![]() Show where a(1) shows up in the equation and where n and r show up in the equation. After the final step, identify the various parts of your equation. Explain that for now you will not simplify 1-3 so the pattern becomes more visible. Ask them which terms cancel in the subtraction. Ask them why you choose to multiply all terms by 3 and where that number came from. 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. Have groups share out in a round-robin format. ![]() They all had things they noticed even if they didn’t understand the process from beginning to end. Our classes erupted in conversation at this point. When you are done, ask students to discuss in their groups what they notice and what they wonder. This builds suspense and anticipation! Do not yet generalize the formula. You can point, but don’t use any words to clarify. At this point, stop talking and start writing the method shown on the board. Tell them you’re going to try to come up with a shortcut and you’ll show your method for finding S(5). Their only job is to try to understand what you are doing. 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. Feel free to modify the document and add your own name. This is where you as the teacher step in, in question 6. ![]() 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. As you are monitoring students listen for students that are able to articulate why there is an (n-1) in the exponent. This also leads students smoothly to question 4 where they have to write an explicit rule for the sequence. 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?). First, we want students to see the repeated multiplication so we have them write the number of crumbs using 2s and 3s only. 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.
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