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

 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

 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?

 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

 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

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, ¾ x n 1 ½ n 1 ¾ n

 How 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

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

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/ ¾ = 3 ¾

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 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?

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.

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

 Write a story problem that matches this expression  Solve the problem using two different strategies

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?

 Read the commentary that goes with this task.  How does the pictorial representation help make sense of the problem?

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

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?