Multiplication and Division, Part 1

Lesson 5

Objective

Relate multiplication and division and understand that division can represent situations of unknown group size or an unknown number of groups.

Materials and Resources

Common Core Standards

Core Standards

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  • 3.OA.A.1 — Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

  • 3.OA.A.2 — Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

  • 3.OA.A.3 — Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

  • 3.OA.B.6 — Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Criteria for Success

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  1. Represent unknown factor problems, including number of groups unknown and group size unknown problems, with division (MP.2). 
  2. Understand that the dividend is the number being divided, the divisor is the number it’s being divided by, and the quotient is the solution in a division equation. 
  3. Understand that the dividend always represents the total, but the divisor and the quotient can either represent the number of groups or the size of each group, depending on the situation.
  4. Understand the relationship between an unknown factor equation and its equivalent division equation (e.g., $$4 \times$$ ___ $$=20$$  corresponds with $$20 \div 4 = $$ ___). 

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Anchor Tasks

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Problem 1

Presley has 10 markers. Her teacher gives her 2 boxes and asks her to put an equal number of markers in each box. 

Anthony has 10 markers. His teacher wants him to put 2 markers in each box until he is out of markers. 

a. Before you figure out what the students should do, write a multiplication sentence to correspond with each context above. 

b. Solve each problem for the missing factor.

Guiding Questions

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References

Illustrative Mathematics Markers in Boxes

Markers in Boxes, accessed on Oct. 10, 2018, 2:32 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by The Match Foundation, Inc.

Problem 2

Suppose that there are 4 tanks and 3 fish in each tank, as shown in the picture below. The total number of fish in this situation can be expressed as 4 x 3 = 12.

a. Describe what is meant in this situation by 12 $$\div$$ 3 = 4. 

b. Describe what is meant in this situation by 12 $$\div$$ 4 = 3. 

Guiding Questions

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References

Illustrative Mathematics Fish Tanks

Fish Tanks, accessed on Oct. 10, 2018, 2:35 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 3

Write a multiplication equation and a division equation to represent each of the following situations. 

a.  Ross has 15 flowers that he wants to make into flower arrangements. Each flower arrangement will use 5 flowers. How many flower arrangements can he make? 

b.  Heidi has 8 apps that she wants to place into rows of 4. How many apps will there be in each row? 

Guiding Questions

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Discussion of Problem Set

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  • What multiplication sentence did you write for #4 Part (a)? What division sentence did you write for Part (b)? If you used an equation to find the answer in Part (c), which one was more helpful for you? If you didn’t use an equation to solve, what strategy did you use? 
  • In #5, explain why all of the answer choices you did not select are incorrect.  
  • In #6, what division equation did you write? What did the dividend, divisor, and quotient represent? Is there another correct division equation you could have written? How would the meaning of the dividend, divisor, and quotient change in that different equation? 
  • What model did you draw for #8? Could you have drawn equal groups? Since the blank represented the number of groups, what did you have to make sure was true about your equal groups model? Could you have drawn an array? Since the blank represented the number of groups, what did you have to make sure was true about your array model?
  • What is the relationship between the quotient in division and the unknown factor in a related multiplication equation?

Target Task

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Cesar says that he can solve the equation $$35 \div 5$$ by thinking about the multiplication problem $$5\times $$ ___$$=35$$. Do you agree or disagree? Explain. 

Mastery Response

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