Multi-Digit Division

Lesson 1

Objective

Solve division word problems with remainders.

Common Core Standards

Core Standards

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  • 4.OA.A.3 — Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Foundational Standards

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  • 3.OA.C.7

  • 3.OA.D.8

Criteria for Success

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  1. Understand that a remainder is the number left over when one number is divided by another. 
  2. Solve division word problems within 100 that involve a remainder, using an array, an area model, or a tape diagram to represent the problem (MP.4).
  3. Check the solution to division problems using inverse operations, multiplying the quotient with the dividend and adding the remainder. 
  4. Understand that the units of the quotient and remainder are different, as the quotient represents either the number of groups or the size of the groups, but the remainder represents what remains of the whole.

Tips for Teachers

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  • Students have likely come across division situations in real life that do not result in a simple whole number quotient (especially considering that problems with 6 as a divisor, for example, will result in a whole number only one time out of six). Because of this, students may choose to interpret the remainder in the problems they encounter in this lesson. You don’t necessarily want to discourage students from doing so, but note that interpretation will happen in Lesson 2.
  • “To compute and interpret remainders in word problems (4.OA.A.3), students must reason abstractly and quantitatively (MP.2), make sense of problems (MP.1), and look for and…search for the structure (MP.7) in problems with similar interpretations of remainders” (PARCC Model Content Frameworks, Mathematics, Grade 3—11).
  • “The result of division within the system of whole numbers is frequently written as:

$$ 84 \times 10 = 8\space \mathrm{R} \space 4$$ and $$44 \div 5 = 8 \space \mathrm{R}\space 4$$

Because the two expressions on the right are the same, students should conclude that $$84\times10$$ is equal to $$44\div5$$, but this is not the case. (Because the equal sign is not used appropriately, this usage is a non-example of Standard for Mathematical Practice 6.) Moreover, the notation $$8\space \mathrm{R} \space 4$$ does not indicate a number. Rather than writing the result of division in terms of a whole-number quotient and remainder, the relationship of whole-number quotient and remainder can be written as:

$$ 84 = 8 \times 10 + 4$$ and $$44 = 8\times 5 + 4$$

(Progressions for the Common Core State Standards in Mathematics, Number and Operations in Base Ten, K-5, p. 16). Thus, you should avoid using the “R” notation immediately following the equal sign.

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Anchor Tasks

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Problem 1

There are 13 students to be split among 4 teams. How many students will be on each team?

Guiding Questions

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References

EngageNY Mathematics Grade 4 Mathematics > Module 3 > Topic E > Lesson 14Concept Development Problem 1

Grade 4 Mathematics > Module 3 > Topic E > Lesson 14 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

Kristy has 22 roses to sell. She arranges 6 roses in each vase. 

  1. How many vases will she be able to fill completely? Will there be any roses left over? 
  2. What unit should Kristy use for the quotient and remainder?
  3. How can you check your answer?

Guiding Questions

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References

EngageNY Mathematics Grade 4 Mathematics > Module 3 > Topic E > Lesson 14Concept Development Problem 3

Grade 4 Mathematics > Module 3 > Topic E > Lesson 14 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 Set & Homework

Discussion of Problem Set

  • Look at #4 and #5. These problems seem very similar, but the units used for their quotients are different. Why is that?
  • Look at #2. How many total tables would be needed to seat everyone? Why? 
  • Look at #3. How many total pies can Caleb make? Why? 

Target Task

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Fifty-three students are going on a field trip. The students are divided into groups of 6 students. How many groups of 6 students will there be? How many remaining children will there be?

References

EngageNY Mathematics Grade 4 Mathematics > Module 3 > Topic E > Lesson 14Exit Ticket

Grade 4 Mathematics > Module 3 > Topic E > Lesson 14 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.

Mastery Response

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