No Matter How You Slice It

Sixth Grade Poster Problem

The Number System

This poster problem creates the need for fraction division using a realistic setting. Through repeatedly solving problems about slicing blocks of cheese using different fractional thicknesses, students will be motivated to use division to make calculations easier.

Understanding fraction division can take a significant amount of time, and it may be valuable to plan enough time for you and your students to get comfortable with the slicing context using other operations (e.g., repeated subtraction) before using division. Teaching this poster problem may take more time than other poster problems, but we think this will be time well used.

Materials:

Consider using a hand-operated slicer (or a mandoline) to demonstrate how the thickness of each slice and the dimensions of the object being sliced determine the number of slices that can be made.

Learning Objectives:

- Interpret and compute quotients of fractions.
- Solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

Common Core State Standards for Mathematics:

Note:

In this poster problem, students need to understand some “slicing” situations. Students will probably develop strategies for solving the problems without division, but we want to go beyond that. First, they need to recognize that division is a good operation to use in similar situations. Then, they need to learn how to perform the division-by-fractions computation.

There are several ways for students to do such a calculation: using common denominators, multiplying by one strategically, and so forth. The danger comes when we tell students to “invert and multiply” too early, and they start doing this without (a) understanding why division is the right operation or (b) understanding why the procedure gives the right answer. See the tune-ups below for more details.

The Lesson Plan:

1. LAUNCH

Start with a question: Has anyone been to the deli and seen how they make slices of meats and cheeses? How does the butcher make the slices so thin and all equally thick? She uses a deli slicer! Let’s watch a video to see how a deli slicer works.

Show Slide #1 (video)

Questions:

- Can you explain how the slicer works? [answers will vary]
- According to the video, what is the thinnest slice this slicer can make? [1/32 of an inch]
- What is the thickest slice this slicer can make? [1/2 inch]
- If you know the length of a block of cheese, can you determine how many slices it can make? [Answer: You need more information than just the length. You also need to know the thickness of each slice. Given a certain size block of cheese, if you make thicker slices, then you will get fewer slices. If you make very thin slices, then you can make more slices.]
- Suppose you get a new block and you know how thick you want your slices. What do you need to know in order to tell how many sandwiches you can make? [The length of the block and how many slices go in a sandwich.

2. POSE A PROBLEM

Start this phase of the lesson with a warm-up problem to cue students to think about division. Show Slide #2 and ask,

“Without solving this problem, what operations or steps would you use to find the number of sandwiches that the chef can make? Why?”

Have students talk in pairs.

Many students will recognize that this is a division problem, but they might struggle to explain why. One common “trick” students use to select the correct operation to solve these problems is to reason that there must be fewer sandwiches than slices, so they should divide 570 by 3 (they might even deduce that subtracting 3 is not small enough). This “division makes smaller” reasoning works when dividing whole numbers. However, as we will see, it is not necessarily true for dividing fractions.

Now let’s solve a similar problem that uses fractions. The chef is using a slicer to make thin slices of cheese for sandwiches. The slicer has settings for different thicknesses ranging from 1/32” to 1/2”. Consider this: How many 1/12” slices can the chef make with a block of cheddar that is 2 inches long?

Show Slide #3.

Ask: “How would you start thinking about this problem? What operation will you use to find the answer?” Again, have students work in pairs.

After giving students time to discuss ideas (one or two minutes should suffice), ask two or three students to share how they thought about this problem. Some students might not have a complete solution; that is OK—partially formed ideas can still contain useful information, and it is helpful to show students it is OK to share an incomplete idea.

Some students will start by noticing that there are 12 copies of a 1/12 inch slice of cheese in each inch of the block, so therefore they can get 12 slices per inch, or 2 × 12 = 24 slices total from the 2-inch long block of cheese.

Slide #4 shows how 1 inch of the block can be partitioned into 12 equally thick slices of cheese.

Some students might notice that this problem parallels the previous question with whole numbers, and say that you should divide 2 by 1/12. If no students make this claim, you should do so.

At this point, do not introduce the standard algorithm for doing fraction division (“Ours not to reason why; just invert and multiply”). Nevertheless, connecting these two solutions is key to understanding this poster problem: dividing 2 by 1/12 has to be the same as multiplying 2 by 12. Let students discuss this and try to explain why both answers are correct.

Let’s consider one way of thinking about the division. Division finds the missing number in multiplication: 15 ÷ 5 = ? is the same as 5 * ? = 15. So 2 ÷ 1/12 is the same as (1/12) * ? = 2. But 2 is 24/12, so I want (1/12) * ? = (24/12). And that’s clearly 24.

Note: Some students might solve the problem by saying that 2 divided by 12 is 24. These students might be thinking: “Two [inches, where each inch is] divided [into] 12 parts is 24 [parts].” That is, the student understands the situation, and that it involves dividing, but he or she is confused about how the numbers in the situation take up their roles in the calculation. So check for understanding and praise it—but ask if it’s really true that 2 ÷ 12 = 24.

Now pass out Handout #1 and remind students to spend ample time solving each question. The problems are ordered in increasing difficulty.

Discussing Handout #1

Question 4 should prompt a discussion of the meaning of the remainder in division of fractions. In this problem, there is 1/20 of an inch of cheese left over after making 3 slices that are 2/5” thick each. Some students might argue that the answer should be 3 and 1/20. This is a good moment to highlight the distinction between calculating the number of inches left over and the number of slices left over. Stress that both ideas are correct, but that 3 1/20 (written as a mixed number) is not, because the 3 is in slices and the 1/20 is in inches. The answer is "3 1/4 slices," or "3 whole slices with 1/20 inch left over.” We will be focusing on the first representation because that’s the result you get when you step up to doing division.

Completing this worksheet and discussing the answers can easily take a full class period. This is OK—fraction division is one of the most challenging topics in the school curriculum, and it is worth the time to delve deeply into the ideas in this section.

An extra—student reality check! Ask: What do you think are realistic thicknesses for slices of cheese you would put in a sandwich?

3. WORKSHOP

Arrange students in pairs or groups. Pass out Handout #2 and explain that they will now create their own own cheese-slicing problems, and create a poster showing how to solve them.

While students work in groups to create posters, encourage students to consider describing the problems using division rather than repeated addition or subtraction, or multiplication guess-and-check.

4. POST, SHARE, COMMENT

Teams display their posters in the classroom, get to know other teams’ posters, and attach questions/comments by way of small adhesive notes (or similar).

Sample Posters:

A

B

C

D

In Poster A, for Problem A, students chose a fraction with a numerator of 1. These fractions are called “unit fractions.” If the length of the block of cheese is a whole number, then dividing by a unit fraction will not leave a remainder. In this example, 3 ÷ (1/4) = 12.

In Poster B (Problem B), the group used repeated addition to solve their problem. It would help if they clearly labeled their solution—but you can still see how they did it (their answer is 27 slices). Their remainder is in inches, not slices.

Poster C shows a group that did their entire problem using a diagram.

Finally, Poster D shows a group that was able to correctly set up and solve a cheese slicing problem using non-trivial numbers. (Note that this problem does use a slice that is more than 1/2 inch, but the setup and solution are sound).

5. STRATEGIC TEACHER-LED DISCUSSION

Select a sequence of posters to use during the teacher-led discussion that will help move all students from their current thinking (often Levels 1–3 below) up to 4 or 5.

Level 1. The group has defined a block of cheese and a thickness, and successfully use a diagram or number line to find the number of slices. There is no calculation.

Level 2. Here, the group uses calculation (possibly in addition to a diagram) but the calculation may be guess and check, repeated addition (as in Poster B), or repeated subtraction—but not division.

Level 3. The group uses division, but its use might be muddled. They might state division incorrectly, e.g., stating that 2 ÷ 12 = 24 (instead of 2 ÷ 1/12). Numbers in the problem may not correspond with a diagram. The remainder might be larger than the thickness of a slice.

Level 4. The group uses more challenging fractions (e.g., 3/8) for the thickness, but the block length is still a whole number. Remainders are in terms of slices. The group describes their computation in terms of division. The diagram corresponds to the problem and solution.

Level 5. The group uses challenging fractions for the slices and non-whole numbers for the block length. The group describes the calculation in terms of division and shows how to perform the calculation. When asked, students can also show how the division calculation corresponds to less-sophisticated techniques.

Connecting across groups

During the discussion of this problem there will be a few key things to highlight.

- Students will choose different numbers, some “friendlier” than others. Be sure to highlight some students who used non-unit fractions.
- Students may struggle to interpret the remainder. Highlight that the remainder could be in inches or in slices—but you have to be clear. Be sure students understand that if you use a calculator or a division algorithm, your remainder is in slices.

Other Directions

Depending on your class, you may want to consider extensions to the discussion like these:

- Remember the example with two inches of cheese and 1/12 inch slices? Is it easier to think of this as 2 ÷ (1/12) = 24 or as 2 ⨉ 12 = 24? (The latter.) Why does the second one work? What would be a situation where thinking about it as division makes it easier rather than more confusing? (Harder numbers that don’t match up, e.g., a block 6 1/4 inches long divided up into slices 2/15 thick.)
- Before, division always made numbers smaller. But now, sometimes division makes the answer bigger than the numbers you started with. What has to be true for that to happen? (Divisor is less than 1.)
- How can you check your work with a calculator? (Find decimal equivalents, but watch out because you’ll be working with approximations.)

6. FOCUS PROBLEM: Same Content in a New Context

Show Slide #5 to the class. Have the students work on the question in pairs. Slide #6 can be used as a discussion guide after students have worked with the problem for awhile.

Slide #7 contains two optional “Explore More!” questions. Slide #8 contains the answers to the “Explore More!” questions on Slide #7.

Project funding provided by The William and Flora Hewlett Foundation and S.D. Bechtel Jr. Foundation.

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