Lesson 5
Representing Subtraction
Let's subtract signed numbers.
5.1: Equivalent Equations
Consider the equation \(2+3=5\). Here are some more equations, using the same numbers, that express the same relationship in a different way:
\(3 + 2 = 5\)
\(5  3 = 2\)
\(5  2 = 3\)
For each equation, write two more equations, using the same numbers, that express the same relationship in a different way.
 \(9+ (\text 1)= 8\)
 \(\text 11+ x= 7\)
5.2: Subtraction with Number Lines

Here is an unfinished number line diagram that represents a sum of 8.
 How long should the other arrow be?
 For an equation that goes with this diagram, Mai writes \(3 + {?} = 8\).
Tyler writes \(8  3 = {?}\). Do you agree with either of them?  What is the unknown number? How do you know?

Here are two more unfinished diagrams that represent sums.
For each diagram:
 What equation would Mai write if she used the same reasoning as before?
 What equation would Tyler write if he used the same reasoning as before?
 How long should the other arrow be?
 What number would complete each equation? Be prepared to explain your reasoning.
 Draw a number line diagram for \((\text8)  (\text3) = {?}\) What is the unknown number? How do you know?
5.3: We Can Add Instead

Match each diagram to one of these expressions:
\(3 + 7\)
\(3  7\)
\(3 + (\text 7)\)
\(3  (\text 7)\)
 Which expressions in the first question have the same value? What do you notice?

Complete each of these tables. What do you notice?
expression value \(8 + (\text 8)\) \(8  8\) \(8 + (\text5)\) \(8  5\) \(8 + (\text12)\) \(8  12\) expression value \(\text5 + 5\) \(\text5  (\text5)\) \(\text5 + 9\) \(\text5  (\text9)\) \(\text5 + 2\) \(\text5  (\text2)\)
It is possible to make a new number system using only the numbers 0, 1, 2, and 3. We will write the symbols for adding and subtracting in this system like this: \(2 \oplus 1 = 3\) and \(2\ominus 1 = 1\). The table shows some of the sums.
\(\oplus\)  0  1  2  3 

0  0  1  2  3 
1  1  2  3  0 
2  2  3  0  1 
3 
 In this system, \(1 \oplus 2 = 3\) and \(2 \oplus 3 = 1\). How can you see that in the table?
 What do you think \(3 \oplus 1\) should be?
 What about \(3\oplus 3\)?
 What do you think \(3\ominus 1\) should be?
 What about \(2\ominus 3\)?
 Can you think of any uses for this number system?
Summary
The equation \(7  5 = {?}\) is equivalent to \({?} + 5= 7\). The diagram illustrates the second equation.
Notice that the value of \(7 + (\text5)\) is 2.
We can solve the equation \({?} + 5= 7\) by adding 5 to both sides. This shows that \(7  5= 7 + (\text 5)\)
Likewise, \(3  5 = {?}\) is equivalent to \({?} + 5= 3\).
Notice that the value of \(3 + (\text5)\) is 2.
We can solve the equation \({?} + 5= 3\) by adding 5 to both sides. This shows that \(3  5 = 3 + (\text 5)\)
In general:
\(\displaystyle a  b = a + (\text b)\)
If \(a  b = x\), then \(x + b = a\). We can add \(\text b\) to both sides of this second equation to get that \(x = a + (\text b)\)
Glossary Entries
 deposit
When you put money into an account, it is called a deposit.
For example, a person added $60 to their bank account. Before the deposit, they had $435. After the deposit, they had $495, because \(435+60=495\).
 withdrawal
When you take money out of an account, it is called a withdrawal.
For example, a person removed $25 from their bank account. Before the withdrawal, they had $350. After the withdrawal, they had $325, because \(35025=325\).