# Multi-Digit Multiplication

## Objective

Explore patterns in multiples of various whole numbers.

## Common Core Standards

### Core Standards

?

• 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.

• 4.OA.B.4 — Find all factor pairs for a whole number in the range 1—100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1—100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1—100 is prime or composite.

## Criteria for Success

?

1. Look for structure (MP.7) to find patterns in multiples of various factors, such as:
1. All multiples of 2 are even numbers (i.e., end in 0, 2, 4, 6, or 8).
2. All multiples of 3 have digits that add up to 3, 6, or 9.
3.  All multiples of 5 end in 0 or 5.
4. All multiples of 6 end in 0, 2, 4, 6, 8 (i.e., are even/divisible by 2) and have digits that add up to 3, 6, or 9 (i.e., are divisible by 3).
5. All multiples of 9 have digits that add up to 9 (including adding the digits of subsequent sums together, e.g., 99 → 9 + 9 = 18 and 1 + 8 = 9).
6. All multiples of 10 end in 0.
2. Make use of structure (MP.7) by using the divisibility rules stated above to determine whether a number larger than 100 is a multiple of 2, 3, 5, 6, 9, or 10.

## Tips for Teachers

?

• This lesson explores divisibility properties for the numbers 1–10. This is slightly beyond the scope of the standard, but because knowing these properties will help students to determine whether numbers are factors or multiples more quickly, it is included here. It is at your discretion to keep or cut.
• Note that the lesson, including the objective, does not contain the language of “divisible” or “divisibility.” This is because it is not language called for in the standards. You may decide to use it, though.

#### Fishtank Plus

• Problem Set
• Student Handout Editor
• Vocabulary Package

?

### Problem 1

1. Take 60 seconds to list as many multiples of 2 as you can.
2. What do you notice about the values in that list? What do you wonder?
3. Is 3,498 a multiple of 2? How do you know?

### Problem 2

1. Make a list of the first fifteen multiples of 3.
2. Which of the numbers in your list are multiples of 6? What pattern do you see in where the multiples of 6 appear in the list?
3. Which numbers in the list are multiples of 11? Can you predict when multiples of 11 will appear in the list of multiples of 3? Explain your reasoning.

#### References

Illustrative Mathematics Multiples of 3, 6, and 7

Multiples of 3, 6, and 7, accessed on Dec. 14, 2018, 2:36 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 Set & Homework

#### Discussion of Problem Set

• What do all multiples of 2 have in common?
• How can you determine if a number is a multiple of 3? What must be true about it?
• What do all multiples of 4 have in common? Is every even number a multiple of 4? What about every other?
• How can you determine if a number is a multiple of 5? What must be true about it?
• How can you determine if a number is a multiple of 6? How is this related to knowing whether a number is a multiple of 2? A multiple of 3?
• How can you determine if a number is a multiple of 9?
• What do all multiples of 10 have in common? How is this related to the work we did in Unit 1? (This question connects two domains in the grade, 4.OA.B and 4.NBT.A.)

?

### Problem 1

What do all multiples of 9 have in common?

### Problem 2

Select the list of numbers that are all multiples of 9.

A. 9, 27, 35, 63

B. 9, 48, 81, 90

C. 18, 36, 45, 64

D. 18, 54, 72, 99

?