# Multi-Digit Division

## Objective

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

## Common Core Standards

### Core Standards

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• 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.

## Criteria for Success

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1. Divide two-, three-, and four-digit numbers by one-digit numbers using a variety of mental strategies, including:
1. Finding partial quotients and adding them together (e.g., $276 \div 6 = (240 + 36) ÷ 6 = (240 \div 6) + (36 \div 6)$), and
2. Finding a compatible or “friendly” number and adjusting from there (e.g., $276 \div 6 = (300 – 24) \div 6 = (300 \div 6) – (24 \div 6)$).
2. Understand and explain why various mental strategies work (MP.3).
3. Understand that partial quotients can be found in more than one way (e.g., $276 \div 6 = (120 \div 6) + (120 \div 6) + (30 \div 6) + (6 \div 6)$).

## Tips for Teachers

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• As Bill McCallum notes on his blog, “the distinction between a strategy and an algorithm is that an algorithm is general, it works in all possible cases, whereas a strategy might be specialized” (Mathematical Musings, Algorithms Grades 2-5).  He goes on to say that the progression towards fluency with any computation begins with strategies, then algorithms (note the plurality), then the standard algorithm. Thus, this lesson attempts to explore strategies students may use for particular computational problems before they jump into generalizable methods and algorithms in later lessons.
• In this unit, students will not be explicitly asked to estimate quotients, since the NBT Progression states that in Grade 5, “estimating the quotients is a new aspect of dividing by a two-digit number” (NBT Progression, p. 18). However, the friendly number strategy mentioned here is essentially dependent on the skill of finding a friendly number (often called a “compatible number”) with which to estimate and adjusting from there. All of the dividends in this lesson are fairly close to the compatible number to which they are adjusted, but not always the number that the dividend would be rounded to. Thus, students will be well set up to use estimation as a strategy to assess the reasonableness of their answer, if you choose to teach students how to estimate quotients. If not, students will depend on using multiplication to check their answer to assess the reasonableness of their answer in this unit, with explicit instruction on estimating quotients coming in Grade 5 Unit 2.
• Problem Set #1 and #2 provide an array of Number Talks, similar to Anchor Task #2. You may decide to use these tasks in a similar way to Anchor Task #2, using them as a whole class and restricting the use of paper and pencil, forcing the use mental math to solve and therefore encouraging the use of the mental strategies discussed in the lesson. Or, if Anchor Tasks #1 and #2 take up much of the lesson block, you can use them as a source for problems for upcoming fluency blocks so that students can gradually develop the use of each strategy.

#### Remote Learning Guidance

If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Task 1 (benefits from discussion). Find more guidance on adapting our math curriculum for remote learning here.

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### Problem 1

Three different students solved the problem $232 \div 8$ below. Study their work and determine whether the strategy they used works.

 Student A Student B Student C

#### Guiding Questions

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#### References

Book: Mindset Mathematics: Visualizing and Investigating Big Ideas, Grade 4 by Jo Boaler, Jen Munson, and Cathy Williams (Jossey-Bass, 2017)pp. 195-202

### Problem 2

 $40\div4$ $80\div4$ $83\div4$ $50\div5$ $150\div5$ $10\div5$ $140\div5$ $30\div3$ $21\div3$ $51\div3$

#### Guiding Questions

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#### References

Book: Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5 by Sherry Parrish (Math Solutions, 2010)pp. 286-292

## Problem Set & Homework

#### Discussion of Problem Set

• How did each scaffolding problem help you solve the final problem in each part of #1?
• Is Jillian’s calculation in #2 correct? How do you know? How did you use her strategy to solve $222\div 6$
• Were there specific strategies that you found yourself relying on repeatedly in #3?
• Which strategies seem to be useful for all computations? Which ones seem to be useful just in specific cases?

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Solve. Show or explain your work.

1.   $87\div4$
2.   $294\div6$
3.   $4,256\div7$

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