Lesson 4 of 8
Objective: SWBAT to conduct research and select a data sample from which they'll construct mathematical models.
This course runs an alternating-day schedule with 75-minute periods, and I see each of my individual classes every other day. This means that in a two-week period, I can count on seeing each class five times.
When you look at this unit, you'll see that it consists of eight lessons. Adding a couple of work periods for students to make sure they've got everything done, I hope to complete this unit over the course of four weeks.
I mention that at the start of this lesson, because this one is very likely to extend into a second class period. That's perfect for right now. If students can complete most of the Moore's Law Investigation today, that will leave tomorrow for them to finish up, and to go back and make sure that work from the first three lessons of the unit is complete.
So to summarize, this and the three lessons preceding it should fill two weeks at my school. You can adapt these activities to fit your bell schedule, and as a rule of thumb, if your students can get through all of this in two weeks, you'll be in good shape.
Opening Quiz (timed)
Today's opener is a timed quiz that serves as a check-in on what students understand so far about the learning targets we've studied in the previous three lessons:
- 3.2: I can distinguish between situations that can be modeled with linear functions and with exponential functions.
- 3.3: I can write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
I use the timed-transitions feature of Powerpoint to make the slides advance automatically at defined intervals, which serves two purposes. First, it allows me to move around the room and check in with how students are doing. Second, it creates some healthy urgency for students to come in and get down to business.
- I describe how to set up such a quiz in this video.
- The U3 Powerpoint presentation is included here.
This structure packs a lot into the first 12 minutes of class! Students are warmed-up, I've got a stack of easily-assessable information about what they know, and we're ready to move on to some big ideas!
Today's investigation is about Moore's Law. A popular summary of Moore's law is to say that computing power doubles every 18-24 months. More specifically, it's the observation that the transistor capacity of a dense integrated circuit will double every two years or so. Here's an introduction on Wikipedia if you're new to this idea.
"Mystery Text" : 2000 iMac vs. 2010 iPhone
As a "Mystery Text," I post this infographic, which continues to be one of my favorites, even though it's probably due for an update here in 2015. To begin, I simply project it on the board and my class what they think. Kids are instantly curious, and they'll have all sorts of questions and observations. The key observation is that over a ten year period, computers got more powerful while shrinking dramatically in size. Most student questions are about what these numbers mean: what are the units of measurement, and how can we interpret them.
And that's today's object. In this and the next lesson, students will learn how to interpret most of these values. When it comes to interpreting parameters, a nice place to start is with a claim, like, "Computing power has grown exponentially over the last few decades," and then to wonder how we might measure such a claim? Should we use storage capacity? CPU clock speed? RAM? Do we account for cost? Do we account for the amount of physical space that computing components take up? And when we ask each of these questions, they're related to this mystery text: how are each of these expressed in the iPhone vs. iMac example?
Part 1: Powers of 2
As we finish the initial conversation, I distribute the Moore's Law Investigation handout, which describes all the steps students should take and lists the questions they should answer today. Students will reference two online articles as they complete this investigation:
- Part 1: Memory Prices - http://www.jcmit.com/memoryprice.htm
- Part 2: Transistor Count - http://en.wikipedia.org/wiki/Transistor_count
In the first part of the assignment, students will look at a historical chart of memory prices (RAM). Moore's Law doesn't refer to the capacity or cost of memory, but the progressive increase in capacity and decrease in cost is remarkable to see. The changes in cost help to generate student interest, and there are all sorts of "I remember when..." stories that we teachers could share. We'll put these numbers in context during the next lesson, when we investigate the cost of storing MP3 files. For today, I'm most interested in the "Size" column in the middle of the chart. We'll use these numbers to understand how memory capacities are always based on powers of 2. Understanding that computing is based on powers of 2 is fundamental. (If you've ever wondered about the discrepancy between the published capacity of your hard drive and how much space your computer claims is on it, it's because of the dual definition of a kilobyte for example as either "1000 bytes" or "2^10 (1024) bytes".)
The purpose of these questions is to get kids thinking about how often computing power doubles, and to recognize both small and larger powers of 2. In doing so, we're approaching an understanding of Moore's law from one direction. Next week, we're also going to be able to connect these ideas to an understanding of logarithmic scale. We won't go too deep into logarithms in this course, but I do want to expose kids some advanced mathematical ideas that might not have seen otherwise. Take a look at questions #1 and #2 to see what I mean.
Students work independently or in groups to complete Part 1. When they're done, they'll have a table that shows that memory capacity has been doubling at regular intervals since the late 1950's.
Part 2: Moore's Law & Transistor Count
If we're talking specifically about Moore's Law, however, we're really talking about transistor count. As they begin Part 2, students conduct some quick research about transistors. Then, they reference the second link (http://en.wikipedia.org/wiki/Transistor_count) to complete this part of the assignment.
The real work begins on problem #9, where students select a sample of the data in the transistor count chart. This "sampling" of a some subset of the data (I'm using the definition loosely here) is crucial to understanding stats. We note that even this wikipedia article provides only a sample of all the computer processors that have existed over the year. In stats, as in science, we have to take samples - it's usually impossible to look at every case!
I tell students that they should try to find at least one data point for every year since 1971. When that's done, it's on to the steps that have become familiar over the last few days: they create a graph (paying particularly close attention to scale), run linear and exponential regressions, and then record their observations for some further subsets of the data. The best sort of work period ensues: I circulate to answer questions, join students on tangents, and we all express and sate our curiosity.
Here are some results for you. This graph on Desmos plots all data points from the Transistor Count chart, letting x = the number of years since 1971. The linear and exponential regression models for both are also graphed there. Students should be able to note quickly that the exponential model looks a whole lot more accurate than the linear model. Because each student might take a slightly different sample of this data, each model might look a little different. I check in with students as they work, and I make sure that kids assess the work of at least one other classmate to see that sampling decisions can lead to different results.
I ran the data for the whole set, however, and here's the exponential model I got:
y = 958.6460285 * 1.427280789^x
where x is the number of years since 1971 and y is the transistor count.
And wouldn't you know it! That factor looks quite a lot like the square root of two, and that means that our model is in agreement with Mr. Moore: the number of transistors does appear to be doubling every two years!