Exploring a Geometric Sequence - Different Dots

Today's lesson builds on the previous lesson of the growing dots pattern. While the previous pattern represented an arithmetic sequence, today's pattern is geometric.  I begin today's class by posting the pattern on the board and giving students a little time to take a look. Next, I ask for a student to come up and draw the next figure in the sequence.  It's ok if students share out how they see the pattern here, the real meat of the task is asking students to write various ways to represent the pattern itself.

I remind students of the work we did yesterday and ask them what they remember about recursive and explicit formulas. I also ask them what makes something an arithmetic sequence. This opening is also outlined in the first few slides of the Growing Growing Dots PowerPoint.

Next, students get to work in small groups on Questions 3 and 4 of Growing, Growing Dots. Students may need reminders to represent their work in tables, graphs and equations. Some issues I watch for as students work: 

• Some students may need help generating a table. They may need a reminder that the time is related to the number of dots. Once they have a table, they may see the doubling pattern in the out column. I ask them how is that similar and/or different from the the pattern we looked at in the last class.  Can they approach the recursive function the same way?

• Some students will see the recursive pattern right away but may need help writing it. Again, I'll refer them to the work in the last class where we added 4 the previous term and ask students if they could figure out how to tweak the function to fit this pattern.

• Students may have trouble scaling their graphs. This pattern grows much faster than the one they looked at yesterday. It would be great to get them to point out that difference. Students may also enjoy using technology like desmos.com to graph this pattern.

• Most students I work with will struggle to write an explicit function. Much of our class discussion will focus on this piece today, so I might let them wrestle with this idea until that time.

There is a lot to discuss in today's lesson! The discussion follows the Growing Growing Dots PowerPoint. I like to begin with the recursive equation and have students share out a table they made.   Students will likely be able to identify that the out column of their table is multiplying by 2 each time. Like we did in the last lesson, I'll ask students to write the recursive function in words (The current number of dots = the previous number of dots x 2), before moving to function notation. 

Next, we explore the difference column in the table. I think it is important here to spend some time showing students how to figure out if there is a common ratio or not.  I find that students have a much easier time seeing a common difference then a common ratio, so we spend some time here really looking at the common ratio and how we can see it.

Once we identify the common ratio, I introduce the idea of a geometric sequence to students. 

Next, we take a look at student generated graphs. We begin to struggle with the idea of an explicit equation for this pattern. We begin to break down the Out column to see how many times the result was multiplied by 2 (see slide #12 in the PowerPoint).  This is a difficult idea for my students. They struggle to see that the multiplying by 2 can happen multiple times and this is what we're counting up for a our exponent.