Mental Math Strategies for Multiplication: 107 Self-Checking Problems, Instant feedback
Ask a class to work out 14 × 9 in their heads and watch what happens. Some students reach for paper to line up the digits. Another one thinks "14 × 10 is 140, and that's one 14 too many: 126." That student didn't memorise a bigger times table. They used a strategy.
Mental math strategies for multiplication turn hard-looking problems into easy ones. Students break numbers into factors, round them, double them, halve them and move them around, all using place value and the properties of multiplication. Along the way they build the number sense that makes estimating and checking answers second nature.
Mental Math Strategies: Multiplication is a self-checking digital version of our classroom slide deck for grades 3 to 5. It has 30 slides, base-ten block explanations you can step through, and 107 problems that check themselves.
Watch it in action
It plays in any web browser, one slide at a time, just like the deck. There's nothing to print, cut or laminate, and students don't need logins.
The mental math strategies for multiplication
Start with the properties
Every strategy below works because of three properties of multiplication:
- Commutative: you can multiply in any order. 4 × 13 = 13 × 4
- Associative: you can group the numbers any way you like. (9 × 5) × 2 = 9 × (5 × 2) = 9 × 10
- Distributive: you can split a number and multiply each part. 6 × 14 = 6 × 10 + 6 × 4 = 60 + 24
In the game, students fill in the missing number that keeps both sides equal. A wrong number shows what it would make instead, so they can see why it doesn't balance.
1. Multiplying by 9, 11 and 12
These are all close to 10, so multiply by 10 and adjust.
- × 9: 17 × 9 = 17 × 10 − 17 = 170 − 17 = 153
- × 11: 36 × 11 = 36 × 10 + 36 = 360 + 36 = 396
- × 12: 8 × 12 = 8 × 10 + 8 × 2 = 80 + 16 = 96
With base-ten blocks it's easy to see: give every group of 9 one more cube to make 14 tens, then take those 14 extra cubes away.
2. Multiplying by 5
5 is half of 10. With an even number, double the 5 and halve the other number: 5 × 18 = 10 × 9 = 90. With an odd number, multiply by 10 and then halve: 5 × 17 = 170 ÷ 2 = 85.
3. Multiplying by 4 and 8: doubling
4 is 2 × 2, and 8 is 2 × 2 × 2. So to multiply by 4, double twice. To multiply by 8, double three times.
- 23 × 4 = 46 × 2 = 92
- 27 × 8 = 54 × 2 × 2 = 108 × 2 = 216
4. Multiplying by 10, 100 and 1000
Times 10, every one becomes a ten and every ten becomes a hundred, so each digit moves one place to the left. Times 100, the digits move two places.
- 32 × 10 = 320 and 32 × 100 = 3200
- 30 × 20 = 3 × 2 × 10 × 10 = 6 × 100 = 600
- 120 × 400 = 12 × 4 × 1000 = 48000
Take the zeros off, multiply the front digits, then put all the zeros back.
5. Factors of 100 and 1000
Some pairs are worth knowing by heart because they make round numbers:
- 100: 10 × 10, 2 × 50, 5 × 20, 4 × 25
- 1000: 10 × 100, 2 × 500, 5 × 200, 4 × 250, 8 × 125, 20 × 50
Once students know them, they can group with the associative property: 2 × 9 × 4 × 125 = (2 × 4 × 125) × 9 = 1000 × 9 = 9000.
6. Splitting (front-end multiplication)
Split the bigger number into hundreds, tens and ones, multiply each part, then add: 4 × 135 = 400 + 120 + 20 = 540.
7. Decomposing to factors
Break a number into factors, then multiply in the easiest order. 24 × 125 = 3 × 8 × 125 = 3 × 1000 = 3000. And 44 × 25 = 11 × 4 × 25 = 11 × 100 = 1100.
8. Rounding
Round one number to a ten or a hundred, multiply, then fix the difference.
- 19 × 12 = 20 × 12 − 12 = 240 − 12 = 228
- 53 × 99 = 5300 − 53 = 5247
Ask students "Did you add a group too many, or too few?" before they fix it.
9. Halving and doubling
Halve one number and double the other. The answer stays the same, so keep going until it's easy: 8 × 15 = 4 × 30 = 2 × 60 = 120, and 32 × 145 = 16 × 290 = 8 × 580 = 4 × 1160 = 2 × 2320 = 4640.
Try these: a free number talk
Put one problem on the board, give students quiet think time, then ask "How did you do it?" Collect two or three different strategies before you reveal the answer.
- 25 × 16
- 19 × 7
- 5 × 48
- 15 × 14
- 8 × 35
- 36 × 11
Answers: 400, 133, 240, 210, 280, 396.
Listen for factors (25 × 4 × 4), rounding (20 × 7 − 7), halving and doubling (5 × 48 = 10 × 24, or 15 × 14 = 30 × 7), doubling three times (35, 70, 140, 280) and multiplying by 11 (360 + 36). Problem 4 is a good one to compare: some students halve and double, others split 14 into 10 + 4.
How the self-checking works
Explanation slides with base-ten blocks
Ten slides explain a strategy step by step. Students press Next step to turn nines into tens, double a group of 23, watch every ten become a thousand, or cut a thousand cube into 8 groups of 125. Then they try two problems of their own.
Worked examples, then practice
Practice slides look just like the deck: the worked examples at the top and practice problems underneath. Students type an answer and press Check.
A wrong answer turns the box red and opens a hint for that strategy, like 29 is close to 30. Work out 32 × 30, then take away one 32.
A second wrong answer shows the worked steps with the last line hidden, so students still finish the problem themselves.
A right answer earns a star, as long as the steps weren't opened first. See the strategy shows one way to solve any problem they've finished.
Mixed practice: any strategy
Four slides mix everything up. Students choose their own strategy, and after solving they see one way it could be done, labelled with the strategy name.
Which strategy did they use?
The last two slides show two students' work on the same problem, like Annie and Amir solving 32 × 29. Students pick the strategy each one used. Annie rounded 29 up to 30; Amir split 29 into 20 + 9. It's a ready-made number talk about why both ways work.
One slide at a time
Like the original deck, it opens on a title slide and moves with Next and Back or the arrow keys. All slides jumps straight to any strategy and shows the stars on each one. Slides always open blank, so the next student on the device never sees the answers.
What's inside
- Properties of multiplication: commutative, associative and distributive
- Multiplying by 9, 11 and 12
- Multiplying by 5, with even and odd numbers
- Multiplying by 4 and 8 by doubling
- Multiplying by 10, 100 and 1000, and by multiples of 10
- Factors of 100 and 1000, and grouping with the associative property
- Splitting, decomposing to factors, rounding, and halving and doubling
- Mixed practice: 24 problems to solve with any strategy
- Which strategy did they use? Compare two students' work
Teaching tips
- Model it first. Project an explanation slide and step through the blocks together before students try "Your turn".
- Say the strategy out loud. "29 is close to 30, so I work out 32 × 30 and take one 32 away."
- Learn the partners. A few minutes on 4 × 25 and 8 × 125 pays off on every factors and grouping slide.
- Compare strategies. On the mixed slides, the same problem can be solved more than one way. Ask which way was quickest, and why.
- Revisit often. One slide makes a five-minute warm up, and stars show which strategies each student has tried.
Standards
- 3.OA.B.5: Apply properties of operations as strategies to multiply
- 3.OA.C.7: Fluently multiply within 100 using strategies
- 3.NBT.A.3: Multiply one-digit whole numbers by multiples of 10
- 4.NBT.A.1: A digit in one place represents ten times what it represents in the place to its right
- 4.NBT.B.5: Multiply using strategies based on place value and the properties of operations
- 5.NBT.A.2: Explain patterns in the number of zeros when multiplying by powers of 10
Quick answers
Do students need accounts? No. It's a link, so there's nothing to sign in to.
What devices does it work on? Any device with a web browser, including Chromebooks, iPads, laptops and the classroom board.
What grades is it for? Grades 3 to 5. Third graders can start with the properties, multiplying by 10, 5, 4 and 8, and the factors of 100. Rounding, halving and doubling with three-digit numbers, and the factors of 1000, stretch older students.
Do I need to check answers? No. Every problem checks itself, and every one has a hint and worked steps.
Get Mental Math Strategies: Multiplication
Give students a strategy for every multiplication problem, and the practice to make it stick.
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