Experience First

Little Red Riding Hood is on her way to grandma’s house but something must be wrong with her pockets, because she keeps dropping crumbs of cake--and in a rather unusual pattern! In this lesson, students learn about geometric sequences by describing the number of crumbs Little Red drops on successive tiles. First, we want students to see the repeated multiplication so we have them write the number of crumbs using 2s and 3s only. Though some students may be ready for exponential reasoning right away, we find that writing it out long-hand reinforces important algebra concepts that students may forget along the way (when do I add exponents and when do I multiply??). This also leads students smoothly to question 4 where they have to write an explicit rule for the sequence. As you are monitoring students listen for students that are able to articulate why there is an (n-1) in the exponent.

We have students find sums by hand in question 5 which should lead to some frustration or annoyance (especially the sum up to the 16th tile!) Students should start to wonder if there is a faster way to do this. This is where you as the teacher step in, in question 6. Feel free to modify the document and add your own name. Have students put their pencils down and tell them you are going to write some things on the board but you are not going to talk. Their only job is to try to understand what you are doing. They can’t write anything down. They can only watch. Tell them you’re going to try to come up with a shortcut and you’ll show your method for finding S(5). At this point, stop talking and start writing the method shown on the board. You can point, but don’t use any words to clarify. This builds suspense and anticipation! Do not yet generalize the formula.

When you are done, ask students to discuss in their groups what they notice and what they wonder. Our classes erupted in conversation at this point. They all had things they noticed even if they didn’t understand the process from beginning to end. Have groups share out in a round-robin format. After ideas have been generated, erase the board and start again, this time annotating verbally as you go and having students write on their paper alongside you. Ask them why you choose to multiply all terms by 3 and where that number came from. Ask them which terms cancel in the subtraction. Ask for common factors. Explain that for now you will not simplify 1-3 so the pattern becomes more visible. After the final step, identify the various parts of your equation. Show where a(1) shows up in the equation and where n and r show up in the equation. Release students to now try question 7.

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Formalize Later

The focus of the debrief is to generalize the explicit formula for the nth term of a geometric sequence and for the nth partial sum as described above. In the Important Ideas, we use the word “finite” to distinguish from the infinite sums they will see tomorrow. This is a new vocabulary word for some of our students.

Just as we did with arithmetic sequences, we want to build connections to previous content as much as possible. While arithmetic sequences represent linear relationships, geometric sequences represent exponential ones, providing a nice opportunity to review some ideas from Unit 3!