in  the diagram is repeated.)
• Display: \(3 \times {?} = 60\)
• “How does this equation help us find the number of pages Jada read?” (We know that Han read 60 pages which is 3 times as many as Jada, because the diagram shows 3 equal sized rectangles for Han and one rectangle for Jada. We just need to know the amount that each rectangle represents, which will be the amount of pages Jada read.)

The purpose of this activity is for students to represent multiplicative comparison situations and solve for an unknown factor or unknown product.

In the synthesis, students make connections between the description, their diagram, and multiplication equations that represent the situation (MP2).

• Groups of 2
• Read directions for activity aloud.
• “What will we do with each problem in this task?” (Write an equation with the unknown represented by a symbol, draw a diagram to show comparison, and answer the question.)

• 4 minutes: independent work time
• 2 minutes: partner discussion about the first problem
• “Share with your partner how you represented the unknown value in your diagram and equation.”
• “Work with your partner on the rest of the problems.”
• 8 minutes: partner work time
• Monitor for students who: 
  • draw and label each comparison.
  • identify and represent the unknown in their equation and diagram.
  • use “times as many” language to describe each comparison.
• If students finish early, they can create their own situation and question and trade with their partner. The partner will write the equation, draw a diagram, and answer the question. 

Students draw a diagram to represent and solve a situation without representing the unknown because it is less intuitive after the solution has been determined. Consider asking, “Which part of the equation were you working to solve?” and “What might you add to your representation to help others understand your thinking?”

• “What information did you use in each situation to know how to draw each diagram?” 
  • The number of repeating rectangles in the larger quantity represents “times as many”
  • The row with one rectangle is the amount being multiplied
  • The row with multiple rectangles represents the larger quantity in the comparison
• “How is the last problem different from the first two?” (Sample response: The unknown value is  how many “times as many”.)