Welcome to content professional development sessions for Grades 35. The focus is Fractions.

1
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.

2
Introduction of Facilitators

• INSERT
• the names and affiliations
• of the facilitators

3
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.

4
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.

5
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.

6
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.

7
Problem 1.1

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

8
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.

9
Problem 1.3

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

10
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.

11
Problem 1.5

• Baluka Bubble Gum comes four pieces to a package.
  
• Three children each chewed a piece from one
  package.

12
Problem 1.6

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

13
Problem 1.7

• Mary asked Jack how much money he had.
• Jack reached into his pocket and pulled out three
  quarters.

14
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.

15
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.

16
Problem 1.10

• Martins Men Store had a big sale 75 off.

17
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.

18
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.

19
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?

20
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.

21
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.

22
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 ___

23
Problem 2.3

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

24
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?

25
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?

26
Problem 2.6

• Ralph is out walking his dog.
• He walks 2/3 of the way around this circular
  fountain.
• Where does he stop?

27
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 ---------gt

28
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?

29
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.

30
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

31
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

32
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

33
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.

34
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

35
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

36
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

37
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

38
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

39
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?

40
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.

41
Problem 4.1

• 16 eggs are how many dozens?
• 26 eggs are how many dozens?

42
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?

43
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?

44
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?

45
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?

46
Problem 4.6

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

47
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?

48
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.

49
Problem 5.1

• How do you know that 6/8 9/12?
• Give as many justifications as you can.

50
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?

51
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

52
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

53
Problem 5.3

• Would you purchase the following poster?
• Why or why not?

54
Reflection

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

55
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.

56
Problem 6.1

• Find three fractions equally spaced between 3/5
  and 4/5.
• Justify your solutions.

57
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.

58
Problem 6.3

• Find three fractions equally spaced between 1/4
  and 1/3.
• Justify your solutions.

59
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.

60
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?

61
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.

62
Problem 7.1

• What number, when added to 1/2, yields 5/4?
• Write at least 5 different answers.

63
Problem 7.2

• Write two fractions whose sum is 5/4.
• Write at least 5 different answers.

64
Problem 7.3

• Write two fractions, each with double-digit
  denominators, whose sum is 5/4.
• Write at least 5 different answers.

65
Problem 7.4

• Which of problems 7.1, 7.2, and 7.3 is the most
  unusual?
• Why?

66
Reflection

• Do your curriculum materials include unusual
  problems?
• Why is it important for students to have
  experience with unusual problems?

67
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.

68
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

69
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

70
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

71
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?

72
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.

73
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?

74
Problem 9.2

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

75
Problem 9.3

• Is a fraction a number?
• Explain.

76
Problem 9.4

• Why are fractions called equivalent rather than
  equal?

77
Reflection

• What knowledge for teachers do these problems
  address?
• Why is this important knowledge for teachers?

78
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.