Lesson 5Representing Subtraction

Learning Goal

Let’s subtract signed numbers.

Learning Targets

I can explain the relationship between addition and subtraction of rational numbers.
I can use a number line to subtract positive and negative numbers.

Lesson Terms

deposit
withdrawal

Warm Up: 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.

$9 + (- 1) = 8$
$- 11 + x = 7$

Activity 1: Subtraction with Number Lines

Problem 1

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

A number line with the numbers negative 10 through 10 indicated. Above the number line, an arrow pointing right starts at 0 and ends at 3. A solid dot is indicated at 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?

Problem 2

Here is one unfinished diagram that represents a sum.

A number line with the numbers negative 10 through 10 indicated. Above the number line an arrow pointing left starts at 0 and ends at -3. A solid dot is indicated at 8.

Here is another unfinished diagram that represents a sum.

For the first image, what equation would Mai write if she used the same reasoning as before? What about the second image?
For the first image, what equation would Tyler write if he used the same reasoning as before? What about the second image?

Problem 3

Draw a number line diagram for $(- 8) - (- 3) = ?$ What is the unknown number? How do you know?

Activity 2: We Can Add Instead

Problem 1

Match each diagram to one of the expressions

Diagram 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.

Diagram 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.

Diagram 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.

Diagram 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.

$3 + 7$
$3 - 7$
$3 + (- 7)$
$3 - (- 7)$

Diagram 1
Diagram 2
Diagram 3
Diagram 4

Problem 2

Which expressions in the first question have the same value? What do you notice?

Problem 3

Complete each of these tables. What do you notice?

expression	value
$8 + (- 8)$
$8 - 8$
$8 + (- 5)$
$8 - 5$
$8 + (- 12)$
$8 - 12$

expression	value
$- 5 + 5$
$- 5 - (- 5)$
$- 5 + 9$
$- 5 - (- 9)$
$- 5 + 2$
$- 5 - (- 2)$

Are you ready for more?

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 $2 ⊖ 1 = 1$ . 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 $3 ⊖ 1$ should be?
What about $2 ⊖ 3$ ?
Can you think of any uses for this number system?

Lesson Summary

The equation $7 - 5 = ?$ is equivalent to $? + 5 = 7$ . The diagram illustrates the second equation.

A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 2, and is labeled with a question mark. A second arrow starts at 2, points to the right, ends at 7, and is labeled "plus 5." There is a solid dot indicated at 7.

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

A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 7, and is labeled "plus 7". A second arrow starts at 7, points to the left, ends at 2, and is labeled "minus 5". There is a solid dot and a question mark labeled at 2.

We can solve the equation $? + 5 = 7$ by adding -5 to both sides. This shows that $7 - 5 = 7 + (- 5)$

Likewise, $3 - 5 = ?$ is equivalent to $? + 5 = 3$ .

A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the left, ends at negative 2, and is labeled with a question mark. A second arrow starts at negative 2, points to the right, ends at 3, and is labeled "5". There is a solid dot indicated at 3.

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

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, ends at 3, and is labeled "plus 3". A second arrow starts at 3, points to the left, ends at negative two, and is labeled "minus 5". There is a solid dot and a question mark labeled at 2.

We can solve the equation $? + 5 = 3$ by adding -5 to both sides. This shows that $3 - 5 = 3 + (- 5)$

In general:

$a - b = a + (- b)$

If $a - b = x$ , then $x + b = a$ . We can add $- b$ to both sides of this second equation to get that $x = a + (- b)$