Comparing and ordering
Comparing two numbers looks like the easiest thing on a maths sheet until the numbers are close. 4,732 and 4,723 differ by nine, agree for three digits, and cannot be told apart by glancing — which is exactly why they are worth practising. The skill being built is reading a number by its places rather than by its shape.
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Two numbers that agree for a while
4,732 __ 4,723
- Both have four digits, so neither is longer.
- Compare from the left. Thousands: 4 and 4, the same. Hundreds: 7 and 7, the same.
- Tens: 3 and 2. Three tens is more than two tens, so the first number is bigger.
- 4,732 > 4,723.
Worth knowing. Stop at the first place where the digits differ. Everything to the right of it makes no difference at all, however big it looks.
Comparing below zero
-9 __ -2
- Both are negative, so the usual rule about bigger digits is going to mislead.
- Picture a number line. -9 sits further to the left than -2.
- Further left means smaller.
- -9 < -2.
Worth knowing. This is the comparison children get wrong most often, because 9 is bigger than 2 and the digits say so. With negatives, the bigger the digit the smaller the number.
Where it usually goes wrong
- Comparing by how long the numbers look rather than by their digits, which fails once both have the same width.
- Ordering negatives by their digits, so -9 is put above -2.
- Reading the symbol backwards — the wide end faces the bigger number.
The Professor checks the length first. A longer whole number is always bigger, and half the comparisons never need a second look.
Practise it
All level pathsQuestions teachers and parents ask
How do you remember which way < points?
The wide end faces the bigger number, and the point faces the smaller one. Some children read it as a mouth that always eats the larger amount. Whichever image sticks, it is worth checking the direction out loud until it stops needing checking.
When is this taught?
Comparing two-digit numbers with >, = and < is grade 1 (1.NBT.B.3) and three-digit numbers grade 2 (2.NBT.A.4). Comparing multi-digit numbers is grade 4, and ordering negative numbers arrives in grade 6 (6.NS.C.7).
Why is -9 smaller than -2?
Because it is further from zero in the negative direction. On a number line -9 sits to the left of -2, and further left is always smaller. A thermometer helps: -9 degrees is colder than -2 degrees.
Is the longer number always bigger?
For whole numbers with no leading zeros, yes — any four-digit number beats any three-digit one. It stops being true for decimals, where 0.5 is bigger than 0.25, and that is worth flagging before decimals arrive.
Where this shows up
Standards
- K.CC · Counting and cardinality
- 1.NBT · Number and operations in base ten
- 2.NBT · Number and operations in base ten
- 4.NBT · Number and operations in base ten
- 4.NF · Number and operations — fractions
- 5.NBT · Number and operations in base ten
- CCSS.MATH.CONTENT.1.NBT.B.3 — Compare two two-digit numbers from the meanings of their tens and ones digits, writing >, = or <.
- CCSS.MATH.CONTENT.2.NBT.A.4 — Compare two three-digit numbers from their hundreds, tens and ones digits, writing >, = or <.
- CCSS.MATH.CONTENT.4.NBT.A.2 — Read, write and compare multi-digit whole numbers using numerals, number names and expanded form.
- CCSS.MATH.CONTENT.6.NS.C.7 — Understand ordering and absolute value of rational numbers, including negative numbers on a number line.