Start at the first number and move along the line. Adding a negative moves left; subtracting moves the opposite way to adding, so subtracting a negative moves right.
A temperature that rises 9 degrees from −4 °C, a lift that goes down 5 floors from level 2, a bank balance below zero: all of them are moves along a number line.
In the last lesson, further left on the number line meant less. Now the line is for working out: start at the first number, then move.
Adding a positive number moves right. In this lesson you find out what adding a negative does, and what subtracting does, including subtracting a negative.
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.
Subtracting a negative
Start at 5. Which way does subtracting move you? Which way does the minus sign on −3 point?
TryTry it on the board and compare the result with your prediction.
Subtract on the line
The blue jump starts at a and lands on R. When you subtract, the faint dashed arrow shows which way adding b would go. Watch it as you make b smaller, past zero.
Try it yourselfYour edits are saved in this browser.
The board shows 5−3. Change b to −3. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Which way to move
Here is how to use the line for each kind of sum.
Worked exampleBefore each step, say what you think comes next.
−2+5Start at the first number, −2.
−2+5=3Adding a positive moves right: 5 to the right lands on 3.
4+(−6)=−2Adding a negative moves left: from 4, left 6 lands on −2.
5−(−3)Subtracting moves the opposite way to adding.
adding −3 is left 3, so subtracting it is right 3Find the way adding would go, then turn round.
5−(−3)=8From 5, right 3 lands on 8. Subtracting −3 is the same as adding 3.
Read Mia's working one line at a time. Which line moves the wrong way?
Adding moves the way the number points; subtracting moves the opposite way. Sort each sum.
Work out 4+(−6).
Mia works out 5−(−3). Click the line where her working goes wrong.
From the first number, does each sum move right or left?
2 + 3
2 + (−3)
2 − 3
2 − (−3)Adding −3 would go left, so subtracting it goes right.
−5 + 4
−2 − 4
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing. First: start below zero and add a positive.
Need a hint?
Start at −6 and move right 4.
Now the number you add is negative.
Need a hint?
Adding a negative moves left.
A positive subtracted from a smaller number.
Need a hint?
Subtracting 5 moves left 5, past zero.
The move from the Predict step.
Need a hint?
Adding −6 would move left. Subtracting goes the other way.
Another hint
From −4, right 6.
Need a hint?
Where does subtracting a number from itself always land?
Try it yourselfYour edits are saved in this browser.
Work out −6+4. Set the board to check.
Work out 3+(−7). Set the board to check.
Work out 2−5. Set the board to check.
Work out −4−(−6). Set the board to check.
Edge case: work out −5−(−5).
PractiseA few on your own, to check you have it.
Mixed sums
Work out −7+3.
Work out 3−8.
Work out −2−(−6).
At 6 am it is −4 °C. By noon it is 9 degrees warmer. What is the temperature at noon?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A subtraction that grows
Use the + and − buttons and both sliders. The answer must be 10, and you must subtract a negative number.
Stretch: find two different subtractions that give −3, one subtracting a positive and one subtracting a negative.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Press − to subtract.
Another hint
To land above where you start, subtracting has to move right. What kind of number does that?
Make a subtraction that gives 10, where the number you subtract is negative.
Use the board. This ticks when you get there.
For teachers
Students add and subtract integers as moves on a number line, find that adding a negative moves left and that subtracting moves the opposite way to adding, so 5 − (−3) = 8 (AC9M7N07, node 2). About 45 minutes.
I can add and subtract integers by moving along a number line.
I can explain why adding a negative number moves left.
I can explain why 5 − (−3) is 8, not 2.
Before class
Do the Run board and the Make task yourself first. Students need the last lesson: further left is less.
The one rule of the lesson: start at the first number; adding moves the way the second number points (right if positive, left if negative); subtracting moves the opposite way. The board draws the "adding" direction faintly above a subtraction, so students can see the turn-round.
The lesson does not teach "two minuses make a plus". If students bring it, ask them to explain it with the line: subtracting −3 turns round a move to the left.
Running the lesson
Predict: take a show of hands. Most will say 2 (5 − 3). Do not correct it yet.
Run: ask students to make b smaller one step at a time from 3 and say the answer before they look: 2, 3, 4, 5, then past zero, 6, 7, 8. Every step of b smaller adds one to the answer. Point at the dashed arrow when b goes negative.
Investigate: model the worked example live. Draw a number line, then for each line say 'start at…, which way, how far' before drawing the jump. For 5 − (−3), draw the dashed 'adding −3' arrow first, then turn round. Reveal the prepared lines afterwards to compare.
Modify: each task changes one thing. Ask students to say the direction before they set the board.
Practise: the subtracted negative question plants Mia's mistake (−8). A student who gets −8 can draw the dashed arrow first.
Make: collect a few answers. Ask: what do all the b values have in common, and why?
When students get stuck
A student says 5 − (−3) = 2. Ask: Which way would 5 + (−3) go? So which way does 5 − (−3) go? Fix: Draw the adding arrow (left 3), then the subtracting arrow the other way (right 3). Then walk the Run board: as b goes 3, 2, 1, 0, −1, −2, −3 the answer goes 2, 3, 4, 5, 6, 7, 8.
A student starts at zero instead of the first number. Ask: Where does the sum start? Fix: Put a finger on the first number before any move.
A student moves the right distance but lands on the wrong side of zero (−6 + 4 = 2). Ask: Count the steps out loud. Where is zero on the way? Fix: Count each step on the line: −5, −4, −3, −2.
Stretch and support
Stretch: Which is bigger, 3 − (−5) or 3 + 5? Explain with the line. Then work out −2 − (−2) − (−2).
Support: Use the board for every question: set a and b, choose + or −, and say the move before reading R. A paper number line from −10 to 10 on every desk helps.
Where it leads
Next in Number: negative fractions and decimals on the same line, then multiplying and dividing integers.
Where it leads: tables of values with negative inputs (Tables from rules in Linear relations has a quick check of this), gradients and intercepts below zero, and changes in temperature, height and money.
What each step is for
subtracting a negative: Committing to an answer for 5 − (−3) before the board shows it makes the move of subtracting a negative a test of their idea.
subtract on the line: Changing b from 3 to −3 while subtracting shows the jump turn round: subtracting moves the opposite way to adding b, which is drawn faintly above.
which way to move: One rule covers all four cases: adding moves the way the number points (right for positive, left for negative), subtracting moves the opposite way. Mia's mistake treats the minus sign of the number as the operation.
one change at a time: One change at a time: crossing zero when adding, adding a negative, subtracting past zero, subtracting a negative, then the edge case of subtracting a number from itself.
mixed sums: Adding and subtracting with negatives in mixed forms and a context; the review question keeps ordering from the last lesson fresh.
a subtraction that grows: Choosing numbers to make a given answer reverses the lesson: from the result back to the move, and it needs subtracting a negative.
Now you can
Add and subtract integers by moving along a number line.