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.edc.org. 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 50- pound 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 https://creativecommons.org/licenses/by-nc-nd/4.0/. To contact the copyright holder email [email protected] 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. 3 2 How many -cup servings are there in 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. http://commoncoretools.me/wp- content/uploads/2013/07/ccssm_progression_NS+Number_2013-07-09.pdf 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 2 3 serving size and have gotten to the point where they are trying to figure out what is ÷ . They 3 4 are using their understanding of unit fractions to figure out how to perform such a division. 3 (1) Anita: We have to divide by . 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? (2) Sam: 4. (3) Anita: And 3? (4) Sam: 12 (5) Anita: And 2? (6) Sam: 8. (7) Anita: And 5? (8) Sam: 20. (9) Dana: You’re always just multiplying by 4. 3 3 (10) Anita: Right. And because is larger, there will be fewer -size pieces in something 4 4 1 than -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. 3 (14) Anita: Well, let’s just do it once more. Let’s try it with 7 ÷ . 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 3 (15) Dana: How about another one, 4 ÷ . 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. 3 (17) Sam: Or 6 ÷ . 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. 3 (20) Dana: Wait a minute! To divide by , we multiply by 4 and divide by 3. That’s the 4 4 same as multiplying by . 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? 3 4 (24) Dana: To divide by , we multiply by . Look at those fractions. I’m sure it’s going to 4 3 2 4 work that way with all fractions. Let’s try ÷ . 3 7 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…] 4 (26) Dana: 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!! 3 (29) Sam: So, now can we go back to the original problem? How many cup servings are 4 2 2 3 there in of a cup of yogurt? We said that’s ÷ and we needed to figure out 3 3 4 how to divide those two fractions. (30) Dana: Yes, and now we have a way to figure out how to divide by 3-fourths.