Week 11 · Area model and partial products · 3 lessons
3
From partial products to the short method
Discover Mathematics P4: Term 1, Unit 7, Lessons 1–5
GoalWrite partial products in columns and then use the short method with regrouping.
New words
partial products · the small products we find by multiplying one place at a timestandard algorithm · the short written method that multiplies from the ones place and carriesregroup · trade ten of one place for one of the next place, also called carrying
Explanation
The boxes of an area model can be written under each other as partial products. To find 4 × 362, multiply each digit by 4 and keep its place value: 4 × 2 = 8, 4 × 60 = 240 and 4 × 300 = 1,200. Add them: 8 + 240 + 1,200 = 1,448. The standard algorithm does the same in a shorter way. Start at the ones: 4 × 2 = 8, write 8. Then 4 × 6 tens = 24 tens: write 4 in the tens place and carry 2 hundreds. Then 4 × 3 hundreds = 12 hundreds, and 12 + 2 = 14 hundreds. Write 14 and the product is 1,448. The carried number is part of the same partial products, so both methods give the same answer. Estimate first: 4 × 360 is about 1,440.
Step
Work
Result
4 × 2
4 × 2 ones
8
4 × 60
4 × 6 tens
240
4 × 300
4 × 3 hundreds
1,200
Add
8 + 240 + 1,200
1,448
Worked examples
Partial product
Value
4 × 2
8
4 × 60
240
4 × 300
1,200
Sum
1,448
Find 4 × 362 with partial products. 4 × 2 = 8; 4 × 60 = 240; 4 × 300 = 1,200. The sum is 1,448. Short method: write 8, then 24 tens gives 4 and carry 2, then 12 + 2 = 14. Same answer: 1,448.
6 × 1,254 = 7,524
Place
Multiply
Write / carry
ones
6 × 4 = 24
4 / 2
tens
6 × 5 + 2 = 32
2 / 3
hundreds
6 × 2 + 3 = 15
5 / 1
thousands
6 × 1 + 1 = 7
7
Find 6 × 1,254 with the short method. 6 × 4 = 24: write 4, carry 2. 6 × 5 = 30, plus 2 is 32: write 2, carry 3. 6 × 2 = 12, plus 3 is 15: write 5, carry 1. 6 × 1 = 6, plus 1 is 7. The product is 7,524. Check with partial products: 24 + 300 + 1,200 + 6,000 = 7,524.
Estimate to check. Menna finds 3 × 498 = 1,594. Round 498 to 500: 3 × 500 = 1,500. The exact answer should be a little less than 1,500, so 1,594 is wrong. The correct product is 3 × 498 = 1,494.
Class activity
Short or long? Choose a 3-digit number and a 1-digit number. A grown-up finds the product with partial products and the child with the short method. Compare. If they differ, find the step where one of you made a mistake.
Practice
1
Find 5 × 236 with the short method.
5 × 6 = 30: write 0, carry 3. 5 × 3 = 15, plus 3 is 18: write 8, carry 1. 5 × 2 = 10, plus 1 is 11. Product 1,180.
