Take away a bigger number from a smaller one by adding zero pairs, and see why subtracting is the same as adding the opposite.
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Positive tokens mean +1, negative tokens mean −1, and one of each makes zero.
In the book, a green token stands for +1 and a red token stands for −1.
In our pictures, a positive token is an outlined circle with , and a negative token is a filled circle with . (The book notes that tokens of different shapes work too.)
Watch 18 positives leave a bag that only had 7.
Problem. Find with tokens. (Textbook p. 28)
Why 7 − 18 is the same as 7 + (−18).
Subtracting a number is the same as adding its additive inverse (its opposite).
a − b = a + (−b), and the opposite of an opposite.
Complete the token steps for Math Talk (b).
Math Talk (b), Textbook p. 28: show 4 − (−12) with tokens.
Start with positives. We must remove negatives, but there are none. So we put in zero pairs. Then we remove the 12 negatives.
Left in the bag: positives. So 4 − (−12) = , which is the same as 4 + .
Five subtractions and opposites to test the zero-pair idea.
Explain in your own words why 7 − 18 = 7 + (−18).
Math Talk (Textbook p. 28). Use tokens to argue that . Describe the tokens for each side and what is left.
What do you start with, what do you add, and what do you remove?
What do you put together, and what cancels?
What is left in each case?