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Comparing and Ordering Integers

Maths · Y7 Integers
Year 7
Lesson Guide
Explanation
Comparing integers comes down to one rule: whichever number sits further left on the number line is the smaller one. Look at negative seven, negative three and two. Negative seven is furthest left, so it is the smallest, and two is furthest right, so it is the largest. This one rule handles every case. Two positives, two negatives, or one of each, it never changes. It also gives you a shortcut: any negative number is smaller than any positive number, because the whole negative side sits left of zero. Temperature is a good way to feel it. A morning at negative seven degrees is colder than one at negative three, and colder means smaller.
Your Turn
Which is smaller, −5 or 1? Check: −5, because every negative number sits left of every positive one.
Watch Out
Ignore the size of the digits when the numbers are negative. 7 is bigger than 3, but −7 is SMALLER than −3.
Explanation
Two negatives are where most mistakes happen, so slow down here. Walking left from zero, the digits count up one, two, three, four, but the values are getting smaller the whole way. That means negative nine is smaller than negative one, even though nine is a bigger digit than one. Negative six is smaller than negative four. Negative twenty is smaller than negative twelve. Here is the quick check you can trust: with two negative numbers, cover the signs, and whichever number looks bigger is actually the smaller one. If you are ever unsure, sketch a short number line and mark both. The picture settles it in seconds and never lies to you.
Your Turn
Which is larger, −2 or −15? Check: −2, because −15 is thirteen steps further left.
Watch Out
A big-looking negative is a small number. −20 is not "more" than −12; it is 8 less.
Explanation
Mathematicians write comparisons with two symbols. The one that looks like a less-than sign means is less than, and the one that opens the other way means is greater than. The trick for remembering which is which is that the wide open end always faces the larger number and the narrow point always faces the smaller one. So negative seven is less than negative three, and you could equally write negative three is greater than negative seven. Both sentences say the same thing; only the order of the numbers has changed. Read the statement out loud from left to right and ask whether it is true. If it sounds wrong, flip the symbol.
Your Turn
Choose the symbol for −12 ? −20. Check: −12 is greater, so you want the greater-than sign pointing its tip at −20.
Watch Out
Read left to right. Writing −20 > −12 says negative twenty is greater, which is false.
Explanation
Ordering a whole set is easier than it looks if you draw the line. Take negative five, three, negative two, zero, seven and negative eight. Mark each one on a number line, then simply read them off from left to right. That gives negative eight, negative five, negative two, zero, three, seven, and that is the ascending order, smallest first. Descending order is the identical list read backwards: seven, three, zero, negative two, negative five, negative eight. Two habits keep you safe here. Count how many numbers you started with and check the same number appear in your answer, and always sort the negatives among themselves before you worry about the positives.
Your Turn
Put 2, −5, 0, −1 in ascending order. Check: −5, −1, 0, 2 — the two negatives first, most negative leading.
Watch Out
Sorting the digits and then adding the signs back gives −5, −1 the wrong way round. Sort the actual values.
Explanation
Real questions rarely use the words larger and smaller. They ask which is coldest, deepest, or most in debt, and you have to translate. Look at these overnight lows. Sydney is six degrees, Orange is negative two, Thredbo is negative eleven and Armidale is negative seven. Coldest means smallest, so read the bars from the one hanging furthest below the zero line: negative eleven, then negative seven, then negative two, then six. Thredbo is coldest. Be careful with words like deepest and lowest, because a deeper dive is a smaller number, and the biggest debt is the most negative balance. Translate the word into smaller or larger first, then use the number line.
Your Turn
Diver A is at −18 m and diver B is at −25 m. Who is deeper? Check: B, because −25 is further below zero and so is the smaller number.
Watch Out
"Deepest" and "coldest" mean SMALLEST, not biggest. Read the word carefully before you order anything.
📌 Key Points — this lesson in a nutshell
  • On a number line, the number further LEFT is always the smaller one.
  • Every negative number is smaller than zero, and smaller than every positive number.
  • For two negatives, the one with the bigger digit is the SMALLER number: −9 < −1.
  • The < and > symbols always open towards the larger number.
  • −7 < −3 and −3 > −7 say exactly the same thing.
  • To order a set, mark every number on a line and read from left to right.
  • Ascending means smallest first; descending is the same list reversed.
  • Check your ordered list has the same count of numbers as the original set.
  • Coldest, deepest and largest debt all mean the SMALLEST number.
  • When in doubt, sketch a quick number line — it settles any comparison.
Step 1: Introduction
Step 1 of 5
One rule decides every comparison −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 −7 −3 2 −7 is left of −3, and −3 is left of 2 so −7 is the smallest and 2 is the largest THE RULE: whichever number sits further LEFT is the smaller one. It works for two positives, two negatives, or one of each — every time. Any negative number is smaller than any positive number, with no working needed. Think temperature −7 °C is a colder morning than −3 °C. Colder = smaller = further left. The trap 7 is bigger than 3, so people assume −7 is bigger than −3. Check the LINE, not the digits.
With negatives, the bigger digit is the SMALLER number −9 −8 −7 −6 −5 −4 −3 −2 −1 0 The further you walk from zero on the LEFT, the smaller the number gets. So the digits count UP while the value counts DOWN. getting smaller −1 or −9 ? the one further from zero on the left is −9 −9 is smaller −4 or −6 ? the one further from zero on the left is −6 −6 is smaller −12 or −20 ? the one further from zero on the left is −20 −20 is smaller −2 or −15 ? the one further from zero on the left is −15 −15 is smaller Quick check for two negatives: ignore the signs, and the BIGGER number wins the title "smaller".
The open mouth faces the LARGER number < means "is less than" −7 < −3 read left to right: "negative seven is less than negative three" > means "is greater than" −3 > −7 read left to right: "negative three is greater than negative seven" Both sentences say exactly the same thing — only the order of the numbers changed. −5 < 0 every negative is less than zero 0 < 4 zero is less than every positive −1 > −9 closer to zero on the left is greater −6 < 2 negative beats positive to the left, always Point the narrow tip at the smaller number and the wide opening at the larger one.
Order −5, 3, −2, 0, 7, −8 from smallest to largest −5 3 −2 0 7 −8 Step 1 — mark every number on a number line. −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 Step 2 — read them off left to right. That IS the order. −8, −5, −2, 0, 3, 7 ASCENDING smallest first, climbing up −8, −5, −2, 0, 3, 7 read the line left to right DESCENDING largest first, coming down 7, 3, 0, −2, −5, −8 the same list, reversed
Overnight lows: which town was coldest? A bar hanging BELOW the zero line is a negative temperature. 0 °C 5 °C −5 °C −10 °C 6 °C Sydney −2 °C Orange −11 °C Thredbo −7 °C Armidale Coldest to warmest: −11, −7, −2, 6 Thredbo is coldest because −11 hangs furthest below the zero line. Not because 11 is a big number — the sign is doing the work.
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