The purpose of today's lesson is for students to explore the idea of arithmetic means. Today's opening is a little longer than usual as students explore a couple of problems at the start of class, briefly share out some strategies, and then get to work on some more challenging problems. We start class today by looking at the first three problems in What Does It Mean?. Students work individually on these problems and once they've had a chance to solve a couple of them, they share out some of their strategies with the whole class. It's important for students to focus here on the question that asks them to describe their method. I want students to begin writing about their math strategy to solve these problems.
Next, students get to work in small groups on the back page. These sequences are more challenging and students may find they need to find a new method to find the mean. Students who have been using guess and check may be especially challenged. If students are reliably subtracting the first term from the last and then dividing by the number of terms, I ask them if they can represent this work algebraically. I also watch for students who don't realize the same number has to be added one more time than the number of missing places. I might ask students to articulate why this is so.
The main idea I want to come out of today's discussion is that we can write an equation in order to find geometric means. I'll have students share out any methods they found besides this one first so we can get to the equation piece last. I might ask a student who has used this method without generalizing to an equation to share out the his/her work. I then ask the whole class how we could write these steps out in words. From here, we work to write an equation. I might say something like, "So we start with the number 50, add the same number 3 times, and then end up at 86. How could we write this as an equation?
Again, we spend some time talking about why we would add the same number 3 times here, rather than 2, which is the number of terms that are missing.
Next, I ask students to use this equation method (if they haven't already) to resolve the problems on the back page.
To end today's class, I ask students to write a clear explanation of how to find the missing terms of an arithmetic mean. If there's time, I might have some students share their explanations verbally with the class. Or, I might type up the responses and share them at the start of the next lesson.
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