Numbers & the Number System • Topic 6 of 9

Ordering Numbers

Ordering extends comparison to a whole set. Ascending order runs smallest to largest; descending order runs largest to smallest. The procedure is the comparison rule applied repeatedly: group by number of digits first (fewer digits means smaller), then within each group compare from the leftmost place rightwards until a difference shows. Take 321, 45, 3210, 456 in ascending order. The two-digit 45 is smallest; among the three-digit numbers 321 < 456 because 3 < 4 at the hundreds place; the four-digit 3210 is largest - giving 45 < 321 < 456 < 3210. Descending is the same logic reversed: for 505, 55, 550, 5005 the four-digit 5005 leads, the two-digit 55 trails, and between 505 and 550 the tens place decides (0 < 5) so 550 > 505, giving 5005 > 550 > 505 > 55. The misconception to watch is ordering by the leading digit alone - placing 35, 123, 87 as 35, 87, 123 by reading only 3, 8, 1 - when 123 is in fact the largest because it has the most digits.

✅ Solved examples

1. Arrange in ascending order: 321, 45, 3210, 456.
Group by digits: 45 (2 digits) smallest, then 321 and 456 (3 digits) with 321 < 456, then 3210 (4 digits) largest. Result: 45, 321, 456, 3210.
2. Arrange in descending order: 505, 55, 550, 5005.
5005 (4 digits) is largest; 55 (2 digits) smallest; among 505 and 550 the tens place gives 550 > 505. Result: 5005, 550, 505, 55.
3. A child orders 35, 123, 87 as 35, 87, 123 by reading first digits 3, 8, 1. What is wrong?
The child ignored digit count. 123 has three digits so it is largest; the correct ascending order is 35, 87, 123.
4. In the ascending arrangement 45, 321, 456, 3210, which number is in the middle two positions if there were a single middle - identify the second-smallest.
The second-smallest is 321 (it follows 45 and precedes 456 and 3210).

✏️ Practice — try these, take hints as needed

1. Arrange in ascending order: 9, 99, 909, 90.
Group by digit count.
One digit, then two, then three.
9, 90, 99, 909
2. Arrange in descending order: 210, 21, 2100, 201.
Most digits first.
Among 210 and 201 compare the tens.
2100, 210, 201, 21
3. Which number is the largest in the set 78, 708, 87, 780?
Two three-digit numbers.
Compare 708 and 780 at the tens.
780
4. Arrange in ascending order: 100, 10, 1000, 1.
Sort by number of digits.
Each has one more zero.
1, 10, 100, 1000

📝 Topic test — 8 questions

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