Math Facilitator Meeting

January 17, 2013

Multiplication and Division of Fractions

and DecimalsSession 1

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?

Something to Think About……

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?

“The Fraction Trajectory”

What place value understanding do students need when describing this number?

Write the number in expanded form

864,352.79

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?

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

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

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

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

5

1

5

2

5

35

17

8

3

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

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

Fractional Parts of a Whole Name the unit fractions that equal one

whole Hexagon

1/3 1/2

1/3 1/6

1/6 1/6 1/6 1/6

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;

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

Identifying Fractional Parts of a Whole What part is red?

15

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.

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

17

1 ¾ x

n 1 ½ n

1 ¾

n

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

Multiplying Unit Fractions

Understand a fraction a/b as a multiple of 1/b

is the product of 5 x ( )

= 5 x

4

5

4

5

4

1

4

1

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 of5

2

5

1

Fractions Greater than One How much is shaded?◦ How could you name the amount as a fraction? ◦ As a whole number and a fraction?

21

(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

4/4 + 4/4 + 4/4 + ¾ = 15/4

1 + 1 + 1 + ¾ = 3 ¾

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?”

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 subsGroup Two had 5 students sharing 4 subsGroup Three had 8 students sharing 7 subsGroup Four had 5 students sharing 3 subsDid each student get a “fair share?”If not, which group ate the least? Most?

How do you know?

Sharing Submarine Sandwiches

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.

Help the Cafeteria Staff

5 x 1/3

•Write a story problem that matches this expression

•Solve the problem using two different strategies

5 ÷ 1/3Write a story problem that matches this expression

Solve the problem using two different strategies

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?

www.Illustrativemathematics.org

Paper Folding Multiplication

Task Reflection….

Read the commentary that goes with this task.

How does the pictorial representation help make sense of the problem?

Task

How can you prove the following:

5 ÷ 2/3 = 5 x 3/2

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)

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

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?