# Multiplicative Thinking

# Stop Saying…

Sometimes we say things in order to help students **do** the math rather than **understand** the math. Those statements can create misconceptions and confusion when the statements no longer hold true at later grades.

# “Multiplication is repeated addition”

#### When does it fall apart?

Solve the following questions using repeated addition:

- 3 x 5
- 5 x 3
- 2 x 1.3
- 1.3 x 2
- 1.3 x 2.4

Which ones could you solve using just repeated addition? Which ones couldn’t you solve using just repeated addition?

#### Why?

Repeated addition is

- an Additive Thinking strategy
- a strategy that helps students bridge to multiplicative thinking

#### Instead say:

Repeated addition is **one** way we can solve this multiplication question.

# “The multiplication symbol means groups of”

#### When does it fall apart?

Solve the following questions by drawing groups of:

- 3 x 5
- 5 x 3
- 2 x 1.3
- 1.3 x 2
- 1.3 x 2.4

Which ones could you solve by drawing “groups of”? Which ones couldn’t you solve drawing “groups of”?

#### Why?

The multiplication symbol can be used to represent different phrases, including but not limited to:

- groups of: 3 groups of 5
- percent of: 30% of 5
- fraction of: 3/5 of 7

#### Instead say:

Drawing “groups of” is **one** way we can solve this multiplication question.

# “Multiplying by 10 is just adding a 0.”

# or “Dividing by 10 is just removing a 0.”

# or “Move the decimal when multiplying or dividing.”

#### When does it fall apart?

Solve the following questions by adding or removing a 0.

- 3 x 10
- 10 x 3
- 10 x 1.3
- 50 ÷ 10
- 4 ÷ 10
- 4.3 ÷ 10

Which ones could you solve by adding/removing a 0? Which ones couldn’t you solve?

#### Why?

*“Multiplying by 10 is just adding a 0.”*If you “add a 0” to solve 10 x 1.3 you would end up with an answer of 1.30. Secondly, “adding a 0” actually means + 0 not appending the number by 0. 1.3+0 = 1.3.

*“Dividing by 10 is just removing a 0.”*

Removing a 0 to solve 4.3 ÷ 10 would be impossible as 4.3 does not have a 0. Secondly, “removing a 0” actually means – 0. 4.3 – 0 = 4.3.

*“Move the decimal when multiplying or dividing.”*Students may struggle with the idea of moving the decimal when solving 4 ÷ 10 because 4 does not have a decimal “showing”. Most teachers will remind students to “stick one in” and then move it. Visualize a place value chart. with ones, tens and tenths. Imagine moving the decimal point to the left or to the right. Did the ones, tens, tenths, etc move? No. “Moving” the decimal point actually messes up the setup of a place value chart. If you move it to the left, your decimal would be to the left of the ones.

#### Instead say:

What’s really happening then? It is imperative that students understand the decimal does not move! Instead, you are shifting the value left or right depending on if you are multiplying or divididing.

Let’s explore a simple example: 3 x 10. Build 3 on the place value chart. Multiply 3 by 10 and place 30 ones in the ones place. Regroup into tens. You now have 3 tens in the tens place. There are 0 ones in the ones place. The answer is 30. (Don’t forget though, that there are still 30 ones altogether, even though you have regrouped.) After a few simple examples, explore the idea by looking at patterns that, when multiplying by 10, the number is “moving to the left” because it is increasing in value by 10 times. When multiplying by 10, 3 ones become 3 tens. 5 tens become 5 hundreds. This process can also be used to explore the idea of dividing by 10. 3 tens become 3 ones. 5 hundreds become 5 tens. This reinforces the multiplicative idea of place value. As you increase to the left, each place value position is 10 times more than the position to its right. As you decrease to the right, each place value position is 10 times less than the position to its left. This reinforces the multiplicative idea of place value.