Fraction
Mania
Lane ESD
February 12, 2016
Common Difficulties
 Next whole numbers are obvious…not so with fractions
 Multiplication makes amounts bigger…except with fractions
 Division makes amounts smaller…except with fractions
 Viewing the numerator and denominator as separate entities,
rather than viewing the fraction as a single number
 Difficulties grasping the “less than 1” idea; then difficulties
extending on a number line past one.
 Not recognizing the role of the whole
From Uncomplicating Fractions by Marian Small (2015)
What do we expect
students to learn?
Kindergarten Focus
 Number bonds, part-part-whole mats
3
5
2
3
2
5
part
part
whole
 Making whole of 10 (“Ten Friends”)
 8 and ?
 4 and ?
 How does this relate to fractions later on?
1
First Grade Focus
 Kindergarten ideas, plus…
 Whole, halves, fourths, quarters
 No symbolic notation, just words
is smaller than
Second Grade Focus
 Halves, fourths, quarters and thirds
 Equal shares don’t have to have the same shape, but must have
identical wholes
 Fractions come in pairs (see “Ten Friends”)
one
fourth
one
third
two
thirds
three
fourths
Fraction Standards
ALL fraction clusters in Grades 3 through 7 fall into
the Priority Clusters!
Look at your grade’s fraction standards.
What are the big ideas that students must master at your
grade level?
Be prepared to contribute to a bullet point list.
2
How will we teach the
fraction standards so
students can learn?
Five Big Ideas
for Understanding
1: THE WHOLE STORY
2: COHERENCE WITH UNITS
3: EQUIVALENT FRACTIONS
4: COMPARING AND ORDERING
5: FRACTION OPERATIONS
Big Idea:
The Whole Story
What fraction is represented by the
shaded area?
1: THE WHOLE STORY
3
Using Pattern Blocks
If
= 1, what is
?
If
= 1, what is
?
If
= 1, what is
?
If
= 1, what is
?
If
= 1, what is
?
If
= 1, what is
?
If
= 1, what is
?
1: THE WHOLE STORY
Tantalizing Tangrams
(p. 26)
If the large square is 1 square unit, what
fraction does each piece represent?
1: THE WHOLE STORY
Pizza Task: Grade 4
A pizza store makes two sizes of pizza shaped as rectangles.
The large pizza is twice as long and twice as wide as the
small pizza.
3
3
Fadia ate 4 of a small pizza and Lori ate 16 of a large pizza.
3
3
Fadia says she ate more pizza because 4 is greater than 16 .
Is she correct? Explain your reasoning.
1: THE WHOLE STORY
4
Big Idea:
Coherence with Units
3 cats + 5 cats = 8 cats
3 pennies + 5 pennies = 8 pennies
3 tens + 5 tens = 8 tens
3 inches + 5 inches = 8 inches
1
1
4
4
1
3 ( inches) + 5 ( inches) = 8 ( inches)
3
4
4
5
8
+4=4
2: COHERENCE WITH UNITS
Wholes and Then Some…
What amount does
the shaded part
show?
1
Can students “see” the denominator within the whole, and
count from there?
2: COHERENCE WITH UNITS
Book Buddies
Central Middle School has a book buddies
program where sixth graders read to
kindergarten students. Each sixth grader
who volunteers is paired with one
kindergarten student who wants a book
2
3
buddy. If 3 of the sixth graders and 5 of
the kindergarten students have book
buddies, what fraction of the combined
kindergarten and sixth grade classes have
book buddies?
2: COHERENCE WITH UNITS
5
Big Idea:
Equivalent Fractions
 Try to avoid thinking
procedurally…just use the strips
 Organize the strips
 Work with partners to determine
all the equivalent fractions that
you can.
 Pick a set of equivalent fractions
and use student language to
prove why they are equal.
3: EQUIVALENT FRACTIONS
Equivalent Work
2
4
Use linking cubes or two-color counters to show why 3 = 6
3
Create all the equivalent fractions you can for where the
5
denominator is less than 20.
3: EQUIVALENT FRACTIONS
Equivalence on the Number Line
 Select and organize any three rows of your fraction tiles. How is
each row like a number line?
 What would the numbers be on either end of the line?
 How would the marks in between be labeled?
 What would the number line look like if you included all three
rows you selected? Draw a quick sketch.
eighths
sixths
fifths
Every fraction has a place in relation to the same whole.
3: EQUIVALENT FRACTIONS
6
Equivalent A-HAs

Equivalent is another name for the same amount, just like:
 20 = 2 tens
or 1 ten and 10 ones
or 4 groups of 5
 Say “simplest form” instead of “reducing” to reinforce the
idea that it’s just another way to say that fraction, not really
smaller.
 Equivalent fractions on the multiplication chart – can you
find them?
3: EQUIVALENT FRACTIONS
Multiplication Chart
3: EQUIVALENT FRACTIONS
Big Idea:
Comparing & Ordering
Use only your tiles to find which is larger:
5
5
or
8
6
3
7
or
4
8
4: COMPARING AND ORDERING
7
Comparing Fractions
 Wholes must be the same
 Math Practice #6 – Attend to
Precision. Encourage students to
find ways other than freehand
drawing to visually compare
fractions.
 Have a drawing lesson
 Teach using benchmarks such as halves
and thirds.
 If it’s close, find another way
 Outlaw circles except for halves and
fourths
4: COMPARING AND ORDERING
Comparing Fractions
 Grade 3:
 Compare with same numerator or same denominator
 Grade 4:
 Compare with different numerators and different denominators by
creating a common numerator or common denominator
1
2
 Compare to benchmark fractions, such as .
 Grade 5:
 Use comparison skills to assess the reasonableness of answers
 Recognize that
2 1 3
3 1
+ = is incorrect because <
5 2 7
7 2
4: COMPARING AND ORDERING
Big Idea:
Fraction Operations
Believe it or not, both of these awesome bands make a contribution to this section.
5: FRACTION OPERATIONS
8
Adding and Subtracting
Use the fraction tiles to find…
a)
3
8
+
7
8
(4.NF.3.a)
b)
5
8
−
1
4
(5.NF.1)
5: FRACTION OPERATIONS
Back to the Unit

3 9 12
+ =
𝟓 𝟓 𝟓
because 3 grapes + 9 grapes = 12 grapes.
 Units must be consistent – you can’t add grapes and bananas.
 If they don’t match, make a smoothie!
 Relate to measurement. You wouldn’t add inches and feet,
right? Or pounds and ounces. One of them has to change.
 Interesting math fact: Adding the numerators and denominators
of two fractions always results in a fraction between the two
original ones. Try it.
5: FRACTION OPERATIONS
9
Rock Around the Fraction Clock
1, 2, 3 o’clock, 4 o’clock rock
5, 6, 7 o’clock, 8 o’clock rock
9, 10, 11 o’clock, 12 o’clock rock
We’re gonna rock around the fraction clock
We gotta take two fractions, add ‘em up
If the D’s the same then we’re in luck
We’re gonna rock around the fraction clock
We’re gonna add some knowledge to our block
We’re gonna rock, rock, rock around the fraction clock
If the D’s are different, then we gotta find
A common D by skip-counting or times
(Repeat chorus)
5: FRACTION OPERATIONS
More than an Algorithm
More than an algorithm…more than an algorithm to meeeeee
5: FRACTION OPERATIONS
Using Models to Add & Subtract
 Fraction Tiles - requires guess and check
 Pattern Blocks - with denominators of 2, 3, and 6
 Grid Model - any denominator within reason:
2
5
1
11
3
15
+ =
Try it:
1
2
+
5: FRACTION OPERATIONS
3
8
Try it:
1
4
+
4
5
Try it:
3 1
-
5 4
10
Using Models to Add & Subtract
Try it:
3 1
7
5 4
20
- =
5: FRACTION OPERATIONS
Improper Fractions
 Changing an improper fraction to a mixed number is considered
simplifying.
 Introduce the idea in 4th grade, work to mastery in 5th grade.
 More than an algorithm – can students see how to convert?
8
5
5
5
3
5
5: FRACTION OPERATIONS
Find the Operation
For each problem, find the answer using the fraction tiles and write an equation
that matches the numbers in the problem.
1
4
1.
How many tiles are in 2 wholes?
2.
How many tiles does it take to make 4 wholes?
3.
Set out two tiles. What tiles make 4 equal parts of what you set out?
4.
What tiles make of a tile?
5.
What tiles make of a tile?
1
5
1
3
1
3
1
2
1
2
1
5
5: FRACTION OPERATIONS
11
Find the Operation
3
The answer is 10. What is the question?
1
The answer is 10. What is the question?
For bonus nerd points (not a 5th grade requirement!)
 The answer is 8. What is the question?
  The answer is 2. What is the question?
5: FRACTION OPERATIONS
Multiplying Fractions
 Math Practice #7 – Look for and make use of structure.
 Commutative Property
3
8
3
8
 If students know a x b = b x a, then 4 x = x 4
3
8
 Some problems make more sense if you ask “What is of 4?”
3
8
 Some problems make more sense if you ask “What is 4 groups of ?”
Write a story situation for each.
3
3
What is 8 of 4?
What is 4 copies of 8?
5: FRACTION OPERATIONS
Brownie Units
Mrs. Logan went to the school bake sale to buy
some brownies. All the pans of brownies were
square. A pan of brownies cost $12. Customers
could buy any fractional part of a pan and pay that
fraction of $12 (For example, ½ of a pan costs ½
of $12.).
Mrs. Logan bought ¾ of a pan that was ⅖ full.
How much did she pay?
2: COHERENCE WITH UNITS
Without doing the math yet,
what steps do you need to
take to solve this problem?
Source: Doing What Works, 2012
12
Find the Unit: Area Model
2
Draw 5 horizontal sections; shade 2 of them ( ).
5
2: COHERENCE WITH UNITS
Find the Unit: Area Model
3 2
3
Draw 4 vertical sections; shade 3 of them ( ). The overlap shows of .
4
4 5
2: COHERENCE WITH UNITS
What’s the Unit?
3 2
6
So of is 6 out of 20 pieces, or .
4 5
20
If the whole pan is $12, how much will six
pieces cost?
20 pieces = $12.00
10 pieces = $6.00
1 piece = $0.60
6 pieces x $0.60= $3.60
𝟑
𝟒
𝟐
𝟔
x 𝟓 = 𝟐𝟎
𝟔
𝟐𝟎
x $12 =
𝟔
𝟐𝟎
x
𝟏𝟐
𝟏
=
𝟕𝟐
𝟐𝟎
𝟏𝟐
𝟔
= 𝟑 𝟐𝟎 =3𝟏𝟎 = 𝟑. 𝟔
2: COHERENCE WITH UNITS
13
Reflect on the Big
Of the FIVE BIG IDEAS, which is
one that you feel your students
need more time to understand
and practice?
Ideas
1: THE WHOLE STORY
2: COHERENCE WITH UNITS
What are some steps you will take
so they have that opportunity?
3: EQUIVALENT FRACTIONS
4: COMPARING AND ORDERING
Which of the FIVE BIG IDEAS
helped you see something you
hadn’t seen before?
5: FRACTION OPERATIONS
Your Turn!
 Find the value of each expression using as many different
methods as you can (include models, algorithms,
conceptual explanations, etc).
 Start with one model for each.
 “Give one-Get one” -- compare work with someone else.
Share the first unique item you each have. Continue.
R.A.F.T.
 Role, Audience, Format, Topic
 Low-prep way to differentiate
 Can be used on a “choice day” or as alternative homework
 “If you have completed 5 Hardest, choose one item from the RAFT”
 Broadens kids’ responsibilities for quality work
Role
Audience
Format
Topic
Fraction
One Whole
Letter
Our relationship.
1/6
5/6
Valentine
You complete me.
Instructions
If you’re going to compare us, at
least do it fairly.
2/5 and 3/4 4th Graders
14
Two-Minute Speed Write
What do I want to GET from today?
15
2
4
3
6
Use linking cubes or two-color counters to show why = . Show your
work here.
3
Create all the equivalent fractions you can for where the
5
denominator is less than 20.
Quick sketch of fraction number line with three denominators:
16
Multiplication Chart
Use fraction tiles to solve. Draw a picture of how you solved.
3
8
5
8
+
7
8
−
1
4
17
Grid Model for Adding and Subtracting (Use 1” Grid Paper)
1
3
Try it: 2 + 8
1
4
Try it: 4 + 5
3
1
Try it: 5 − 4
Answer: ________
Notes:
Answer: ________
Notes:
Answer: ________
Notes:
18
Find the operation.
For each problem, find the answer using the fraction tiles and write an equation
that matches the numbers in the problem.
𝟏
1. How many 𝟒 tiles are in 2 wholes?
Answer: ______________
Equation: _________________________________
𝟏
2. How many 𝟓 tiles does it take to make 4 wholes?
Answer: ______________
Equation: _________________________________
𝟏
3. Set out two 𝟑 tiles. What tiles make 4 equal parts of what you set out?
Answer: ______________
𝟏
Equation: _________________________________
𝟏
4. What tiles make 𝟑 of a 𝟐 tile?
Answer: ______________
𝟏
Equation: _________________________________
𝟏
5. What tiles make 𝟐 of a 𝟓 tile?
Answer: ______________
Equation: _________________________________
6. Write your own: ______________________________________________________
Answer: ______________
Equation: _________________________________
19
What’s the Story?
Write a story situation for each expression.
3
What is 8 of 4?
3
What is 4 copies of 8?
20
1: THE WHOLE STORY
2: COHERENCE WITH UNITS
Reflection on the
3: EQUIVALENT FRACTIONS
FIVE BIG IDEAS
4: COMPARING AND ORDERING
5: FRACTION OPERATIONS
Of the FIVE BIG IDEAS, which is one that you feel your students need more time
to understand and practice?
What are some steps you will take so they have that opportunity?
Which of the FIVE BIG IDEAS helped you see something you hadn’t seen before?
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Name ______________________________
Fractions and Decimals R.A.F.T.
Challenge Options
As you work, keep the following in mind:





Include an accurate mathematical explanation
Provide an example or solution within your piece
Keep the format consistent
Final work should be neat and easy to read and understand
Bonus points for creativity, presentation and effort.
Date
Role
Audience
Format
Topic
Tenths &
Hundredths
4th graders
Instructions
How to read me right!
Fraction
One whole
Letter
Our relationship
1/3
4/5
Love Letter
There’s only one way we
can be together
Decimal Point
Decimal Point
Warning
Straighten up or we’re
through adding!
Cartoonist
4th graders
Comic strip
Fractions and Decimals:
The Secret Twins of Math
1/6
5/6
Valentine
You complete me
Mixed Number
Improper
Fraction
Instructions
How you can be like me.
Disney
Job application
for one of Snow
White’s dwarves
I’m smaller than all those
others…1/2, 1/3, 1/4,
1/5, etc.
Speech (written)
If you’re going to
compare us, at least do it
fairly.
1/10
3/4 and 2/5
th
4 graders
KEEP THIS PAGE IN CASE YOU NEED IT AGAIN DURING THIS UNIT!
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