# Addition and Subtraction of Fractions/Decimals

## Objective

Subtract fractions from fractions greater than 2 with unlike denominators.

## Common Core Standards

### Core Standards

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• 5.NF.A.1 — Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

• 5.NF.A.2 — Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

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• 4.NF.A.1

• 4.NF.A.2

• 4.NF.B.3

## Criteria for Success

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1. Find common units for fractions with unlike denominators by finding equivalent fractions using multiplication or division.
2. Understand that there is more than one possibility for the common unit used, and use that to optionally find the least common denominator.
3. Assess the reasonableness of an answer using number sense and estimation (MP.1).
4. Subtract two fractions, including mixed numbers, with unlike denominators that require regrouping whose whole is greater than 2, simplifying and writing the sum as a mixed number, if applicable.
5. Solve one-step word problems involving the subtraction of two fractions with unlike denominators whose whole is more than 2 (MP.4).

## Tips for Teachers

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• “Calculations with mixed numbers provide opportunities for students to compare approaches and justify steps in their computations (MP.3)” (NF Progression, p. 13). In general, given the Grade 4 instruction on this content, it’s unlikely that students will rewrite mixed numbers as “improper” fractions and subtract but instead will regroup just one whole and subtract (which is “an analogue of what students learned when…subtracting numbers...: decomposing a unit of the minuend into small used… Instead of decomposing a ten into 10 ones…, a one [is] decomposed into” fractional units, such as 3 thirds) (NF Progression, p. 13). For the problems that require regrouping, you should at least go through the strategies of (1) regrouping a whole to subtract (e.g., $9\frac{1}{12}-\frac{7}{12}=8\frac{13}{12}-\frac{7}{12}=8\frac{6}{12}$) and (2) subtracting the wholes, then regrouping to subtract (e.g., $14\frac{7}{18}-12\frac{13}{18}=2\frac{7}{18}-\frac{13}{18}=1\frac{23}{18}-\frac{13}{18}=1\frac{12}{18}$) since these are universal strategies, whereas going down over a whole and converting to an unknown-addend problem and using an addition strategy are more computation-specific. The computation-specific possibilities are listed below.
• For some problems in this lesson, students may use a computation-specific strategy. For example, students might think of a computation as an unknown-addend problem and use an addition strategy to solve (including the simplifying strategy of making a whole, e.g., to solve $2\frac{1}{5}-1\frac{1}{2}=2\frac{2}{10}-1\frac{5}{10}$, a student might add $1\frac{5}{10}+\frac{5}{10}=2$ and $2+\frac{2}{10}=2\frac{2}{10}$, so the difference is $\frac{5}{10}+\frac{2}{10}=\frac{7}{10}$). They may also subtract like units, but then use the simplifying strategy of going down over a whole, e.g. $2\frac{1}{5}-1\frac{1}{2}=2\frac{2}{10}-1\frac{5}{10}=1\frac{2}{10}-\frac{5}{10}=1\frac{2}{10}-\frac{2}{10}-\frac{3}{10}=1-\frac{3}{10}=\frac{7}{10}$.

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### Problem 1

1. Estimate the following differences.

a.   ${2{1\over2}-1{1\over5}}$

b.   ${2{1\over5}-1{1\over2}}$

1. Solve for the actual differences in #1 above. Are your answers reasonable? Why or why not?

#### Guiding Questions

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#### References

EngageNY Mathematics Grade 5 Mathematics > Module 3 > Topic C > Lesson 12Concept Development

Grade 5 Mathematics > Module 3 > Topic C > Lesson 12 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by The Match Foundation, Inc.

### Problem 2

1. Estimate the following differences. Determine whether the actual difference will be more or less than the estimated difference.

a.   ${6{1\over2}-5{2\over3}}$

b.   ${7{3\over4}-2{6\over7}}$

1. Solve for the actual differences in #1 above. Are your answers reasonable? Why or why not?

#### Guiding Questions

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### Problem 3

1. Estimate the following differences. Determine whether the actual differences will be more or less than the estimated differences.

a.  ${5{1\over3}-2{5\over6}}$

b.  ${8{1\over6}-3{3\over4}}$

1. Solve for the actual differences in #1 above. Are your answers reasonable? Why or why not?

#### Guiding Questions

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## Discussion of Problem Set

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• Look at #5. What fractions did you come up with that had a difference of ${2{1\over5}}$?
• Look at #7. What is the difference of your two fractions? Was anyone able to come up with a smaller difference? What if you used fractions greater than 1 for the fractional part of each mixed number? Why do you think it is that we don’t usually write numbers in that way?

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### Problem 1

Solve. Show or explain your work.

${5{1\over2}-1{1\over3}}$

#### References

EngageNY Mathematics Grade 5 Mathematics > Module 3 > Topic C > Lesson 12Exit Ticket, Question #1

Grade 5 Mathematics > Module 3 > Topic C > Lesson 12 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

### Problem 2

Solve. Show or explain your work.

The length of Eagle Trail is $6\frac{5}{6}$ miles. The length of Bear Trail is $2\frac{7}{8}$ miles. What is the difference in length between Eagle Trail and Bear Trail?

#### References

Massachusetts Department of Elementary and Secondary Education Spring 2016 Grade 5 Mathematics TestQuestion #3

Spring 2016 Grade 5 Mathematics Test is made available by the Massachusetts Department of Elementary and Secondary Education. © 2017 Commonwealth of Massachusetts. Accessed Dec. 5, 2017, 3:53 p.m..

Modified by The Match Foundation, Inc.

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