Unit 2, Lesson 6
How Much for One?
Accelerated 6
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This Math Talk focuses on division by a two-digit number. It encourages students to think about the numbers in a computation problem and to rely on what they know about numbers in base-ten, patterns, division with remainders, and the relationship between multiplication and division to mentally find quotients. To divide larger numbers prompts students to look for and make use of structure (MP7).
Notice how students handle a remainder in a problem, which may depend on their prior experiences with division. When students begin finding unit price, they will need to be able to interpret non-whole-number quotients in either decimal or fraction form.
Reveal one problem at a time. For each problem,
Keep all previous problems and work displayed throughout the talk.
Find the value of each quotient mentally.
To involve more students in the conversation, consider asking:
If students express the result of the last two divisions with “2 with a remainder of 6” and “20 with a remainder of 6,” respectively, ask them if the 6 could be divided by 12, or remind them that they divided 6 by 12 in a preceding problem.
At the end of discussion, if time permits, ask a few students to share a story problem or context that
To Gather
Rulers
In this activity, students practice finding unit prices, using different reasoning strategies, and articulating their reasoning. They also learn about the term “at this rate.”
For each problem, select 1–2 students or groups to share their solution and strategy later.
In the digital version of the activity, students can choose to use an applet to create a double number line to explore the problems.
Display the problem and read it aloud: Pizza costs $1.25 per slice. At this rate, how much will 6 slices cost?
Ask students what they think “at this rate” means in the question. Ensure that they understand that “at this rate” means we know that equivalent ratios are involved:
The something is the thing we are trying to figure out, and “at this rate” tells us that the two ratios in this situation are equivalent.
Another way to understand “at this rate” in this context is “at this price per unit” and that the price per unit is the same no matter how many items or units are purchased.
Keep students in the same groups. Explain that they will work together to solve some shopping problems and can use double number lines if they wish.
Four bags of chips cost $6.
At a used book sale, 5 books cost $15.
Neon bracelets cost $1 for 4.
Invite previously selected students or groups to share their work. Highlight 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.
The unit price is the price of 1 thing—for example, the price of 1 ticket, 1 slice of pizza, or 1 kilogram of peaches.
If 4 movie tickets cost $28, then the unit price would be the cost per ticket. We can create a double number line to find the unit price.
This double number line shows that the cost for 1 ticket is $7. We can also find the unit price by dividing,
Help us improve by sharing suggestions or reporting issues.
Frame the task in shopping terms. Say that when most of us go shopping, we often see prices for multiple items or units (for example, 2 bottles for $3, or $1.99 for 3 pounds). Sometimes, however, we want to know how much it costs to buy a quantity different from what is posted on the price tag (for example, 5 bottles or 8 pounds of something). Tell students they will explore ways to solve such problems.
Arrange students in groups of 2–4. Give students a few minutes of quiet think time for the first problem about avocados. Provide access to rulers in case students choose to draw double number lines. Then pause for a discussion.
Invite students to share their approach for each part of the problem, recording or displaying their reasoning for all to see. Focus the discussion on how they reasoned about the price of 9 avocados. If no students mentioned finding the price of a single avocado, or dollars per avocado, ask how it could be determined. Consider using a double number line diagram to illustrate students’ explanations.
If any students say that the price for an avocado is $2, ask what the cost of 8 avocados would be in that case. A double number line diagram can also help students see that 2 is the number of avocados that can be bought with 1 dollar, rather than the number of dollars for 1 avocado.
Encourage students to continue thinking about the use of “per” as they reason about the remaining problems about prices.
Answer each question and explain or show your reasoning. If you get stuck, consider drawing a double number line diagram.
Eight avocados cost $4.
Twelve large bottles of water cost $9.
A 10-pound sack of flour costs $8.
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.
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 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.