In this activity, students return to the context of height and depth to continue making sense of adding signed numbers. They 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 length of the arrows in the number line diagrams correspond to the magnitude of the addends. Using repeated reasoning, students see that when the addends have opposite signs, the longer arrow (the number with the larger magnitude) determines the sign of the sum (MP8).

In the digital version of the activity, students use an applet to represent a mountaineer climbing up or down a cliff. The applet allows students to set the starting elevation and the elevation change. The digital version may reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.

The purpose of this discussion is to emphasize how to tell whether the sum of two numbers will be positive, negative, or 0 based on the magnitude of the addends. Begin by displaying both of the tables from the Task Statement, and invite students to share their responses, recording them for all to see. Discuss the following:

• “What addition equations can be written to describe the mountaineer's elevation in the first row of each table?” (Print:  and . Digital:  and .)
• “What patterns do you notice?” (Adding two positive numbers results in another positive number, and adding two negative numbers results in another negative number.)
• Invite students to share their equations for row B in the first table and rows C and D from the second table. Record their responses for all to see, and ask, “What patterns do you notice?” (All these equations are adding a positive number and a negative number. The sum can be positive, negative, or 0.)
• “When adding two numbers with different signs, how can we tell what the sign of the sum will be?” (The sum will have the sign of the addend with the larger magnitude. If the magnitudes of both numbers are the same, the sum will be 0.)

In this activity, students find sums of signed numbers. No scaffolding is given, and students may use any strategy to find the sums.

Monitor for students who use different strategies to find the sums, such as:

• Drawing a diagram or number line.
• Reasoning about the magnitude and sign of the numbers.

Invite selected students to share their solutions and reasoning to the first three problems. Help students connect the different reasoning strategies by asking the following questions:

• “What do the different strategies have in common? How are they different?”
• “Did anyone solve the problem the same way but would explain it differently?”

In this activity, students use the structure of the number line and the relative positions of numbers 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 locations on a number line. Students see that even though each group may have worked with different lengths for and , they still get the same answers to the final set of questions because everyone started with . Students use a number line to support their reasoning about comparisons between signed numbers.

Identify a group that has a small difference between the lengths of their two objects and a group that has a large difference between the lengths of their two objects.

The goal of this discussion is for students 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. Begin by displaying the number lines from the previously selected groups for all to see (one with a small difference between and and one with a large difference).

Discuss the following questions:

• “Which points are in the same relative position?” (For example,  is greater than and less than in both number lines.)
• “Which points are in different relative positions?” (For example, may be greater than on one diagram but less than on the other.)

Then invite students to indicate whether they used <, >, or = and explain their reasoning for each part of the last question. Point out the different positions on the number lines that students refer to during their explanations.

If time allows, extend the discussion by highlighting the fact that all the comparisons in the Task Statement had the same answer for every group. Ask students to invent another comparison that would have a different answer for some of the groups than others.