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. As students develop the skill of calculating slope or average rate of change, they look for and express regularity in repeated reasoning (MP8).

The purpose of the discussion is to clarify the meaning of slopes.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they calculate the slope between two points. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Select students to share their solutions, including how they arrived at the value. Here are some questions for discussion:

• “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 rises only 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.)
• “What is an example 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.)

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. As students interpret the values in the context of this situation, they reason abstractly and quantitatively (MP2). 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.

The purpose of the discussion is to interpret slope in situations and begin to recognize trends. Select students to share their solutions and reasoning. Here are some questions for discussion:

• "In terms of the and variables, how can you describe a slope of in terms of rate of change? How can you describe a slope of -3 in terms of rate of change?” (A slope of means that for every increase of 3 in the variable, the variable increases by 2. A slope of -3 means that for every increase of 1 in the variable, the  variable decreases by 3.)
• “Are the years for the greatest and least slopes in Michigan the same years as for the United States?” (No. The greatest slope in Michigan is the same as for the US, but the least slope in Michigan was between 2010 and 2011.)