Lesson 12 | Multi-Digit Division | 4th Grade Mathematics

Objective

Solve two-step word problems involving all four operations, including those involving interpreting the remainder.

Common Core Standards

Core Standards

The core standards covered in this lesson

4.NBT.B.6 — Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Number and Operations in Base Ten

4.NBT.B.6 — Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

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.

Operations and Algebraic Thinking

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

The foundational standards covered in this lesson

4.NBT.B.4

Number and Operations in Base Ten

4.NBT.B.4 — Fluently add and subtract multi-digit whole numbers using the standard algorithm.
4.NBT.B.5

Number and Operations in Base Ten

4.NBT.B.5 — Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
3.OA.D.8

Operations and Algebraic Thinking

3.OA.D.8 — Solve two-step word problems using the four operations. 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. This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Make sense of a three-act task and persevere in solving it (MP.1).
Solve two-step word problems involving all four operations (MP.4).
Represent word problems using equation(s) with a letter standing for the unknown quantity (see Tips for Teachers) (MP.2).
Interpret the remainder in the context of a problem (MP.1).
Assess the reasonableness of an answer (MP.1).

Tips for Teachers

Suggestions for teachers to help them teach this lesson

As noted in Units 1 and 2, it isn’t especially clear whether students are expected to interpret, understand, and/or write one equation that represents a two- or multi-step word problem. As the Progressions state, “some problems might easily be represented with a single equation, and others will be more sensibly represented by more than one equation or a diagram and one or more equations” (OA Progression, p. 29). Thus, it is left to the teacher to decide what “easily” and “sensibly” means for your students, and when to make the transition to representing some two- and multi-step word problems with one step of representation/one equation, which you could do here or wait for subsequent lessons and units to do so. As some general guidance, it seems reasonable to expect every student to be able to write an equation for all one-step problems, two-step problems where the unknown is isolated on one side of the equal sign in the equation without needing to manipulate it, and some multi-step problems with the same parameters by the end of the year. For example, see #2 of Mathematics Common Core Sample Questions Grade 4, #5c of MCAS Spring 2016 Grade 4 Mathematics Test, and #11 of PARCC Math Spring 2017 Grade 4 Released Items.
“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… search for the structure (MP.7) in problems with similar interpretations of remainders” (PARCC Model Content Frameworks, Mathematics, Grades 3–11).
This lesson provides an opportunity to connect two domains in the grade, 4.OA and 4.NBT, since students will be solving two-step word problems involving addition, subtraction, multiplication, and division and assessing the reasonableness of their answers using rounding, thus connecting standards 4.OA.3, 4.NBT.3, 4.NBT.4, 4.NBT.5, and 4.NBT.6.
Let students work on each problem in the Problem Set independently and circulate to see whether students are solving correctly. If not, come back together to discuss how/what to draw on a tape diagram, then allow them to try again on their own. Encourage students to use a strategy of their choice to assess the reasonableness of their answer.

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Anchor Tasks

Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

Problem 1

Act 1: Look at the following image -

a. What do you notice? What do you wonder?

b. How long will it take Joao to finish the book? Make an estimate.

Guiding Questions

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Problem 2

Act 2: Use the following information to determine how long it will take Joao to finish his book -

Diary of a Wimpy Kid is 221 pages long.
Joao has already read 15 pages.
He only reads the book during independent reading and reads nine pages during each hour-long block.

Guiding Questions

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Problem 3

Act 3: It will take Joao 23 blocks of independent reading to finish the book.

Was your answer reasonable? Why or why not?

Guiding Questions

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Problem 4

Act 4 (the sequel):

Joao loves The Diary of a Wimpy Kid and convinces Ms. Glynn to buy as many as she can for his classmates to read. Ms. Glynn has $150 left in her classroom budget to buy them. Each book costs $8. How many copies can she get?

Guiding Questions

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Problem Set

Problem Set

Answer Keys

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Discussion of Problem Set

How did you interpret the remainder in #2, 3, 5, 6, 7, and 8?
What equation did you write in #4, Part A? How did you use your equation to help you with Part B?
What equation did you write in #7? How did you use your equation to help you find the number of tables needed for all of the students? Is the solution to that equation the answer, or does it still require some interpretation?
How many tickets could Jalen have sold? Why is there more than one correct answer?
About how many empty parking spaces will the garage have in #9, Part C? Can you use that rounded answer to assess the reasonableness of your answer? Why or why not?

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Tanya collects sports trading cards. She has 451 baseball cards and 386 basketball cards. She wants to put all her trading cards into an album. Each page of the album holds 4 cards.

Write an equation or equations to find the fewest number of pages she could use in the album. Use a letter for the unknown value in your equation(s).
What is the fewest number of pages she could use in the album to hold all her trading cards?
Show or explain your work.

Student Response

An example response to the Target Task at the level of detail expected of the students.

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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

Extra Practice Problems

Answer Keys

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Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Lesson 11

Lesson 13

Lesson Map

Topic A: Understanding and Interpreting Remainders

Solve division word problems with remainders.

4.OA.A.3

Solve division word problems that require the interpretation of the remainder.

4.OA.A.3

Topic B: Division of up to Four-Digit Whole Numbers by One-Digit Whole Numbers

Divide multiples of 10, 100, and 1,000 by one-digit numbers.

4.NBT.B.6

Divide two-, three-, and four-digit numbers by one-digit numbers using a variety of mental strategies.

4.NBT.B.6

Solve two-digit dividend division problems with no remainder or a remainder in the ones place with smaller divisors and quotients.

4.NBT.B.6

Solve two-digit dividend division problems with a remainder in the tens and/or ones place with smaller divisors and quotients.

4.NBT.B.6

Solve two-digit dividend division problems with a remainder in any place with larger divisors and quotients.

4.NBT.B.6

Solve three-digit dividend division problems with a remainder in any place.

4.NBT.B.6

Solve four-digit dividend division problems with a remainder in any place.

4.NBT.B.6

Solve two-, three-, and four-digit dividend problems, including the special cases of having a 0 in the quotient or dividend, and assess the reasonableness of the quotient.

4.NBT.B.6

Topic C: Multi-Step Word Problems and Patterns

Apply the formulas for area and perimeter in real-world and mathematical problems involving all four operations.

4.MD.A.3 4.OA.A.3

Solve two-step word problems involving all four operations, including those involving interpreting the remainder.

4.NBT.B.6 4.OA.A.3

Solve multi-step word problems involving all four operations.

4.MD.A.3 4.NBT.B.6 4.OA.A.3

Identify and extend growing number patterns.

4.OA.C.5

Identify and extend growing shape patterns.

4.OA.C.5

Identify and extend repeating shape patterns.

4.OA.C.5

Multi-Digit Division

Objective

Common Core Standards

Core Standards

Number and Operations in Base Ten

Operations and Algebraic Thinking

Foundational Standards

Number and Operations in Base Ten

Number and Operations in Base Ten

Operations and Algebraic Thinking

Criteria for Success

Tips for Teachers

Anchor Tasks

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem 4

Guiding Questions

Problem Set

Discussion of Problem Set

Target Task

Student Response

Additional Practice

Extra Practice Problems

Word Problems and Fluency Activities

Lesson Map

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