# Lesson 7

Slopes of Segments

These materials, when encountered before Algebra 1, Unit 4, Lesson 7 support success in that lesson.

## 7.1: Math Talk: Evaluating Fractions (10 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for subtracting and dividing. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to find the slope of the line through 2 given points.

### Launch

Display one problem at a time. Give students quiet think time for each problem, and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

### Student Facing

Evaluate mentally.

$$\frac{102 - 96}{45 - 42}$$

$$\frac{\text{-}8 - 4}{6 - 2}$$

$$\frac{31 - 18}{5 - 10}$$

$$\frac{4 - 9}{12 - 18}$$

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{.5in}}$$’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?”

## 7.2: Connect the Dots (15 minutes)

### Activity

In this activity, students find the slope of a line that goes through 2 given points. In the associated Algebra 1 lesson, students find the average rate of change over an interval. By finding a slope using 2 coordinate pairs, students are focusing on the mechanics of computing the value and can focus more fully on the context and meaning in the Algebra 1 lesson.

### Student Facing

1. Find the slope of the line that connects the given points.
1. $$(0,0)$$ and $$(3,2)$$
2. $$(4,2)$$ and $$(10,7)$$
3. $$(1,\text{-}2)$$ and $$(2,5)$$
4. $$(\text{-}3,4)$$ and $$(\text{-}5,\text{-}2)$$
5. $$(8,3)$$ and $$(10,\text{-}9)$$
2. For each pair of points, find the slope of the line that goes through the 2 points.

1. $$A$$ and $$B$$
2. $$A$$ and $$D$$
3. $$B$$ and $$C$$
4. $$C$$ and $$D$$

### Activity Synthesis

The purpose of the discussion is to clarify the meaning of slopes. Select students to share their solutions including how they arrived at the value. Ask students:

• “If line A has a slope of 2 and line B has a slope of 5, describe how the graphs of these two lines would look different.” (Line B would be more steep. That is, for every increase of 1 in the horizontal direction, the line rises 5 units in the vertical direction for line B, while line A only rises 2 units.)
• “What does it mean for the graph of a line when it has a negative slope?” (It means the graph is decreasing. In other words, for every increase of 1 in the horizontal direction, the line falls some amount in the vertical direction.)
• “Can you think of a situation that connects 2 variables with a negative slope?” (The distance to home as a function of time when someone is walking home.)

## 7.3: Ups and Downs (15 minutes)

### Activity

In this activity, students consider the slope of segments in a situation. Students interpret the meaning of slope in the situation, then find the segments where the slope is greatest and least. In the associated Algebra 1 lesson, students examine the average rate of change in situations. The work of this activity supports students by allowing them to consider only segments between two points connected by a segment so they can focus on greater trends in the Algebra 1 lesson.

### Launch

If necessary, explain the meaning of the unemployment percentage.

### Student Facing

Year Michigan United States
2003 7.2 6
2004 7 5.5
2005 6.8 5.1
2006 7 4.6
2007 7 4.6
2008 8 5.8
2009 13.7 9.3
2010 12.6 9.6
2011 10.4 8.9
2012 9.1 8.1
2013 8.8 7.4
2014 7.2 6.2
2015 5.4 5.3
1. What do the slopes of the segments mean?
2. Find the slope of the segment between 2004 and 2005 for unemployment in Michigan.
3. Between what 2 years is the slope for the United States unemployment percentage greatest?
1. Explain your reasoning using the graph.
2. Explain your reasoning using the table.
4. Between what 2 years is the slope for the United States unemployment percentage the least? Explain or show your reasoning.

### Student Response

• "In terms of the $$x$$ and $$y$$ variables, how can you describe a slope of $$\frac{2}{3}$$ in terms of rate of change? How can you describe a slope of -3 in terms of rate of change?” (A slope of $$\frac{2}{3}$$ means that for every increase of 3 in the $$x$$ variable, $$y$$ increases by 2. A slope of -3 means that for every increase of 1 in the $$x$$ variable, $$y$$ decreases by 3.)