Dividing Fractions—Servings of Yogurt

[Pages:23]Dividing Fractions--Servings of Yogurt

About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection questions, and student materials. While the primary use of Illustrations is for teacher learning about the SMP, some components may be used in the classroom with students. These include the mathematics task, student dialogue, and student materials. For additional Illustrations or to learn about a professional development curriculum centered around the use of Illustrations, please visit mathpractices.. About the Dividing Fractions--Servings of Yogurt Illustration: This Illustration's student dialogue shows the conversation among three students, with experience multiplying fractions and dividing whole numbers by fractions, trying to answer how many 3/4 cup servings of yogurt fit in 2/3 of a cup. They try several examples of dividing a whole number by a unit fraction (1/4) and then reason that if they are dividing by 3/4 instead of 1/4, the answer should be 1/3 the size. Next they try this same reasoning on examples where the dividend is a fraction, and find an answer to the original problem. Highlighted Standard(s) for Mathematical Practice (MP) MP 1: Make sense of problems and persevere in solving them. MP 7: Look for and make use of structure. MP 8: Look for and express regularity in repeated reasoning. Target Grade Level: Grade 6 Target Content Domain: The Number System, Number & Operations--Fractions Highlighted Standard(s) for Mathematical Content 6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving

division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ? (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ? (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ? (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? 5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ? (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ? (1/5) = 20 because 20 ? (1/5) = 4. 5.NF.A.2 Interpret a fraction as division of the numerator by the denominator (a/b = a ? b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? Math Topic Keywords: fractions, division, multiplication, unit fractions

? 2016 by Education Development Center. Dividing Fractions--Servings of Yogurt is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. To view a copy of this license, visit . To contact the copyright holder email mathpractices@

This material is based on work supported by the National Science Foundation under Grant No. DRL-1119163. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Dividing Fractions--Servings of Yogurt

Mathematics Task

Suggested Use This mathematics task is intended to encourage the use of mathematical practices. Keep track of ideas, strategies, and questions that you pursue as you work on the task. Also reflect on the mathematical practices you used when working on this task.

How many 3 -cup servings are there in 2 of a cup of yogurt?

4

3

Task Source: Common Core Standards Writing Team. (2013, July 4). Progressions for the Common Core State Standards in Mathematics (draft). The Number System, 6?8. Tucson, AZ: Institute for Mathematics Education, University of Arizona.

Dividing Fractions--Servings of Yogurt

Student Dialogue

Suggested Use The dialogue shows one way that students might engage in the mathematical practices as they work on the mathematics task from this Illustration. Read the student dialogue and identify the ideas, strategies, and questions that the students pursue as they work on the task.

Students have already learned how to multiply two fractions and how to divide a whole number by a unit fraction and a unit fraction by a whole number. They are currently learning how to divide two fractions. They have been given a division-of-fractions problem with a context about serving size and have gotten to the point where they are trying to figure out what is 2 ? 3 . They

34 are using their understanding of unit fractions to figure out how to perform such a division.

(1) Anita:

(2) Sam: (3) Anita: (4) Sam: (5) Anita: (6) Sam: (7) Anita: (8) Sam: (9) Dana:

We have to divide by 3 . Let's start by figuring out how to divide by one quarter. 4

[Pauses to think. Then, to Sam...] Oh, this is going to be easy. How many quarters are in 1? 4. And 3? 12 And 2? 8. And 5? 20. You're always just multiplying by 4.

(10) Anita:

Right. And because 3 is larger, there will be fewer 3 -size pieces in something

4

4

than 1 -size pieces. Exactly a third as many. So how many three-fourths are there 4

in 5?

(11) Sam:

Umm, well.... There are 20 fourths in 5. Done. And 3 fourths is three times as big as 1 fourth so.... fewer three-fourths will fit in 5. A third of them in fact. A third of 20. That's how many three-fourths are in 5.

Dividing Fractions--Servings of Yogurt

(12) Dana:

So to find how many three-fourths there are in any number, we multiply by 4 first to find out how many fourths there are and then we divide by 3 to find out how many three-fourths.

(13) Sam:

Exactly, we multiply by 4 and divide by 3. Great! That should work for any number.

(14) Anita:

Well, let's just do it once more. Let's try it with 7 ? 3 . There are 28 fourths in 7; 4

that's easy, just multiply by 4. And then we divide by 3 to find how many 3 28

fourths. . 3

(15) Dana:

How about another one, 4 ? 3 . Multiply 4 by 4, that's 16. Divide by 3, that's 4

16 . 3

(16) Anita: That's a little bit more than 5. That sounds about right.

(17) Sam:

Or 6 ? 3 . Multiply by 4 is 24 and divide by 3 is 8. 4

(18) Dana: We can check it. 8 times 3 fourths is 24 fourths. That's 6, so it works.

(19) Anita: Or you can think of 3 quarters of 8 is 6.

(20) Dana:

3 Wait a minute! To divide by , we multiply by 4 and divide by 3. That's the

4 same as multiplying by 4 .

3

(21) Sam: So can we go back to the original problem now?

(22) Dana: No, wait Sam! This is even more important than our silly problem! I think that method is going to work for all fractions! I don't even like yogurt.

(23) Sam: What's going to work?

(24) Dana:

To divide by 3 , we multiply by 4 . Look at those fractions. I'm sure it's going to

4

3

work that way with all fractions. Let's try 2 ? 4 . 37

Dividing Fractions--Servings of Yogurt

(25) Sam: Eeeuw! Can't we work up to that with some more whole numbers?

[Again, they first check out how many sevenths in 1, in 2, in 5, and conclude they're always multiplying by 7. Then they check out how many four-seventh pieces in each of those, and decide it must be one-fourth as many as there were sevenths. A good quarter of an hour later...]

(26) Dana:

4 It really does work! To divide by , we multiply by 7 and then divide by 4, so we

7 7 are multiplying by . It really does work!!! 4

(27) Sam: Dana, you're way too excited! Take it easy!

(28) Dana: Don't you see? We invented a way to divide by any fraction, and we know why it works!!

(29) Sam:

So, now can we go back to the original problem? How many 3 cup servings are 4

there in 2 of a cup of yogurt? We said that's 2 ? 3 and we needed to figure out

3

34

how to divide those two fractions.

(30) Dana:

Yes, and now we have a way to figure out how to divide by 3-fourths. Multiply by 4. 3

(31) Anita:

Well, two-thirds times four is eight-thirds. But we're not multiplying by 4, we're

4

8

multiplying by , so now we need to divide by 3. That's... 8-ninths.

3

3

(32) Sam: Why ninths?

(33) Anita:

Because dividing by 3 is like taking a third of it. We want 1-third times 8-thirds.

[she writes 1 ? 8 ]. That's 8 .

33

9

(34) Sam:

Oh right. We're just multiplying fractions. So if 2 ? 3 is 8 , that means there are 34 9

8 servings in 2 cups of yogurt.

9

3

Dividing Fractions--Servings of Yogurt

(35) Dana:

8

3

of a serving--not quite a full serving--sounds about right. A -cup serving is

9

4

2 more than the cup we have, but not by much. You would get just under a full

3

3 -cup serving from 2 of a cup of yogurt. But why didn't you just use the simple

4

3

method we found? To compute 2 ? 3 we can simply do 2 ? 4 . That's easier.

34

3 3

(36) Sam:

This is making me hungry. Even 3 of a cup isn't a lot of yogurt, and we're only 4

8 getting of that?! I'd want more!

9

(37) Dana: Eeeuw!

Dividing Fractions--Servings of Yogurt

Teacher Reflection Questions

Suggested Use These teacher reflection questions are intended to prompt thinking about 1) the mathematical practices, 2) the mathematical content that relates to and extends the mathematics task in this Illustration, 3) student thinking, and 4) teaching practices. Reflect on each of the questions, referring to the student dialogue as needed. Please note that some of the mathematics extension tasks presented in these teacher reflection questions are meant for teacher exploration, to prompt teacher engagement in the mathematical practices, and may not be appropriate for student use.

1. What evidence do you see of students in the dialogue engaging the Standards for Mathematical Practice?

2. How do students use repeated calculations in lines 1?10 and later in lines 14?17? How does completing several calculations that all look similar help the students?

3. Think about the path students take in the Student Dialogue as they work through the problem. If you were overhearing this as it was taking place, are there places you would want to intervene? Places you would want to follow up on afterward? Ideas you would particularly want to share with the class, or alternatives you would want (later or immediately) to share with Anita, Dana, and Sam?

4. Based on the Student Dialogue, what issues might teachers want to keep in mind as they help engage students in MP 8: Look for and express regularity in repeated reasoning?

5. How did Anita multiply 2 ? 4 in lines 31 and 33? What is the logic behind Anita's 33

approach?

6. In line 35, Dana presents an argument to show that the result of their computation, 8 , is 9

reasonable. What generalization can be made about when the result of a division computation is, as in this case, between 0 and 1? When can the result of a division be larger than one of the numbers that are being divided? When can the result of a division be larger than both of the numbers? Most importantly, which, if any, of these generalizations do you think are important for students to learn or make on their own?

7. Why are there two different notations--fractions and decimals--at all?! What are some of

the advantages and disadvantages of representing each notation? As part of your answer, you

might consider the following:

A)

Which

feels

easier

to

compute:

3? 5 7 11

or

the

decimal

equivalent

0.428571? 0.454545 ?

Dividing Fractions--Servings of Yogurt

B) Which feels easier to compare in magnitude: 3 and 5 or their decimal equivalents 7 11

0.428571 and 0.454545 ?

C)

Which feels easier to compute:

3 +1 10 2

or

0.3+ 0.5?

D)

Both

0.3333...

and

3 10

+3 100

+ 3 + 3 + ... 1000 10000

mean

the

same

thing.

What

are

the

advantages and disadvantages of each?

8. How can a visual model be used to represent 2 ? 3 from the original task, which the 34

students return to in line 29?

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