What changes when a factor changes
Add 1 to a factor and count the new squares.
One number goes up by 1
Take 23 × 27 = 621. Now make the second number 27 → 28.
By how much does the product increase?
What this lesson covers
The idea
Increasing b by 1 increases ab by a; increasing both a and b by 1 increases ab by a + b + 1; increasing a by 1 and decreasing b by 1 gives ab + b – a – 1.
One number goes up by 1
Take 23 × 27 = 621. Now make the second number 27 → 28.
By how much does the product increase?
- By 1
- By 23
- By 27
Grow the rectangle
A rectangle with a squares across and b squares up has ab squares. Tap each change. Green squares are added and red squares are taken away.
Count the new strips
Increasing b by 1 increases ab by a. Increasing both a and b by 1 increases ab by a + b + 1. Increasing a by 1 and decreasing b by 1 gives ab + b − a − 1.
For a = 23 and b = 27, the product ab = 621 changes like this:
23 = a · 27 = b 51 = 23 + 27 + 1 3 = 27 − 23 − 1
The same rules hold when the letters stand for negative integers too.
- Change | New product | Up by
- b + 1 | 644 | 23
- a + 1 | 648 | 27
- both + 1 | 672 | 51
- a + 1, b − 1 | 624 | 3
Notes
Increase b by 1: ab grows by a. Increase both by 1: it grows by a + b + 1. Increase a by 1 and decrease b by 1: ab + b − a − 1.
Check yourself
18 × 35 = 630. Without multiplying, find 18 × 36.
Answer: 648
b goes up by 1, so the product goes up by a = 18: 630 + 18 = 648.
25 × 40 = 1000. Both numbers go up by 1. By how much does the product increase?
Answer: 66
25 + 40 + 1 = 66. And 26 × 41 = 1066 = 1000 + 66.
Why is the increase a + b + 1 when both a and b go up by 1?
9 × 6 = 54. The first number goes up by 1 and the second goes down by 1: 10 × 5. By how much does the product change?
Answer: -4
10 × 5 = 50 and 50 − 54 = −4. Also b − a − 1 = 6 − 9 − 1 = −4: the product goes down by 4.
- Two strips of a and b squares, and one extra corner square — correct. Yes! The row and the column each add a strip and they meet in one corner square that both missed.
- Only the two strips, a and b. The two strips leave a gap in the corner. That corner square is the extra +1.
- Each number grows by 1, so 1 + 1 more. The strips are as long as the other factor, not 1 square long.