Grade by Grade
B. Number · Operations: Numbers to 100 000, Decimals, Fractions, Ratios and Rates

Multiplying 2-Digit by 2-Digit Numbers: The Algorithm

B2.6 B2.1 A1About 45 minutes
Learning goals
  • I can multiply a 2-digit number by a 2-digit number using the standard algorithm.
  • I can explain why the second row starts with a zero.
  • I can connect each row of the algorithm to the area model.
  • I can solve and check multiplication story problems.

Words to know

standard algorithm
A short, step-by-step way to multiply in columns.
placeholder zero
The 0 at the end of the second row. It shows we are multiplying by tens.
regroup
Carry a ten (or hundred) to the next place when a product in one place is 10 or more.

Let's start with a comparison

In the last lesson you found 23 × 14 = 322 with an area model: 200 + 30 + 80 + 12.

The area model is great for understanding, but writing four parts takes time. Mathematicians found a shorter way, called the standard algorithm. It does the same math with only two rows.

How the algorithm works

To multiply 46 × 37:

  1. Multiply by the ones digit of 37 (that's 7): 46 × 7 = 322. This is row 1.
  2. Multiply by the tens digit of 37 (that's 3 tens, or 30): 46 × 30 = 1380. Write a 0 in the ones place first, because you're multiplying by tens. This is row 2.
  3. Add the rows: 322 + 1380 = 1702.
6473×2230831+2071
The standard algorithm for 46 × 37. Row 1: 46 × 7 = 322. Row 2: 46 × 30 = 1380. Add: 1702.

Connecting to the area model

Each row of the algorithm is one row of an area model.

  • Row 1 (46 × 7) is the bottom strip: 40 × 7 + 6 × 7 = 280 + 42 = 322.
  • Row 2 (46 × 30) is the top strip: 40 × 30 + 6 × 30 = 1200 + 180 = 1380.

The algorithm just adds two parts at a time while you go.

406301200180728042not to scale
An area model for 46 × 37. Top strip (× 30): 1200 + 180 = 1380. Bottom strip (× 7): 280 + 42 = 322. These are the two rows of the algorithm.
Worked example

Example 1: Use the algorithm

Solve 58 × 24.

  1. Estimate: 60 × 20 = 1200, or 60 × 25 = 1500. The answer is between those.
  2. Row 1 (× 4): 4 × 8 = 32. Write 2, regroup 3 tens. 4 × 5 tens = 20 tens, plus 3 = 23 tens. Row 1 = 232.
  3. Row 2 (× 20): Write 0 in the ones place. 2 × 8 = 16. Write 6, regroup 1. 2 × 5 = 10, plus 1 = 11. Row 2 = 1160.
  4. Add: 232 + 1160 = 1392. It's between 1200 and 1500. ✓

58 × 24 = 1392

Worked example

Example 2: A story problem

A school orders 24 boxes of pencils. Each box holds 36 pencils. How many pencils is that?

  1. We need 36 × 24.
  2. Row 1: 36 × 4 = 144.
  3. Row 2: 36 × 20 = 720.
  4. Add: 144 + 720 = 864.
  5. Check with an estimate: 40 × 20 = 800. ✓

864 pencils

Worked example

Example 3: Fix a mistake

Tia says 23 × 15 = 138. Her work shows row 1 = 115 and row 2 = 23. What went wrong?

  1. Row 1: 23 × 5 = 115. That's correct.
  2. Row 2 should be 23 × 10 = 230, because the 1 in 15 means 1 ten. Tia forgot the placeholder zero.
  3. Add: 115 + 230 = 345.
  4. Estimate: 20 × 15 = 300. 345 makes sense; 138 doesn't.

23 × 15 = 345

Algorithm tips

  • Don't forget the zero at the start of row 2. You are multiplying by tens.
  • When you regroup in row 2, cross out or erase the row 1 carries first so you don't mix them up.
  • Estimate first, and compare at the end.
  • Stuck? Go back to the area model. It always works, and it helps you understand.
Just for fun

Zero is supposed to mean "nothing", but in row 2 it is a superhero. Forgetting that one little 0 turned Tia's 230 into 23, and her answer shrank to less than half of what it should be!

Hands-on activity

Two ways, same answer

You need: paper, a pencil and a coin to flip.

  1. Roll or choose two 2-digit numbers. (You can write the digits 1 to 9 on scraps of paper and pick 4.)
  2. Flip the coin. Heads: you use the area model, and your grown-up uses the algorithm. Tails: switch.
  3. Compare answers. Do they match? Find the two strips of the area model inside the two rows of the algorithm.
  4. Do 4 rounds. Which method do you like better for which numbers? Why?

Math talk: "Where is the 1380 in the area model for 46 × 37?"

Your turn

1

34 × 21 = ?

2

52 × 13 = ?

3

67 × 25 = ?

4

43 × 36 = ?

5

89 × 12 = ?

6

A league has 28 teams with 15 players each. How many players?

7

99 × 99 = ?

More practice: warm-up

1

11 × 10 = ?

2

12 × 20 = ?

3

25 × 4 = ?

4

32 × 3 = ?

5

41 × 2 = ?

More practice: on your own

1

76 × 54 = ?

2

38 × 47 = ?

3

64 × 64 = ?

4

An orchard has 35 rows of apple trees with 28 trees in each row. How many trees?

5

Is 62 × 38 = 2356 reasonable? Estimate to decide.

Challenge

1

Find the missing digit: 4□ × 21 = 945.

2

Solve 57 × 33 with the area model and the algorithm. Show where the algorithm's rows are in the area model.

3

A sugar bush has 48 maple trees with taps. Suppose each tree gives 45 L of sap in a season. How many litres of sap is that? Is your answer more or less than 2000 L?

Check your understanding

Quiz

1.

What is 23 × 31?

2.

In the algorithm for 47 × 26, what is the second row?

3.

What is 54 × 12?

4.

Why does the second row start with a 0?

5.

What is 35 × 35?

6.

Is 62 × 38 = 2356 reasonable?

For parents and teachers

Big idea: B2.6 asks Grade 5 students to multiply two-digit numbers by two-digit numbers using the area model and algorithms, and to make connections between the two methods. The standard algorithm is fast, but your child should be able to say what each row means (for 46 × 37: "46 times 7" and "46 times 30").

Common difficulties:

  • Forgetting the placeholder zero in the second row, which makes the answer far too small.
  • Mixing up regrouped digits from the two rows. Cross out the first row's carries before starting the second.
  • Basic fact errors (often 6s, 7s, 8s). A quick fact review helps.

How to help: If your child learned a different algorithm, that's fine; the curriculum asks for algorithms in general. Ask "How do you know your answer is reasonable?" every time. Comparing two methods and choosing one builds A1 reasoning and reflecting skills.