Teaching 2-digit by 2-digit multiplication and teaching 3-digit by 2-digit multiplication
Concrete: students use centimeter tiles or other objects to make simple arrays. Next, they make 2-digit by 2-digit arrays (if there are enough materials!). Sometimes we use base-10 blocks.
Pictorial: students use graph paper to draw simple arrays. Then students use graph paper to make 2-digit by 2-digit arrays. Next, students use blank paper and draw the arrays by breaking down the 2-digit numbers.
Abstract: students use the Distributive Property to multiply. (Also called the “partial products” multiplication strategy.) *Note: students do not have to master step 9. Mastering step 8 still means achieving the fourth-grade multiplication benchmark.)
Repeat the cycle for 3-digit by 2-digit multiplication.
Making arrays with tiles (concrete)
Simple array: 5 x 6
Students can count each tile separately to find the product, count by 5s, or count by 6s. 5 x 6 = 30
Multi-digit array: 12 x 13
Students break up the multi-digit array into easier numbers:
Now there are four arrays:
10 x 10 = 100
10 x 2 = 20
3 x 10 = 30
3 x 2 = 6
Students add the four arrays together to find the final product: 100 + 20 + 30 + 6 = 156
Therefore, 12 x 13 = 156.
Making arrays with graph paper (pictorial)
12 x 13 = ?
The 12 breaks into easier numbers: 10 and 2
The 13 breaks into easier numbers: 10 and 3
Next, students find the area of each rectangle
Finally, students add all 4 numbers.
100 + 20 + 30 + 6 = 156
So 12 x 13 = 156
The Partial Products multiplication strategy (The Distributive Property - abstract)
12 x 13 = ?
10 x 10 = 100 (tens place x tens place)
10 x 3 = 30 (tens place x ones place)
2 x 10 = 20 (ones place x tens place)
2 x 3 = 6 (ones place x ones place)
100 + 30 + 20 + 6 = 156, so 12 x 13 = 156