Lesson 8

This number talk encourages students to think about the numbers in a computation problem and rely on what they know about structure, patterns, and division with remainders to mentally solve a problem.

Only one problem is presented to allow students to share a variety of strategies for division. Notice how students handle a remainder in a problem, which may depend on their prior experiences with division. Students may write it as “r_” and struggle with either fraction or decimal notation. In the next lesson, when students begin finding unit price, they will need to be able to interpret the remainder in either decimal or fraction form.

Display the problem for all to see. Give students 2 minutes of quiet think time and ask them to give a signal when they have an answer and a strategy. Follow with a whole-class discussion.

Students may struggle to interpret the remainder as a decimal or fraction and may instead write r 6.

Invite students to share their strategies. Record and display student explanations for all to see. Ask students to explain if or how the dividend or divisor impacted their choice of strategy and how they decided to write their remainder. To involve more students in the conversation, consider asking:

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

At the end of discussion, if time permits, ask a few students to share a story problem or context that \(246\div 12 = 20.5\) would represent.

This activity continues the work on ratios involving one unit of something. Students determine the prices of grocery items and learn to use the term unit price to describe cost per unit. To determine unit prices, students may:

• Divide the cost by the number of items
• Use discrete diagrams
• Use a double number line

As students work, monitor for students using different methods. 

If students choose to draw a double number line diagram, remind them to label each number line and to circle the ratio where they find the answer.

Some students may have difficulty with the answers not being integers. Either fractions or decimals are acceptable. Fractions provide the most direct route, but decimals are common for working with dollars and cents. Also, students may use the larger numbers as the dividend, simply because they are larger. Encourage students to check the reasonableness of their answers.

Select students who used unique methods to share their reasoning, as listed in the narrative. If no one used double number lines, represent one of the statements with a double number line diagram and display it for all to see. Although double number lines are not required in the task, their use in the context of problem situations helps students see their merits and illustrates how they might be used in other problems, especially as students transition from unit prices to constant speed and other contexts. Draw connections between the double number line strategy and the dividing by the numbers of items strategy.

Tell students that each “cost per one” unit being sold—avocado, pound, or bottle—is an example of a unit price. Ask them to name as many kinds of unit prices they can and to think of a situation where they might be used, starting with the list from the task:

• Cost per avocado
• Cost per pound
• Cost per bottle

Other possibilities include cost per liter, cost per ounce, cost per jelly bean, and so on.

In this task, students practice finding unit prices, using different reasoning strategies, and articulating their reasoning. They also learn about the term “at this rate.”

As students work, observe their work and then assign one problem for each group to own and present to the class. (The problems can each be assigned to more than one group). Have them work together to create a visual display of their problem and its solution.

The first and third questions involve using decimals to represent cents. If the decimal point is forgotten, remind students that the cost of the bracelet is less than one dollar, and the cost of the chips is in between one and two dollars.

Watch for students working in cents instead of dollars for the bracelets. They may come up with an answer of 275 cents. For these students, writing 25 cents as \$0.25 should help, or consider reminding them of the avocados from a previous activity, which had a unit price of \$0.50.

Invite each group to present its assigned problem. After each group presents, highlight the group’s strategy, accurate uses of the terms “at this rate” and “per,” and the ways in which a double number line might have been used when working with unit price.