Partial products is the way of multiplying that shows the work: break each number into its place values, multiply every piece by every piece, add. It is the Grade 4 method in the US (Common Core 4.NBT.5 asks for "strategies based on place value" and the area model), it is taught before the standard algorithm because it explains the standard algorithm, and it is searched by parents who learned only the column method and cannot follow the homework. The method, the area model and the comparison are below, with the printable practice.
Download the partial products practice sheet (PDF) (the method and the area model worked on page 1, a practice set in both layouts with answers on page 2; two pages, free).
The method
2-digit by 1-digit: 24 × 3. Split 24 into 20 + 4. Multiply each part by 3: - 20 × 3 = 60 - 4 × 3 = 12 Add: 60 + 12 = 72.
2-digit by 2-digit: 23 × 45. Split both: 23 = 20 + 3, 45 = 40 + 5. Every part times every part (four products): - 20 × 40 = 800 - 20 × 5 = 100 - 3 × 40 = 120 - 3 × 5 = 15 Add: 800 + 100 + 120 + 15 = 1,035.
Written in a column, each partial product on its own line, with the biggest first so the adding is tidy:
``` 23 × 45
800 (20 × 40) 100 (20 × 5) 120 (3 × 40) 15 (3 × 5)
1035 ```
The area model: the same method drawn
Draw a rectangle 23 wide and 45 tall. Cut the width into 20 and 3, the height into 40 and 5. Four smaller rectangles appear, and each one's area is a partial product: 20 × 40 = 800 at the top left, 3 × 40 = 120 at the top right, 20 × 5 = 100 at the bottom left, 3 × 5 = 15 at the bottom right. The whole rectangle's area, 1,035, is the sum. The drawing is why the method is taught: a child who can draw the box knows why 23 × 45 has four parts and why the tens-times-tens part is the big one.
More worked examples
- 36 × 7. 30 × 7 = 210; 6 × 7 = 42; 252.
- 58 × 6. 50 × 6 = 300; 8 × 6 = 48; 348.
- 31 × 24. 30 × 20 = 600; 30 × 4 = 120; 1 × 20 = 20; 1 × 4 = 4; 744.
- 68 × 29. 60 × 20 = 1,200; 60 × 9 = 540; 8 × 20 = 160; 8 × 9 = 72; 1,972. (Check by rounding: 70 × 30 = 2,100, close.)
- 124 × 6 (3-digit). 100 × 6 = 600; 20 × 6 = 120; 4 × 6 = 24; 744.
- 47 × 50. 40 × 50 = 2,000; 7 × 50 = 350; 2,350. (A number ending in 0 gives only two products.)
Partial products against the standard algorithm
| Partial products | Standard algorithm | |
|---|---|---|
| What it shows | every product by place value, visible | the same products, combined as you go with carrying |
| Lines of work for 2 × 2 digits | four products plus the sum | two lines plus the sum |
| Where errors happen | dropping zeros (writing 8 for 800) | carrying wrongly; forgetting the zero on the second line |
| When it is taught | Grade 4 (and Grade 3 for 2 × 1 digit) | Grade 5, once the place-value idea is secure |
| Best use | understanding; checking; mental maths (25 × 12 = 250 + 50) | speed with big numbers |
The standard algorithm is partial products with the adding done early: in 23 × 45 the first line, 23 × 5 = 115, is 100 + 15 combined, and the second line, 23 × 40 = 920, is 800 + 120 combined. A child who can do both and explain that they are the same has finished the topic.
The mistakes that lose marks
- Dropping the zeros: 20 × 40 is 800, not 8. Write the full numbers.
- Missing a product: 2-digit by 2-digit always has four (unless one number ends in 0).
- Multiplying tens by tens and ones by ones only (20 × 40 and 3 × 5), skipping the two cross products.
- Lining the products up wrongly when adding; put the biggest first and keep the columns straight.
- Splitting 45 as 4 + 5 instead of 40 + 5.
Where to get help
If the method is not sticking from a page, a 1-on-1 online class fixes it faster than another worksheet: one tutor with a master's degree, one child, on a shared whiteboard, in your time zone (7 am before school, after 4 pm and weekend mornings, US time; after-school slots for the UK, Canada, Australia and the UAE), for about US$10 to $13 an hour. The first 30-minute class is free: how 1-on-1 online tutoring works.
Related pages
Questions parents ask
1What are partial products?
2How do you multiply with partial products?
3Why do schools teach partial products instead of the standard method?
4What is the area model?
5What is the most common mistake with partial products?
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Shobha
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Teaching maths, science and English to children online since 2018. 500+ students, 20+ tutors, families in nine countries.