5.2.4: Representing Subtraction

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Illustrative Mathematics
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Lesson

Let's subtract signed numbers.

Exercise $\PageIndex{1}$: 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:

$3+2=5\qquad\qquad 5-3=2\qquad\qquad 5-2=3$

For each equation, write two more equations, using the same numbers, that express the same relationship in a different way.

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

Exercise $\PageIndex{2}$: Subtraction with Number Lines

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

Figure $\PageIndex{1}$: A number line, 21 evenly spaced tick marks. Tick marks begin at negative 10 and count up by 1 to the final tick mark at 10. Above the number line is an arrow that begins at 0 and stops at 3. There is a dot on the tick mark labeled 8.

How long should the 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?

2. Here are two more unfinished diagrams that represent sums.

Figure $\PageIndex{2}$: Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1s. An arrow, pointing to the left, goes from 0 to negative 3.

Figure $\PageIndex{3}$: Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1s. There is a point at negative 8. An arrow, pointing to the right, goes from 0 to 3.

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.

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

Exercise $\PageIndex{3}$: We can Add Instead

1. Match each diagram to one of these expressions:

$3+7\qquad\qquad 3-7\qquad\qquad 3+(-7)\qquad\qquad 3-(-7)$

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?

Table $\PageIndex{1}$
expression	value
$8+(-8)$
$8-8$
$8+(-5)$
$8-5$
$8+(-12)$
$8-12$

Table $\PageIndex{2}$
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 $2\ominus 1=1$. The table shows some of the sums.

Table $\PageIndex{3}$
$\oplus$	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.

Figure $\PageIndex{8}$: 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.

Figure $\PageIndex{9}$: 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 $?+5=3$.

Figure $\PageIndex{10}$: 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 $-2$.

Figure $\PageIndex{11}$: 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 $x=a+(-b)$

Glossary Entries

Definition: 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 $435+60=495$.

Definition: 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 $350-25=325$.

Practice

Exercise $\PageIndex{4}$

Write each subtraction equation as an addition equation.

$a-9=6$
$p-20=-30$
$z-(-12)=15$
$x-(-7)=-10$

Exercise $\PageIndex{5}$

Find each difference. If you get stuck, consider drawing a number line diagram.

$9-4$
$4-9$
$9-(-4)$
$-9-(-4)$
$-9-4$
$4-(-9)$
$-4-(-9)$
$-4-9$

Exercise $\PageIndex{6}$

A restaurant bill is $59 and you pay $72. What percentage gratuity did you pay?

(From Unit 4.3.1)

Exercise $\PageIndex{7}$

Find the solution to each equation mentally.

$30+a=40$
$500+b=200$
$-1+c=-2$
$d+3,567=0$

Exercise $\PageIndex{8}$

One kilogram is 2.2 pounds. Complete the tables. What is the interpretation of the constant of proportionality in each case?

Table $\PageIndex{4}$
pounds	kilograms
$2.2$	$1$
$11$
$5.5$
$1$

______ kilogram per pound

Table $\PageIndex{5}$
kilograms	pounds
$1$	$2.2$
$7$
$30$
$0.5$

______ pounds per kilogram

(From Unit 2.1.3)

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

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

pounds	kilograms
\(2.2\)	\(1\)
\(11\)
\(5.5\)
\(1\)

kilograms	pounds
\(1\)	\(2.2\)
\(7\)
\(30\)
\(0.5\)

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Table \(\PageIndex{3}\)
\(\oplus\)	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3