Lesson 3
Changing Elevation
3.1: That's the Opposite (5 minutes)
Warmup
The purpose of this warm up is to think about how we might define opposites in different contexts.
Launch
Arrange students in groups of 2. Give students 2 minute of quiet work time followed by 1 minute of partner discussion, then follow with wholeclass discussion.
Student Facing

Draw arrows on a number line to represents 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.


What’s the opposite?

Running 150 feet east.

Jumping down 10 steps.

Pouring 8 gallons into a fish tank.

Student Response
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Activity Synthesis
Ask a few students to share their responses. Tell students that there are many situations involving changes in quantities where we can represent the opposite action with the opposite value. In this lesson, we are going to investigate several situations like this.
3.2: Cliffs and Caves (15 minutes)
Activity
The purpose of this activity is to see how to tell, from the equation, whether the sum will be positive, negative, or 0, without having to draw a number line diagram every time.
In this activity, students return to the context of height and depth to continue making sense of adding signed numbers. They use the structure of number line diagrams (MP7) to examine a variety of situations, which include starting in the positives, starting in the negatives, moving away from 0, moving towards 0, crossing over 0, and ending exactly on 0. From this variety of situations, students see that the lengths of the arrows in the number line diagrams give important information about the situation. This opens up the discussion of comparing the magnitude of the addends. When the addends have opposite signs, the longer arrow (the number with the larger magnitude) determines the sign of the sum. \(400 + (\text150) = 250\) and \(\text200 + 150 = \text50\).
This activity purposefully gives many problems to figure out, but only asks students to draw a diagram for 3 rows. Watch for students who think they need to draw a number line for every problem. Encourage them to look for and generalize from regularity in the problems (MP8) so that they don't need to always draw a diagram.
For students using the digital version of the materials, be sure they do not use the applet to represent and answer every question and bypass good thinking. Encourage students to figure out an answer for each question first, and then using the applet to check their work.
In preparation for dealing with subtraction of signed numbers, some of the problems involve determining how much the temperature changed, given the initial and final temperatures.
Launch
Explain that a mountaineer is someone who climbs mountains, and a spelunker is someone who explores caves and caverns. Arrange students in groups of 2. Give students 3 minute of quiet work time followed by partner discussion after the first question. Then have students work on the second question and follow with wholeclass discussion.
Supports accessibility for: Conceptual processing; Visualspatial processing
Student Facing
Explore the applet and then answer the questions.

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.


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 20 15 down B 20 10 up C 20 20 up D 20 25 up E 20 50 
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.


What does the expression \(\text45 + 60\) tell us about the spelunker? What does the value of the expression tell us?
Student Response
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Launch
Explain that a mountaineer is someone who climbs mountains, and a spelunker is someone who explores caves and caverns. Arrange students in groups of 2. Give students 3 minute of quiet work time followed by partner discussion after the first question. Then have students work on the second question and follow with wholeclass discussion.
Supports accessibility for: Conceptual processing; Visualspatial processing
Student Facing

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.


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.

What does the expression \(\text75 + 100\) tell us about the spelunker? What does the value of the expression tell us?

Student Response
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Anticipated Misconceptions
Some students may struggle with working backwards to fill in the change when given the final elevation. Ask them "What do you do to 400 to get to 50?" for example.
Activity Synthesis
The most important thing for students to get out of this activity is how to tell from the equation whether the sum will be positive, negative, or 0, without having to draw a number line diagram every time.
 Write all of the equations where the two addends have the same sign. The ask, “What happens when the two addends have the same sign?”
 Write all of the equations where the two addends have the opposite sign. The ask, “What happens when the two addends have opposite signs and . . .
 the number with the larger magnitude is positive?”
 the number with the larger magnitude is negative?”
 Ask, “How can you tell when the sum will be zero?”
Design Principle(s): Optimize output (for generalization)
3.3: Adding Rational Numbers (10 minutes)
Activity
In this activity, no scaffolding is given and students use any strategy to find the sums. Monitor for students who reason in different ways about the sums.
Launch
Arrange students in groups of 2. 2 minutes of quiet work time, followed by partner and wholeclass discussion.
Student Facing
Find the sums.
 \(\text 35 + (30+ 5)\)
 \(\text 0.15 + (\text 0.85) + 12.5\)
 \(\frac{1}{2} + (\text \frac{3}{4})\)
Student Response
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Student Facing
Are you ready for more?
Find the sum without a calculator.
\(\displaystyle 10 + 21 + 32 + 43 + 54 + (\text 54)+ (\text 43)+ (\text 32)+ (\text 21) +(\text 10)\)
Student Response
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Activity Synthesis
Select students to share their solutions. Sequence beginning with diagrams, then with more abstract reasoning. Help students connect the different reasoning strategies.
Design Principle(s): Support sensemaking; Maximize metaawareness
3.4: School Supply Number Line (10 minutes)
Optional activity
The purpose of this activity is to understand that even without knowing the actual numbers, knowing how the signs and magnitudes of two numbers compare is enough to determine whether their sum will be positive or negative.
In this activity, students use the relative position of numbers on the number line to compare them (MP7). Instead of working with specific given numbers, they start with two different, arbitrary lengths. They use these lengths to label various points on a number line. Students see that even though each group may have worked with different lengths, they should still get the same answers to the final set of questions, because everyone started with \(a > b\). Students use their new insights to explain comparisons between signed numbers (MP3).
Identify a group that has a small difference between the lengths of their two objects, as well as a group that has a large difference between the lengths of their two objects, to contrast their number lines during the wholeclass discussion.
Launch
Arrange students in groups of 2. Provide each group a blank strip of receipt tape. The length of tape has to be longer than four times the length of their longer object. Ensure that each group has access to two objects of appropriate length.
Supports accessibility for: Conceptual processing; Memory
Design Principle(s): Optimize output (for justification); Maximize metaawareness
Student Facing
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.
\(a\)
\(b\)
\(2a\)
\(2b\)
\(a + b\)
\(\texta\)
\(\textb\)
\(a + \textb\)
\(b + \texta\)

Complete each statement using <, >, or =. Use your number line to explain your reasoning.
 \(a\) _____ \(b\)
 \(\texta\) _____ \(\textb\)
 \(a + \texta\) _____ \(b + \textb\)
 \(a + \textb\) _____ \(b + \texta\)
 \(a + \textb\) _____ \(\texta + b\)
Student Response
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Anticipated Misconceptions
Some students might forget which symbol means greater than or less than.
Some students may be confused between comparing the value of the expression and the magnitude of the expression. Explain that the number to the left on a number line has the lesser value, even if it may have the greater magnitude (farther away from zero).
Activity Synthesis
The most important thing for students to understand is that even without knowing the actual numbers, knowing how the signs and magnitudes of two numbers compare is enough to determine whether their sum will be positive or negative.
Display the number lines from the selected groups (one with a small difference between \(a\) and \(b\) and one with a large difference).
Discuss:
 Which points are in the same relative position? (For example, \(a + b\) is greater than \(2b\) and less than \(2a\) in both cases.)
 Which points are in different relative positions? (For example, \(2b\) may be greater than \(a\) on one diagram but less than \(a\) on the other.)
For each part of the last question, ask students to indicate whether they used <, >, or =. Have students explain their reasoning until they come to an agreement. It may be helpful to point out the different positions on the number lines that students refer to during their explanations.
If desired, you can extend the discussion by highlighting the fact that all the comparisons in the task statement have the same answer for every group, but this didn't have to be the case. Ask students to invent another comparison that would have a different answer for some of the groups than others. For students who are struggling, ask them to connect positive and negative numbers and addition and subtraction to the previous contexts using MLR 7 (Compare and Connect).
Lesson Synthesis
Lesson Synthesis
Main learning points:
 To add two numbers with the same sign, we add their magnitudes (because the arrows point in the same direction) and keep the same sign for the sum.
 To add two numbers with different signs, we subtract their magnitudes (because the arrows point in the opposite direction) and use the sign of the number with the larger magnitude for the sum.
 When we add a number and its opposite, the sum is zero. These are called additive inverses.
Discussion questions:
 What is the opposite of 5? of 8? of \(\frac13\)? of 0.6?
 What is the sum of a number and its opposite?
 Explain how to add two numbers with the same sign. With different signs.
3.5: Cooldown  Add 'Em Up (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
The opposite of a number is the same distance from 0 but on the other side of 0.
The opposite of 9 is 9. When we add opposites, we always get 0. This diagram shows that \(9 + \text9 = 0\).
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.
To find the sum, we add the magnitudes and give it the correct sign. For example, \((\text5) + (\text4) =\text  (5 + 4)\).
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, \((\text5) + 12 = +(12  5)\).