The purpose of this lesson is for students to use what they have already learned about arithmetic and geometric sequences to understand linear and exponential functions.
I begin class with a Warm Up question where students have 3 minutes or so to recall what they know about sequences. I ask them to respond to the following prompt:
Students may need a reminder that in order to be a sequence, the In column of values have to increase consecutively. This is a key piece for them to understand the difference between sequences and functions that are not considered sequences.
I let them know that in today's lesson, we are going to zoom out a little bit to look at a broader category of functions. In today's class we will be looking at constant differences and constant ratios over equal intervals. I ask students to come up with an example of table that might show this (starting with equal differences over equal intervals).
Next, we read through Sorting Out the Change together. I let students get to work and circulate around the room. Issues I watch for:
We start today's discussion by having students share out a question they think represents equal differences over equal intervals. I have a small group present their work and other students can challenge or agree with the group's ideas. I explain to students that this type of pattern represents a linear function.
Next, we do the same with a question that shows equal factors over equal intervals. I explain to students that this type of pattern represents an exponential function.
It is worth reminding students here that arithmetic and geometric sequences are kinds of linear and exponential functions, but that not all functions are sequences. I remind students that in order to be a sequence, the x values must increase by consecutive terms.
The remainder of the discussion we spend looking at the other questions and getting explanations and/or ideas from students about whether or not they show equal differences or equal factors or neither. Many of the questions can lead to rich discussions among students.
Because the key idea of today's lesson is for students to zoom out and take a look at functions and compare them to sequences, I want to end today's class with a reflection that will help students remember this idea. I ask students to complete an exit ticket in response to the following prompt:
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