Addition and Subtraction of Fractions/Decimals

Lesson 9

Math

Unit 4

5th Grade

Lesson 9 of 15

Objective


Subtract fractions from fractions greater than 2 with unlike denominators.

Common Core Standards


Core Standards

  • 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.

Foundational Standards

  • 4.NF.A.1
  • 4.NF.A.2
  • 4.NF.B.3

Criteria for Success


  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. 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.
  4. Assess the reasonableness of an answer using number sense and estimation (MP.1).
  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


  • “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. Students may also use computation-specific strategies, which 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 mental 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 mental 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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Anchor Tasks


Problem 1

a.   Estimate which two whole numbers each of the following differences will be in between.

  1. $${2{1\over2}-1{1\over5}}$$
  2. $${2{1\over5}-1{1\over2}}$$

b.   Solve for the actual differences in Part (a) above.

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 Fishtank Learning, Inc.

Problem 2

a.   Estimate the following differences. Determine whether the actual difference will be more or less than the estimated difference. Be prepared to explain your reasoning.

  1. $${6{1\over5}-5{11\over12}}$$
  2. $${5{3\over10}-2{1\over2}}$$
  3. $$8\frac{7}{9}-3\frac{11}{12}$$

b.   Solve for the actual differences in Part (a) above. Are your answers reasonable? Why or why not?

Guiding Questions

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Problem 3

Anton and Emmy are competing in the long jump. Anton has a long jump of $$6\frac{1}{4}$$ feet. This is $$1\frac{5}{6}$$ feet longer than Emmy’s long jump. How far, in feet, is Emmy’s long jump?

Guiding Questions

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Problem Set


Answer Keys

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Discussion of Problem Set

  • 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?

Target Task


Problem 1

Solve. Show or explain your work.

$${5{1\over2}-3{1\over7}}$$

Problem 2

It takes Joseph $$5\tfrac{5}{6}$$ hours to finish his book. It takes Annette $$3\tfrac{7}{8}$$ hours to finish her book. What is the difference in time between how long it took Joseph and Annette to finish their books?

Student Response

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Additional Practice


The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

Answer Keys

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Word Problems and Fluency Activities

Word Problems and Fluency Activities

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Lesson 8

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Lesson 10

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Addition and Subtraction of Fractions

Topic B: Addition and Subtraction of Decimals

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