I love the day in calculus that students figure out properties of a definite integral using the Calculus Nspired activity Definite Integral. When I first started using Math Nspired activities, I often copied the student handout. I’ve used them long enough now that there are only a few that I still copy. Instead, I will show some of the questions from the student handout at the front of the room as a guide for student exploration, or I will just use the questions myself to guide our whole class discussion.
We practice look for and express regularity in repeated reasoning. What changes and you move a and b? What stays the same?
Students explore individually first and write what they notice. While they share with a partner, I monitor their conversation, select what they notice that needs to be brought to the attention of the whole class, and sequence their findings in our whole class discussion.
The functions in the activity are carefully chosen – one is odd, and one is even – so that students can begin to generalize what will be true for definite integrals of odd and even functions as well as all functions.
I used to tell my students properties of definite integrals.
Now they tell me.
(At least those who notice the word NOT.)
And so the journey continues, providing students opportunities to figure out the mathematics we want them to know without just telling it to them.