Changing temperature of cooling liquid
Suggested approach part i
Showing just the first model, introduce the graph as that of a cooling liquid and its temperature measured during an experiment. What statements can students make? What questions do they have? (Expected responses include starting temperature, duration of observations etc).
You could ask students what they expect would happen to the temperature if we watched for another 800 seconds. Ask about the rate of cooling and what units we should use.
Suggested approach part ii
Draw students' attention to the linear rate of cooling. What would be a more realistic graph shape? Hide model 1 and show model 2. What do students think? What's the same and what's different about the two models?
Suggested approach part iii
Showing model 2 and point at time, ask students to consider what is happening at various positions of the point. E.g. what time are we at? What is the temperature? At this instant point in time, is the liquid cooling faster or slower than it was earlier? What is the average rate of cooling over the whole 800 seconds? Give me a time that it was cooling faster than the average rate of cooling. Etc.
Suggested approach part iv
Depending upon students' familiarity with the use of tangents to estimate the rate of change in such contexts, show and discuss measure at point in time. How do we use this to quantify the rate of change? What's the rate of change at t = 100s? Is the rate of cooling quicker or slower than the average rate of change?
At what time does the instantaneous rate of change happen to be the same value as the average rate of change?