Multiplication and Division, Part 2

Lesson 15

Math

Unit 3

3rd Grade

Lesson 15 of 23

Objective


Use the distributive property as a strategy to multiply by units of 8 and 9, including the subtractive use of the distributive property.

Common Core Standards


Core Standards

  • 3.OA.B.5 — Apply properties of operations as strategies to multiply and divide. Students need not use formal terms for these properties. Example: Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) Example: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)
  • 3.OA.C.7 — Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
  • 3.OA.D.9 — Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Foundational Standards

  • 3.NBT.A.2
  • 3.OA.A.1
  • 3.OA.A.2

Criteria for Success


  1. Make use of structure by using the distributive property to replace the 9 in a multiplication problem with 10 – 1 (or optionally the 8 in a multiplication problem with 10 – 2), multiplying those values separately by the other factor, then subtracting those products to give the overall product (MP.7). 
  2. Look for structure in the multiplication table to see the pattern between the 1s facts, the 9s facts, and the 10s facts (and optionally the 2s facts, 8s facts, and 10s facts) and explain that pattern using the distributive property (MP.3, MP.7).

Tips for Teachers


As mentioned in Lesson 14, Lesson 15 explores the use of the distributive property to solve facts involving 8 and 9, but instead of additive uses of the distributive property (e.g., $$8\times 4 = (5\times4)+(3\times4)$$), it will explore subtractive uses of the distributive property (e.g., $$9\times6 = (10\times6)-(1\times6)$$). Because students are more likely to use the former strategy for 8s than 9s and the latter strategy for 9s than 8s, today’s lesson should focus more on 9s facts.

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Anchor Tasks


Problem 1

a.   Solve.

$$3\times 10$$

$$3\times 1$$

$$3\times 9$$

b.   What do you notice? What do you wonder?

Guiding Questions

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Problem 2

a.   Solve.

  1.  

$$(10\times 7)-(1\times 7)$$

$$(10-1)\times 7 $$

$$9\times 7$$

  1.  

$$(6\times 10)-(6\times 2)$$

$$6\times (10-2)$$

$$6\times 8$$

b.   Use the reasoning in Part (a-i) to explain how you could use $$5\times 10$$ to solve $$5\times 9$$

c.   Use the reasoning in Part (a-ii) to explain how you could use $$4\times 10$$ to solve $$4\times 8$$

Guiding Questions

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Problem 3

Use the relationship between 9 and 10 to decompose 9 in the following expression and use those values to find the following product.

$$6\times 9$$

Guiding Questions

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Problem Set


Answer Keys

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Discussion of Problem Set

  • Think about the strategy used to solve #2(c). How could a similar strategy be used to solve $$6\times 8$$ instead of $$6\times 9$$
  • How was today’s strategy illustrated on the multiplication table? 
  • In Lesson 14 and Lesson 15, we explored two different ways to solve 9s facts. How are these strategies similar? How are they different?

Target Task


Hector solves $$9\times 8$$ by finding $$(10\times8)-(1\times8)$$. Gabriella solves $$9\times 8$$ by finding $$(5\times8)+(4\times8)$$. Who is correct, Hector, Gabriella, both of them, or neither of them? Show or explain your reasoning. 

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 14

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Lesson 16

Lesson Map

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

Topic A: Introduction to The Properties of Operations

Topic B: Multiplication and Division by 6 and 7

Topic C: Multiplication and Division by 8 and 9

Topic D: Multiplication and Division by Values Greater than 10

Topic E: Two-Step Word Problems and Patterns in Arithmetic

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