In the previous post in this series, we broached the subject of *equivalence classes*. We did so in the context of looking at an addition table and noticing how each value in the table occurred multiple times. We drew a blue line to connect all the entries that had the same value in it, and this blue line now marks a particular equivalence class. What all the entries in that equivalence class has in common is the value of the sum. Same sum, same equivalence class; different sum, different equivalence class. The value of the sum is the *property *that links the entries in the equivalence class together.

In this post, I’m going to develop a similar theme with multiplication and multiplication tables. In the diagram on the left, I’ve drawn a simplified multiplication table, only going up to 5. In the middle, I’ve repeated all that information but this time drew some contour lines connecting entries with the same value for the multiplication result (the product). Compared to the addition case in the prior post, the contour lines are no longer straight. Only some of the contour lines are shown, but you can get a good idea about the others, especially since 1.5, 2.5, 3.5 and 4.5 are marked also. Finding the product associated with any point in that middle diagram is directly analogous with how you find the altitude on a topographical map such as the one shown on the right.

In such a topographical map, the contour lines represent locations of equal altitude. You can see contour lines marked for elevations of 400 ft and 350 ft (this is a US Geologic Survey map) with 4 contour lines separating them. From this you can conclude that adjacent contour lines are 10 ft in latitude apart. The altitude of any point on this map can be estimated fairly accurately by looking at contour lines below and above.

It’s the same with the product. In the middle diagram, the contour lines are drawn 2.5 apart, from 0 to 25, though the contour line for 0 is not strictly on the diagram itself, but rather just beyond the top left corner. You could draw contour lines 1 apart instead, and this certainly would remove the need for estimating if what is being multiplied is two whole numbers (from 1 to 5). Of course, the need for estimating would rear its head in the same way, when you wanted to use the map for multiplying numbers that weren’t whole numbers. And you could. The middle diagram is a simple (and somewhat crude) example of a class of computational devices called nomographs, of which the Wikipedia entry provides some interesting examples. Now largely replaced by electronic devices, for centuries nomographs represented the most sophisticated computing devices available. (A particularly versatile and useful one, for centuries, is the slide rule. I still have the slide rule I bought for college, in those days obligatory for math/science/engineering students. The real geeks of the day had them hanging from their belt.)

I like the contour line analogy. I included an activity based on this once for a teacher in-service on exploring math using software (Fathom, in this case) – some notes are here. Thanks for the great series.

Dan, thank you. There is considerable similarity between your rainbows and my contour lines.

Though I have the Fathom/Tinkerplots/GeometerSketchPad package, I rely mostly on Visio for making my diagrams, sometimes enhanced with PrintScreen applied to an Excel spreadsheet. For this one, I played with WolframAlpha, then annotated it using Visio.

I find that most of the work in writing clear blog posts goes into designing clear diagrams.

I like the contour lines. A suggestion: add zero and make the (0,0) point the lower left corner, rather than the upper left. That way, it corresponds with the coordinate plane and your contour lines become the first-quadrant graphs of xy = p for various products p.

I recommend it for your addition tables too: the blue lines you draw would then become the graphs of x + y = s for various sums s. Interestingly you included zero there, but not for multiplyin’.

(Then later you can introduce negative coordinates… like you did in part 31!… and much later, the graph of f(x,y) = xy…)

Thanks for the cool blog.

Bowen, thanks.

Conventions for multiplication tables are different from conventions for graphs, and both are largely arbitrary. For this, I wanted it to connect with multiplication tables as people are likely to have seen them before. To make the contour figure in the center, I did start from z=xy using WolframAlpha and then flipped it.

I started at 0 for addition and at 1 for multiplication mostly because I wanted the representative members of the equivalence class to show at the edges, mostly for later work with subtraction.

Sounds like your interest is mostly in the graphs, and using multiplication tables to lead to graphs, whereas the direction I’m heading here is the algebraical idea of dividing out an equivalence relation.

I’m familiar with some of the work of EDC, by the way, and have worked with people in the DMI group, Deborah Schifter and Amy Morse. I also met Al Cuoco many years ago. How did you find this blog, and how does it relate to the work you do?

Hey, was led here from some comments you made on Dan Meyer’s blog. I read a lot of the posts but not everything… yet…

I was interested in the connections between addition & multiplication tables and graphs since it forms some of the initial ideas and connections in the Algebra 1 book of the high school series we published in 2008, with Al Cuoco as lead author.

The first activity in the book uses the addition and multiplication tables oriented with (0,0) in the lower left; the goal is for students to learn what rules can be used to fill in missing entries and justify those rules. A little later, we extend the tables for negative entries and ask students to use the rules they’ve built to perform the extension — among other things, this gives a mathematical justification for why the product of two negatives should be positive.

The orientation of the tables helps students get used to coordinate notation, and when we look at simple graphs like x + y = 7 and xy = 12 we draw connections back to the tables. Much later in the book we look at using both tables to solve sum-product problems (find two numbers whose sum is 7 and product is 12) that help students with quadratic factoring.

Thanks for the clarification and it’ll be interesting to see your thoughts on subtraction (and fractions, today). One thing I’ve seen helpful to students regarding equivalent fractions is the use of two rows of the multiplication table — for example, the “2” and “5” row give a huge pile of equivalent fractions to 2/5, and can be used to explain why 0/0 is an indeterminate form.

Thanks and keep writing great stuff.

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