1. Multiplication and Division of Fractions Thursday, June 18, 2015

2. The Meaning of Multiplication  When working a multiplication problem, it is important to understand what the factors represent in the problem. Consider the picture below: What multiplication problem is represented by this picture? 5 x 2 = 10 What does the 5 represent?Number of groups What does the 2 represent?Number in each group

3. The Meaning of Multiplication  While multiplication is commutative (that is A x B = B x A), we will stick to the convention illustrated in the previous slide for interpreting the factors in a multiplication problem: A x B = C A (First factor): Number of groups B (Second factor): Number in each group

4. Multiplication 

5. How could you model these questions using fraction strips?  

6. Consider the following example: The Lance Elementary School asked the fifth grade students to help the art teacher design some tile murals for the new art room. One of the murals is going to have ¾ of the design as red tiles and ½ of those red tiles will have flowers on them. What fraction of the tiles will be red and have flowers on them? Draw a picture to solve this problem. Using An Area Model to Illustrate Multiplication of Fractions

7. The Multiplying Game  This is a game that can help students practice with multiplying fractions, along with comparing fractions.  This game also continues to force students to think about fractions, in general, particularly the role of the numerator and denominator.

8. Relationship between Multiplication and Division  We know that every division problem can be written as a multiplication problem. For example, 10 ÷ 5 = ? could be expressed as ? X 5 = 10 or 5 x ? = 10

9. Vocabulary  Dividend  Divisor  “How Many Groups?” problem  “How Many in Each Group?” problem Two Interpretations for 8 ÷ 2 = 4

10. Interpretations/Models of Division 1. “How Many Groups?” Problem (also called the measurement model of division or the subtractive model of division) Dividend = total number of objects Divisor = number of objects in each group Quotient = number of groups Using a “How Many Groups?” problem to show 18 ÷ 6 = 3

11. Interpretation/Models of Division 2. “How Many in Each Group?” Problem (also called the partitive or sharing model of division) Dividend = total number of objects Divisor = number of groups to divide the objects into Quotient = number of objects in each group Using a “How Many in Each Group?” problem to show 18 ÷ 6 = 3

12.  When working with fractions, the “How Many Groups?” interpretation to division is the most helpful.  With this interpretation, we ask, “How many groups of size x are in y?” where x and y may be fractions or whole numbers.  Using this interpretation, we can try to make sense of what it means to divide fractions.  We can use our fraction strips to help us understand and see division of fractions.

13. Division 

14. More Division with Fraction Strips 

15. Using an Area Model to divide a proper fraction by a proper fraction (How Many Groups?) 1. Draw an area model to represent the dividend. 2. Divide your area model into equal-sized sections using the denominator or your divisor. 3. Circle the number of sections you have based on the numerator of your divisor. 4. Fit as many of the shaded squares into the circled section as you can (you may possibly fill up a complete section or sections). 5. The solution is the fraction of the shaded section you have. 6. Note: This drawing works best when the dividend is smaller than the divisor, but can still be shown if the dividend is larger.

16. Let’s Practice  Consider the How Many Groups? division problem: Create a story problem that seeks to answer the following question: How many groups of 2/3 are in ¼? Then, draw an area model to determine the solution.

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18. 1 cup of blue icing What do we need to do first?

19. 1 cup of blue icing We have 13 thirds total. Now what?

20. 1 cup of blue icing Now, we need to divide the cup of blue icing into 13 parts to spread onto each third of the cake.

21. Paralleling to the Algorithm  Now, let’s consider how we can better understand why the algorithm of “invert and multiply” works with division of fractions using our previous two examples.

22. Dividing by 2 vs. Dividing by 1/2  Create a story problem that would be used to solve the following problem: ¾ ÷2  Create a story problem that would be sued to solve the following problem: ¾ ÷ ½  Are these two problems the same? Is dividing by 2 the same thing as dividing by ½?