On a number line, the number further left is less. So −7 is less than −2, even though 7 is more than 2.
Thredbo at −9 °C, Canberra at −3 °C: which is colder? Below zero, the number that looks bigger is the smaller one.
Numbers below zero are negative: −1,−2,−3 and so on. Together with zero and the positive whole numbers they are the integers. On a number line, negatives sit to the left of zero, with −1 next to zero.
In this lesson you find out how to tell which of two integers is less, and write it with < ("is less than") or > ("is greater than").
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Which is greater?
Think of temperatures: is −7 °C or −2 °C warmer?
TryTry it on the board and compare the result with your prediction.
Two points on the line
Move A with slider a and B with slider b (or the − and + buttons). The board writes the comparison under the line. Try a few pairs first: two positives, a positive and a negative, then −7 and −2.
Try it yourselfYour edits are saved in this browser.
Put A at −7 and B at −2. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Further left is less
Here is how to put integers in order, using where each one sits on the number line.
Worked exampleBefore each step, say what you think comes next.
−2,−8,3,0Put these in order from least to greatest.
place each on the lineNegatives go left of zero: −2 is 2 left, −8 is 8 left.
−8 is furthest leftFurthest left is least, so −8 comes first.
−8,−2,0,3Read the line from left to right.
−8<−2Between two negatives, the one further from zero is less.
Read Ari's working one line at a time. Which line orders the numbers by their size alone?
Sort each number: is it less than −3, or greater?
Ari puts four numbers in order. Click the line where the reasoning goes wrong.
Less than −3, or greater than −3?
−6
−1
0
−10
2
−4−4 is one place left of −3.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing. First: drag P to a number between −8 and −4.
Need a hint?
Which way from −4 is less: left or right?
Another hint
Go left of −4, but not as far as −8.
The same idea in degrees. Warmer means further right.
Need a hint?
Warmer is further right. Is it above or below zero?
Another hint
Right of Canberra (−3), but still left of 0.
Zero is greater than every negative number.
Need a hint?
Which way is less: left or right? Is zero left or right of −1?
When is the number further from zero the smaller one?
Try it yourselfYour edits are saved in this browser.
Drag P to a number that is less than −4 and greater than −8.
Use the board. This ticks when you get there.
Drag P to a temperature warmer than Canberra but still below zero.
Use the board. This ticks when you get there.
Drag P to a number less than −1. Then try zero: is it less or greater than −1?
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Compare and order
Review: what number is 4 places to the left of 1 on a number line?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A surprising pair
Find a pair where the less number has the bigger size without its sign. The sentence under the line tells you which is less.
Stretch: write three temperatures below zero, then put them in order from warmest to coldest.
Try it yourselfYour edits are saved in this browser.
Need a hint?
P must be further left than Q.
Another hint
Try putting P below zero, further from zero than Q is: P at −6 and Q at −2, or Q at 3.
Place P and Q so that P<Q, but P is further from zero than Q.
Use the board. This ticks when you get there.
For teachers
Students place integers on a number line, find that further left is less (so −7 < −2), and compare and order integers with < and > (AC9M7N07, node 1). About 40 minutes.
I can put a set of integers, including negatives, in order from least to greatest.
I can write a comparison between two integers with < or >.
I can explain why −10 is less than −1, even though 10 is more than 1.
Before class
Do the Run board and the Make task yourself first. Students should be able to find negative numbers on a number line (Year 6) and know what a temperature below zero means.
The one idea of the lesson is "further left is less". The common mistake is to compare negatives by their size without the sign (−8 > −2 because 8 > 2), and it is planted only after the method, in Ari's working.
The lesson says "further from zero" for size without the sign, not "absolute value". The primer has the word if you want it.
Running the lesson
Predict: take a show of hands. Students who chose −7 usually say 7 is bigger. Do not correct it yet: the board will.
Run: ask students to try two positives first, then a positive and a negative, then −7 and −2. Ask: what decides which is less every time?
Investigate: model the worked example live. Draw a line on the board, place −2, −8, 3 and 0 one at a time, saying 'left of zero, 8 places' and so on, then read left to right. Reveal the prepared lines afterwards to compare.
Modify: each task changes one thing. Before the temperature task ask: which way is warmer?
Practise: the sign question plants the reversed sign. A student who picks < can place both on the line and say which is further left.
Make: collect a few pairs. Ask: could P be positive? (No: a positive further from zero than Q would be greater than Q.)
When students get stuck
A student says −8 is greater than −2. Ask: Which is colder, −8 °C or −2 °C? Fix: Place both on the line. Further left is less, and −8 is further left. Say it as 'eight below zero' and 'two below zero'.
A student puts negatives in the order −1, −2, −3 when asked for least to greatest. Ask: Which of these is furthest left on the line? Fix: Point at the line and read it from left to right: −3, −2, −1.
A student mixes up < and >. Ask: Which number is greater? Which way does the wide end of the sign face? Fix: The wide end faces the greater number: −6 > −9 reads '−6 is greater than −9'.
Stretch and support
Stretch: Which is greater, −0.5 or −5? Place both on the line. Then: is there an integer between −3 and −2?
Support: Use the Run board for every comparison: set a and b to the two numbers and read the sentence. A paper number line from −10 to 10 on every desk helps.
Where it leads
Next: adding and subtracting integers, as moves along the same line.
Where it leads: plotting points in all four quadrants, reading graphs with negative values, and comparing negative fractions and decimals (AC9M7N04).
What each step is for
which is greater?: Committing to an answer for −7 and −2 before using the line makes the board a test of their idea about negatives, not a demonstration.
two points on the line: Placing −7 and −2 and reading the sentence shows the rule: further left is less, for negatives as well as positives.
further left is less: Further left is less, so the order of negatives runs the opposite way to their sizes without the sign: −8 is less than −2. Ari's mistake orders by the size alone.
one change at a time: One change at a time: a number between two negatives, the same idea in temperatures, then the edge case where the number further from zero is the smaller one.
compare and order: Comparing with < and >, ordering, and a context, mixed; the review question keeps locating integers from Year 6 fresh.
a surprising pair: Building a pair where the less number is the one further from zero reverses the lesson: from the rule to an example that shows it.
Now you can
Put a set of integers, including negatives, in order from least to greatest.
Write a comparison between two integers with < or >.
Explain why −10 is less than −1, even though 10 is more than 1.