1. Math Facilitator Meeting January 17, 2013
Multiplication and Division of Fractions and Decimals
Session 1

2. Something to Think About……
How do we help students develop conceptual understanding of operations with decimals and fractions?
How does our work with multiplication and division of whole numbers relate to decimals and fractions?
What is “flexibility” with fractions and decimals?
Why is flexibility in working with decimals and fractions important for solving problems?

3. “The Fraction Trajectory”
Look at the fraction standards from grades
1-5
What standards are new at each grade level?
With a partner, make a list of the concepts that should be mastered before learning to reason with multiplying and dividing decimals and fractions
What other standards are important in building that relationship?

4. 864,352.79
What place value understanding do students need when describing this number?
Write the number in expanded form
Story about factors of 18 and 180- not seeing the relationship, but just add zero!!!!!!

5. Common Misconceptions when Multiplying and Dividing Fractions
Multiplication does not always make things bigger
Multiplication is not “just” repeated addition
The meaning of “times” 3 x 4 = 4 x 3. Are they the same? (think about groups)
Is 3/4 of a group of 3 the same as 3 groups of 3/4?

6. Common Misconceptions continued
Translating multiplication expressions 5 x 6 could be 5 groups of 6 or 5 taken 6 times.
We need pictorial representations when it comes to fractions!!- the idea of 1/2 taken 1/4 times makes no sense. 1/2 a group of 1/4 makes more sense.
If students can connect multiplication equations to real things, it will help them make sense of problems

7. More Misconceptions…
Students shouldn’t be focused on just the numbers, but make sense of the magnitude of the fractions. Example: 3 1/2 x 3 1/2 The answer can’t be more than 4 x 4 or less than 3 x 3.
There is a real connection between multiplication and division of fractions (they are not just opposites)
Example: 10 x 1/2 is the same thing as 10 ÷ 2
and 10 ÷ 1/2 is the same thing as 10 x 2

8. A Fraction Represents…
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts;
Understand a fraction a /b as the quantity formed by a parts of size 1/b
CCSS talks about fractions with different language and emphases than we are used to – but the meanings of fractions and their symbols have not changed.
ASK: Talk in groups – What examples of unit fractions might we use or hear in everyday life? ½ hour; 1/2 of an hour; half time in some sports; measurement – ½ inch etc Pie sliced into 4 parts - etc.
Defining fractions in terms of unit fractions avoids a major conceptual problem, namely, establishing that a fraction is a number.
A second advantage of defining fractions in terms of unit fractions is that this approach separates the study of the numerator and the denominator of a fraction. Numerators are standard counting numbers and are used to count unit fractions. 8

9. Unit Fractions A unit fraction is a proper fraction with
a numerator of 1 and a whole number denominator
is the unit fraction that corresponds
to or to or to
As there are 3 one-inches in 3 inches, there are 3 one-eighths in
The challenge for thinking about rational number is that children have to coordinate different units. For example, in the brownie problem we have serving, pan, and pans (2 pans might be the whole, rather than 1 pan being the whole as in our problem. 1/8 of both pans may be the whole serving rather than 1/8 of one pan as in our problem).
The label of unit in a unit fraction can be connected to the fraction in two ways.
Just as unit means one single thing, a unit fraction consists of a value of one of the fractional parts designated by the denominator.
And just as we talk about units when we measure, a unit fraction can be thought as an indication of the size of the fractional piece. Thus comparing 3 fifths to 3 sevenths, fifths of a whole are larger than sevenths of the same whole, so that 3/5 is greater than 3/7. 9

10. Unit Fractions
Unit fractions are formed by partitioning a whole into equal parts and naming fractional parts with unit fractions 1/3 +1/3 = 2/3
1/5 + 1/5 + 1/5 = ?
Unit fractions are the basic building blocks of fractions, in the same sense that the number 1
is the basic building block of whole numbers
We can obtain any fraction by combining a sufficient number of unit fractions 1 b

11. Fractional Parts of a Whole
If the yellow hexagon represents one whole, how might you partition the whole into equal parts? Name the fractional parts with unit fractions

12. Fractional Parts of a Whole
Name the unit fractions that equal one whole Hexagon
1/ /2
1/3
1/6
1/ / / /6
Handout: - sheet of hexagon shapes - equivalent to the Hexagon - Materials: pattern blocks
Tell participants to Build and label each trapezoid on a hexagon. Label fractions - Continue with other pattern blocks.
Example: 1/3 + 1/ 3 + 1/3 = 3/3 or 1 whole OR count 1/3 , 2/3, 3/3 = 1 whole . 12

13. Fractional Parts of a Whole
Two yellow hexagons = 1 whole
How might you partition the whole into equal parts?
Name the unit fraction for one triangle; one hexagon;
One hexagon is ½ of the whole and the unit fraction is one half. The trapezoid is ¼ of the whole and the unit fraction is 1/4; rhombus is 1/6 of the whole and the unit fraction is 1/6 ; triangle is 1/12 of the whole and the unit fraction is 1/12 13

14. Fractional Parts of a Whole
What is the value of the red trapezoid, the green triangle and the yellow hexagon?
Show and explain your answer
What names the unit fraction? Explain.
One trapezoid has a value of 1 ½ ; the hexagon has a value of 3; and the triangle has a value of ½. 14

15. Identifying Fractional Parts of a Whole
What part is red?
Have participants build this figure with pattern blocks and determine what fractional part of the figure is red. (6/20 or 3/10) Then ask what fractional part is blue? (4/20 or 1/5) Green? (4/20 or 1/5) Yellow? (6/20 or 3/10) Ask for what relationships they see within the figure. For example, the fractional part that is red is equivalent to the fractional part that is yellow, and the fractional part that is green is equivalent to the fractional part that is blue. The “common denominator” or the unit for comparison here is the green triangle. Fourth graders are not expected to find a common denominator, but for this activity they do need to see the relationships of the pieces – that two triangles are in each blue rhombus, three triangles are in the red trapezoid, and six triangles are in the yellow hexagon. Likewise, two red trapezoids are equivalent to one yellow hexagon. Looking for relationships in an activity like this provides experiences that can lead children to recognizing equivalent fractions. The concept of fraction equivalence will be explored later in this module.
Tell participants that one student said that 2/9 of the figure is red. Ask them what mistake the student is making. What does the student not understand about fractional parts?
A similar activity is in the handout. Point out that children could build their own figures and figure out the fractional parts. 15 15

16. Create the whole if you know a part…
If the blue rhombus is ¼, build the whole.
If the red trapezoid is 3/8, build the whole.
Unit fraction is important here. One must know the unit fraction in order to build the whole.
Do part to whole activity with pattern blocks.
Have DMP part to whole activities in handout. 16

17. Fractions in Balance Problems
Find the missing values.
Figures that are the same size and shape must have the same value.
Adapted from Wheatley and Abshire, Developing Mathematical Fluency, Mathematical Learning, 2002
1 ¾ x
1 ¾
The balance format brings algebraic thinking into the work with number, and helps to develop meaning for fractions and operations with fractions. When using this format in working with fractions, students are more likely to develop solution strategies that are meaningful to them, rather than following a formula.
In the top balance, the x = 3 ½. Participants may think: = 2 ¾ + ¾ = ½ + ½ + ¼ + ¼ = ½ + ½ + ½ = 1 ½; ½ = 3 ½ .
In the second balance, since 1 = ½ + ½ , the block 1 ½ = 3 sets of ½ so each of the three cylinders, ”n” = ½ .
Before leaving this slide, be sure that participants see that activities like this involve both algebra and number/operations so that more than one strand can be addressed at the same time.
n ½ n n 17 17

18. 5/4
How many different ways can you model 5/4?

19. Multiplying Unit Fractions
Understand a fraction a/b as a multiple of 1/b
is the product of 5 x ( )
= 5 x

20. Multiplying Unit Fractions
Understand a multiple of a/b as a multiple of 1/b,
and use this understanding to multiply a fraction
by a whole number
3 sets of is the same as 6 sets of

21. Fractions Greater than One
How much is shaded?
How could you name the amount as a fraction?
As a whole number and a fraction?
(¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ ) = 15/4
(4 x 1/4) + (4 x 1/4) + (4 x 1/4) + (3 x 1/4) = 15/4
4/ / / ¾ = 15/4
¾ = 3 ¾
Children need experience naming fractional amounts that are greater than one. They often think that fractions like 7/4 are impossible. Most of their previous experience has probably been with dividing one whole into fractional parts. What real life situations could help them think about fractional parts in more than one whole? 21 21

22. Early Introduction of Multiplication and Division of Fractions…..
Young Mathematicians at Work:
How is multiplication and division connected to fractions?
What is meant by “there are two wholes when dividing fractions?”

23. Sharing Submarine Sandwiches
The cafeteria made lunches for the fourth graders going on a field trip. They were in four different groups so the number of sandwiches differed. The sandwiches were all the same size.
Group One had 4 students sharing 3 subs
Group Two had 5 students sharing 4 subs
Group Three had 8 students sharing 7 subs
Group Four had 5 students sharing 3 subs
Did each student get a “fair share?”
If not, which group ate the least? Most? How do you know?

24. Help the Cafeteria Staff
Next trip we want to guarantee that each student will receive 2/3 of a sub
Using large paper, create a chart for the cafeteria to help them know how many subs to make for up to 15 students
What patterns do you notice?
What strategy could cafeteria workers use for any number of students?
If you knew there were 8 subs made, how could you figure out how many students could each get 2/3 sub?
Model this situation using numbers and symbols.

25. 5 x 1/3 Write a story problem that matches this expression
Solve the problem using two different strategies

26. 5 ÷ 1/3 Write a story problem that matches this expression
Solve the problem using two different strategies

27. Painting a Wall..
Nicholas is helping to paint a wall at a park near his house as part of a community service project. He had painted half of the wall yellow when the park director walked by and said, This wall is supposed to be painted red.”
Nicholas immediately started painting over the yellow portion of the wall. By the end of the day, he had repainted 5/6 of the yellow portion red. What fraction of the entire wall is painted red at the end of the day?

28. Paper Folding Multiplication

29. Task Reflection…. Read the commentary that goes with this task.
How does the pictorial representation help make sense of the problem?

30. Task
How can you prove the following:
5 ÷ 2/3 = 5 x 3/2

31. Problem-Based Number Sense Approach
Keep the following guidelines in mind when developing computational strategies for fractions:
Begin with simple contextual problems
Connect the meaning of fraction computation with whole-number computation
Let estimation and informal methods play a big role in the development of strategies
Explore each of the operations with models
(Van de Walle, Karp, & Bay-Williams, 2010, p.310)
Go over these ideas briefly. 31

32. Making Sense of Fractions
We must go beyond how we were taught and teach how we wish we had been taught.
Miriam Leiva, NCTM Addenda Series, Grade 4, p. iv 32

33. So why don’t we just teach it that way?
Reflection……
Multiplying and Dividing fractions is so easy when you just use the procedure.
Multiplication: multiply numerator x numerator and denominator x denominator.
Division: Just invert the second fraction and multiply.
So why don’t we just teach it that way?