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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.

to support your implementation of the Mathematics
Standards.

2008 May 29         Fractions: Grades 3-5: slide 1
Introduction of Facilitators

INSERT
the names and affiliations
of the facilitators

2008 May 29            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.

2008 May 29            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.
2008 May 29       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.

2008 May 29      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.

2008 May 29       Fractions: Grades 3-5: slide 6
Problem 1.1
John told his mother that he would be home in
45 minutes.

2008 May 29       Fractions: Grades 3-5: slide 7
Problem 1.2
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.

2008 May 29       Fractions: Grades 3-5: slide 8
Problem 1.3
Mr. Albert has 3 boys to 4 girls in his history
class.

2008 May 29        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.
2008 May 29       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.

2008 May 29      Fractions: Grades 3-5: slide 11
Problem 1.6
There were 12 men and 3/4 as many women at
the meeting.

2008 May 29     Fractions: Grades 3-5: slide 12
Problem 1.7

Jack reached into his pocket and pulled out
three quarters.

2008 May 29       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.

2008 May 29      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.

2008 May 29      Fractions: Grades 3-5: slide 15
Problem 1.10
Martin’s Men Store had a big sale – 75% off.

2008 May 29      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.

2008 May 29      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.

2008 May 29       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?
2008 May 29        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.

2008 May 29       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.
2008 May 29        Fractions: Grades 3-5: slide 21
Problem 2.2
The large rectangle represents one whole that has been divided
into pieces.
A                       E

F

B    C      D
G         H

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 ___
2008 May 29              Fractions: Grades 3-5: slide 22
Problem 2.3
What is the sum of your eight fractions? What
should the sum be? Why?

2008 May 29      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.
How does the shape of the cake influence your
2008 May 29          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?

2008 May 29       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?

start

2008 May 29       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 --------->
2008 May 29              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?

2008 May 29       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.

2008 May 29       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
2008 May 29           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

2008 May 29               Fractions: Grades 3-5: slide 31
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

2008 May 29          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.
2008 May 29          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

2008 May 29     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

2008 May 29     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

2008 May 29     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

2008 May 29       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

2008 May 29     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?

2008 May 29       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.

2008 May 29      Fractions: Grades 3-5: slide 40
Problem 4.1
16 eggs are how many dozens?

26 eggs are how many dozens?

2008 May 29     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?
2008 May 29       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?
2008 May 29      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?

2008 May 29        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?

2008 May 29        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?

2008 May 29      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?

2008 May 29       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.
2008 May 29      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.

2008 May 29       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?
2008 May 29      Fractions: Grades 3-5: slide 50
Problem 5.2
Blue table: 6 children                                Red table: 4 children

(a) blue table: 12 cupcakes
red table: 12 cupcakes

(b) blue table: 12 cupcakes
red table: 8 cupcakes

2008 May 29               Fractions: Grades 3-5: slide 51
Problem 5.2
Blue table: 6 children                                Red 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
2008 May 29               Fractions: Grades 3-5: slide 52
Problem 5.3
Would you purchase the following poster?
Why or why not?

2008 May 29      Fractions: Grades 3-5: slide 53
Reflection
Why is it so important to keep track of the unit
for fractions?

2008 May 29       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.

2008 May 29      Fractions: Grades 3-5: slide 55
Problem 6.1
Find three fractions equally spaced between 3/5
and 4/5.

2008 May 29        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.

2008 May 29      Fractions: Grades 3-5: slide 57
Problem 6.3
Find three fractions equally spaced between 1/4
and 1/3.

2008 May 29        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.

2008 May 29      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?
2008 May 29        Fractions: Grades 3-5: slide 60
Problem Set 7
The focus of Problem Set 7 is variations on

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

When you are done, share your solutions with
others.
2008 May 29      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.

2008 May 29        Fractions: Grades 3-5: slide 62
Problem 7.2
Write two fractions whose sum is 5/4.

Write at least 5 different answers.

2008 May 29        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.

2008 May 29        Fractions: Grades 3-5: slide 64
Problem 7.4
Which of problems 7.1, 7.2, and 7.3 is the most
“unusual”?

Why?

2008 May 29       Fractions: Grades 3-5: slide 65
Reflection
“unusual” problems?

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

2008 May 29          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.
2008 May 29      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
2008 May 29      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
2008 May 29      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
2008 May 29       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?

2008 May 29      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.
2008 May 29       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?

2008 May 29       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.

2008 May 29       Fractions: Grades 3-5: slide 74
Problem 9.3
Is a fraction a number?

Explain.

2008 May 29       Fractions: Grades 3-5: slide 75
Problem 9.4
Why are fractions called equivalent rather than
equal?

2008 May 29       Fractions: Grades 3-5: slide 76
Reflection
What knowledge for teachers do these

Why is this important knowledge for teachers?

2008 May 29      Fractions: Grades 3-5: slide 77
Implementing the K-8 Mathematics
Standards will require a deeper focus of

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

2008 May 29       Fractions: Grades 3-5: slide 78

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