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TJ: CA II.7 The Derivatives of cf and f+g

This is a long activity but I found that after initially floundering with how to start an investigation to develop a derivative rule that students really got into this. They really enjoyed being able to check their rules in Mathematica. One thing that you have to keep on top of is that they are understanding what they are checking and that each group really is carrying out a proper investigation. One way that it might help to highlight this is to put some examples of investigations up and ask students if that investigation is really checking the rule. I would only do this after all of the students have completed their own investigations. The problem some groups run into is not seeing the difference between the two sides of the equality that they are looking at - so for instance they define and f and a g but they never define a function h=f+g. So they are never comparing a h' (calculated by Mathematica and we are trying to figure out how it might have done that) to f'+g'. They also struggle with the algebraic proof. Some students though like that they are able to get the result by themselves - again though it is important to talk about what we are trying to show. Although this causes problems for students it is also a great place to talk about mathematical process. Other than those issues all went well with developing the rules and I think students leave this activity feeling pretty convinced that these rules are true.
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