1. Thinking Multiplicatively: From Arrays to the ‘Area Model’
(and beyond!)

2. Let’s think about 15

3. Arrays and Multiplication
3 x 5 = 15
“15 is still 15…”
It’s 3 x 5, OR…
It’s 5 x 3.

4. Arrays and Multiplication15
15 = (3 x 3) + (2 x 3)
15 = (4 x 3) + (1 x 3)

5. Arrays and Factors
The factors of 15 are:
3 and 5
1 and 15

6. Arrays and Factors The factors of 24 are: 1 and 24 2 and 12 3 and 8

7. Moving from Arrays to Area Grids3 5 3 5
It’s still 15.

8. …but what about Larger Numbers?
17 x 8?
17 (= ) 8
10 x 8 = 80
7 x 8 = 56
= 136
So, 17 x 8 = 136

9. Area Grids for Larger Numbers
14 x 23? 23 = 14
10 x 20
10 x 3 = 10 + 4
4 x 20
4 x 3

10. Area Grids for Larger Numbers
14 x 23? 23
10 x 20 = 200
10 x 3 = 30 14
4 x 20 = 80
4 x 3 = 12
322
So, 14 x 23 = 322

11. Do we really need the ‘Grid’ part?
34 x 46? 30 6 40 4 23
30 x 40
30 x 6
4 x 40
4 x 6

12. Do we really need the ‘Grid’ part?
34 x 46? 30 6 40 4
30 x 40
30 x 6
4 x 40
4 x 6

13. …Now it’s the ‘Area Model’!
34 x 46? 30 6 40 4 23
30 x 40
30 x 6
4 x 40
4 x 6

14. The boxes DON’T even have to be ‘to scale’!23 40 6
1 200
180
160 24 30
1200
30 x 40
30 x 6
4 x 40
4 x 6
180 4
160 24
1 564
So, 34 x 46 = 1 564

15. Still thinking about the Area Model…
Remember the ‘normal’ algorithm? ×

16. 4 7 3 5 × 1 4 2 1 0 3 5 + 1 6 4 5 What’s different here?×
Which way? Why? What works best for students?
WHY WOULD WE DO THIS!?!??

17. Starting with the Area Model…× 40
+ 7 30
1200
210
+ 5
200 35
Which way? Why? What works best for students?
CHALLENGE: Try the ‘flipped’ version of the ‘normal’ algorithm yourself!

18. Enrichment: The Japanese Method
Hundreds
Tens
Ones 2 3 8 12
23 x 14
2 hundreds + 11 tens + 12 ones
= 3 hundreds + 2 tens + 2 ones = 322

19. Enrichment: Square Numbers
There’s a reason ‘square numbers’ are called ‘squares’!
FUN FACT:
Square numbers can only ever end in the digits
0, 1, 4, 6, 9
or in ’25’ (Why?)
A question for students.

20. Enrichment: Square Roots
A question for students.
Not that kind of ‘Square Root’!

21. Enrichment: Square Roots
Take the square number ’36’ …
What is the ‘root’ of the square?
A question for students.
The square root of 36 is 6

22. Enrichment: The Area Model for Squaring Numbers
What is 282 ? (28 x 28) 20 8
400 64
160
(20 + 8)2
20 + 8
(20 + 8)2 = (20 x 8) + (20 x 8) + 82
= x (20 x 8) + 82 =
= 784

23. Enrichment: The Area Model for Squaring Numbers
What if we represented 20 as ‘a’ and 8 as ‘b’?
(a+b)2
a + b a b a2 b2 ba ab
(a + b)2 = a2 + 2ab + b2

24. Enrichment: The Area Model for Squaring Numbers
Get it?