The goal of this discussion is to continue to connect graphical representations and function notation. Either reveal the correct responses, or invite students to share their responses. Focus on the last question, drawing slope triangles on the graph as needed. This can be connected back to the grade 8 understanding of slope based on the side lengths of similar right triangles. The important point to emphasize is that for a given line, we can choose any two points to calculate its slope. Here are some questions for discussion:

• “What pairs of points were used to calculate the slope in this work?” ( and , and and .)
• “Does it matter which points are chosen? Explain your reasoning.” (No, any two points from a linear situation will give the same rate of change or slope.)

The purpose of this discussion is to see that, in nonlinear situations, the average rate of change is not constant but is still a useful calculation. Reintroduce the term average rate of change as what we’d typically call the rate of change between any two points in a function that is not perfectly linear. Here are some questions for discussion:

• “How was the process for calculating the average rate of change between two points similar or different to calculating the slope between two points in the earlier activity?” (The process was the same, but the result was not constant in this activity.)
• “Why would we calculate the slope of a linear situation?” (The slope tells us something consistent about a linear situation. That every time the -value changes by a specific amount, the -value changes by a specific amount.)
• “What does calculating the average rate of change between and tell us about this situation? Why would that be useful to know?” (The average rate of change between and tells us how the temperature changed on average over that time period. This is useful to know because the average rate of change shows larger trends in the data without being influenced by unusual points in the middle.)