I start today's class by reminding students about the work we did yesterday on arithmetic means. I tell students that today we'll be doing the same things with geometric means. I ask students how they think the method for finding geometric means might be similar to finding arithmetic means? Next, I ask students to predict how they think the methods might be different.
We take a look at Geometric Meanies and go over what today's task is. I ask students to pay special attention to the questions that ask them if there answer is the only one. I try to present this piece to them as a puzzle. Can there be more than one common ratio. I also remind them that yesterday we were able to write equations to find arithmetic means. Do they think we might be able to do something similar with today's work?
I let students get to work in small groups. If groups work quickly, I ask them to spend some time describing their method for finding geometric means. I also encourage them to write an equation if they haven't already.
Like in the Arithmetic Means lesson, the focus of today's discussion is on how to write and solve an equation in order to find the geometric means. Some students may need to write the situation out in words before they are ready to write an equation. We might start with an example like the third table and I'll elicit a sentence from students along the lines of "so we start with 6 and multiply by the same number 4 times to get to 96." How can we write this as an equation? Some students will struggle with how to notate repeated multiplication, so I might fill in the table with the x's so they can see that we have 6 * x * x * x * x = 96. Of course, I want to elicit that x times itself 4 times is x^4.
Some students may also need guidance around how to solve x^4 = 16. Students may need the reminder that they are looking for a number that when multiplied by itself 4 times equals 16. Some students may need a calculator as an aid here.
The second part of our discussion focuses on whether or not there can be two different answers for geometric means. If students are stuck I might say something like, "so 2 times itself 4 times is 16. Is there another number times itself that also equals 16." We then spend some time looking at the examples we worked on and when there will be two answers and when there will be just one, depending on whether or not there are an odd or an even number of x's. The second table gives an example of an odd number of means that will only have one correct answer.
Now that we've had a full discussion on how to find geometric means and when there is more than one answer, I ask students to summarize their work. Even though they may have already written an explanation earlier in class, I ask them to go back and modify their work knowing what they know. Students are often quick with their explanations and I want to get them in the habit of writing thorough explanations. I ask a few students to share out their summaries and other students can listen to see if anything is missing or should be modified.
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