Lesson 6Subtracting Rational Numbers

Let's bring addition and subtraction together.

Learning Targets:

  • I can find the difference between two rational numbers.
  • I understand how to subtract positive and negative numbers in general.

6.1 Number Talk: Missing Addend

  1. Solve each equation mentally.

    247 + c = 458

    c + 43.87 = 58.92

    \frac{15}{8} + c = \frac{51}{8}

  2. Rewrite each addition equation as a subtraction equation.

6.2 Expressions with Altitude

A mountaineer is changing elevations. Write an expression that represents the difference between the final elevation and beginning elevation. Then write the value of the change. The first one is done for you.

“Mountaineer” by Orca via Pixabay. Public Domain.
beginning
elevation
(feet)
final
elevation
(feet)
difference
between final
and beginning
change
+400 +900 900 - 400 +500
+400 +50
+400 -120
-200 +610
-200 -50
-200 -500
-200 0

Are you ready for more?

Fill in the table so that every row and every column sums to 0. Can you find another way to solve this puzzle?

-12 0 5
0 -18 25
25 -18 5 -12
-12 -18
-18 25 -12
-12 0 5
0 -18 25
25 -18 5 -12
-12 -18
-18 25 -12

6.3 Does the Order Matter?

  1. Find the value of each subtraction expression.
    A B
    3 - 2 2 - 3
    5 - (\text-9) (\text-9) - 5
    (\text-11) - 2 2 - (\text-11)
    (\text-6) - (\text-3) (\text-3) - (\text-6)
    (\text-1.2) - (-3.6) (\text-3.6) - (\text-1.2)
    (\text-2\frac12) - (\text-3\frac12) (\text-3\frac12) - (\text-2\frac12)
  2. What do you notice about the expressions in Column A compared to Column B?
  3. What do you notice about their values?

Lesson 6 Summary

When we talk about the difference of two numbers, we mean, “subtract them.” Usually, we subtract them in the order they are named. For example, the difference of +8 and \text-6 is 8 - (\text-6) .

The difference of two numbers tells you how far apart they are on the number line. 8 and -6 are 14 units apart, because 8 - (\text-6) = 14 :

A number line with the numbers negative 10 through 10 indicated. Two solid dots are on the number line located at, negative 6 and 8. An arrow starts at negative 6, points to the right, ends at 8, and is labeled "positive 14."

Notice that if you subtract them in the opposite order, you get the opposite number:

A number line with the numbers negative 10 through 10 indicated. Two solid dots are on the number line located at, negative 6 and 8. An arrow starts at 8, points to the left, ends at negative 6, and is labeled "negative 14."

(\text-6)-8 = \text-14

In general, the distance between two numbers a and b on the number line is |a - b| . Note that the distance between two numbers is always positive, no matter the order. But the difference can be positive or negative, depending on the order.

Lesson 6 Practice Problems

  1. Write a sentence to answer each question:

    1. How much warmer is 82 than 40?

    2. How much warmer is 82 than -40?

  2. A company produces screens of different sizes. Based on the table, could there be a relationship between the number of pixels and the area of the screen? If so, write an equation representing the relationship. If not, explain your reasoning.

    square inches of screen number of pixels
    6 31,104
    72 373,248
    105 544,320
    300 1,555,200
    1. What is the difference in height between 30 m up a cliff and 87 m up a cliff? What is the distance between these positions?

    2. What is the difference in height between an albatross flying at 100 m above the surface of the ocean and a shark swimming 30 m below the surface? What is the distance between them if the shark is right below the albatross?

  3. Find each difference.

    1. (\text-5) - 6

    2. 35 - (\text-8)

    1. \frac25 - \frac35

    2. \text-4\frac38 - (\text-1\frac14)

  4. A family goes to a restaurant. When the bill comes, this is printed at the bottom of it:

    Gratuity Guide For Your Convenience:
    15% would be $4.89
    18% would be $5.87
    20% would be $6.52

    How much was the price of the meal? Explain your reasoning.

  5. Which is a scaled copy of Polygon A? Identify a pair of corresponding sides and a pair of corresponding angles. Compare the areas of the scaled copies.