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Welcome

Welcome to content professional development sessions for Grades 3-5. The focus is Fractions.

Fractions in Grades 3-5 lays critical foundation for proportional reasoning in Grades 6-8, which in turn lays critical foundation for high school algebra.

The goal is to help you understand this mathematics better to support your implementation of the Mathematics Standards.

Fractions: Grades 3-5: slide 1

Introduction of Facilitators

INSERT

the names and affiliations

of the facilitators

Fractions: Grades 3-5: slide 2

Introduction of Participants

In a minute or two:

1. Introduce yourself.

2. Describe an important moment in your life that contributed to your becoming a mathematics educator.

3. Describe a moment in which you hit a “mathematical wall” and had to struggle with learning.

Fractions: Grades 3-5: slide 3

Overview

Some of the problems may be appropriate for students to complete, but other problems are intended ONLY for you as teachers.

As you work the problems, think about how you might adapt them for the students you teach.

Also, think about what Performance Expectations these problems might exemplify.

Fractions: Grades 3-5: slide 4

Problem Set 1

The focus of Problem Set 1 is representing a single fraction.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.

Fractions: Grades 3-5: slide 5

Problem Set 1

Think carefully about each situation and make a representation (e.g., picture, symbols) to represent the meaning of 3/4 conveyed in that situation.

Fractions: Grades 3-5: slide 6

Problem 1.1

John told his mother that he would be home in 45 minutes.

Fractions: Grades 3-5: slide 7

Problem 1.2

Melissa had three large circular cookies, all the same size – one chocolate chip, one coconut, one molasses.

She cut each cookie into four equal parts and she ate one part of each cookie.

Fractions: Grades 3-5: slide 8

Problem 1.3

Mr. Albert has 3 boys to 4 girls in his history class.

Fractions: Grades 3-5: slide 9

Problem 1.4

Four little girls were arguing about how to share a package of cupcakes.

The problem was that cupcakes come three to a package.

Their kindergarten teacher took a knife and cut the entire package into four equal parts.

Fractions: Grades 3-5: slide 10

Problem 1.5

Baluka Bubble Gum comes four pieces to a package.

Three children each chewed a piece from one package.

Fractions: Grades 3-5: slide 11

Problem 1.6

There were 12 men and 3/4 as many women at the meeting.

Fractions: Grades 3-5: slide 12

Problem 1.7

Mary asked Jack how much money he had.

Jack reached into his pocket and pulled out three quarters.

Fractions: Grades 3-5: slide 13

Problem 1.8

Each fraction can be matched with a point on the number line.

3/4 must correspond to a point on the number line.

Fractions: Grades 3-5: slide 14

Problem 1.9

Jaw buster candies come four to a package and Nathan has 3 packages, each of a different color.

He ate one from each package.

Fractions: Grades 3-5: slide 15

Problem 1.10

Martin’s Men Store had a big sale – 75% off.

Fractions: Grades 3-5: slide 16

Problem 1.11

Mary noticed that every time Jenny put 4 quarters into the exchange machine, three tokens came out.

When Mary had her turn, she put in twelve quarters.

Fractions: Grades 3-5: slide 17

Problem 1.12

Tad has 12 blue socks and 4 black socks in his drawer.

He wondered what were his chances of reaching in and pulling out a sock to match the blue one he had on his left foot.

Fractions: Grades 3-5: slide 18

Reflection

Even a “simple” fraction, like 3/4, has different representations, depending on the situation.

How do you decide which representation to use for a fraction?

How can we help students learn how to choose a representation that fits a given situation?

Fractions: Grades 3-5: slide 19

Problem Set 2

The focus of Problem Set 2 is representing different fractions.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.

Fractions: Grades 3-5: slide 20

Problem 2.1

Represent each of the following:

a. I have 4 acres of land. 5/6 of my land is planted in corn.

b. I have 4 cakes and 2/3 of them were eaten

c. I have 2 cupcakes, but Jack as 7/4 as many as I do.

Fractions: Grades 3-5: slide 21

Problem 2.2

The large rectangle represents one whole that has been divided into pieces.

Identify what fraction each piece is in relation to the whole rectangle. Be ready to explain how you know the fraction name for each piece.

A ___B ___C ___D ___E ___F ___G ___H ___

Fractions: Grades 3-5: slide 22

Problem 2.3

What is the sum of your eight fractions? What should the sum be? Why?

Fractions: Grades 3-5: slide 23

Problem 2.4

Mom baked a rectangular birthday cake.

Abby took 1/6.

Ben took 1/5 of what was left.

Charlie cut 1/4 of what remained.

Julie ate 1/3 of the remaining cake.

Marvin and Sam split the rest.

Was this fair?

How does the shape of the cake influence your answer?

Fractions: Grades 3-5: slide 24

Problem 2.5

If the number of cats is 7/8 the number of dogs in the local pound, are there more cats or dogs?

What is the unit for this problem?

Fractions: Grades 3-5: slide 25

Problem 2.6

Ralph is out walking his dog.

He walks 2/3 of the way around this circular fountain.

Where does he stop?

Fractions: Grades 3-5: slide 26

Problem 2.7

Ralph is out walking his dog.

He walks 2/3 of the way around this square fountain.

Where does he stop?

START --------->

Fractions: Grades 3-5: slide 27

Reflection

Why is it important for students to connect their understanding of fractions with the ways they represent fractions?

How do you keep track of the unit (that is, the value of 1) for a fraction?

How can you help students learn these things?

Fractions: Grades 3-5: slide 28

Problem Set 3

The focus of Problem Set 3 is unitizing.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.

Fractions: Grades 3-5: slide 29

Describing Unitizing

Unitizing is thinking about different numbers of objects as the unit of measure.

For example, a dozen eggs can be thought of as:

12 groups of 1,6 groups of 2,

4 groups of 3, 3 groups of 4,

2 groups of 6, 1 group of 12

Fractions: Grades 3-5: slide 30

Applying Unitizing

4 eggs is 1/3 of a dozen since it is 1 of the 3 groups of 4

4 eggs = 1 (group of 4)

12 eggs = 3 (group of 4)

so

4 eggs / 12 eggs = 1 (group of 4) / 3 (group of 4)

= 1/3

Fractions: Grades 3-5: slide 31

Thinking about the Unit

4 eggs can be thought of as a unit which measures thirds of a dozen.

2/3 of a dozen = 2 groups of 4 eggs = 8 eggs

5/3 of a dozen = 5 groups of 4 eggs = 20 eggs

Fractions: Grades 3-5: slide 32

Usefulness of Unitizing

Skill at unitizing (that is, thinking about different units for a single set of objects) helps develop flexible thinking about “the unit” for representing fractions.

Flexible thinking is a critical skill in understanding fractions deeply and in developing a base for proportional reasoning.

Fractions: Grades 3-5: slide 33

Problem 3.1

Can you see ninths? How many cookies will you eat if you eat 4/9 of the cookies?

O O O O O O

O O O O O O

O O O O O O

Fractions: Grades 3-5: slide 34

Problem 3.2

Can you see twelfths? How many cookies will you eat if you eat 5/12 of the cookies?

O O O O O O

O O O O O O

O O O O O O

Fractions: Grades 3-5: slide 35

Problem 3.3

Can you see sixths? How many cookies will you eat if you eat 5/6 of the cookies?

O O O O O O

O O O O O O

O O O O O O

Fractions: Grades 3-5: slide 36

Problem 3.4

Can you see thirty-sixths? How many cookies will you eat if you eat 14/36 of the cookies?

O O O O O O

O O O O O O

O O O O O O

Fractions: Grades 3-5: slide 37

Problem 3.5

Can you see fourths? How many cookies will you eat if you eat 3/4 of the cookies?

O O O O O O

O O O O O O

O O O O O O

Fractions: Grades 3-5: slide 38

Reflection

Was it easy for you to think about different units for “measuring” the size of a set of objects?

How can we help students think about different units for a set?

Fractions: Grades 3-5: slide 39

Problem Set 4

The focus of Problem Set 4 is more unitizing.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.

Fractions: Grades 3-5: slide 40

Problem 4.1

16 eggs are how many dozens?

26 eggs are how many dozens?

Fractions: Grades 3-5: slide 41

Problem 4.2

You bought 32 sodas for a class party.

How many 6-packs is that?

How many 12-packs?

How many 24-packs?

Fractions: Grades 3-5: slide 42

Problem 4.3

You have 14 sticks of gum.

How many 6-packs is that?

How many 10-packs is that?

How many 18-packs is that?

Fractions: Grades 3-5: slide 43

Problem 4.4

There are 4 2/3 pies left in the pie case.

The manager decides to sell these with this plan:

Buy 1/3 of a pie and get 1/3 at no extra charge.

How many servings are there?

Fractions: Grades 3-5: slide 44

Problem 4.5

There are 5 pies left in the pie case.

The manager decides to sell these with this plan:

Buy 1/3 of a pie and get 1/3 at no extra charge.

How many servings are there?

Fractions: Grades 3-5: slide 45

Problem 4.6

Although “unitizing” is a word for adult (and not children), how might work with unitizing help children understand fractions?

Fractions: Grades 3-5: slide 46

Reflection

Would it be easy for students to think about different units for “measuring” the size of a set of objects?

How can we help them learn that?

Fractions: Grades 3-5: slide 47

Problem Set 5

The focus of Problem Set 5 is keeping track of the unit.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.

Fractions: Grades 3-5: slide 48

Problem 5.1

How do you know that 6/8 = 9/12?

Give as many justifications as you can.

Fractions: Grades 3-5: slide 49

Problem 5.2

Ten children went to a birthday party.

Six children sat at the blue table, and four children sat at the red table.

At each table, there were several cupcakes.

At each table, each child got the same amount of cake; that is they “fair shared.”

At which table did the children get more cake?

How much more?

Fractions: Grades 3-5: slide 50

Problem 5.2

Blue table: 6 childrenRed table: 4 children

(a) blue table: 12 cupcakes

red table: 12 cupcakes

(b) blue table: 12 cupcakes

red table: 8 cupcakes

Fractions: Grades 3-5: slide 51

Problem 5.2

Blue table: 6 childrenRed table: 4 children

(c) blue table: 8 cupcakes

red table: 6 cupcakes

(d) blue table: 5 cupcakes

red table: 3 cupcakes

(e) blue table: 2 cupcakes

red table: 1 cupcake

Fractions: Grades 3-5: slide 52

Problem 5.3

Would you purchase the following poster?

Why or why not?

Fractions: Grades 3-5: slide 53

Reflection

Why is it so important to keep track of the unit for fractions?

Fractions: Grades 3-5: slide 54

Problem Set 6

The focus of Problem Set 6 is in between.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.

Fractions: Grades 3-5: slide 55

Problem 6.1

Find three fractions equally spaced between 3/5 and 4/5.

Justify your solutions.

Fractions: Grades 3-5: slide 56

Problem 6.2

We know that 3.5 is halfway between 3 and 4, but is 3.5/5 halfway between 3/5 and 4/5?

Explain.

Fractions: Grades 3-5: slide 57

Problem 6.3

Find three fractions equally spaced between 1/4 and 1/3.

Justify your solutions.

Fractions: Grades 3-5: slide 58

Problem 6.4

We know that 3.5 is halfway between 3 and 4, but is 1/3.5 halfway between 1/4 and 1/3?

Explain.

Fractions: Grades 3-5: slide 59

Reflection

How do you know when fractions are equally spaced?

Is it important for students in Grades 3-5 to be able to do determine this?

Where would this idea appear in the K-8 Mathematics Standards?

Fractions: Grades 3-5: slide 60

Problem Set 7

The focus of Problem Set 7 is variations on fraction tasks.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.

Fractions: Grades 3-5: slide 61

Problem 7.1

What number, when added to 1/2, yields 5/4?

Write at least 5 different answers.

Fractions: Grades 3-5: slide 62

Problem 7.2

Write two fractions whose sum is 5/4.

Write at least 5 different answers.

Fractions: Grades 3-5: slide 63

Problem 7.3

Write two fractions, each with double-digit denominators, whose sum is 5/4.

Write at least 5 different answers.

Fractions: Grades 3-5: slide 64

Problem 7.4

Which of problems 7.1, 7.2, and 7.3 is the most “unusual”?

Why?

Fractions: Grades 3-5: slide 65

Reflection

Do your curriculum materials include “unusual” problems?

Why is it important for students to have experience with “unusual” problems?

Fractions: Grades 3-5: slide 66

Problem Set 8

The focus of Problem Set 8 is modifying fractions.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.

Fractions: Grades 3-5: slide 67

Problem 8.1

What happens to a fraction if

(a) the numerator doubles

(b) the denominator doubles

(c) both numerator and denominator double

(d) both numerator and denominator are halved

(e) numerator doubles, denominator is halved

(f) numerator is halved, denominator doubles

Fractions: Grades 3-5: slide 68

Problem 8.2

What happens to a fraction if

(a) the numerator increases

(b) the denominator increases

(c) both numerator and denominator increase

(d) both numerator and denominator decrease

(e) numerator increases, denominator decreases

(f) numerator decreases, denominator increases

Fractions: Grades 3-5: slide 69

Problem 8.3

The letters a, b, c, and d each stand for a different number selected from {3, 4, 5, 6}.

Solve these problems and justify each answer.

(a) Write the greatest sum: a/b + c/d

(b) Write the least sum: a/b + c/d

(c) Write the greatest difference: a/b - c/d

(d) Write the least difference: a/b - c/d

Fractions: Grades 3-5: slide 70

Reflection

Which of these problems could be presented to students as “mental math” problems?

Which of these problems would students need to explore over a long period of time?

Fractions: Grades 3-5: slide 71

Problem Set 9

The focus of Problem Set 9 is reflection on thinking.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.

Fractions: Grades 3-5: slide 72

Problem 9.1

Write a division story problem appropriately solved by division so that the quotient has a label different from the labels on the divisor and the dividend.

What does “divisor” mean?

What does “dividend” mean?

Fractions: Grades 3-5: slide 73

Problem 9.2

Write a story problem appropriately solved by division that demonstrates that division does not always make smaller.

Fractions: Grades 3-5: slide 74

Problem 9.3

Is a fraction a number?

Explain.

Fractions: Grades 3-5: slide 75

Problem 9.4

Why are fractions called equivalent rather than equal?

Fractions: Grades 3-5: slide 76

Reflection

What knowledge for teachers do these problems address?

Why is this important knowledge for teachers?

Fractions: Grades 3-5: slide 77

Closing Comments

Implementing the K-8 Mathematics Standards will require a deeper focus of mathematics ideas at each grade.

Personal understanding of these ideas will make the implementation process easier.

Fractions: Grades 3-5: slide 78