You know what fact families are? In elementary and middle school, this is increasingly common terminology to express relationships among basic number facts. A sample fact family:
Fact family thinking is intended to encourage connections between operations and their inverses. Addition and subtraction fact families work the same way as multiplication and division fact families:
I have written a lot about logarithms this past year and I keep wondering about ways to get students’ minds wrapped around the relationships that logarithms are intended to express, and to focus less on the notation.
Here is a fact family for an exponential relationship.
![\sqrt[4]{81}=3](https://s0.wp.com/latex.php?latex=%5Csqrt%5B4%5D%7B81%7D%3D3&bg=ffffff&fg=333333&s=0&c=20201002)
Some things I notice:
- We’re not used to thinking about roots, logs and exponentials all at the same time. Maybe we should be.
- Exponentiation is not commutative, so we only have three facts in the family, not four. That seems useful to know.
- Off the top of my head, I can’t think of another non-commutative binary operation (besides those listed out above), so I can’t test the proposition that:
- All non-commutative binary operations will necessarily have TWO inverse operations in their fact families.
Overthinking my teaching 
Matrix multiplication is not commutative. There is no commonly accepted notation for matrix “division”. People use multiplication and the inverse matrix.
Nice.
I guess I’m thinking about binary operations on numbers. The other non-numerical example that occurred to me is composition of functions. In that circumstance, we don’t ever ask students to invert, do we? Except by inspection so that we can apply the chain rule in Calculus.
And by invert I mean invert the operation, not invert the function. We ask students to invert functions all the time.
But just as with matrix multiplication, if we compose with an inverse function, we end up decomposing.
If
then 
And just as with exponents, logs and roots, it is not the case that
We’re not used to thinking about roots, logs and exponentials all at the same time. Maybe we should be.
You know, this is so true that it startled me. In College Algebra I usually point out the fact that to write an exponential model to fit two points, you have to solve a power equation, and to write a power model to fit two points, you have to solve an exponential equation. However, I’d never thought to relate simple statements about roots with logarithmic and exponential statements the way you do here. What a great observation. I imagine this kind of exercise will be helpful to my students the next time through.
I’ve always thought of roots, exponentiation, and logarithms together. It is surprising to me to hear that they are treated as separate concepts in some algebra classes. I suppose there is a natural order to introducing the concepts (starting with exponentiation with integer exponents, then with rational exponents to get roots, then logarithms), but they do fit together so nicely it seems a shame to teach them separately. It is like teaching multiplication and division separately (which I know is done in some elementary math classes).
gasstation:
In some algebra classes? Are you kidding? Find me a textbook that even comes close to showing these facts together to students:
Maybe that book exists; maybe several do. But it’s certainly not the norm.
And I’ll push back on your sequencing argument a bit too. Writing that second fact in the list above using rational exponents
obscures the relationship I’m trying to point to, rather than highlighting it.
Oh! And we talk all the time about a logarithm is an exponent, but I think the claim that a root is a base is much more rare.
Quick lazy research: “logarithm is an exponent” yields 323 Google results; “root is a base” yields 383. But none of the “root is a base” results on the first page is referring to the mathematical relationship we’re discussing here. They are linguistic and culinary.
I looked in the only Algebra book I have in the house: Richard Rusczyk’s Introduction to Algebra published by the Art of Problem Solving. Although he uses squaring and square roots from fairly early in the book, Chapter 19 is “Exponents and Logarithms” and Section 20.1 is “Radicals”, so the main concepts are indeed grouped together in this text.
The textbook is intended, I believe, mainly for more advanced younger students—at least, that seems to be the focus of a lot of their publications and classes. Overall, I’ve been very impressed with their texts, and very unimpressed with most of the texts from major publishers.
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In some sense “most” binary operations are non-commutative. (For binary operations on a finite set, this can be made very precise.) Here’s two noncommutative binary operations off the top of my head:
Tetration and other hyperoperations:
http://en.wikipedia.org/wiki/Tetration
http://en.wikipedia.org/wiki/Hyper_operator
and
a*b = a weighted average of a and b
Tetration! I had considered such an operation in my private college algebra musings, but had no idea it had a name. Lovely, and thanks Barry. So Tetration must have two inverse operations. If 2*3 is 2^2^2 (i.e. 16), then we need one operation (°) such that 16°2=3 and another different one (@) such that 16@3=2. I suppose it shouldn’t be too difficult to demonstrate that these are not the same operation.
Ton of fun. Thanks.
Thanks for these insights. I like the idea that “4) All non-commutative binary operations will necessarily have TWO inverse operations in their fact families.” My students are struggling with understanding why 2^x and the xth root of 2 are not inverses (afterall, x^2 and the square root of x ARE inverses… well close enough).
Your post got me thinking along the lines of another simple noncommutative example – subtraction. If f(x) = x – 3 (i.e. “input minus three) then I can see that the inverse function is clearly g(x) = x + 3 (i.e. “input plus three”). But if I start with f(x) = 3 – x (i.e. “three minus input”), I cannot concur that the inverse will follow the same pattern as before (i.e. “three plus input”). In fact, here I must think of the original function in a completely different way in order to arrive at an inverse through introspection (“f says ‘take the opposite of the input and add three’, so the inverse will need to say ‘subtract three and then take the opposite’. g(x) = -(x – 3)”
I guess, we don’t really have language for describing 2^x in a completely different way; thus the need for creating one to define the inverse with logarithms.