The pattern keeps going
Multiplication patterns continue below zero.
A times table that goes down
4 × 3 = 12, 3 × 3 = 9, 2 × 3 = 6, 1 × 3 = 3, 0 × 3 = 0. Every time the multiplier goes 1 down, the product goes down by 3.
The multiplier goes below zero: (−1) × 3. If the pattern keeps going, what is it?
What this lesson covers
The idea
For each unit decrease in the multiplier, the product decreases by a positive multiplicand and increases by the magnitude of a negative multiplicand, and this pattern continues when the multiplier becomes negative.
A times table that goes down
4 × 3 = 12, 3 × 3 = 9, 2 × 3 = 6, 1 × 3 = 3, 0 × 3 = 0. Every time the multiplier goes 1 down, the product goes down by 3.
The multiplier goes below zero: (−1) × 3. If the pattern keeps going, what is it?
- −3
- 3
- 0
Keep the pattern going
Each row starts from the one above it. Choose the next product. Does the pattern stop at zero, or go on?
The same step, below zero too
When the multiplicand is 3, every step down in the multiplier takes 3 away from the product. This goes on below zero: (−1) × 3 = −3, (−2) × 3 = −6.
When the multiplicand is −3, every step down adds 3: 0 × (−3) = 0, then (−1) × (−3) = 3, (−2) × (−3) = 6.
For each unit decrease in the multiplier, the product decreases by a positive multiplicand and increases by the magnitude of a negative multiplicand. This pattern continues when the multiplier becomes negative.
- Multiplicand | Each unit decrease in the multiplier
- positive (3) | product decreases by 3
- negative (−3) | product increases by 3
Notes
For each unit decrease in the multiplier, the product decreases by a positive multiplicand and increases by the magnitude of a negative multiplicand. The pattern continues below zero.
Check yourself
4 × (−5) = −20, 3 × (−5) = −15, 2 × (−5) = −10, 1 × (−5) = −5, 0 × (−5) = 0. Keep the pattern going. What is (−1) × (−5)?
Answer: 5
The product goes up by 5 each row: −20, −15, −10, −5, 0, then 5.
5 × 6 = 30, 4 × 6 = 24, 3 × 6 = 18, and so on. Keep going: what is (−2) × 6?
Answer: -12
18, 12, 6, 0, −6, −12. So (−2) × 6 = −12.
The multiplicand is negative. The multiplier goes down by 1 each row. What happens to the product?
Use the pattern for the multiplicand −4: 4 × (−4) = −16, 3 × (−4) = −12, … 0 × (−4) = 0. Continue it to find (−3) × (−4).
Answer: 12
0, then (−1) × (−4) = 4, (−2) × (−4) = 8, (−3) × (−4) = 12.
- It decreases by the multiplicand. That is the pattern for a positive multiplicand. A negative one makes the product go up.
- It stays the same. The product changes in every row, by the size of the multiplicand.
- It increases by the size of the multiplicand — correct. Yes! For −3 each row adds 3. This goes on below zero.