Developing an Understanding of Fraction Division

1. Developing an Understanding of Fraction Division Presented by Lisa Couey and Katie Mayfield June 5, 2013

2. Common Core State Standard • CCSS 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. • CCSS 5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. • CCSS 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. • CCSS NF.B.7c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

3. Common Core State Standard • CCSS 6.NS.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 examples, 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 ¾ 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?.

4. Vocabulary • dividend • divisor • quotient • numerator • denominator

5. “Ours is not to reason why, just invert and multiply”. How do we move our students beyond a basic algorithm to a true understanding of fraction division? First we must understand fraction division ourselves. Models, drawings, and story writing can assist us in developing our understanding, and in developing our students’ understanding.

6. Measurement Division Problems • Measurement division problems identify the size of the group in a division problem, but not the number of groups. (In the following problem, 3 is the size of the group; 4 is the unknown number of groups) • Sam has 12 tacos, and he wants to give 3 to each of his friends. How many friends can Sam share tacos with?

7. Partitive Division Problems • Sam has 24 tacos left and he wants to give these to 4 of his friends. How many tacos will he give to each friend? • In partitive division problems, the number of the groups is identified, but not the size of the groups. (4 is the number of groups; 6 is the unknown size of the groups)

8. Cookie Problem • At your table, remove the cookies from the bag. If these cookies represent of the batch of cookies, what was the total number of cookies in the batch? Justify your answer by drawing a picture, or explain your reasoning in words.

9. Using Stories and Illustrations • Write and illustrate a story for the following problem: 2 ÷

10. One Possible Story and Illustration for the Problem • For the 6th grade class parties, Sally’s mom brought 2 gallons of punch. The teacher knew she needed of a gallon of juice per class. • There are 3 sets of in 2 .

11. References • Cengiz, N. & Rathouz, M., (2011). Take a bite out of fraction division. Mathematics teaching in the middle school. (17)3. lcouey@stone.k12.ms.us kmayfield@stone.k12.ms.us