Compare without adding
Follow the gap, not the sums.
Marbles for Raja and Joy
Which is greater: 1023 + 125 or 1022 + 128? Do not add! Think of two friends with marbles: Raja has 1023 and gets 125 more, Joy has 1022 and gets 128 more.
Who has more marbles now?
What this lesson covers
The idea
Expressions can often be compared without finding their values, by reasoning about how their numbers differ, such as by imagining a situation that the expressions describe.
Marbles for Raja and Joy
Which is greater: 1023 + 125 or 1022 + 128? Do not add! Think of two friends with marbles: Raja has 1023 and gets 125 more, Joy has 1022 and gets 128 more.
Who has more marbles now?
- Raja, he started with more
- Joy, she gets more today
- They have the same
Follow the gap
The meter shows who is ahead and by how much. First tap the gap at the start. Then tap where the gap is after today. Then choose the sign. You never have to add the big numbers.
Imagine a situation
1023 + 125 and 1022 + 128: Raja starts 1 ahead. But Joy gets 3 more than Raja today. So Joy ends 2 ahead. 113 − 25 and 112 − 24: Raja starts 1 ahead. But Raja loses 1 more than Joy. So they end equal.
Expressions can often be compared without finding their values, by reasoning about how their numbers differ, for example by imagining a situation the expressions describe.
Notes
Expressions can often be compared without finding their values, by reasoning about how their numbers differ, such as by imagining a situation.
Check yourself
Which is greater: 500 + 49 or 499 + 52? Reason, do not add.
Compare 80 − 17 and 79 − 15 by reasoning.
Raja has 300 marbles and gets 50 more. Joy has 298 and gets 53 more. By how many marbles does Joy end up ahead?
Answer: 1
Raja starts 2 ahead, Joy gets 3 more: Joy ends 3 − 2 = 1 ahead. Check: 350 and 351.
- 499 + 52 — correct. Yes! The first starts 1 ahead, but the second adds 3 more. So the second is greater by 2.
- 500 + 49. 500 is 1 more than 499, but 52 is 3 more than 49. The second gains more than the first starts ahead.
- They are equal. The first starts 1 ahead and the second adds 3 more. 3 is not 1, so they are not equal.
- 80 − 17 > 79 − 15. 80 is 1 more than 79. But 17 is also 2 more than 15, so the first loses more. It ends behind.
- 80 − 17 < 79 − 15 — correct. Yes! 80 is 1 more than 79, but 17 is 2 more than 15. Losing 2 more beats a 1 head start, so the first is smaller.
- 80 − 17 = 79 − 15. The head start is 1 but the extra loss is 2. They would be equal only if the extra loss were also 1.