Adding integers

When you reach for it

Many quantities come with a direction. A temperature can be above zero or below it, a game score can be points ahead or points behind, a floor can be above the street or below it. The Integers record both the size and the direction in one number: 5 for five above, -5 for five below. As soon as two of these quantities are combined, you need to add integers.

When both numbers point the same way, adding them is the addition you already know, with a sign attached. The new case is when they point opposite ways. A lift going up 7 floors from level -3, a player on -27 points who wins a hand worth 50, a cave explorer at -90 meters who climbs 35 meters: in each one, the second number partly undoes the first. The sum is what is left after the two cancel as far as they can.

This lesson takes three days. Part 1 reads a sum as a move along the number line. Part 2 shows the same sums with counters, where the canceling can be seen as pairs that are worth nothing. Part 3 turns both pictures into a way of adding any integers without drawing, using absolute value, and uses it on sums of several numbers, expressions and problems.

The procedure

Part 1: adding as moves on the number line

Read an integer as a move along the number line. A positive number is a move to the right, a negative number a move to the left, and the size of the number, its absolute value, is how far the move goes. So +4 means “four to the right” and -4 means “four to the left”.

To add a + b, start at a and make the move b. The point where you land is the sum. This is exactly what addition meant before negative numbers: 2 + 3 was “start at 2 and go 3 to the right”. The only new thing is that a move can now go left.

The three lines in the figure show the three ways a sum can go.

• -1 + (-4): start at -1 and move 4 to the left. The start and the move point the same way, so the sum is farther from zero than either: -5.
• 3 + (-7): start at 3 and move 7 to the left. The first 3 steps bring you back to 0, and the other 4 carry you past it to -4.
• -5 + 8: start at -5 and move 8 to the right. Five steps reach 0, and 3 more land on 3.

Two special moves are worth naming. Adding 0 is a move of no length, so a + 0 = a. And adding the opposite of a number, the same distance on the other side of zero, always lands on 0: starting at 6 and moving 6 to the left lands on 0. For every integer a,

When a negative number follows a plus sign, it is written in brackets, 3 + (-7), so that the two signs do not run together. The bracket changes nothing about the value.

Part 2: adding with counters

A second picture makes the canceling visible. Let a white counter marked + stand for 1 and a shaded counter marked - stand for -1. Any integer can be shown by a pile of counters: 4 is four white counters, -6 is six shaded ones.

One white counter and one shaded counter together are worth 1 + (-1) = 0. Such a pair is a zero pair, and it is the counter version of a + (-a) = 0. Since a zero pair is worth nothing, it can be removed from a pile, or added to one, without changing what the pile is worth.

To add two integers, put both piles together and count what they are worth.

• Same signs. All the counters are the same color, so there are no zero pairs. Count them all: -3 + (-2) is five shaded counters, worth -5.
• Different signs. Line the two colors up against each other and ring each zero pair. The pairing stops when the smaller pile runs out. What is left over is all one color, the color of the bigger pile, and it is the sum.

In the figure, -6 + 4 makes four zero pairs, and two shaded counters are left, so -6 + 4 = -2. The number left over is 6 - 4, the bigger pile minus the smaller, and it is negative because the bigger pile was the negative one.

The two pictures say the same thing. On the number line, the steps of the move that bring you back toward zero are the steps that cancel; with counters, they are the zero pairs. In -6 + 4, the move of 4 to the right undoes four of the six steps that -6 is from zero, and four zero pairs cancel four of the six shaded counters.

Part 3: adding without a model

Nobody counts out 38 white counters and 65 shaded ones, but the counters tell you what would happen. Write |a| for the absolute value of a, its distance from zero. That is also the number of counters in the pile for a.

Two integers with the same sign. There are no zero pairs, so the counters simply add up. The size of the sum is |a| + |b|, and the sign is the sign the two numbers share. For example,

Two integers with different signs. The smaller pile is canceled completely, and an equal number from the bigger pile goes with it. What is left is the larger absolute value minus the smaller one, and it has the sign of the number with the larger absolute value, the one farther from zero. For example, in 42 + (-67) the negative number is farther from zero, 67 - 42 = 25, and

Two opposites. If |a| = |b| and the signs differ, every counter is in a zero pair and the sum is 0, which is neither positive nor negative.

More than two integers. Addition of integers keeps the properties of addition that whole numbers have. The commutative property, a + b = b + a, lets you change the order of the numbers, and the associative property, (a + b) + c = a + (b + c), lets you change which two you add first. So in a long sum you may collect all the positive numbers into one total and all the negative numbers into another, and then combine the two totals. Adding from left to right gives the same answer; collecting first just means only one sum with different signs is left to do.

Brackets and the order of operations work exactly as before: work out what is inside the brackets first, then multiply, then add.

Worked examples

Examples 1 to 3 go with Part 1, the number line. Examples 4 and 5 go with Part 2, the counters. Examples 6 to 10 go with Part 3, adding without a model.

Boundaries

Counters show integers only. A counter is worth 1 or -1, so no pile of counters is worth -2.5. The number line has no such limit, and the rules of Part 3 hold for every signed number, fractions and decimals included: -2.5 + 4 = 1.5 by the same “subtract the absolute values, take the sign of the one farther from zero”. That extension is adding rational numbers.

The sign rule for different signs needs one number to be farther from zero. When the two numbers are equally far from zero they are opposites, and the sum is 0, which has no sign at all. Only then is “the sign of the larger” without meaning.

Adding no longer always makes a number bigger. With whole numbers, a sum is never less than either of the numbers added. With integers, a + b is greater than a exactly when b is positive, equal to a when b = 0, and less than a when b is negative: 5 + (-2) = 3, which is less than 5.

The reordering and regrouping belong to addition. The commutative and associative properties used in Part 3 are properties of addition, and they fail for subtraction: 5 - 3 = 2 but 3 - 5 = -2. A long sum may be rearranged freely only once every subtraction in it has been written as adding a negative number, which is the step the next lesson takes.

Where this goes next

The question in Example 3, which move takes you from one integer to another, is subtraction. Subtracting integers turns every subtraction into an addition of the opposite number, so that the rules of this lesson do all the work, and the reordering and regrouping of Part 3 become available for any mix of plus and minus signs.

The same rules then carry into signed fractions and decimals, into longer expressions with negative numbers, and into algebra. Collecting like terms such as 7x + (-10x) = -3x is adding integers with a variable attached, which is combining like terms. Solving x + 9 = -4 means finding the missing move from 9 to -4, exactly as in Example 3, and that is the start of solving one-step equations.

Exercises

Part 1: moves on the number line

Use a number line to find each sum.

1. -4 + 6
2. 5 + (-8)
3. -3 + (-6)
4. -7 + 7
5. On a straight road running east and west, positions east of the town hall are positive and positions west of it are negative, measured in kilometers. A bus is at -2. It drives 9 kilometers east. Where is it now?
6. What number must be added to 4 to give -3?

Part 2: counters and zero pairs

Use counters to find each sum, and say how many zero pairs you made.

7. -5 + 2
8. 6 + (-1)
9. -4 + (-4)
10. A pile has 7 shaded counters and 3 white counters. Write the addition it shows, and find what it is worth.
11. Describe two different piles of counters that are each worth 3, and show that each is worth 3.

Part 3: two integers without a model

12. 27 + (-41)
13. -63 + 18
14. -35 + (-48)
15. 150 + (-95)
16. -212 + 212

Part 3: several integers and expressions

17. 8 + (-15) + 11 + (-6)
18. -25 + (-12) + 40 + (-9)
19. -6 + 3(-2 + 5)
20. Evaluate y + (-17) when (a) y = -8 and (b) y = 22.
21. Evaluate -m + 6 when (a) m = 10 and (b) m = -10.
22. Find the integer n for which n + (-15) = -6.

Part 3: problems

23. A cave explorer is at -90 meters, that is, 90 meters below the cave entrance. She climbs 35 meters and then goes down a shaft of 25 meters. At what height is she now?
24. A market stall records its profit each day, a loss counting as a negative profit. Over four days it records a loss of 40 dollars, a profit of 65 dollars, a loss of 15 dollars and a profit of 30 dollars. What is its total profit over the four days?

Reasoning

25. Explain, using the number line or counters, why the sum of two negative integers is always negative.
26. Without working out the sum, decide whether -347 + 289 is positive or negative, and explain how you know. Then find it.
27. The sum of two integers is 0. What must be true about the two integers? Explain with the number line.
28. Someone says that adding a number to a always gives an answer bigger than a. For which numbers is that true? Give an example where it fails.
29. Use moves on the number line to explain why -9 + 4 and 4 + (-9) land on the same point.