P4 Multiplication and Division Algorithms

Large calculations become manageable when every digit stays in its place. Keep the exchange or partial product visible so another reader can follow your method.

Parents: let your child predict the size of the answer, then reveal one written step at a time in the lab.

By the end you will be able to multiply up to four digits by one digit, multiply three digits by two digits with aligned partial products, and divide four digits by one digit using long division.

Four digits by one digit

Multiply from the ones place. If a product has two digits, write its ones digit in that column and exchange its tens to the next column. In 2,408 × 3, the zero tens place is still part of the algorithm.

🔬

Written multiplication: 2,408 × 3

🤔 Predict first: What is 2,408 × 3?

7,2247,0246,224

Multiply 2,408 by 3

$$\begin{array}{r r r r r}\text{carry}&1&&2&\\&2&4&0&8\\\times&&&&3\\\hline&7&2&2&4\\\end{array}$$

$$\text{3}\times\text{8}=24\text{: write 4, carry 2}$$

$$\text{3}\times\text{0}+2=2$$

$$\text{3}\times\text{4}=12\text{: write 2, carry 1}$$

$$\text{3}\times\text{2}+1=7$$

$$2,408\times3=7,224$$

Three digits by two digits

Split the two-digit multiplier into ones and tens. Multiply 236 by 4 to get 944, then multiply 236 by 20 to get 4,720. The second partial product starts one place to the left because it is tens.

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Written multiplication: 236 × 24

🤔 Predict first: Which aligned partial-product total is correct for 236 × 24?

5,6645,4406,564

Add aligned partial products

$$\begin{array}{r r r r r}&&2&3&6\\\times&&&2&4\\\hline&&9&4&4\\&4&7&2&0\\\hline&5&6&6&4\\\end{array}$$

$$236\times4=944$$

$$236\times20=4,720$$

$$944+4,720=5,664$$

Four digits by one digit division

Long division is a sequence: divide the current part, subtract its product, then bring down the next digit. If the current part is too small, its remainder joins the next place. In 8,576 ÷ 4, the 5 leaves one remainder, which joins 7 to make 17.

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Long division: 8,576 ÷ 4

🤔 Predict first: What is 8,576 ÷ 4?

2,1442,1142,214

quotient2144

48576

8

bring down05

4

bring down17

16

bring down16

16

remainder0

8576 ÷ 4 = 2144

Check: 4 × 2144 = 8576

Practise with worked steps

Practice 1. Calculate 2,408 × 3. Show every exchange, including the zero tens place.

Show the complete multiplication solution

Four-digit by one-digit

$$\text{3}\times\text{8}=24\Rightarrow\text{write 4, carry 2}$$

$$\text{3}\times\text{0}+2=2$$

$$\text{3}\times\text{4}=12\Rightarrow\text{write 2, carry 1}$$

$$\text{3}\times\text{2}+1=7$$

$$2,408\times3=7,224$$

Practice 2. Calculate 236 × 24 with two labelled partial products.

Show the aligned partial-product solution

Three-digit by two-digit

$$\begin{array}{r r r r r}&&2&3&6\\\times&&&2&4\\\hline&&9&4&4\\&4&7&2&0\\\hline&5&6&6&4\\\end{array}$$

$$236\times4=944$$

$$236\times20=4,720$$

$$944+4,720=5,664$$

Practice 3. Calculate 8,576 ÷ 4 and check the answer.

Show the sequential division solution

quotient2144

48576

8

bring down05

4

bring down17

16

bring down16

16

remainder0

8576 ÷ 4 = 2144

Check: 4 × 2144 = 8576

Watch out — easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. What is 2,408 × 3?

   7,2247,0246,224

2. What is 236 × 24?

   5,6645,4406,564

3. What is 8,576 ÷ 4?

   2,1442,1142,214

4. Which is the correct first partial product for 312 × 23?

   6,2409367,176

5. A division gives quotient 1,206 and remainder 2 when dividing by 7. What dividend does it represent?

   8,4448,4408,449

6. Which written method keeps 236 × 20 aligned?

   Write 4,720 below 944 with both ones columns alignedWrite 472 below 944 with no zero placeholderWrite 4,720 above 236

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