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.

  1. \(9+ (\text- 1)= 8\)
  2. \(\text- 11+ x= 7\)

5.2: Subtraction with Number Lines

  1. Here is an unfinished number line diagram that represents a sum of 8.

    Number line. 
    1. How long should the other arrow be?
    2. For an equation that goes with this diagram, Mai writes \(3 + {?} = 8\).
      Tyler writes \(8 - 3 = {?}\). Do you agree with either of them?
    3. What is the unknown number? How do you know?
  2. Here are two more unfinished diagrams that represent sums.

    Number line. 


     

    Number line. 


     

    For each diagram:

    1. What equation would Mai write if she used the same reasoning as before?
    2. What equation would Tyler write if he used the same reasoning as before?
    3. How long should the other arrow be?
    4. What number would complete each equation? Be prepared to explain your reasoning.
  3. Draw a number line diagram for \((\text-8) - (\text-3) = {?}\) What is the unknown number? How do you know?

5.3: We Can Add Instead

  1. Match each diagram to one of these expressions:

    \(3 + 7\)

    \(3 - 7\)

    \(3 + (\text- 7)\)

    \(3 - (\text- 7)\)

     

    1. A number line with the numbers negative 10 through 10 indicated. An arrow starts at 3, points to the right, and ends at 10. A second arrow starts at 0, points to the right, and ends at 3.
    2. A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the left, and ends at negative 7. A solid dot is indicated at 3.
    3. A number line with the numbers negative 10 through 10 indicated. An arrow starts at 3, points to the left, and ends at negative 4. A second arrow starts at 0, points to the right, and ends at 3.
    4. A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at 7. A solid dot is indicated at 3.
  2. Which expressions in the first question have the same value? What do you notice?
  3. Complete each of these tables. What do you notice?

    expression value
    \(8 + (\text- 8)\)
    \(8 - 8\)
    \(8 + (\text-5)\)
    \(8 - 5\)
    \(8 + (\text-12)\)
    \(8 - 12\)
    expression value
    \(\text-5 + 5\)
    \(\text-5 - (\text-5)\)
    \(\text-5 + 9\)
    \(\text-5 - (\text-9)\)
    \(\text-5 + 2\)
    \(\text-5 - (\text-2)\)


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
  1. In this system, \(1 \oplus 2 = 3\) and \(2 \oplus 3 = 1\). How can you see that in the table?
  2. What do you think \(3 \oplus 1\) should be?
  3. What about \(3\oplus 3\)?
  4. What do you think \(3\ominus 1\) should be?
  5. What about \(2\ominus 3\)?
  6. 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.

A number line. 

Notice that the value of \(7 + (\text-5)\) is 2. 

A number line. 

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 \({?} + 5= 3\).

A number line. 

Notice that the value of \(3 + (\text-5)\) is -2.

A number line. 

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 \(350-25=325\).