I just cleaned up an existing page on my Web site: Constant Sums, Constant Products, “an untraditional approach to traditional topics”. This is a mega-unit, spanning content from middle school, all the way to what one might call “teachers’ mathematics.” Take a look at it on a summer day when you have a little time to devote to it.
The unit is based on material from Algebra: Themes, Tools, Concepts (free download: ATTC). I’ll comment on the various parts of the unit in this post. (You can also find teacher notes and solutions in the ATTC Teachers’ Edition, which is also freely available on my Web site.)
Part 1: x + y = S, xy = P
These are the foundational lessons of the unit. The first two both start with reasonably realistic “real world” contexts. (Note for math book nerds: “Constant Sums” is in part an homage to a textbook author who specialized in fun real-world connections. Can you recognize him?)
– Constant Sums: One possible side use of this lesson is to derive and justify the rules of signed number addition, which presumably should extend the line into quadrants 2 and 4 (and in the case of a negative sum, quadrant 3.)
– Constant Products: One argument for including the constant product graph in middle school is to help kids know early on that not all graphs are lines, and moreover that not all curves are parabolas.
– Analyzing Graphs: The key idea of the whole unit is in here. It was suggested by my co-author Anita Wah. My first reaction was that this was too abstract, and at first it sure felt that way. However as we got more comfortable with it, it turned out to make for great discussions in our Math 1 classes.
– The GeoGebra Applet is a complement to the worksheets, and can help structure some discussion of “Analyzing Graphs” as well as preview some ideas that are useful in Parts 3 and 4 below.
– We ended up making the Five Representations writing assignment a regular end-of-year feature of Math 1. It connects the ideas from this unit with the Lab Gear representation of trinomial factoring, and makes for a great review activity. It helps students consolidate and elevate their understanding, and for some students, it offers one more chance to grasp some of these ideas if they didn’t fully master them the first time around. Each student is supposed to choose their own trinomial, and one way they can create the graphics is by using some of the links on the “Five Representations” page.
Part 2: ax + by = c
As it turns out, constant sums provide a great jumping off point to introduce the standard form for the equation of arbitrary lines.
Part 3: Geometric Connection / Optimization
Rectangles with constant perimeter are related to constant sum relationships. The first two lessons pursue this topic with some area maximization problems. Rectangles with constant area are related to constant product relationships. This section’s final lesson explores a perimeter-minimizing problem.
Part 4: Quadratic Connections
– Find the Dimensions can work as a way to tie many ideas together for algebra students (probably Algebra 2), or as a sort of introduction to the next item.
– A New Path to the Quadratic Formula came to me some years ago when grading the “Five Representations” assignment. Basically, it uses that kind of thinking to proving the quadratic formula, with no recourse to either completing the square, or parabolas. You can work through the ideas yourself using the worksheet as a guide, or read the article (shared on my site with permission from The Mathematics Teacher), or follow the argument step by step using the slides or the GeoGebra file, or some combination thereof. I have done this many times with teachers, and a few times in precalculus classes.
Well, that’s it. I hope you can find something in here to use in your classes, but even if you don’t, I hope you had your mind blown by this proof of the quadratic formula!
Note: most of the information in this blog post, plus many more links, can be found on my website.