Experience First

Today we want students to practice writing a four-step proof by induction with decreasing amounts of scaffolding. First, have groups complete question 1, which only requires writing expressions for various parts of the proof. Differentiating between and writing expressions for   a     , S   , and S      are all critical sub skills of a proof by induction and this tends to be one of the  biggest challenges for students. By isolating this skill, students can have more success later.

For questions 2-4, send groups of 3-4 students to whiteboards or other non-permanent vertical surfaces. Have each group member use a different colored marker and take turns writing the various parts of the proof. The first person writes the base case, the second person writes the induction hypothesis, the third person writes the two expressions in the induction step, and the fourth person shows algebraically that these expressions are equivalent, and then writes the conclusion sentence. One affordance of using whiteboards is that the teacher has an easy visual of where all the students are at and the students can look around the room for ideas if they get stuck. You can use this time to conference with various groups and give feedback on their proofs.

Once students move on to the next proof, have them switch roles.

One modification is to have one entire group write the base case, the next group write the induction hypothesis, and so on, then switching roles for the subsequent problems. Use this option if you feel your students are not yet ready to do one of these steps individually. You could also use this format for question 2, and then use the original format for questions 3 and 4 to ensure more individual accountability and decrease scaffolding.

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Formalize Later

When monitoring groups, make sure students don’t get too lost in the algebraic manipulations and can still articulate why each part of the proof is important. Nevertheless, the algebra portion of these proofs is excellent review of adding rational functions, making common denominators, factoring, expanding, and order of operations!

Question 4 is different from all previous problems. You may have to review what it means to be a factor of a number and how this can be written in the induction hypothesis.

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