Lesson 21 | Multiplication and Division, Part 2 | 3rd Grade Mathematics

Objective

Identify patterns in single or multiple rows/columns of the multiplication table.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Operations and Algebraic Thinking

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.

Operations and Algebraic Thinking

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

The foundational standards covered in this lesson

3.OA.B.5

Operations and Algebraic Thinking

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.)

Criteria for Success

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

Look for structure to identify patterns in specific rows or columns of the multiplication table (e.g., the twos row/column consists of all even numbers) and explain why those patterns exist (MP.7).
Look for structure to see that the distributive property can be observed in the multiplication table (e.g., the square corresponding with $$7\times4$$ will contain the sum of the values in the squares corresponding with $$5\times4$$ and $$2\times4$$) (MP.7).
Look for structure to see that the associative property can be observed in the multiplication table (e.g., the square corresponding with $$8\times6$$ will be double the value in the square corresponding with $$4\times6$$) (MP.7).
Make use of structure to use known products to find unknown products using the properties of operations (MP.7).

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Students may benefit from the interactive version of the multiplication table shown in Anchor Tasks #1, #2, and #3, which can be found and is explained here on Steve Wyborney's blog.
If you use the interactive version of the multiplication table for the Anchor Tasks, be sure to either provide students with a copy of this Multiplication Table or display one for the whole class for use on the Problem Set and Target Task.

Lesson Materials

Template: Multiplication Table (1 per student) — These are needed for the Problem Set and Target Task. See Tips for Teachers for more information.

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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

Choose any one row or column in the multiplication table below. What do you notice about the values in that row/column? What do you wonder?

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

Guiding Questions

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Problem 2

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

What do you notice about the three rows that are highlighted? What do you wonder?

Guiding Questions

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Problem 3

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

What do you notice about the two rows that are highlighted? What do you wonder?

Guiding Questions

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Problem 4

In each row, write down at least two multiplication facts you can figure out because you know the given multiplication fact in the left column. Be prepared to share your reasoning.

If I know...	Then I know...
$$3\times5$$	$$3\times6$$, $$6\times5$$, $$3\times10$$, $$4\times5$$
$$2\times4$$
$$4\times10$$
$$7\times 2$$
$$5\times8$$

Guiding Questions

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References

Illustrative Mathematics Patterns in the Multiplication Table — Grade 3 Unit 4 Lesson 9 Activity 2

Patterns in the Multiplication Table, accessed on Aug. 25, 2022, 12:04 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 Fishtank Learning, Inc.

Problem Set

Problem Set

Answer Keys

Unlock the answer keys for this lesson's problem set and extra practice problems to save time and support student learning.

Discussion of Problem Set

Were there any rows in the multiplication table in which you found a new pattern we haven’t yet explored? Which row and what was the pattern?
How are the nines facts related to the tens facts? How can you see this in the multiplication table? How can you use this to help solve nines facts?
How can you use your multiplication table to solve $$16\times7$$ despite the multiplication table not going up to $$16$$?
How can you create a row for $$19$$? What existing rows would you use to construct it?

Target Task

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

Problem 1

Look at the 4 and 8 rows in the multiplication table. What relationship do you see? Why does this relationship exist?

Problem 2

Explain how the multiplication table shows the following relationship.

$$7\times8=(5\times8)+(2\times8)$$

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 20

Lesson 22

Lesson Map

Topic A: Introduction to The Properties of Operations

Study commutativity to find known facts of 6, 7, 8, and 9.

3.OA.B.5 3.OA.D.9

Understand the zero and identity properties of multiplication.

3.OA.A.4 3.OA.B.5 3.OA.C.7 3.OA.D.9

Understand the role of parentheses and apply to solving problems.

3.OA.B.5 3.OA.D.8

Introduce the distributive property of multiplication.

3.OA.B.5 3.OA.D.9

Introduce the associative property of multiplication.

3.OA.B.5 3.OA.D.9

Topic B: Multiplication and Division by 6 and 7

Build fluency with multiplication and division facts using units of 6.

3.OA.A.4 3.OA.C.7

Build fluency with multiplication and division facts using units of 7.

3.OA.A.4 3.OA.C.7

Use the associative property as a strategy to multiply by units of 6.

3.OA.B.5 3.OA.C.7 3.OA.D.9

Use the distributive property as a strategy to multiply by units of 6.

3.OA.B.5 3.OA.C.7 3.OA.D.9

Use the distributive property as a strategy to multiply by units of 7.

3.OA.B.5 3.OA.C.7 3.OA.D.9

Solve one- and two-step word problems involving units up to 7.

3.OA.A.3 3.OA.D.8

Topic C: Multiplication and Division by 8 and 9

Build fluency with multiplication and division facts using units of 8 and 9.

3.OA.A.4 3.OA.C.7

Use the associative property as a strategy to multiply by units of 8 and 9.

3.OA.B.5 3.OA.C.7 3.OA.D.9

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

3.OA.B.5 3.OA.C.7 3.OA.D.9

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

3.OA.B.5 3.OA.C.7 3.OA.D.9

Solve one- and two-step word problems involving units up to 9.

3.OA.A.3 3.OA.D.8

Topic D: Multiplication and Division by Values Greater than 10

Multiply one-digit whole numbers by multiples of 10.

3.NBT.A.3 3.OA.A.4 3.OA.B.5

Multiply one-digit whole numbers by two-digit whole numbers using the associative and distributive properties.

3.NBT.A.3 3.OA.A.4 3.OA.B.5

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

Solve two-step word problems involving all four operations.

3.OA.D.8

Solve two-step word problems involving all four operations, including multiplying one-digit whole numbers by multiples of 10.

3.NBT.A.3 3.OA.D.8

Identify patterns in single or multiple rows/columns of the multiplication table.

3.OA.C.7 3.OA.D.9

Identify other patterns in the multiplication table.

3.OA.D.9

Identify arithmetic patterns and explain them using properties of operations.

3.OA.D.9

Multiplication and Division, Part 2

Objective

Common Core Standards

Core Standards

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Foundational Standards

Operations and Algebraic Thinking

Criteria for Success

Tips for Teachers

Lesson Materials

Anchor Tasks

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem 4

Guiding Questions

References

Problem Set

Discussion of Problem Set

Target Task

Problem 1

Problem 2

Student Response

Additional Practice

Extra Practice Problems

Word Problems and Fluency Activities

Lesson Map

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x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

x	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100