GRADE 5 • UNIT 3
Table of Contents
Addition and Subtraction of Fractions
Lessons
Topic 1: Equivalent Fractions
1-2
Lesson 1: Make equivalent fractions with the number line, the area model, and numbers.
Lesson 2: Make equivalent fractions with sums of fractions with like denominators.
Topic 2: Making Like Units Pictorially
3-7
Lesson 3: Add fractions with unlike units using the strategy of creating equivalent fractions.
Lesson 4: Add fractions with sums between 1 and 2.
Lesson 5: Subtract fractions with unlike units using the strategy of creating equivalent fractions.
Lesson 6: Subtract fractions from numbers between 1 and 2.
Lesson 7: Solve two-step word problems.
Topic 3: Making Like Units Numerically
8-12
Lesson 8: Add fractions to and subtract fractions from whole numbers using equivalence and the
number line as strategies.
Lesson 9: Add fractions making like units numerically.
Lesson 10: Add fractions with sums greater than 2.
Lesson 11: Subtract fractions making like units numerically.
Lesson 12: Subtract fractions greater than or equal to one.
Topic 4: Further Applications
13-16
Lesson 13: Use fraction benchmark numbers to assess reasonableness of addition and
subtraction equations.
Lesson 14: Strategize to solve multi-term problems.
Lesson 15: Solve multi-step word problems; assess reasonableness of solutions using benchmark
numbers.
Lesson 16: Explore part to whole relationships.
Vocabulary
Review Familiar Terms and Symbols1
> - greater than
< - less than
= - equal to
Between (e.g., 1/2 is between 1/3 and 3/5, 3 is between 2 and 5)
Denominator – the number on the bottom (names the fractional unit: fifths in 3 fifths, which is
abbreviated to the 5 in 3/5)
Equivalent fraction – fractions that name the same amount or part using different units (e.g., 3/5 = 6/10)
Equivalent fractions represent the same amount of area of a rectangle and the same point on the
number line.
2 2 × 4
8
=
=
3 3 × 4 12
Fraction – a number that names a part of a whole or part of a group (e.g., 3 fifths or 3/5)
Fraction greater than or equal to 1 - (e.g., 7/2 = 3 1/2, an abbreviation for 3 + 1/2)
Fraction written in the largest possible unit (e.g., 3/6 = 1 x 3 / 2 x 3 = 1/2 or 1 three out of 2 threes = ½)
Fractional unit (e.g., the fifth unit in 3 fifths denoted by the denominator 5 in 3/5)
Hundredth - (1/100 or 0.01)
One tenth of (e.g., 1/10  250)
Number sentence – (e.g., word form: “Three plus seven equals ten.” or standard form: “3 + 7 = 10.”)
Numerator – the number on the top (names the count of fractional units: 3 in 3 fifths or 3 in 3/5)
Tenth - (1/10 or 0.1)
Whole unit (e.g., any unit that is partitioned into smaller, equally sized fractional pieces – called “units”)
New Terms for 5th Grade Unit 3
Benchmark fraction (e.g., 1/2 is a benchmark fraction when comparing 1/3 and 3/5)
Like denominators - the numbers on the bottom of the fractions, the units, are the same (e.g., 1/8 and
5/8)
Rectangular fractional model
Partition – to make equal parts
Unlike denominators – the numbers on the bottom of the fractions, the units, are not the same (e.g., 1/8
and 1/7)
1
These are terms and symbols students have used or seen previously.
Lesson by Lesson Suggestions
Lesson 1: Make equivalent fractions using a number line, area model, and numbers
Equivalent fractions represent the same amount of area of a rectangle and the same point on the
number line.
For this lesson students fold a strip of paper in half.
The paper is used to create a number line labeling the
beginning as 0-halves (0/2), the middle as 1-half (1/2)
and the end as 2-halves (2/2); which equals 1 whole.
Then, rectangular models are used to display 1-half
and equivalent fractions. The shaded portion of the
model always shows the fraction we are representing.
In the example we have 1/2 = 2/4 = 3/6 = 4/8.
This is done by partitioning or breaking the pieces into
smaller equal pieces (fourths, sixths, and eights) using
horizontal lines.
Below are examples of thirds and fourths.
In this example 2/3 of the
original square is shaded
(2 columns). Then the 3
columns are partitioned
horizontally to create
ninths showing 6-ninths
shaded.
To show fractions greater than a whole the
number line will continue past a whole, in this
example 4-fourths (4/4) represents a whole
and the number lines continues to include 5-fourths.
Using a model two rectangles are needed to show more than a
whole. The first model is completely shaded to show the whole or
4-fourths, and the second model shows 1 additional fourth
shaded.
4/4 + 1/4 = 5/4 or 5-fourths
The fourths are then partitioned into eighths to make an
equivalent fraction. 5/4 = 10/8
Lesson 2: Make equivalent fractions with sums of fractions with like denominators
Adding fractions with the same denominator.
Example 1: Addition expressions can be shown on a number line.
1
3
+
1
3
=
2
“ 1 third + 1 third = 2 thirds”
3
This can also be written as a multiplication
2 x
sentence:
1
3
=
2
3
Example 2: Sums can be made using parts.
3
8
3
1
7
8
8
8
+ + =
or
3
1
7
( 2 x 8) + 8 = 8
“ 3 eighths + 3 eighths + 1 eighth = 7 eighths”
“2 copies of 3 eighths + 1 eighth = 7 eighths”
6 2 2 2
= + + =1+1+1=3
2 2 2 2
Example 3: Fractions greater than 1 whole can be expressed as the sum of a whole and a fraction.
Multiple wholes can also be used.
Lessons 3 - 7: Add and subtract fractions with unlike units (denominators)
Students will replace different fractional units with an equivalent fraction or like unit.
Note: Models will be used for lessons 3-6, therefore the least common denominator will not be used until
future lessons when models are no longer needed and students are able to add and subtract fractions
using only numbers.
Students use the familiar rectangular fraction model to add and subtract fractions with unlike
denominators.
First, draw a rectangle and partition it with vertical lines as you would a tape diagram, representing the
first fraction with a bracket and shading. Then partition a second equally sized rectangle with horizontal
lines to show the second fraction. Next, partition both rectangles with matching lines to create like
(equivalent) units.
} 1/3
}
1/2
This strategy pictorially proves 3 sixths are equal to 1 half and 2 sixths are equal to 1 third. Students
practice making these models extensively until they internalize the process of making like units. Students
use the same systematic drawing for addition as they do for subtraction. In this manner, students are
prepared to generalize with understanding to multiply the numerator and denominator by the same
number. The topic closes with lesson 7 devoted to solving two-step word problems involving addition and
subtraction of fractions.
Examples:
Lesson 6: Subtract fractions from numbers between 1 and 2 wholes.
*To be successful with lesson 6 students must understand how to take a fraction away from a
whole. If this skill is not mastered please practice this fluency drill until mastery is reached.
Say a subtraction expression. Have the student say the answer.
Example “1 – 1 half.”
1 – 1 third
1 – 2 thirds
1 – 2 fifths
1 – 4 fifths
Answer:
Answer:
Answer:
Answer:
Answer:
1 half
2 thirds
1 third
3 fifths
1 fifth
Once students are fluent with taking a fraction away from a whole you can use either of
the following methods when subtracting a fraction from a number greater than a whole.
Examples:
In Method 1 the half is taken from the whole first and then the third that was part of the original number
is added on. 1 – 1 half = 1 half or 3 sixths + 2 sixths = 5 sixths.
In Method 2 the half is taken from the total of the whole and the part. 8 sixths - 3 sixths = 5 sixths.
In each method we use models to show like units and get the same final answer. Students are
encouraged to use the method they are most comfortable with.
Additional Example:
Lessons 8 - 12: Fraction Addition and Subtraction using numbers
These lessons also use the number line when adding and subtracting fractions greater than or equal to 1
so that students begin to see and manipulate fractions in relation to larger whole numbers and to each
other. The number line takes fractions into the larger set of whole numbers.
Examples:
Numbers
Models
Number Line
Notice the wholes are added first then the fraction added last.
1 2
1×3
2×4
3
8
11
+ =(
)+(
)=
+
=
4 3
4×3
3×4
12 12 12
Start on the number line at one whole
and jump back to ¾, this shows that we
subtracted ¼. It may be helpful to think
about 1 whole as 4/4. 4/4 – ¼ = ¾.
Students will also be asked: “Between what two whole numbers will the sum of 1 3/4 and 5 3/5 lie?”
3
3
___ < 14 + 5 5 < ___
This leads to understanding of and skill with solving more interesting problems, often embedded within
multi-step word problems: In this example the numbers are added and subtracted using a numerical
strategy. Additional examples of this strategy are shown below.
Cristina and Matt’s goal is to collect a total of 3 ½ gallons of sap from the maple trees. Cristina collected 1
¾ gallons. Matt collected 5 3/5 gallons. By how much did they beat their goal?
1
3
3
1
3×5
3×4
1 × 10
gal + 5 gal − 3 gal = 3 + (
)+(
)−(
)
4
5
2
4×5
5×4
2 × 10
= 3+
15 12 10
17
+
−
= 3
gal
20 20 20
20
Cristina and Matt beat their goal by 3
17
/20 gallons.
Word problems are part of every lesson. Students are encouraged to draw bar diagrams, which allow
analysis of the same part–whole relationships. You will notice in the word problem example above a
numerical expression using equivalent fractions is used. Examples of this are shown on the next page.
Examples of adding and subtracting numerically:
ADDITION
Each fraction is re-written using a common
“unit” or denominator. Then, the numerators
are added and the final sum is simplified.
Students will NOT always be asked to simplify.
*Method 2 uses the least common denominator of
30 instead of the product of the denominators (60).
Some students may start to realize you can use a
smaller number as the denominator than the
product of the two denominators. This is fine and
should only be done when the student is
comfortable with using the least common
denominator.
SUBTRACTION
You can solve it as
2 fifths plus 3/4 .
First take the 3/5
from 1 to get 2
fifths and add the
3 fourths.
You can subtract the
fractional units, and then add
back the whole number.
This works because 3 fifths is
less than 3 fourths, so you
change only the fractional
units to twentieths.
The whole number
can be represented
as 4 fourths and
added to 3 fourths
to equal 7 fourths.
Then, subtract using
equivalent fractions.
Lessons 13 - 16: Further Application of Fraction Addition and Subtraction
In Topic D, students strategize to solve multi-term problems and more intensely assess the
reasonableness both of their solutions to word problems and their answers to fraction
equations.
In lesson 13 students are estimating the value of expressions involving
sums and differences with fractions. “Will your sum be less than or
greater than one half? One? How do you know?” This will help students
build reasoning while using fractions.
In lesson 14, students look for number relationships before calculating, for example, to use the
associative property or part−whole understanding. Looking for relationships allows them to see
shortcuts and connections that are so often bypassed in the rush to get the answer.
In lesson 15, students solve multi-step word problems and actively assess the reasonableness of
their answers. In Lesson 16, they explore part−whole relationships while solving a challenging
problem: “One half of Nell’s money is equal to 2 thirds of Jennifer’s.” This lesson challenges the
underlying assumption of all fraction arithmetic—that when adding and subtracting, fractions
are always defined in relationship to the same whole amount.
Students will be expected to write explanations to prove their understanding. From the word
problem example provided previously in this guide this is an example of a complete explanation:
Cristina and Matt’s goal is to collect a total of 3 ½ gallons of sap from the maple trees. Cristina collected 1
¾ gallons. Matt collected 5 3/5 gallons. By how much did they beat their goal?
1
3
3
1
3×5
3×4
1 × 10
gal + 5 gal − 3 gal = 3 + (
)+(
)−(
)
4
5
2
4×5
5×4
2 × 10
= 3+
15 12 10
17
+
−
= 3
gal
20 20 20
20
Cristina and Matt beat their goal by 3
17
/20 gallons.
“I know my answer makes sense because the total amount of sap they collected is going to be
about 7 and a half gallons. Then, when we subtract 3 gallons, that is about 4 and a half. Then,
1 half less than that is about 4. 3 17/20 is just a little less than 4.”
Recommended Resources
IXL skills covered in this unit:
L.1 Fractions review
L.2 Equivalent fractions
L.3 Reduce fractions to lowest terms
L.4 Convert between improper fractions and mixed numbers (review of 4th grade skill)
L.5 Least common denominator (use if students are struggling with reducing/simplifying fractions)
L.6 Graph and compare fractions on number lines
L.7 Compare fractions using benchmarks
L.8 Compare fractions and mixed numbers
L.9 Put fractions in order
M.1 Decompose fractions multiple ways
M.2 Add and subtract fractions with like denominators using number lines
M.3 Add and subtract fractions with like denominators
M.4 Add and subtract fractions with like denominators: word problems
M.5 Add and subtract mixed numbers with like denominators
M.6 Add fractions with unlike denominators using models
M.7 Add up to 4 fractions with denominators of 10 and 100
M.8 Add fractions with unlike denominators
M.9 Subtract fractions with unlike denominators using models
M.10 Subtract fractions with unlike denominators
M.11 Add and subtract fractions with unlike denominators: word problems
M.12 Add 3 or more fractions with unlike denominators
M.13 Add 3 or more fractions: word problems
M.14 Compare sums and differences of unit fractions
M.15 Complete addition and subtraction sentences with fractions
M.16 Inequalities with addition and subtraction of fractions
M.17 Estimate sums and differences of mixed numbers
M.18 Add mixed numbers with unlike denominators
M.19 Subtract mixed numbers with unlike denominators
M.20 Add and subtract mixed numbers: word problems
M.21 Add and subtract fractions in recipes
M.22 Complete addition and subtraction sentences with mixed numbers
M.23 Inequalities with addition and subtraction of mixed numbers
**This site has a video providing guidance for every homework page.**

http://www.oakdale.k12.ca.us/ENY_Hmwk_Intro_Math
(Click on 5th Grade – Select the Module – Select the lesson)

Fractions Number Line & Area Models:
 http://vimeo.com/71599250
(Modeling equivalent fractions, adding fractions with unlike units/denominators.)

Add & Subtract Fractions with unlike units/denominators

https://www.youtube.com/watch?v=pmJHyJ0zpw4

https://www.youtube.com/watch?v=WrvDWD9HvOs (addition only)

https://www.youtube.com/watch?v=YjEwB3dqe1A (subtraction only)