Lesson 3Changing Elevation

Learning Goal

Let’s solve problems about adding signed numbers.

Learning Targets

I understand how to add positive and negative numbers in general.

Warm Up: That’s the Opposite

Problem 1

Draw arrows on a number line to represent these situations:

The temperature was -5 degrees. Then the temperature rose 5 degrees.
A climber was 30 feet above sea level. Then she descended 30 feet.

Problem 2

What’s the opposite?

Running 150 feet east.
Jumping down 10 steps.
Pouring 8 gallons into a fish tank.

Activity 1: Cliffs and Caves

Problem 1

Explore the applet and then answer the questions.

open applet in presentation mode

A mountaineer is climbing on a cliff. She is 200 feet above the ground. If she climbs up, this will be a positive change. If she climbs down, this will be a negative change.

Complete the table.
starting
elevation
(feet)
change
(feet)
final
elevation
(feet)
A
$+ 200$
$75$ up
B
$+ 200$
$75$ down
C
$+ 200$
$200$ down
D
$+ 200$
$+ 25$
Write an addition equation and draw a number line diagram for B. Include the starting elevation, change, and final elevation in your diagram.

	starting elevation (feet)	change (feet)	final elevation (feet)
A	$+ 200$	$75$ up
B	$+ 200$	$75$ down
C	$+ 200$	$200$ down
D	$+ 200$		$+ 25$

Print Version

A mountaineer is climbing on a cliff. She is 400 feet above the ground. If she climbs up, this will be a positive change. If she climbs down, this will be a negative change.

Complete the table.
starting
elevation
(feet)
change
(feet)
final
elevation
(feet)
A
$+ 400$
$300$ up
B
$+ 400$
$150$ down
C
$+ 400$
$400$ down
D
$+ 400$
$+ 50$
Write an addition equation and draw a number line diagram for B. Include the starting elevation, change, and final elevation in your diagram.

	starting elevation (feet)	change (feet)	final elevation (feet)
A	$+ 400$	$300$ up
B	$+ 400$	$150$ down
C	$+ 400$	$400$ down
D	$+ 400$		$+ 50$

Problem 2

A spelunker is down in a cave next to the cliff. If she climbs down deeper into the cave, this will be a negative change. If she climbs up, whether inside the cave or out of the cave and up the cliff, this will be a positive change.

Complete the table.
starting elevation
(feet)
change
(feet)
final elevation
(feet)
A
$- 200$
$150$ down
B
$- 200$
$100$ up
C
$- 200$
$200$ up
D
$- 200$
$250$ up
E
$- 200$
$- 500$
Write an addition equation and draw a number line diagram for C and D. Include the starting elevation, change, and final elevation in your diagram.

	starting elevation (feet)	change (feet)	final elevation (feet)
A	$- 200$	$150$ down
B	$- 200$	$100$ up
C	$- 200$	$200$ up
D	$- 200$	$250$ up
E	$- 200$		$- 500$

Problem 3

What does the expression $- 75 + 100$ tell us about the spelunker? What does the value of the expression tell us?

Activity 2: Adding Rational Numbers

Find the sums.

$- 35 + (30 + 5)$
$- 0.15 + (- 0.85) + 12.5$
$\frac{1}{2} + (- \frac{3}{4})$

Are you ready for more?

Find the sum without a calculator.

$10 + 21 + 32 + 43 + 54 + (- 54) + (- 43) + (- 32) + (- 21) + (- 10)$

Activity 3: School Supply Number Line

Problem 1

Your teacher will give you a long strip of paper.
Follow these instructions to create a number line.

Fold the paper in half along its length and along its width.
Unfold the paper and draw a line along each crease.
Label the line in the middle of the paper 0. Label the right end of the paper $+$ and the left end of the paper $-$ .
Select two objects of different lengths, for example a pen and a gluestick. The length of the longer object is $a$ and the length of the shorter object is $b$ .
Use the objects to measure and label each of the following points on your number line.
$a$
$b$
$2 a$
$2 b$
$a + b$
$- a$
$- b$
$a + (- b)$
$b + (- a)$

Problem 2

Complete each statement using <, >, or =. Use your number line to explain your reasoning.

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

Lesson Summary

The opposite of a number is the same distance from 0 but on the other side of 0.

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the left, and ends at negative 9. There is a solid dot indicated at negative 9. A second arrow starts at 0, points to the right, and ends at 9. There is a solid dot indicated at 9.

The opposite of -9 is 9. When we add opposites, we always get 0. This diagram shows that $9 + (- 9) = 0$ .

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at 9. A second arrow starts at 9, points to the left, and ends at 0. there is a solid dot indicated at 0.

When we add two numbers with the same sign, the arrows that represent them point in the same direction. When we put the arrows tip to tail, we see the sum has the same sign.

Two identical number lines with the numbers negative 10 through 10 indicated. On the top number line, an arrow starts at 0, points to the right, and ends at 4. A second arrow starts at 4, points to the right, and ends at 7. There is a solid dot indicated at 7. On the bottom number line, an arrow starts at 0, points to the left, and ends at negative 5. A second arrow starts at negative 5, points to the left, and ends at negative 9. There is a solid dot indicated at negative 9.

To find the sum, we add the magnitudes and give it the correct sign. For example, $(- 5) + (- 4) = - (5 + 4)$ .

On the other hand, when we add two numbers with different signs, we subtract their magnitudes (because the arrows point in the opposite direction) and give it the sign of the number with the larger magnitude. For example, $(- 5) + 12 = + (12 - 5)$ .