Transcript Sense Making With Fractions Through Common

Sense Making With Fractions Through
Common Core State Standards
Steele Creek Elementary – March 27, 2013
Warm Up: Connor ran in a race on
Saturday.
After completing 2/3 of the race, he had run 3/4
mile.
How long was the whole race?
Show your work……..
Fractions: Teaching and Learning
What makes fractions so hard for students?
•
taught too abstractly with limited models
•
•
•
•
Students tend not to make connections between
models and between models and symbols
taught with rote memorization of procedures
not taught in meaningful contexts
more attention to algorithms rather than to
developing number sense and reasoning
Fraction and Decimal Standards
When you look at the highlighted standards for
grades 3-5 you will notice a strong emphasis on
the following:
Visual Fraction Models
Line plot/ number lines
Equations
Symbols
CCSS Standards for Mathematical Practice
1. Make sense of problems and persevere in
solving them
6. Attend to precision
2. Reasoning
and
3. Explaining
4. Modeling with
mathematics
and
5. Using tools
7.Seeing and using
structure
and
8. Generalizing
Adapted from work of William McCallum
Fraction MODELS and REPRESENTATIONS
• Area/Region Models
• Linear or Measurement Models
• Set Models
Model
added in
4th grade
• Symbols (with meaning)
3
4
Models
introduced
in 3rd
grade
7
8
1
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
1
3
2
 5 is the unit fraction that corresponds to
or to
5
5
17
or to
5
 As there are 3 one-inches in 3 inches, there are 3
3
one-eighths in
8
Unit Fractions
 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
 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 = ?
 We can obtain any fraction by combining a
sufficient number of unit fractions
1
b
Unit Fractions
 The numerator 3 of ¾ shows that 3 is the number
you get by combining 3 of the 1/4 ’s together when
the whole is divided into 4 equal parts
 A fraction such as
5/3 shows combining 5 parts
together when the whole is divided into 3 equal
parts – best shown on a number line
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; one trapezoid and one rhombus
Fractional Parts of a Whole
 One blue rhombus = 1 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?
14
Create the whole if you know a part…
 If the blue rhombus is 2/3, build the whole.
 If the red trapezoid is 3/8, build the whole.
If you know a fractional part, can you make the whole?
Set model
If this is two fifths of a set, make the whole set.
If you know a fractional part, can you make the whole?
Make the whole line if this is one third.
Make the whole shape if this is three fourths.
c
c
c
c
c
c
Fractions Greater than One
 How much is shaded?
(¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ ) = 15/4
4/4
+
4/4
+
4/4
+ ¾
= 15/4
1
+
1
+
1
+ ¾
= 3¾
4 x 1/4 + 4 x 1/4 + 4 x 1/4 + 3 x 1/4 = 15 x 1/4
18
NUMBER LINES as Fraction Models
Understanding Number Lines
 Number lines represent the order of numbers and
their magnitude
 Numbers to the right of any given number are
greater in value; numbers to the left of any given
number are less in value
 Once two numbers are marked on the number line,
the location of all other numbers is fixed
Shaughnessy (2011)
Fractions on a Number Line
 Parallel number lines support students in
identifying equivalent fractions
Unit Fractions on a Number Line
 Fractions allow for more precise measurement of
quantities , including fractional parts greater
than 1 whole.
Close to…
• Name a fraction close to 1 but not more than 1.
• Name a fraction that is even closer to 1 than
that.
• Why do you believe it is closer?
• Name a fraction that is even closer than the
previous fraction.
• Again…
Comparing Fractions
Which is greater? Explain.
3
5
8 or 8
5
5
9 or 6
2
3
3 or 4
Which is greater
3/8 or 2/6?
Explain how you
know..
1/4 or 1/2
Can 1/4 be larger than 1/2?
How could a student justify their
answer to this question?
Another way to compare fractions
 Draw two squares of equal size.
 Partition the first one in half vertically and shade in half of the
square
 Partition the second one in thirds vertically and shade in one
third of the square.
 Now, partition the first square into thirds horizontally and the
second square in half horizontally.
Comparing ½ and 1/3
How does this model help build understanding of “finding common
denominators?”
How will this model help students add and subtract fractions with unlike
denominators?
Fraction Addition and Subtraction
Begin with informal exploration
Juan and Tiana were each eating the same kind of
candy bar. Juan had 3/4 of his bar left. Tiana still
had 2/3 of her bar left. Who had the most candy
left? How do you know?
How much candy did the two children have
together?
Using nothing other than simple drawings, how would you
solve this problem without using an algorithm and finding
common denominators?
Try to think of two different methods.
Using Estimation
 Estimate the answer to 12/13 + 7/8
A. 1
B. 2
C. 19
D. 21
• Only 24% of 13 year olds answered correctly
• Equal numbers of students chose the other
answers
NAEP
Fractions in Balance Problems
 Find the missing values.
1¾
n
1¾
n
n
x
1½
Figures that are the same size and shape must have the same value.
Adapted from Wheatley and Abshire, Developing Mathematical Fluency, Mathematical Learning, 2002
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Addition and Subtraction with Mixed Numbers
• A separate algorithm for adding and subtracting
mixed numbers is not necessary.
• Include mixed numbers in all fraction addition and
subtraction activities.
• Let students solve these problems
in ways that make sense to them. 3 ½ + 5 ¾ = ?
• Students tend to work with the
whole numbers first.
9 1/3 – 7 2/3 = ?
Fraction Computation
A problem-based number sense approach
 Begin with simple tasks in contexts.
 Connect the meaning of fraction computation with
whole-number computation.
 Develop strategies using estimation and informal
methods.
 Use models to explore each of the operations.

Adapted from Van de Walle and Lovin, Teaching Student-Centered Mathematics, 2006
Multiplying Unit Fractions
 Understand a fraction a/b as a multiple of 1/b

1
5
is the product of 5 x ( )
4
4
5
4
1
=5x 4
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
2
1
3 sets of 5 is the same as 6 sets of 5
Multiple Solution Strategies
 Solve word problems involving multiplication of a
fraction by a whole number
 At your table, solve in 2 ways…

5
If each person at a party will eat 8 of a pound
of roast beef, and there will be 5 people at the
party, how many pounds of roast beef will be
needed? Between what 2 whole numbers does
your answer lie?
Multiplication with Fractions
How would you solve?
Carolyn has 14 cookies to share with her three
friends. How many cookies will Carolyn and each
of her friends get?
 A sharing problem
 Dividing by 4 (Charlotte + three friends) is the same as
multiplication by ¼.
 Or think of the 14 cookies as the whole. How many in ¼?
 Cookies are used because they can be subdivided.
Multiplying Fractions: No Subdividing of Unit Parts
How would you model and solve?
 Rita used 1/10 of a bottle of vanilla flavoring for a cookie
recipe, leaving 9/10 of the bottle. If she then used 2/3 of
what was left in a cake recipe, how much of the whole bottle
did she use?
 Alexander used 2 ½ tubes of red paint in his picture. Each
tube holds 4/5 ounce of paint. How many ounces of red
paint did Alexander use?
Explain your reasoning.
Multiplying Fractions: Subdividing Unit Parts
How would you model and solve without an algorithm?
 Raj had 2/3 of his bedroom left to paint. After
lunch, he painted 3/4 of what was left. How
much of the whole room did Raj paint after
lunch?
 Olivia was sharing a jar of lemonade with her
sister. Olivia drank 2/5 of the jar; then her sister
drank 2/3 of what was left. How much of the jar
of lemonade did her sister drink?
The type of model can impact students’
understanding of their solution.
Modeling the Process
3
3
x
5
4
means “3/5 of a set of ¾”
Make ¾, then take 3/5 of it.
Why does extending the lines (the dotted part) help?
Consider
 How many fifths are in two wholes?
 How
would you begin to think about this
question?
 Create at least two representations to show
your solution
 What operation is represented by this
problem?
 Look for patterns in the next slides.
Connecting What We Know
 Consider 1 ÷ ½ = ?
 To determine how many of the unit fractions of
the Divisor (1/2) are in the Dividend (1), think
about it as:
 How many one-halves are in 1?
 How
is this the same as thinking of 36 ÷9 as
“How many nines are in 36?”
Words to Symbols
 Write an equation for each situation:
A
grocer has 10 pounds of coffee beans. If he
sells the beans in ½ pound bags, how many
will he have to sell?
 If
you have a spool with 6 feet of ribbon, and
you need 1 ½ foot long pieces for a craft
project , how many can you make?
Building Understanding
 How many one-sixths are in 2?
2÷⅙ = ?
 How many one-halves are in 3?
3÷½ = ?
 How many one-fifths are in 2?
2÷⅕ = ?
 What patterns do you see?
 How might these patterns help develop a
method for dividing by fractions?
More Brownies
You have 1/3 of a pan of brownies left after last
night’s party. If you and three friends share what
is left of the brownies, how much of the whole
pan of brownies will each of you get to eat?
 Write an equation to solve this problem
 Solve the problem using models and share your
method with table partners
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
Resources:
Marilyn Michue
Elementary Math Curriculum Resource
[email protected]
980-343-2792
http://www.smarterbalanced.org
http://www.illustrativemathematics.org
K-5 Math Teaching Resources
http://www.parcconline.org