Integers and negative numbers
Numbers below zero, opposites, and ordering integers on a number line
About four lessons of 45 to 60 minutes
Numbers keep going below zero, and you already use them
You have met negative numbers long before this lesson, you just did not write them down. When the weather app shows a frosty morning at 6 below, when a lift button says B2 for two floors under the ground floor, when a diver drops below the surface of the sea, when you owe a friend five dollars, you are living below zero.
Zero is not the end of the number line. It is the middle. Today we push the number line to the left of 0 and give those below-zero amounts their proper names: negative numbers. By the end you will place any whole number, above or below zero, on a line, name its opposite, and say which of two integers is greater and exactly why.
- A winter morning at 6 below zerosix degrees colder than 0, a temperature of -6
- The lift button marked B2two floors below the ground floor, floor -2
- A diver 8 meters under the seaeight below sea level, an elevation of -8
- Owing a friend $5five dollars in the hole, a balance of -5
What students will be able to do
Students will understand negative numbers as amounts on the opposite side of zero from positive numbers, place positive and negative integers on a number line, name the opposite of an integer, and compare and order integers by reasoning about position on the line and about real contexts such as temperature and elevation.
- I can explain what a negative number means using a real situation like temperature, elevation or money.
- I can place positive and negative integers on a number line.
- I can name the opposite of any integer, and I know 0 is its own opposite.
- I can compare two integers and say which is greater, because it sits further right on the line.
- I can order a set of integers from least to greatest.
Standards this unit teaches
- 6.NS.C.5Common Core (US)Use positive and negative numbers
Understand that positive and negative numbers describe quantities with opposite directions or values, such as temperature above and below zero or elevation above and below sea level, and use 0 to mean the point where the two directions meet.
- 6.NS.C.6Common Core (US)Integers on the number line
Understand a rational number as a point on the number line, extend the line and the coordinate axes to include negative numbers, and recognise opposite signs as locations on opposite sides of 0, with the opposite of the opposite of a number being the number itself.
- 6.NS.C.7Common Core (US)Order and compare integers
Understand ordering of rational numbers, including negatives. Interpret a statement such as -5 is less than -2 as saying -5 sits to the left of -2 on a number line, and interpret order in real contexts.
- AC9M5N01Australian Curriculum v9 (ACARA)Integers and the number line (Year 5)
Recognise situations that use integers, including money, and locate them on a number line and as coordinates on a Cartesian plane. In the Australian sequence, meeting integers on the number line begins at Year 5.
- AC9M7N07Australian Curriculum v9 (ACARA)Comparing and ordering integers (Year 7)
Compare and order integers, and solve problems that add and subtract positive and negative whole numbers. The comparing and ordering work in this unit reaches toward this Year 7 descriptor.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Integer
- any whole number and its opposite, positive, negative or zero, with no fraction part
- Negative number
- a number less than zero, written with a minus sign such as -3
- Positive number
- a number greater than zero, such as 3 (sometimes written +3)
- Opposite
- the number the same distance from 0 but on the other side, such as -4 and 4
- Zero
- the point where positive and negative meet, neither positive nor negative
- Number line
- a line with numbers in order, now stretched to the left of 0 for negatives
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. What a negative number means
ConcreteStart with a real situation the class can feel: temperature. Draw a thermometer on the board, or lay a number line flat and call it a thermometer tipped on its side. Mark 0 as freezing. Warmer than freezing counts up: 1, 2, 3 above zero. Colder than freezing has to keep going below 0, so we count into new numbers written with a minus sign: -1, -2, -3. A temperature of -6 is six degrees colder than freezing.
The big idea is direction. A negative number and a positive number point opposite ways from the same starting point, 0. Up the thermometer is positive, down is negative. Above sea level is positive, below is negative. Money you have is positive, money you owe is negative. Zero is the meeting point, and it is neither positive nor negative.
Say each negative out loud carefully: -3 is 'negative three', not 'minus three'. The minus sign here is not an instruction to subtract, it is a label that says this number lives on the left of zero. Keeping that distinction clear now prevents a very common muddle later.
- Which is colder, a temperature of -2 or -7? How does the thermometer show it?
- A submarine is at an elevation of -30 meters. What does the minus sign tell you?
- Is 0 a positive number, a negative number, or neither?
2. Placing integers on the number line
PictorialMove from the thermometer to a plain horizontal number line with 0 in the middle. To the right of 0 are the positive integers 1, 2, 3 and on. To the left of 0, as a mirror image, are the negative integers -1, -2, -3 and on. The spacing is equal, exactly as it always was, the line just did not stop at 0.
To place an integer, start at 0 and count steps: to the right for a positive, to the left for a negative. To reach -6, take six equal steps to the left of 0. The distance from 0 is the same as for 6, only the direction is opposite.
This is the single most useful model in the whole unit. Almost every question about negatives, comparing them, ordering them, later adding and subtracting them, is answered by asking where each number sits on this line and which way you move.
Place -7, -3 and 5 on a number line from -10 to 10.
- Draw the line with 0 in the middle and equal steps each side.
- For -7, count 7 equal steps to the left of 0 and mark the point.
- For -3, count 3 steps to the left. For 5, count 5 steps to the right.
Answer: -7 sits farthest left, then -3, then 0, then 5 on the right. All three are equal steps from 0 in the labelled direction.
- How many steps from 0, and in which direction, is -4?
- Two numbers are the same distance from 0 but on opposite sides. What can you say about them?
3. Opposites and the meaning of the sign
PictorialEvery number has an opposite: the number the same distance from 0 but on the other side. The opposite of 4 is -4, and the opposite of -4 is 4. On the line they are a matching pair reflected across 0, like a number and its reflection in a mirror standing at 0.
The opposite of the opposite brings you home. Start at 4, its opposite is -4, and the opposite of -4 is 4 again. Two flips land you back where you began, so the opposite of the opposite of a number is the number itself.
Zero is the one number that is its own opposite. It sits exactly on the mirror, so reflecting it does not move it. That is another reason 0 is neither positive nor negative: it is the balance point of the whole line.
What is the opposite of -6? And the opposite of the opposite of -6?
- The opposite of -6 is the number the same distance from 0 on the other side, which is 6.
- Now take the opposite of that: the opposite of 6 is -6.
- So two opposites in a row return the start.
Answer: The opposite of -6 is 6, and the opposite of the opposite of -6 is -6 itself.
- What is the opposite of 9? What is the opposite of -9?
- Which number is its own opposite, and why?
4. Comparing two integers
AbstractNow the question that trips everyone up: which integer is greater? Whole-number habits say 7 beats 2, so -7 must beat -2. On the number line the truth is the reverse. The rule is simple and always works: the number further to the right is greater, and the number further to the left is less.
Compare -5 and -2. On the line, -5 is further left than -2, so -5 is the smaller number: -5 is less than -2. Think of temperature: -5 degrees is colder, so it is lower, so it is less. The bigger the digit after the minus sign, the further left you go and the smaller the number.
Any positive number beats any negative number, because every positive sits to the right of 0 and every negative to the left. And 0 sits above every negative and below every positive. This is where a diagram beats a rule learned by heart: read the position, do not just read the digits.
Put the correct sign, greater than or less than, between each pair: -8 and -3, then -4 and 1.
- On the line, -8 is further left than -3, so -8 is the smaller. Write -8 is less than -3.
- -4 is left of 0 and 1 is right of 0, so 1 is greater. Write -4 is less than 1.
- Check with a real reading: -8 degrees is colder than -3 degrees, and any above-zero temperature beats a below-zero one.
Answer: -8 is less than -3, and -4 is less than 1.
- Which is greater, -1 or -10? Explain using the line.
- Is every negative number less than every positive number? Why?
5. Ordering integers and reading real situations
AbstractPut comparing to work in two ways: ordering a whole set of integers, and reading a real situation where the sign carries meaning. To order a set, place every number on the line in your head and read them off left to right, from least to greatest.
A lift is a perfect story for signed movement. Ground floor is 0, floors above are positive, basement levels are negative. Riding from floor 3 down to floor -2 is a move of five floors downward. Down on the line is to the left, so the number gets smaller.
Watch the language of contexts. In elevation, lower is more negative. In money, a larger debt is a more negative balance. In temperature, colder is lower and therefore less. In each case the same number-line reasoning decides the order, only the story around it changes.
Order these temperatures from coldest to warmest: 2, -5, 0, -1, 4.
- Picture each on the line: -5 is farthest left, then -1, then 0, then 2, then 4.
- Coldest is the least, which is farthest left.
- Read them off left to right.
Answer: From coldest to warmest: -5, -1, 0, 2, 4.
- Order -3, 5, -8, 0 from least to greatest.
- A diver at -12 meters rises to -4 meters. Did the elevation number get bigger or smaller, and did the diver go up or down?
Common misconceptions and how to address them
MisconceptionA number with a bigger digit is bigger, so -7 is greater than -2.
Why it happens: Students carry whole-number thinking straight across the minus sign, where 7 beats 2.
How to address it: Go to the line every time: -7 is further left than -2, so it is less. Reinforce with temperature, -7 degrees is colder than -2 degrees, so it is lower, so it is less. Further left means less.
MisconceptionThe minus sign in -3 means subtract, so it is an instruction, not part of the number.
Why it happens: The same symbol is used for subtraction and for marking a negative, and students meet subtraction first.
How to address it: Say '-3' as 'negative three' and point out it is a single number, a point on the line. Contrast '5 - 3' (an instruction to take away) with '-3' (a place on the line left of 0). The label sign and the operation sign look alike but do different jobs.
MisconceptionZero is a positive number, or zero is a negative number.
Why it happens: Students want every number sorted into one of the two piles.
How to address it: Show that 0 is the meeting point that separates the two directions. It is the balance line the positives and negatives mirror across, so it belongs to neither side. It is its own opposite.
MisconceptionNegative numbers are smaller as you move right, because they count down.
Why it happens: Reading -1, -2, -3 aloud sounds like counting up, so students think the line runs the other way in the negatives.
How to address it: Keep one rule for the whole line, no exceptions: right is always greater, left is always less. Sweep a finger left from 0 and say the values getting smaller, -1, -2, -3, so -3 is less than -1.
MisconceptionThe opposite of a number is found by making the digit smaller, so the opposite of 4 is 3.
Why it happens: Students confuse opposite (a reflection across 0) with one less.
How to address it: Define opposite as the mirror point across 0: same distance, other side. The opposite of 4 is -4, not 3. Fold the line at 0 and the two opposites land on each other.
MisconceptionA bigger debt is a bigger number, so owing $10 is greater than owing $2.
Why it happens: In everyday talk a big debt sounds like a big number, which clashes with the maths where -10 is less than -2.
How to address it: Separate the size of the debt from the value of the balance. The debt of $10 is larger, but the balance -10 is less than -2 because it is further from having money, further left on the line. Name which one you mean.
Guided practice (with answers)
1. Place -6 on a number line from -10 to 10.
Answer: Count 6 equal steps to the left of 0. It sits between -8 and -4, level with the sixth step.
2. What is the opposite of -9?
Answer: 9. It is the same distance from 0, on the other side.
3. Which is greater, -4 or -1?
Answer: -1. It is further right on the line, so it is greater. -4 is colder and lower, so it is less.
4. Order from least to greatest: 3, -2, -7, 0.
Answer: -7, -2, 0, 3. Read them left to right on the line.
5. A basement lift goes from floor 1 to floor -3. Up or down, and how many floors?
Answer: Down 4 floors. On the line that is a move of 4 to the left, from 1 to -3.
6. True or false: every negative number is less than 0.
Answer: True. Every negative sits to the left of 0 on the line, and left means less.
Independent practice worksheets
ChalkBee does not yet have a dedicated integer worksheet generator, so set the closely related number-line, comparing and place-value worksheets whose answer keys are computed in code and never wrong. They rehearse the exact skills this unit rests on: reading a value from its position, and ordering numbers by size.
Differentiation
- Keep a large number line on the wall with 0 in the middle so every answer is checked against a visible line.
- Anchor to one context at a time, temperature first, before mixing in elevation and money.
- Give a partly labelled line so the student only counts steps left or right from 0.
- Use the phrase further left means less on every comparison until it is automatic.
- Introduce distance from 0 (absolute value) informally: -7 and 7 are both 7 from 0, so they are the same distance but different numbers.
- Order a longer mixed set including numbers far apart, such as -20, 15, -3, 0, 8.
- Bridge to adding on the line: start at -3 and move 5 right, where do you land?
- Have students write their own real-world signed-number story and a comparison question for a partner.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling placing an integer, naming an opposite, and comparing.
1. Place -5 on a number line from -10 to 10.
Answer: Five equal steps to the left of 0, level with the fifth step, between -6 and -4.
2. What is the opposite of 7?
Answer: -7, the same distance from 0 on the other side.
3. Write the correct sign between -6 and -1 (greater than or less than), and explain.
Answer: -6 is less than -1, because -6 sits further left on the number line.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 what a negative means, on the thermometer (section 1), Lesson 2 placing integers and opposites (sections 2 to 3), Lesson 3 comparing (section 4), Lesson 4 ordering and contexts plus the exit ticket (section 5 and assessment).
- Language to keep saying: 0 is the middle, right is greater and left is less, negative three not minus three, opposite means the mirror across 0. These four phrases pre-empt most of the misconceptions.
- The number-line diagrams all run from -10 to 10 so the mirror image across 0 is obvious. Sweep a finger from 0 outward in both directions when you introduce each idea.
- Hold the line firmly between the label sign (a negative number) and the operation sign (subtraction). Reading negatives aloud as negative, never minus, does most of the work.
- Curriculum note and a US and AU alignment: the US teaches integers, opposites and ordering together in Grade 6 (6.NS.C.5, 6.NS.C.6, 6.NS.C.7). ACARA has no single Year 6 descriptor for this. It introduces locating integers on a number line at Year 5 (AC9M5N01) and brings comparing, ordering and integer operations at Year 7 (AC9M7N07), so in an Australian setting the placing work here suits Year 5 or 6 and the comparing and ordering work is an early look at Year 7.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.