**High School Number Sense Lesson 58: Using FOIL Method to Multiply Two 3-Digit Numbers**

One of the biggest mistakes I've made teaching number sense was to teach FOILing early to my first class. My purpose was to give my students a "fall-back" method of multiplication, if the quicker method didn't come readily to mind. What happened to those early students was this: each time they saw a multiplication problem, they assumed it had to be FOILed. It was difficult to break them of that habit. There are

**dozens**of easier 2-by-2 digit multiplication skills, depending on the numbers involved.

Well today, we will attack FOILing 3-digit numbers. This concept appeared

**10 times**last year, with a median placement at

**question # 58**.

**Please Like our Facebook Page (**

**https://www.facebook.com/numberdojo/**

**) if you want to see new posts on your wall. I will reward you with a free concept index or flashcard file of your choice. Thank you!**

**Number Dojo Level: TBD**

The secret to solving these problems quickly is to recognize how the traditional 3-by-3 multiplication really works. Let's look at 123 x 456, represented vertically:

x 4 5 6

------------

**9 different multiplication problems**, and you're adding them all together. In other words, you're multiplying

**each digit in 123**by

**each digit in 456**, and adding the results somehow. Specifically, you're doing it in this order (starting from the bottom right):

- 6 x 3
- 6 x 2
- 6 x 1
- 5 x 3
- 5 x 2
- 5 x 1
- 4 x 3
- 4 x 2
- 4 x 1

x 4 5 6

------------

7 3 8

6 1 5

4 9 2

----------------

- calculation 1 above: (6 x 3)
- calculations 2 & 4 above: (6 x 2) + (5 x 3)
- calculations 3, 5, & 7 above: (6 x 1) + (5 x 2) + (4 x 3)
- calculations 6 & 8 above: (5 x 1) + (4 x 2)
- calculation 9 above: (4 x 1)

- | (units digits)
- X (units digits times tens digits)
- Ӝ (units by hundreds, tens by tens, hundreds by units)
- X (tens digits by hundreds digits)
- | (hundreds digits

**How to Solve:**

Now, to make it easier to see horizontally, let's look at it as

**123 x abc**. Multiply and add in this order:

- 3c
- 2c + 3b
- 1c + 2b + 3a
- 1b + 2a
- 1a

**5**, making the products and additions much easier. (You won't see a

**687 x 978**, for example).

**Example 1: 123 x 456 = ___**

- We'll start with this one just because everyone has been in suspense. Multiply 3c: 3 x 6 =
**18**. Write the**8**and regroup the**1**. - 2c + 3b: (2 x 6) + (3 x 5) = 12 + 15 (plus the regrouped 1) =
**28**. Write the**8**in front and regroup the**2**. - 1c + 2b + 3a: (1 x 6) + (2 x 5) + (3 x 4) = 6 + 10 + 12 (plus the regrouped 2) =
**30**. Write the**0**in front and regroup the**3**. - 1b + 2a: (1 x 5) + (2 x 4) = 5 + 8 (plus the regrouped 3) =
**16**. Write the**6**in front and regroup the**1**. - 1a: 1 x 4 (plus the regrouped 1) =
**5**; write this in front. The answer is**56088**.

**Example 2: 123 x 231 = ___**

- 3c: 3 x 1 =
**3**; write this down. - 2c + 3b: (2 x 1) + (3 x 3) = 2 + 9 =
**11**; write the**1**in front and regroup the**1**. - 1c + 2b + 3a: (1 x 1) + (2 x 3) + (3 x 2) = 1 + 6 + 6 (plus the regrouped 1) =
**14**; write the**4**in front and regroup the**1**. - 1b + 2a: (1 x 3) + (2 x 2) = 3 + 4 (plus the regrouped 1) =
**8**; write this in front. - 1a: 1 x 2 =
**2**; write this in front. The answer is**28413**.

**Example 3: 213 x 314 = ___**

- 3 x 4 =
**12**; write the**2**and regroup the**1**. - (1 x 4) + (3 x 1) = 4 + 3 (plus the regrouped 1) =
**8**; write this in front. - (2 x 4) + (1 x 1) + (3 x 3) = 8 + 1 + 9 =
**18**; write the**8**in front and regroup the**1**. - (2 x 1) + (1 x 3) = 2 + 3 (plus the regrouped 1) =
**6**; write this in front. - 2 x 3 =
**6**; write this in front. The answer is**66882**.

**Example 4: 315 x 224 = ___**

- 5 x 4 =
**20**; write the**0**and regroup the**2**. - (1 x 4) + (5 x 2) = 4 + 10 (plus the regrouped 2) =
**16**; write the**6**in front and regroup the**1**. - (3 x 4) + (1 x 2) + (5 x 2) = 12 + 2 + 10 (plus the regrouped 1) =
**25**; write the**5**in front and regroup the**2**. - (3 x 2) + (1 x 2) = 6 + 2 (plus the regrouped 2) =
**10**; write the**0**in front and regroup the**1**. - 3 x 2 = 6 (plus the regrouped 1) =
**7**; write this in front. The answer is**70560**.

**Example 5: 361 x 215 = ___**

- This was question #53 on the TMSCA State test last year. 1 x 5 =
**5**; write this down. - (6 x 5) + (1 x 1) = 30 + 1 =
**31**; write the**1**in front and regroup the**3**. - (3 x 5) + (6 x 1) + (1 x 2) = 15 + 6 + 2 (plus the regrouped 3) =
**26**; write the**6**in front and regroup the**2**. - (3 x 1) + (6 x 2) = 3 + 12 (plus the regrouped 2) =
**17**; write the**7**in front and regroup the**1**. - 3 x 2 = 6 (plus the regrouped 1) =
**7**; write this in front. The answer is**77615**.

**Here's a free worksheet to help you practice FOIL3by3Digit:**

foil3by3digit.pdf |

**Up Next for High School: AddSeqFibon**