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

Let’s think about 15

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

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

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

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

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

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

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

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

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

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

…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

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

Still thinking about the Area Model… Remember the ‘normal’ algorithm? 4 7 3 5 × 2 1 4 3 5 1 0 + 1 6 4 5

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

Starting with the Area Model… 4 7 3 5 × 1 4 2 1 0 3 5 + 1 6 4 5   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!

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

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.

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

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

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 = 202 + (20 x 8) + (20 x 8) + 82 = 202 + 2 x (20 x 8) + 82 = 400 + 320 + 64 = 784

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

Enrichment: The Area Model for Squaring Numbers Get it?