Lesson 13

This Estimation Exploration prompts students to practice making a reasonable estimate based on experience and known information. In this case, it is not practical to count the paletas, but students could reason about groups of paletas by color, or estimate the complete rows and columns of paletas and extend their estimate to the whole set. Some students might also make an estimate based on their familiarity with how paletas are usually arranged in cases.

• Groups of 2
• Display the image. 
• “These are ice pops called paletas. They originated in Mexico and are typically made with many different fruits.”
• Ask students to estimate without counting.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “Is anyone’s estimate less than 30? Greater than 80?”
• “How did you know that 30 (or another number) would be too low and 80 (or another number) too high?”
• “Based on this discussion does anyone want to revise their estimate?”

In this activity, students recall what they know about division from grade 3. The context allows students to connect lived experiences to the math of the activity. By inviting students to consider treats that they enjoy in their homes or neighborhoods, they share experiences and foster connections that build community.

The first question gives students an opportunity to co-craft mathematical questions based on a situation before answering a question based on a division equation. Students divide a two-digit number by a one-digit divisor, as they did in grade 3, in a way that makes sense to them. The activity synthesis highlights different representations students made and relates them to the situation. The term dividend is re-introduced in this lesson to describe a number being divided into equal groups.

This activity uses MLR5 Co-craft Questions. Advances: writing, reading, representing

• Groups of 2
• “What are some of your favorite treats or snacks from home?”
• 30 seconds: quiet think time
• 1 minute: partner discussion

MLR 5: Co-craft Questions

• Display the opening paragraph and the first question.
• “Write a list of mathematical questions that could be asked about this situation.”
• 2 minutes: independent work time
• 2–3 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “How are these questions alike?” (The questions involve multiplying or dividing.) “How are they different?” (The questions can be answered using different operations.)
• “Let’s look at the next question in the activity.”

• 3–4 minutes: quiet work time

If students are unsure how to write a division question for \(84 \div 7\), consider asking:

• “If you were to act out the meaning of this expression, what would you do?”
• “If you were to explain its meaning, what would you say?”
• “In what type of situations would these types of actions take place?”

• Display two questions that students wrote for the equation \(84 \div 7 = {?}\) .
• “What does the 84 represent in both problems?” (The amount being divided into equal groups.) 
• “In mathematics, the number being divided is known as the dividend.”
• Invite students to share their strategies for the last question. Highlight strategies that show equal-size groups and reasoning that relates multiplication and division.

In this activity, students continue to use any strategy to solve division problems in context and to recall the two interpretations of division. One problem involves finding how many in each group and the second involves finding the number of groups. Students work with two- and three-digit dividends and encounter division that results in a number with a remainder. They consider what the leftover means in the given context.

During the synthesis, highlight that both multiplication and division can be used to reason about the solutions, and elicit equations can be written to represent students' reasoning about equal-size groups. Introduce remainders as “leftovers” or quantities that remain after dividing a number into equal groups. 

• Groups of 2
• Display images of snacks in the activity.
• “Take a look at the images. What do you notice? What do you wonder?”
• Share responses.
• “Gulab jamuns are sweet treats that are popular in India, Pakistan, and their neighboring countries in South Asia.”
• “Breadsticks that are covered with chocolate, strawberry cream, or other flavors are popular snacks in Japan, Taiwan, and other East Asian countries.” 
• Ask a student to read the first problem aloud.
• “The problems in this activity involve treats that students enjoy from different places around the world.”

• 6–8 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who reason in terms of partial products or partial quotients and record their thinking as equations.
• Monitor for the different ways students describe and decide what to do with the 2 leftover breadsticks.

• Invite students to share their response for the first problem.
• If no students wrote equations to show their thinking, ask: “What equations can represent how you found the solution to the first problem?“
• Record both multiplication and division equations for all to see. For example:
  • \(5 \times 10 = 50\) and \(5 \times 7 = 35\), so \(5 \times 17 = 85\).
  • \(50 \div 5 = 10\) and \(35 \div 5 = 7\), so \(85 \div 5 = 17\).
• Invite other students to discuss the second problem.
• “When we divide, sometimes we have leftovers that are not enough to make a new group or not enough to put an additional item to each group. We call the leftovers remainders.”
• “What was the remainder when 110 breadsticks were divided by 6?” (2 breadsticks were leftover)
• “How are the questions in the two situations—gulab jamuns and breadsticks—alike?” (They involved division into equal groups.)
• “How are they different?” (The first looks for the number of groups. The second looks for the size of a group. The second involves some leftovers.)