B. Number · Operations: Numbers to 10 000, Decimal Tenths, Multiplying and DividingMultiplying 2- and 3-Digit Numbers by 1-Digit Numbers
B2.5 B2.1About 45 minutes
Learning goals- I can estimate a product.
- I can use an array or an area model to multiply.
- I can multiply a 2- or 3-digit number by a 1-digit number using partial products.
- I can solve multiplication story problems.
Words to know
area model
A rectangle split into parts to show multiplication. Each part is a smaller multiplication.
partial product
The answer to one part of a bigger multiplication. For 6 × 47, the partial products are 6 × 40 = 240 and 6 × 7 = 42.
expanded form
A number written as the sum of its place values, like 254 = 200 + 50 + 4.
estimate
A close guess, found by rounding to friendly numbers.
Let's start with a story
A Grade 4 class is buying juice boxes. Each case has 24 juice boxes. They buy 4 cases. How many juice boxes is that?
That's 4 × 24. It's not one of our facts, but we can split 24 into 20 + 4 and use facts we know.
Use the splitting rule
Split the bigger number by place value. Multiply each part. Then add the parts.
4 × 24 = 4 × 20 + 4 × 4 = 80 + 16 = 96.
The pieces 80 and 16 are called partial products.
The area model
Drawing every dot takes a long time. An area model is a quicker picture. Draw a rectangle. The short side is the 1-digit number. Split the long side by place value. Each part of the rectangle is one partial product.
Worked exampleExample 1: An area model
- Estimate: 47 is about 50. 6 × 50 = 300.
- Split 47 into 40 + 7.
- 6 × 40 = 240 (6 × 4 tens = 24 tens).
- 6 × 7 = 42.
- Add the partial products: 240 + 42 = 282.
6 × 47 = 282. Close to 300. ✓
An area model for 6 × 47. The long side is split into 40 and 7. 240 + 42 = 282.Worked exampleExample 2: A 3-digit number
- Estimate: 254 is about 250. 3 × 250 = 750.
- Split 254 into 200 + 50 + 4.
- 3 × 200 = 600.
- 3 × 50 = 150.
- 3 × 4 = 12.
- Add: 600 + 150 + 12 = 762.
3 × 254 = 762. Close to 750. ✓
An area model for 3 × 254. The parts are 600, 150 and 12. 600 + 150 + 12 = 762.Writing partial products in a list
You can also write the partial products in a column and add them:
- 3 × 200 = 600
- 3 × 50 = 150
- 3 × 4 = 12
- Total: 762
Always estimate first so you can tell if your answer makes sense.
Worked exampleExample 3: A story problem
A school buys 8 boxes of pencils. Each box has 125 pencils. How many pencils?
- Estimate: 8 × 100 = 800, so the answer is more than 800.
- Split 125 into 100 + 20 + 5.
- 8 × 100 = 800, 8 × 20 = 160, 8 × 5 = 40.
- Add: 800 + 160 + 40 = 1000.
Zero in the middle
For 3 × 408, split 408 into 400 + 8 (there are 0 tens). 3 × 400 = 1200 and 3 × 8 = 24. Total: 1224.
Use your facts
If you're not sure of a fact like 6 × 7, use your strategies from the facts lesson. The big multiplication is just a few small facts put together.
Just for funAn octopus has 8 arms. So 23 octopuses have 23 × 8 = 184 arms. That's a lot of high-fives!
Hands-on activityArea models on grid paper
You need: grid paper (or draw a grid with a ruler), coloured pencils.
- Draw a rectangle 4 squares tall and 13 squares long. Count all the squares, or count them by rows: that's 4 × 13.
- Draw a line after the 10th column. Colour the 4 × 10 part one colour and the 4 × 3 part another colour.
- Write the partial products: 4 × 10 = 40 and 4 × 3 = 12. Add: 52. Does it match your count?
- Now draw an area model (without the grid) for 5 × 36 and for 4 × 123. Solve them.
Answers to check: 5 × 36 = 150 + 30 = 180. 4 × 123 = 400 + 80 + 12 = 492.
Your turn
63. 3 × 20 = 60 and 3 × 1 = 3. 60 + 3 = 63.
128. 4 × 30 = 120 and 4 × 2 = 8. 120 + 8 = 128.
230. 5 × 40 = 200 and 5 × 6 = 30. 200 + 30 = 230.
266. 7 × 30 = 210 and 7 × 8 = 56. 210 + 56 = 266.
628. 2 × 300 = 600, 2 × 10 = 20, 2 × 4 = 8. 600 + 20 + 8 = 628.
608. 4 × 100 = 400, 4 × 50 = 200, 4 × 2 = 8. 400 + 200 + 8 = 608.
7A store has 6 boxes of 24 crayons. How many crayons?
144 crayons. 6 × 20 = 120 and 6 × 4 = 24. 120 + 24 = 144.
More practice: warm-up
36. 3 × 10 = 30 and 3 × 2 = 6.
44. 4 × 10 = 40 and 4 × 1 = 4.
More practice: on your own
456. 8 × 50 = 400 and 8 × 7 = 56. 400 + 56 = 456.
234. 9 × 20 = 180 and 9 × 6 = 54. 180 + 54 = 234.
1224. 3 × 400 = 1200 and 3 × 8 = 24. 1200 + 24 = 1224.
1410. 6 × 200 = 1200, 6 × 30 = 180, 6 × 5 = 30. 1200 + 180 + 30 = 1410.
5A school bus holds 48 people. How many people can 7 buses hold?
336 people. 7 × 40 = 280 and 7 × 8 = 56. 280 + 56 = 336.
Challenge
1Find 9 × 999 in a smart way.
8991. 999 is 1 less than 1000. 9 × 1000 = 9000, then take away one group of 9: 9000 − 9 = 8991.
2Estimate, then solve 5 × 846.
Estimate: 5 × 800 = 4000 (a bit more). Exact: 5 × 800 = 4000, 5 × 40 = 200, 5 × 6 = 30. 4000 + 200 + 30 = 4230.
3An arena has 6 sections. Each section has 375 seats. Can 2200 people each get a seat?
Yes. 6 × 300 = 1800, 6 × 70 = 420, 6 × 5 = 30. 1800 + 420 + 30 = 2250 seats, which is 50 more than 2200.
Quiz
4. Which partial products solve 5 × 63?
5. What is the best estimate for 8 × 197?
Big idea: The curriculum (B2.5) asks Grade 4 students to solve problems that multiply 2- or 3-digit numbers by 1-digit numbers (and by 10, 100 and 1000) using tools such as arrays. The focus is on understanding: splitting by place value (the distributive property) and adding partial products. A shorter algorithm is not required in Grade 4.
Common difficulties:
- Multiplying only the ones (6 × 47 = 6 × 7 = 42, then just writing 4 in front). Ask: "What is 6 times the 40?"
- Place-value slips (6 × 40 = 24). Say "6 times 4 tens is 24 tens, or 240."
- Forgetting a partial product when there are three parts.
How to help: Use shopping: "Juice boxes come in packs of 24. How many in 3 packs?" Ask for an estimate first, then the exact answer.