Modeling Multiplication and Division of Fractions 1 1

So what’s new about fractions in Grades 6-8? SOL 6.4 The student will demonstrate multiple representations of multiplication and division of fractions. 2

Thinking About Multiplication The expression… We read it… It means… It looks like… Make the connection to the earlier discussion of the variety of meanings of multiplication (slide 8) and think about which make sense for fractions. Repeated Addition: makes sense if fraction is multiplied by a whole number (2 ∙ 1/3 = 1/3 + 1/3), but not for fraction times fraction Groups-Of: makes sense if fraction is multiplied by whole number (2 groups or sets of 1/3) Rectangular Array: makes sense for fraction multiplied by whole number (2 rows with 1/3 apple in each row) Rate: “Dave walked for 2 hours at a rate of 1/3 mile per hour” or “Dave walked ½ hour at a rate of 1/3 mile per hour” Comparison: “Susie has 2 cookies, and Ralph has 1/3 as many cookies as Susie” Combinations: Area: “I need to cover a bulletin board that measures ½ yard long and 1/3 yard wide; how many square yards of fabric are needed?” 3 3

Thinking About Multiplication The expression… We read it… It means… It looks like… 2 times 3 two groups of three 2 times two groups of one-third times one-half group of one-third Make the connection to the earlier discussion of the variety of meanings of multiplication (slide 8) and think about which make sense for fractions. Repeated Addition: makes sense if fraction is multiplied by a whole number (2 ∙ 1/3 = 1/3 + 1/3), but not for fraction times fraction Groups-Of: makes sense if fraction is multiplied by whole number (2 groups or sets of 1/3) Rectangular Array: makes sense for fraction multiplied by whole number (2 rows with 1/3 apple in each row) Rate: “Dave walked for 2 hours at a rate of 1/3 mile per hour” or “Dave walked ½ hour at a rate of 1/3 mile per hour” Comparison: “Susie has 2 cookies, and Ralph has 1/3 as many cookies as Susie” Combinations: Area: “I need to cover a bulletin board that measures ½ yard long and 1/3 yard wide; how many square yards of fabric are needed?”

Making sense of multiplication of fractions using paper folding and area models Enhanced Scope and Sequence, 2004, pages 22 - 24 Take participants through the enhanced scope and sequence activity. Spend no more than 15 minutes on this… Participants will need blank paper and something to write with. 5

The Importance of Context Builds meaning for operations Develops understanding of and helps illustrate the relationships among operations Allows for a variety of approaches to solving a problem

Contexts for Modeling Multiplication of Fractions The Andersons had pizza for dinner, and there was one-half of a pizza left over. Their three boys each ate one-third of the leftovers for a late night snack. How much of the original pizza did each boy get for snack? Ask participants to draw a picture to solve the problem. One possible pictorial solution is on the following slide.

One-third of one-half of a pizza is equal to one-sixth of a pizza. Which meaning of multiplication does this model fit? We start with one-half of one pizza. That is, in effect, the “whole” amount we have to work with. Each boy gets one-third of that amount, so we divide the half into three equal portions (dividing into three equal shares is the same as finding one-third of the amount). Since the question refers to the share of the original pizza, we have to determine what each of those three pieces is in relation to the original whole. The dotted segments extend across to show that each boy had 1/6 of the original pizza.

Another Context for Multiplication of Fractions Andrea and Allison are partners in a relay race. Each girl will run half the total distance. On race day, Andrea stops for water after running of her half of the race. What portion of the race had Andrea run when she stopped for water?

Students need experiences with problems that lend themselves to a linear model.

Another Context for Multiplication of Fractions Mrs. Jones has 24 gold stickers that she bought to put on perfect test papers. She took of the stickers out of the package, and then she used of that half on the papers. What fraction of the 24 stickers did she use on the perfect test papers?

Problems involving discrete items may be represented with set models. One-third of one-half of the 24 stickers is of the 24 stickers. What meaning(s) of multiplication does this model fit? Problems involving discrete items may be represented with set models.

What’s the relationship between multiplying and dividing? Multiplication and division are inverse relations One operation undoes the other Division by a number yields the same result as multiplication by its reciprocal (inverse). For example: This slide serves as a transition between multiplication and division, and should be reached by the 30 min. mark for this segment… 13 13

Meanings of Division For 20 ÷ 5 = 4…   Divvy Up (Partitive): “Sally has 20 cookies. How many cookies can she give to each of her five friends, if she gives each friend the same number of cookies? - Known number of groups, unknown group size Measure Out (Quotitive): “Sally has 20 minutes left on her cell phone plan this month. How many more 5-minute calls can she make this month? - Known group size, unknown number of groups Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA Publishing, 1998.

Sometimes, Always, Never? When we multiply, the product is larger than the number we start with. When we divide, the quotient is smaller than the number we start with. These questions are posed to get participants to consider what really happens during the operations of multiplication and division, and to recognize that the ways the two numbers in each expression are used is different (one is the multiplier, the other is a quantity or amount).. 15 15

“I thought times makes it bigger...” When moving beyond whole numbers to situations involving fractions and mixed numbers as factors, divisors, and dividends, students can easily become confused. Helping them match problems to everyday situations can help them better understand what it means to multiply and divide with fractions. However, repeated addition and array meanings of multiplication, as well as a divvy up meaning of division, no longer make as much sense as they did when describing whole number operations. Using a Groups-Of interpretation of multiplication and a Measure Out interpretation of division can help: Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA Publishing, 1998.

“Groups of” and “Measure Out” 1/4 x 8: “I have one-fourth of a box of 8 doughnuts.”   8 x 1/4: “There are eight quarts of soda on the table. How many whole gallons of soda are there?” 1/2 x 1/3: “The gas tank on my scooter holds 1/3 of a gallon of gas. If I have 1/2 a tank left, what fraction of a gallon of gas do I have in my tank?” 1¼ x 4: “Red Bull comes in packs of four cans. If I have 1¼ packs of Red Bull, how many cans do I have?” 3½ x 2½: “If a cross country race course is 2½ miles long, how many miles have I run after 3½ laps? 3/4 ÷ 2: “How much of a 2-hour movie can you watch in 3/4 of an hour?” *This type may be easier to describe using divvy up. 2 ÷ 3/4: “How many 3/4-of-an-hour videos can you watch in 2 hours?” 3/4 ÷ 1/8: “How many 1/8-sized (of the original pie) pieces of pie can you serve from 3/4 of a pie?” 2½ ÷ 1/3: “A brownie recipe calls for 1/3 of a cup of oil per batch. How many batches can you make if you have 2½ cups of oil left?”

Thinking About Division The expression… We read it… It means… It looks like… 20 ÷ 5

Thinking About Division The expression… We read it… It means… It looks like… 20 ÷ 5 20 divided by 5 20 divided into groups of 5; 20 divided into 5 equal groups… How many 5’s are in 20? 20 divided by 20 divided into groups of … How many ’s are in 20? 19 19

Thinking About Division The expression… We read it… It means… It looks like… one-half divided by one-third divided into groups of … How many ’s are in ? ? Is the quotient more than one or less than one? How do you know?

Contexts for Division of Fractions The Andersons had half of a pizza left after dinner. Their son’s typical serving size is pizza. How many of these servings will he eat if he finishes the pizza?

pizza divided into pizza servings = 1 servings If 1/3 pizza is one serving, how many servings are in the ½ pizza we are starting with in this problem? Since 1/3 is less than 1/2, our solution will be more than one serving. Since the ½ pizza we have is less than 2/3 (or two servings), the solution will be less than 2 servings. We can portion off one full serving, leaving 1/6 of the original pizza. That sixth is ½ of a serving. We have a total of 1 ½ servings in our half pizza, that are each 1/3 pizza in size. serving 22

Another Context for Division of Fractions Marcy is baking brownies. Her recipe calls for cup cocoa for each batch of brownies. Once she gets started, Marcy realizes she only has cup cocoa. If Marcy uses all of the cocoa, how many batches of brownies can she bake?

1 cup cup 0 cups Three batches (or cup) Two batches (or cup) 1 batches One batch (or cup) How many thirds are in one half? Measurement situations can be represented with a linear model, even if length isn’t being measured. 24

Another Context for Division of Fractions Mrs. Smith had of a sheet cake left over after her party. She decides to divide the rest of the cake into portions that equal of the original cake. How many cake portions can Mrs. Smith make from her left-over cake?

What could it look like? The blue amount represents the leftover ½ sheet cake. The green shaded amount shows 1/3 of the original cake. You can see that one complete 1/3 cake portion exists and then ½ of another portion. Another way to think: common denominators. We know that ½ is 3-sixths and 1/3 is 2-sixths. How many 2/6 are in 3/6? There is one 2/6 and then ½ of another 2/6, for a total of 1 ½. With this diagram you can see that the “little bit more” is really one sixth… so why isn’t “one sixth” a part of the answer? The answer is 1½ … where is that one half in the model? 26

What does it look like numerically? Connect the discussion regarding the pictorial representation to the numerical solution. Toggle back and forth between the two slides pointing out where the common denominator 6 comes from , as well as where the 1 ½ comes from. 27

What is the role of common denominators in dividing fractions? Ensures division of the same size units Assist with the description of parts of the whole Connect concrete to abstract by giving participants some division problems to solve using the common denominator method. 28

the traditional algorithm? What about the traditional algorithm? If the traditional “invert and multiply” algorithm is taught, it is important that students have the opportunity to consider why it works. Representations of a pictorial nature provide a visual for finding the reciprocal amount in a given situation. The common denominator method is a different, valid algorithm. Again, it is important that students have the opportunity to consider why it works. Invariably, the traditional algorithm will come up even if the teacher has not introduced it himself. Teachers need to be ready to discuss it. Remember that one focus of the 2009 standards is to model multiple representations for division of fractions. Time considering the algorithm is worthwhile but it’s not the only way to divide fractions. The next slides attempt to give teachers some ideas about how to help students think about that traditional algorithm. 29 29

the traditional algorithm? What about the traditional algorithm? Build understanding: Think about 20 ÷ . How many one-half’s are in 20? How many one-half’s are in each of the 20 individual wholes? Experiences with fraction divisors having a numerator of one illustrate the fact that within each unit, the divisor can be taken out the reciprocal number of times. 30

the traditional algorithm? What about the traditional algorithm? Later, think about divisors with numerators > 1. Think about 1 ÷ . How many times could we take from 1? We can take it out once, and we’d have left. We could only take half of another from the remaining portion. That’s a total of . In each unit, there are sets of . This is a difficult idea to understand and could take several days in itself. The gist is that we should always be teaching for understanding, and we want students to understand the procedures they are using to solve problems. Multiplying by the reciprocal is not the only method for dividing fractions. 31 31

Multiple Representations Instructional programs from pre-k through grade 12 should enable all students to – Create and use representations to organize, record and communicate mathematical ideas; Select, apply, and translate among mathematical representations to solve problems; Use representations to model and interpret physical, social, and mathematical phenomena. Participants will fold a sheet of paper into four quadrants, then represent a given problem contextually, graphically, and numerically. Assign each table a different expression to work with. ¾ ÷1/2 2/3 ÷ ¼ 5/6 ÷ 2/6 2/3 ÷ 1/6 1/6 ÷ 3/6 12/3÷ 1/2 It is important that everyone come back together and have some debriefing. from Principles and Standards for School Mathematics (NCTM, 2000), p. 67. 32

Using multiple representations to express understanding Given problem Check your solution Contextual situation Solve numerically Solve graphically Participants will fold a sheet of paper into four quadrants, then represent a given problem contextually, graphically, and numerically. Assign each table a different expression to work with. ¾ ÷1/2 2/3 ÷ ¼ 5/6 ÷ 2/6 2/3 ÷ 1/6 1/6 ÷ 3/6 12/3÷ 1/2 It is important that everyone come back together and have some debriefing. 33

understanding of division of Using multiple representations to express understanding of division of fractions 34