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Lesson 6 of 10

The Other Side of Zero · Lesson 6 of 10

Subtracting Integers with Tokens

“Remove signed tokens and use zero pairs to explain every kind of integer subtraction.”

Learning Objectives

• Represent subtraction as removing a specified kind of token. • Add zero pairs when there are too few tokens to remove. • Explain why added zero pairs preserve the starting value. • Find differences with positive or negative subtracted numbers. • Connect token subtraction to adding an inverse.

Taking away a signed collection

With tokens, subtraction can return to the familiar idea of taking away. The first integer tells you which collection to put down. The second integer tells you which tokens must be removed. Removing a positive number means removing positives; removing a negative number means removing negatives.

For +5 − (+4), start with five positive tokens and remove four positive tokens. One positive remains, so the result is +1. For −7 − (−5), start with seven negative tokens and remove five negative tokens. Two negative tokens remain, giving −2. In each case we remove the kind of tokens named by the subtracted number.

Example — Removing positives

Problem
Show (+5) − (+4).

  1. 1.Put down five + tokens.
  2. 2.Remove four + tokens because the subtracted number is +4.
  3. 3.One + token remains, so the result is +1.
Example — Removing negatives

Problem
Show (−7) − (−5).

  1. 1.Put down seven − tokens.
  2. 2.Remove five − tokens because the subtracted number is −5.
  3. 3.Two − tokens remain: −7 − (−5) = −2. This is also −7 + 5 = −2.
Starting value −7−−−−−−−Tokens removed: −5−−−−−Tokens left: −2−−Each + token is +1; each − token is −1.
Removing five negatives— The negative collection becomes less negative because five negative contributions have been removed.

What if there are too few tokens?

The subtraction +5 − (+6) asks us to remove six positives, but the simple starting picture contains only five. This does not make the subtraction impossible. We can draw a different collection for the same starting value by adding a zero pair.

One zero pair adds a positive token and a negative token together. The pair’s total is zero, so the starting value remains +5. The new picture has six positive tokens and one negative token. We now have enough positive tokens to carry out the removal.

Example — A difference below zero

Problem
Use tokens for (+5) − (+6).

  1. 1.Start with five + tokens. Add one zero pair: there are now six + tokens and one − token, with value 6 − 1 = +5.
  2. 2.Remove the six positive tokens requested by +6.
  3. 3.One negative remains, so +5 − (+6) = −1. Adding a zero pair changed the representation, not the starting number.
Starting collection+++++After one zero pair++++++−After removal of +6−Each + token is +1; each − token is −1.
Making a needed positive token— Before removal, 6 positives and 1 negative still represent +5. Removing six positives leaves −1.

We could add two zero pairs instead. The collection would then have seven positives and two negatives. Removing six positives leaves one positive and two negatives; cancelling a pair gives the same −1. Extra complete zero pairs do not change the result. The smallest sufficient number is usually easiest to draw.

Subtracting a negative from a positive

Consider +4 − (−6). The starting collection has four positives and no negatives. To remove six negatives, add six zero pairs. They supply the needed six negatives and also six extra positives. The value before removal is still +4, because those added pairs total zero.

Example — Removing a negative contribution

Problem
Show +4 − (−6) with tokens.

  1. 1.Start with four + tokens.
  2. 2.Add six zero pairs. Now there are ten positives and six negatives, with total value 10 − 6 = +4.
  3. 3.Remove all six negatives. Ten positives remain, so +4 − (−6) = +10.
Original +4++++After six zero pairs++++++++++−−−−−−After removing −6++++++++++Each + token is +1; each − token is −1.
Why subtracting −6 adds 6— Adding six zero pairs preserves +4. Taking away their negatives leaves the six added positives with the original four.

This picture explains the familiar conversion: subtracting a negative number is adding the corresponding positive number. The extra positives left behind are not a trick. They are the other halves of zero pairs whose negative halves were removed.

Negative starting values

A negative starting collection also works. For −5 − (−7), the simple picture has only five negatives, so add two zero pairs. For −3 − (+5), the simple picture has no positives, so add five pairs. Always ask which kind of token is needed and how many are missing.

Example — Removing more negatives than you first have

Problem
Find (−5) − (−7).

  1. 1.Begin with five negative tokens.
  2. 2.Add two zero pairs to make seven negatives and two positives. The value is still −5.
  3. 3.Remove seven negatives. The two positives left give +2. Check the equivalent addition: −5 + 7 = +2.
Example — Removing positives from a negative number

Problem
Find (−3) − (+5). How many zero pairs are needed?

  1. 1.Begin with three negative tokens and no positives.
  2. 2.Add five zero pairs, making five positives and eight negatives. Their value is 5 − 8 = −3.
  3. 3.Remove the five positives. Eight negatives remain, giving −8. Five pairs are the minimum needed for this starting picture.
Starting integerNumber subtractedMinimum added pairs with a single-sign startResult
+9+123−3
−2−64+4
−3+1010−13
+8−77+15

Both models agree: tokens show what remains after removal, while the number line shows the equivalent movement or change. For instance, −3 − 5 = −8 can also be read as −3 + (−5) = −8. Once the reasoning is clear, the inverse rule gives the answer without needing to draw every token.

Preserve the value before removing anything

Adding only a positive token changes the starting integer. Add complete zero pairs whenever you need more tokens, and then remove precisely the sign and number named in the subtraction.

Quiz

Quick check

For −7 − (−5), which tokens must be removed?

Quick check

What is +5 − (+6)?

Quick check

Starting with only four positives, how many zero pairs are needed for +4 − (−6)?

Quick check

What is −5 − (−7)?

Quick check

Why can you add zero pairs?

Quick check

What is −3 − (+5)?

Practice Problems

Practice Problems
  1. Use tokens to find +10 − (+7), −8 − (−4), and −9 − (−4).
  2. Find +9 − (+12), −5 − (−7), and −2 − (−6). State how many zero pairs are needed with a single-sign starting picture.
  3. Evaluate −5 − (−7), +10 − (+13), −7 − (−9), +3 − (+8), −2 − (−7), and +3 − (+15).
  4. Evaluate −3 − (+10), +8 − (−7), −5 − (+9), −9 − (+10), +6 − (−4), and −2 − (+7).
  5. Draw +4 − (−6) and explain each stage.
  6. Draw −3 − (+5) using six zero pairs instead of five. Explain why the result is still −8.
  7. A student adds two negative tokens to make enough negatives for a subtraction. Explain why this changes the starting value.
  8. Show −2 − (−2), −2 − 0, and 0 − (−2) with tokens.

+3, −4, −5.

Key Takeaways

Key Takeaways

• Subtraction removes tokens of the sign named by the second integer. • Zero pairs can supply missing tokens without changing the value. • Subtracting negatives leaves an equivalent positive change. • Subtracting positives gives an equivalent negative change. • Tokens and number lines support the same addition-of-the-inverse rule.

Previous · Lesson 5

Adding Integers with Zero Pairs

Next · Lesson 7

Integers in Money, Height, Time, and Temperature