Mathematics Year 7: Adding and subtracting integers
Mathematics, Year 7 (Portugal): Add by moving on the number line, turn every subtraction into an addition, and know which properties hold.
Adding is walking on the line
Adding a positive number moves you right on the number line; adding a negative moves you left. Start at 3 and add −5: you go 5 units left and land on −2, so 3 + (−5) = −2. In a story: 3 € in your pocket and a 5 € expense leave you owing 2 €. Order does not matter: −4 + 7 = 7 + (−4) = 3 (commutative), and you can group as you like: (−4 + 7) + (−5) = −4 + (7 + (−5)) = −2 (associative).
Subtracting is adding the opposite
Subtracting 5 from 3 is the same as adding the opposite of 5: 3 − 5 = 3 + (−5) = −2. So every subtraction becomes an addition. But subtraction is not commutative: 3 − 5 = −2 and 5 − 3 = 2. And it is not associative: (10 − 4) − 3 = 3 but 10 − (4 − 3) = 9. Addition has both properties, so to regroup freely, first rewrite everything as an addition.
A bank statement
A balance starts at 0 € and has these movements, in euros: +45, −30, +12, −50, +28, −15. Left to right: 45 + (−30) = 15; 15 + 12 = 27; 27 + (−50) = −23; −23 + 28 = 5; 5 + (−15) = −10. Or by grouping: positives 45 + 12 + 28 = 85, negatives (−30) + (−50) + (−15) = −95, and 85 + (−95) = −10. Both paths give −10 €: the commutative and associative properties let you choose the simpler one.
Subtracting does not always make smaller
The reasonable mistake comes from natural numbers, where subtracting always made the number smaller. Here, subtracting a negative makes it bigger: 3 − (−4) = 3 + 4 = 7. A real case: the temperature went from −6 °C to 5 °C. The rise is 5 − (−6) = 5 + 6 = 11 degrees, not −1 and not 1. Another picture: if someone takes away a debt of 4 €, you end up 4 € better off.
Lifts, accounts and thermometers
Adding and subtracting integers happens in daily life: a lift moving between floors above and below the ground floor, a bank statement with credits and debits, the change in temperature between morning and afternoon, game points that can drop below zero. In each case you choose whether to go left to right or to group the positives and the negatives first.
Test what you learned
The temperature is −3 °C and then drops another 4 degrees. Which addition gives the new temperature?
- a) −3 + 4 = 1
- b) −3 + (−4) = −1
- c) −3 + (−4) = −7
- d) 3 + 4 = 7
Correct answer: c) −3 + (−4) = −7 — Yes: a drop of 4 is adding −4. From −3 you go 4 units left and reach −7.
Compare (10 − 4) − 3 with 10 − (4 − 3). What is true?
- a) Both equal 3, because subtraction is associative
- b) The first is 3 and the second is 9: the grouping changes the result
- c) The first is 9 and the second is 3
- d) Both equal 9
Correct answer: b) The first is 3 and the second is 9: the grouping changes the result — Yes: (10 − 4) − 3 = 6 − 3 = 3, but 10 − (4 − 3) = 10 − 1 = 9. Subtraction is not associative.
The temperature went from −6 °C to 5 °C. How many degrees did it rise?
- a) −1, from 5 + (−6)
- b) 1, from 6 − 5
- c) −11
- d) 11, from 5 − (−6) = 5 + 6
Correct answer: d) 11, from 5 − (−6) = 5 + 6 — Yes: subtracting −6 is adding its opposite, 6. From −6 to 0 is 6 degrees, and from 0 to 5 is 5 more: 11.
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