![]() What’s the average of the numbers 2, 4, 6, 8, and 10? What’s the average of the numbers 2, 4, 8, 16, 32? Is the strategy for both the same? Why or why not? If I only told you how many she dropped on the 13th tile and the fact that the number of crumbs triples, could you figure out how many she dropped on the 10th tile? The 1st tile? How would you do it? How is the number of crumbs changing? Is this an arithmetic sequence? What does the 2 represent in your table? What does the 3 represent? This reasoning may seem obvious to us, and should feel intuitive to students, but we find that students often resort to using formulas and procedures that are actually le ss efficient, instead of making use of structure to solve a problem. Knowing the number of crumbs dropped on the 13th tile is sufficient for determining the number of crumbs dropped on the 15th tile because compared to the 13th tile, the number of crumbs on the 15th tile will be 3x3 or 9 times greater. In question 5, we return to yesterday's idea that any term of a sequence can be an “anchor point” for determining other terms. 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 bread-and in a rather unusual pattern! In this lesson, students learn about (or review) geometric sequences by describing the number of crumbs Little Red drops on successive tiles.
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