Download Units counted Units to make 1

Fraction Concepts
SPRING 2014
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Week 1
PARTITIONING, ITERATING, AND COMPARING USING BAR MODELS
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Day 1: Paper Folding and Bar Models
Materials needed: Everyone should have four different colored paper strips of the same size (either 3x12in. or 2x8in.)
1. Here is a bar model that shows 0 to 1.
1
2
2. Fold one paper strip into two equal parts. This makes 2 ( units).
1
3. On the first bar model draw a dotted line to show how you folded the paper strip into units. Now, label your
2
drawing as shown.
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Day 1: Paper Folding and Bar Models (cont.)
1
• Think about the fraction 2 . A fraction is a special kind of number that describes the
space between whole numbers (e.g. 0 to 1, 1 to 2).
• The way we write a fraction is very important as each part tells us something about
the number.
• The bottom part of a fraction is called the denominator and the top part of a fraction
is called the numerator. To understand fractions it is often best to look at the
denominator (bottom part) first. Here is what the different parts of a fraction mean:
Numerator
Denominator
How many of the units I am counting in do I have?
How many units (pieces) will it take to make 1.
1
• Based on these descriptions, discuss with a partner what the fraction 2 means. Use as
many of the words above.
2
3
• Now use the words above to describe 2 ? What about 2 ?
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Units counted
Units to make 1
(unit size)
Day 1: Partner Activity Example Statements
Example Descriptions
Discuss with a partner
Based on these descriptions, what does the fraction
1
mean?
2
2
What about 2 ?
1
“It takes 2 (2 units) to make 1 and you have counted only
1
1 of these 2 units.”
1
“It takes 2 (2 units) to make 1 and you have counted 2 of
1
2
these 2 units. That means 2 is the same as 1.”
3
What about 2 ?
1
“It takes 2 (2 units) to make 1 and you have counted 3 of
1
3
these 2 units. That means 2 is more than 1.”
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Day 1: Paper Folding and Bar Models (cont.)
1.
2.
3.
4.
1
Fold another (second) paper strip into four equal parts. This will make 4 ( units).
4
Use dotted lines on the next bar model to show how you folded this new paper strip into fourths.
Label the parts of the bar model.
Now, your bar model for fourths should look like the one shown.
5. Use what you know about denominators and numerators to describe your bar model for fourths to a partner.
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Day 1: Partner Activity Example Statements
Example Descriptions
Discuss with a partner
1
What does the fraction 4 mean?
1
“It takes 4 (4 units) to make 1 and you have counted only
1
1 of these 4 units.”
3
What about 4 ?
1
“It takes 4 (4 units) to make 1 and you have counted 3 of
1
these 4 units.”
5
What about 4 ?
1
“It takes 5 (4 units) to make 1 and you have counted 5 of
1
5
these 4 units. That means 4 is more than 1.”
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Day 1: Paper Folding and Bar Models (cont.)
1
1. Fold a new (third) paper strip into eight equal parts. This will make 8 (8 units).
2. Partition (split) a bar model into eighths in the same way you modeled halves and
fourths.
3. Label the parts of the bar model.
4. Now your bar model should look the one shown.
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Day 1: Paper Folding and Bar Models (cont.)
Using your bar model drawings and folded paper strips, answer the following questions:
1
4
1.
Which unit fraction is larger than ?
2.
How many different ways can you make fractions that are the same size as
3.
How many units will be the same as 3 ( units)?
1
8
1
4
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1
2
?
Day 2:Bar Models
7. Draw a bar model and label it 0 to 1 as shown*.
1
1
a. Partition the bar model into 3 equal parts. This will make thirds and show 3 (3 units). Label each 3 unit
0
1 2
3
and show where 3 , 3, 3, and 3.
Make sure your bar model looks like this:
*If necessary, students can use paper strips and fold them into thirds prior to drawing their bar models
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Day 2: Bar Models
Using your bar model for thirds to explain your answers, discuss the following questions with a partner:
b. How many units make 1?
𝟏
“It takes 3 units to make 1. That means 3 (𝟑 units) = 1.”
c. What is the name for the size of each unit fraction?
𝟏
“Each unit fraction in the bar model is a 𝟑 unit. These are called thirds.”
d. If we shaded one unit fraction, what would be the number name of the shaded part?
𝟏
𝟏
“We would have shaded 1 (𝟑 unit) which is the number 𝟑.”
e. What would be the number name if we shaded two unit fractions?
𝟏
𝟐
“We would have shaded 2 (𝟑 units) which is the number 𝟑.”
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Day 2: Bar Models
8. Draw these two bar models to represent the number 1.
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Day 2: Bar Models
1
a. Partition one bar model into 3 (3 units).
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Day 2: Bar Models
1
b. Partition the second bar model into 6 (6 units). To do this, think about how many units
make 1 (which is the unit size) as well as how thirds might be related to sixths.
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Day 2: Bar Models
1
b. Partition the second bar model into 6 (6 units). To do this, think about how many units
make 1 (which is the unit size) as well as how thirds might be related to sixths.
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Day 2: Bar Models
c. Using your bar models, discuss all of the relationships you can see between thirds and sixths with a
partner. (Write some of them down.)
Examples: Which unit fraction is larger? What are some fractions that are the same if you made
1
1
1
them with thirds and sixths? How many sixths are the same as 3 ? How much of 3 would 6 cover?
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Day 2: Bar Models
9. Draw five different bar models that represent 1.
1 1 1
1
1
a. Use each bar model to shade 2 , 4 , 8 , 3 and 6.
10. Compare the different sizes of each unit fraction.
a. Which unit fraction is the largest?
b. Which unit fraction is the smallest?
c. Why?
d. Order the unit fractions from least to greatest.
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Day 3: Making 1
Materials needed: Blank paper (turned to a landscape orientation and a single cube for each student.)
1. Place the cube on the left side of your paper and trace around it to make a unit. Then, remove the cube.
1
a. Label the unit as 4 as shown below and be sure to show where 0 is, too.
1
4
0
1
4
1
b. If this unit is the unit fraction 4 where would 1 be? Place a mark where you think 1 is. Then use the cube to
1
iterate 4 units until you get to 1. Was your prediction correct?
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Day 3: Making 1 (cont.)
2. Use the cube to draw this unit on your paper and label it as shown.
1
8
0
1
8
1
a. If this unit is the unit fraction 8 , mark where you think the following numbers are:
2
8
4
8
1
7
8
b. Now, use the cube to find whether your predictions were correct or not. Make sure to iterate
1
the cube as a unit of and to label your model correctly.
8
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Day 3: Making 1
Now you will practice using the cube
as different unit fractions and then
find many numbers. Build models
using the cube and label it for the
following sets.
If the cube is the unit fraction….
1
5
Find where these numbers are…
1
2
5
4
5
6
5
1
4
1
2
4
3
4
5
4
1
6
1
2
6
3
6
5
6
1
2
3
4
3
1
3
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2
Day 4: Iterating Unit Fractions
Follow the example below to create numbers by iterating the given unit fraction. Make sure you include a visual (bar
model) and try to give an explanation that is similar to the example provided.
Unit
fraction
Number to make
by iterating
Visual (bar model)
Explanation
1
1
4
“I iterated the unit fraction four
4
times to make 1. That means 1 = 4
4
1 or 4
1
4
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𝑝𝑖𝑒𝑐𝑒𝑠 =
4
."
4
Day 4: Iterating Unit Fractions (cont.)
Unit
fraction
Number to make
by iterating
Visual (bar model)
Explanation
1
1
4
4
1 or
4
1
3
5
3
1
8
7
8
1
6
12
6
“I iterated the unit fraction four
4
times to make 1. That means 1 = 4
1
4
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𝑝𝑖𝑒𝑐𝑒𝑠 =
4
."
4
Day 5: Comparing Fractions
For the following pairs of numbers, use bar
models to show which is larger. Describe how
you know your bar models are accurate and
your answer is correct by using as many words
as you can from the word bank. An example is
provided.
WORD BANK
denominator
unit size
numerator
unit fraction
count
partition (partitioned)
iterate (iterated)
“I first partitioned one
bar model into thirds and
the other into halves.
𝟏
Then, I iterated 2 (𝟑
Example: Which number is larger?
2
1
or
3
2
𝟐
units) and found that 𝟑 is
𝟏
larger than one 𝟐 unit.”
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Day 5: Comparing Fractions
WORD BANK
For the following pairs of numbers, use bar
models to show which is larger. Describe how
you know your bar models are accurate and
your answer is correct by using as many words
as you can from the word bank.
denominator
unit size
numerator
unit fraction
count
partition (partitioned)
iterate (iterated)
Which number is larger?
2
4
a. or
1
3
3
6
4
5
f.
or
b.
2
4
g.
4
6
or
2
3
or
4
5
c.
3
4
h.
5
6
or
2
3
d.
or
4
5
i.
3
5
5
8
2
or 3
3
or 4
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e.
j.
3
5
7
8
3
or 4
5
or 6
Week 2
NUMBER LINES, COMPOSING AND DECOMPOSING AND
INTRODUCING ADDITION AND SUBTRACTION
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Day 6: Number Lines
1. Draw the segment as shown below. Label it to make it a number line from 0 to 1.
0
1
a. Partition the number line into two equal parts.
b. What is the name for the units you made?
c. How many of these units make 1?
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Day 6: Number Lines
1. Draw the segment as shown below. Label it to make it a number line from 0 to 1.
1
2
1
unit
2
unit
0
1
a. Partition the number line into two equal parts.
b. What is the name for the units you made?
𝟏
𝟐
unit fractions
c. How many of these units make 1?
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Day 6: Number Lines
1. Draw the segment as shown below. Label it to make it a number line from 0 to 1.
1
2
1
unit
2
unit
0
1
a. Partition the number line into two equal parts.
b. What is the name for the units you made?
𝟏
𝟐
unit fractions
𝟏
𝟐
c. How many of these units make 1? It takes 2(𝟐 units) to make 1 so 𝟐 = 𝟏
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Day 6: Number Lines (cont.)
0
1
2. Draw a new number line from 0 to 1 and label as shown above.
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Day 6: Number Lines (cont.)
0
1
0
2
1
2
1
a. Partition the number line into 2(2 units) and label it as shown.
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2
2
Day 6: Number Lines (cont.)
0
1
0
2
1
2
1
b. Use your number line to show 4(4 units)=1.
1
c. How many fourths are equivalent to ?
2
d. What are some other relationships between fourths and halves?
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2
2
Day 6: Number Lines (cont.)
0
0
2
1
1
4
1
2
0
4
3
4
2
2
2
4
4
4
1
b. Use your number line to show 4(4 units)=1.
1
𝟏
𝟏
c. How many fourths are equivalent to ?
2( units) =
2
𝟒
𝟐
d. What are some other relationships between fourths and halves?
𝟏
𝟏 𝟏
𝟒
𝟐
=
(
unit)
=
=𝟏
𝟒
𝟐 𝟐
𝟒
𝟐
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𝟐
𝟒
=
𝟏
𝟐
Day 6: Number Lines (cont.)
0
0
2
0
4
1
1
4
1
2
2
4
3
4
2
2
4
4
e. Use what you know about fractions to model eighths on your number line. Label your number
line accurately and discuss all of the relationships you see between halves, fourths and eighths
with a partner.
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Day 6: Number Lines (cont.)
0
2
0
4
00
8
1
8
1
4
2
8
3
8
1
2
5
8
2
4
4
8
3
4
6
8
7
8
2
2
4
4
81
8
e. Use what you know about fractions to model eighths on your number line. Label your number
line accurately and discuss all of the relationships you see between halves, fourths and eighths
with a partner.
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Day 6: Number Lines (cont.)
0
1
3. Draw a new number line from 0 to 1.
a. Partition the number line into thirds.
2
b. Mark the location 3
c. Use what you know about denominators, numerators and unit fractions to explain the fraction
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2
3
Day 6: Number Lines (cont.)
0
1
0
3
1
3
𝟐
2
3
3
3
𝟏
“I know that 𝟑 is composed of 2(𝟑 units).”
𝟏
“The denominator of thirds means it will take 3(𝟑 units) to make 1.”
𝟐
𝟏
“The numerator 2 in the fraction 𝟑 means that we have counted 2 units of 𝟑.”
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Day 6: Number Lines (cont.)
0
1
0
3
1
3
2
3
3
3
d. Partition your number line into sixths. To do so, think about how thirds are related to sixths as
well as what the numerator and denominator tell you.
1
2
e. How many 6 units are the same as (equivalent) to 3 ?
?
6
=
2
3
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Day 6: Number Lines (cont.)
0
1
0
3
1
6
0
6
1
3
3
6
2
3
2
6
5
6
4
6
3
3
6
6
d. Partition your number line into sixths. To do so, think about how thirds are related to sixths as
well as what the numerator and denominator tell you.
1
2
e. How many 6 units are the same as (equivalent) to 3 ?
?
6
=
2
3
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Day 7: Number Lines and Unit Fractions
0
0
4
1
1
4
2
4
3
4
2
4
4
1. Draw this number line from 0 to 2.
a. Partition the segment from 0 to 1 into fourths using the strategies you learned
yesterday.
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Day 7: Number Lines and Unit Fractions
0
0
4
1
1
4
2
4
3
4
4
4
2
5
4
6
4
7
4
8
4
1. Draw this number line from 0 to 2.
a. Partition the segment from 0 to 1 into fourths using the strategies you learned
yesterday.
b. Continue partitioning the segment from 1 to 2 into fourths.
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Day 7: Number Lines and Unit Fractions
0
0
4
1
1
4
2
4
3
4
1
2
4
4
5
4
5
6
4
7
4
1
8
4
5
c. Use the unit fraction 4 to count from 0 to 4. It should take 5(4 units) to make 4.
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Day 7: Number Lines and Unit Fractions
1
4
+
+
1
4
+
1
4
+
0
0
4
1
4
+
1
4
1
1
4
2
4
3
4
1
2
4
4
5
4
5
6
4
7
4
1
8
4
5
c. Use the unit fraction 4 to count from 0 to 4. It should take 5(4 units) to make 4.
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Day 7: Number Lines and Unit Fractions (cont.)
For each of the following numbers, draw a number line from 0 to 2 and show how to iterate unit fractions to
create the given number.
a.
6
4
e.
10
8
b.
f.
5
3
15
8
c.
6
3
g.
11
6
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d.
7
6
g.
13
12
Day 8: Number Lines and Tree Diagrams
1. Draw four number lines that are
partitioned into eighths and that also
shows all of the halves and fourths
between 0 and 1. (see example).
2. Use these number lines to show how the
5
number 8 can be decomposed into
different groupings shown in the tree
diagrams.
a.
b.
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c.
d.
Day 8: Number Lines and Tree Diagrams(cont.)
a.
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Day 8: Number Lines and Tree Diagrams (cont.)
b.
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Day 8: Number Lines and Tree Diagrams (cont.)
1
3. Explain how the distance of in the model is correct?
𝟏
𝟏
𝟒
𝟖
4
𝟏
Because = 2( units), that means moving 2( units) on the
𝟖
𝟏
number line is the same distance as moving one unit.
𝟒
c.
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Day 8: Number Lines and Tree Diagrams (cont.)
1
4. Explain how the distance of in the model is correct?
𝟏
𝟏
𝟐
𝟖
2
𝟏
Because = 4( units), that means moving 4( units) on the
d.
𝟖
𝟏
number line is the same distance as moving one unit.
𝟐
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Day 9: Number Lines, Tree Diagrams and
Equations
1. Draw four number lines that are
partitioned into sixths and thirds. (see
example)
2. Use these number lines to show how the
2
number 3 can be decomposed into
different groupings shown in the tree
diagrams.
3. Write an equation (number sentence)
that matches each tree diagram and your
1
1
2
number lines. For example, 3 + 3 = 3.
For each equation, write the numbers as
shown in the tree diagram and also as
fractions in the same unit.
a.
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b.
c.
d.
Day 9: Number Lines, Tree Diagrams and
Equations
a.
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Day 9: Number Lines, Tree Diagrams and
Equations
b.
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Day 9: Number Lines, Tree Diagrams and
Equations
c.
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Day 9: Number Lines, Tree Diagrams and
Equations
d.
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Day 10: Examining Incorrect Answers
Each of the following problems includes answers from three different students; Mark, Sasha and Israel. For
each problem, one student is correct. The other two have provided incorrect answers.
1. Use a model to determine which answer is correct.
2. Think about why the two students with incorrect answers may have thought their answers were correct.
What was the cause of their wrong answer?
5
5
4
3
2
Mark:
Sasha:
Israel:
a. +
4
8
5
4
4
2
3
b. +
1
3
Mark:
3
6
Sasha: 1
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Israel:
2
6
Day 10: Examining Incorrect Answers
Each of the following problems includes answers from three different students; Mark, Sasha and Israel. For
each problem, one student is correct and the other two have provided incorrect answers.
1. Use a model to determine which answer is correct.
2. Think about why the two students with incorrect answers may have thought their answers were correct.
What was the cause of their wrong answer?
4
8
c. +
3
8
5
6
1
6
d. +
Mark:
12
8
Mark:
6
6
Sasha:
7
16
7
Israel:
8
Sasha:
6
12
5
Israel:
12
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