What You'll Learn
- Recall products in the 7, 8, and 9 tables through 12 × 12
- Derive uncertain facts from easier ×5, ×10, doubling, and neighboring facts
- Use commutativity, digit patterns, and inverse division to verify products
- Choose an efficient strategy instead of relying on an answer-shaped guess
Why This Matters
Hard-table fluency supports division, fractions, area, ratios, and mixed expressions. Decomposing 9 as 10 − 1 or 7 as 5 + 2 also develops the distributive reasoning later used in algebra.
How Multiplication: 7, 8, and 9 Works
Stage 16 built products from easier tables. Multiplication counts equal groups, so 7 × 8 means seven groups of eight. Commutativity gives 7 × 8 = 8 × 7; reversing factors keeps the product.
Because 8 = 2 × 2 × 2, multiplying by 8 means doubling three times. For 6 × 8, follow 6 → 12 → 24 → 48. This extends the doubling work from Stages 6, 15, and 16, and explains why multiples of 8 are even.
Nine is one less than ten, so n × 9 = n × 10 − n. Thus 7 × 9 = 70 − 7 = 63. For factors 1 through 10, the product’s digits sum to 9. Use that only as a check: 45 also has digit sum 9 but is not 7 × 9.
Build seven from known groups: n × 7 = n × 5 + n × 2. Therefore 8 × 7 = 40 + 16 = 56. A neighboring fact also works: 6 × 7 = 42, then one more group gives 7² = 49, linking to Stage 20 squares.
One product generates a fact family. From 8 × 9 = 72 follow 9 × 8 = 72, 72 ÷ 8 = 9, and 72 ÷ 9 = 8. Stage 18 uses these inverses. Deriving and checking an uncertain product is safer than guessing.
Mental Walkthroughs
Read the strategy, then prove you can use it: every example ends with a matching problem on new numbers that you answer and check right here. Answer it and press New numbers for as many fresh problems as you like.
Build a seven fact from five and two
8 × 7
- 1 Split seven groups into five groups and two groups: 8 × 7 = 8 × 5 + 8 × 2.
- 2 Use earlier facts: 8 × 5 = 40 and 8 × 2 = 16.
- 3 Combine the partial products: 40 + 16 = 56.
8 × 7 = 56
Your turn — try it on new numbers
7 × 8 = ?
Triple-double for eight
7 × 8
- 1 Double 7 once to get 14.
- 2 Double 14 to get 28, then double 28 to get 56.
- 3 Three doublings multiply by 2 × 2 × 2, which is 8.
7 × 8 = 56
Your turn — try it on new numbers
10 × 9 = ?
Use ten minus one group
12 × 9
- 1 Find twelve groups of ten: 12 × 10 = 120.
- 2 Subtract one group of 12 because nine is one less than ten: 120 − 12 = 108.
12 × 9 = 108
Your turn — try it on new numbers
8 × 9 = ?
Step from a square
8 × 9
- 1 Recall the square fact 8 × 8 = 64.
- 2 Nine groups contain one more group of 8 than eight groups.
- 3 Add 64 + 8 = 72 and check that 7 + 2 = 9.
8 × 9 = 72
Your turn — try it on new numbers
5 × 8 = ?
Watch for These Missteps
Blending 7 × 8 with 8 × 8
Anchor 8 × 8 = 64, then remove one group of 8: 64 − 8 = 56.
Treating the nine digit-sum as proof
A digit sum of 9 is only a check. Derive the product with ×10 minus one group first.
Adding the factor only once when doubling
Multiplying by 8 needs three doublings. Mark n → 2n → 4n → 8n.
Multiplication: 7, 8, and 9 in Everyday Life
A theater has 9 rows of 8 seats, so 9 × 8 = 72. The same array model describes tiled floors, cartons, team rotations, and rectangular area.
Pro Tips
- • Say the derivation aloud: “eight fives plus eight twos.”
- • Mix reversed facts such as 7 × 9 and 9 × 7.
- • Check a product by dividing by one factor.
- • Practice weak facts briefly and accurately.
Practice Exercises
Work each problem mentally, then type the answer and check it. Nothing is marked correct until you enter a value that actually equals the result, and every problem keeps its mental-steps note in reach if you get stuck.
- 1. 7 × 8 = ?
- 2. 9 × 6 = ?
- 3. 8 × 8 = ?
- 4. 7 × 7 = ?
- 5. 12 × 9 = ?
- 6. 8 × 11 = ?
- 7. 9 × 9 = ?
- 8. 7 × 12 = ?
- 9. 8 × 6 = ?
- 10. 9 × 4 = ?
- 11. 7 × 9 = ?
- 12. 8 × 12 = ?
- 13. 9 × 11 = ?
- 14. 7 × 6 = ?
- 15. What factor makes 8 × ? = 72?
- 16. What factor makes 7 × ? = 77?
- 17. One box has 9 rows of 7 beads. How many beads?
- 18. Which is greater, 8 × 9 or 7 × 10, and by how much?
Drill Stage 17 with unlimited problems
Type your answer and press Enter. Every problem is generated on the spot and uses the 7, 8 and 9 multiplication tables. Get it wrong and the steps below show how that exact problem is solved mentally.
- Correct
- 0 / 0
- Accuracy
- —
- Best streak
- 0
- Time
- 0:00
Problem 1
7 × 7 = ?
Stage 17 Questions
Must every 7, 8, and 9 fact be memorized?
Recall helps, but derivation comes first. Rebuilding from ×5, ×10, or doubling supplies a dependable recovery method.
Why are 7 × 8 and 8 × 7 equal?
Rotating a seven-by-eight array changes its orientation but not its 56 objects. This is commutativity.
Does the nine pattern continue beyond 9 × 10?
The rule n × 9 = n × 10 − n always continues, although the simple two-digit pattern changes.