Most math teachers start their year with algebra review. I like the idea of “activating prior knowledge“, as it’s known in ed school, but I never want to revisit material as review. It’s so….boring. Similarly, others “reteach” students if they didn’t understand it the first time and again, no, I don’t do that.

The trick is to wrap the review material in something new, something small. It’s wrapping, after all. For example, suppose the kids don’t really get Power Laws 1, 2, and 3 the first time you teach them, even though you went through them in insane detail and taught them both method and meaning. But you give them a quiz, and half the class is like, what means this exponent stuff? so you grit your teeth, yell at them, flunk most of them on that quiz, and go onto another topic for a week or so. Then one morning write ¾ on the board and ask “How would I write this with exponents?” and through the explanation you take them back through all of the power laws.

But I’m not here to write about power laws, although if you want advice on the best way to teach them, even if it takes longer, there’s no better tutorial than Ben Orlin’s Exponential Bait and Switch.

I’m here to explain how I integrate what we usually call “algebra review” into my course, while additionally teaching them some conceptual stuff that, in my experience, helps them throughout the course. Namely, teach them the difference between evaluating and solving functions for specific values.

**Evaluate**–what is widely recognized as “plugging in”. Given an input, find the output. Evaluate is Follows P E MD AS rules–well, technically P F MD AS, but who can say that? Note–I am pretty sure that “evaluate” is a formal term, but google isn’t helpful on this point.

**Solve**–well, technically it’s “plugging in for y”, but no one really thinks of it that way. Given an output, find the input(s). Follows the rules of Johnny Depp’s younger brother, SA MD E P. (I hope I retire before I have to update that cultural reference). And really, it’s SA MD F P, but again, who can say that?

Things that get covered in Evaluate/Solve:

- Remind everyone once more that addition/subtraction and multiplication/division run left to right, not one before the other. SAMDEP reinforces that, as I put the S first for the mnemonic.
- “Evaluate”–Evaluating purely arithmetic expressions is middle school math. At this stage of the game, the task is “evaluate the equation with a given value of x”.
- “Solve” –Solving is, functionally, working backwards, to undo everything that has been done to the input. Right now, they know how to “undo” arithmetic and a few functions. They’ll be expanding that understanding as the course moves forward.x
- Hinted at but not made explicit yet: not all equations are written in function format. I believe that, given an equation like 3x + 2y = 12 or x
^{2}+ y^{2}=25, the terminology is “given x=4, solve for y” or “given y=3, solve for x”, but I’m not enough of a mathie to be sure. Feel free to clarify in the comments. - As I move into functions, this framework is helpful for understanding that evaluating a function must have one and only one answer, whereas solving a function given an output can have more than one input. It’s also useful to start capturing the differences between absolute value and quadratics, which aren’t one to one, and lines and radicals, which are.
- The “PE” in PEMDAS and SAMDEP stands for exponent, but in fact the laws must be followed for every type of function: square root, absolute value, trigonometry, logs, and so on. Informally, the “E” means “do the function” or “undo the function”, depending on whether evaluating or solving. So evaluating y=4|x-5| -6 with x=1 means subtract 5 from 1 (the “parenthesis), then take the absolute value (the “exponent”), then multiply by 4 and subtract 6. Solving the same equation would be adding 6, dividing by four, then undoing the absolute value to create two equations, then adding five in each one. (This is more complicated in text than explaining it with calculations on a promethean.)
- YOU CAN’T DISTRIBUTE OVER ANYTHING EXCEPT MULTIPLICATION. This one is important. Kids will change 2(x-1)
^{2}to (2x-2)^{2}to 4x-4 with depressing speed and while many of them will make the last mistake in perpetuity, I’ve found that I can break them of the first, which also helps with 3|x+5| not turning into|3x+15|. For some reason, they never distribute over a square root, but plenty will try to turn 3cos(3x) into cos(9x).

Here’s a bit of the worksheet I built.

I have found this prepares the groundwork for an indepth introduction to functions, which is my first unit. So when they’ve finished Evaluate and Solve, followed by Simplify ( more on that later), the functions unit:

- Definition of a function
- Function notation
- Function Transformation
- Parent functions–I start with quadratic, square root, absolute value, and reciprocal, and some of my approach is at the bottom of this article.

So by the end of the unit the students can graph f(x) = 2(x-1)^{2 }– 8 , as well as find f(3) and a if f(a)=10, and understand that the x and y intercepts, if they exist, are at f(0) and f(x)=0. They can also do the same for a square root or reciprocal function. Then I do a linear unit and a quadratic unit in depth.

Function notation, particularly f(a)= [value], is much easier for the students to understand once they’ve worked “evaluate” and “solve” with x and y.

This also helps the students read graphs for f(4) or f(z)=7.

Back in January, a Swedish guy living in Germany, as he describes himself, read the vast majority of my blog and then summarized his key takeaways and some critiques. His 6 takeaways are a pretty good reading of my blog, but he’s completely dismissive of my teaching and pedagogy, saying I’m mathematically naive and often, due to my ignorance, end up creating more confusion teaching needless information to my students. He explicitly refers to The Evolution of Equals and The Product of Two Lines, but I suspect he’d feel similarly about The Sum of a Parabola and a Line and Teaching With Indirection.

I’m really sure my students aren’t confused. I get pretty decent feedback from real mathematicians. There are legit differences between teachers on this point that approach religious wars, so there’s that.

Besides, these sort of lessons do two things simultaneously. They give weaker kids the opportunity to practice, and the top kids get a dose of the big picture.

Yes, it’s been a while since I’ve written. Trying to fix that.

September 23rd, 2019 at 5:28 am

who knows if this is actually helpful, but the best way to deal with exponents without getting confusing in my eyes is just to multiply them out. (x+2) squared is just (x+2)(x+2), which helps explain why you can’t just distribute coefficients or treat exponents like coefficients

in general if you’re willing to do this at the low level it helps explain the power laws, though I guess using this as a crutch could work against the whole point of exponents to begin with

September 23rd, 2019 at 4:03 pm

When we refer to exponents, we’re usually referring to pure multiplication. Squaring x+2 is referred to as binomial multiplication.

September 23rd, 2019 at 8:36 pm

Uh, sure. But either way, it’s the same basic process; I’m sure you’d agree that 3^2 is (3)(3) and (x+2)^2 is (x+2)(x+2). My point is that it’s helpful to think of it in that way, at least on a basic level, to understand what’s actually going on (and thus why you can’t distribute coefficients or treat exponents like coefficients). At least, I think it’s helpful.

September 24th, 2019 at 8:13 pm

The power laws are far beyond repeated multiplication, so while everyone does think of it that way, it’s not helpful in moving to the more complex laws.

October 1st, 2019 at 4:57 am

I find “changing the subject” to be a good way to revise solving in a different setting.

July 17th, 2020 at 4:32 am

Hi! Just finding your blog and am reading through the different posts, and enjoying it a lot. A slight quibble:

3|x+5| is in fact equivalent to |3x + 15| but this wouldn’t be the case with -3. So while you shouldn’t necessarily point this out to students proactively so as not to confuse them, it wouldn’t be proper to mark down a kid who distributed in that particular instance (or any always positive number/expression instead of 3). (You can see this because |a|*|b| =|ab| for any real a,b.)

July 17th, 2020 at 6:42 pm

Glad you enjoy it. I would never mark down a kid for distributing over a function, but it’s *incredibly* important kids understand they can’t distribute over a function. Just because somehting might be true doesn’t mean it’s not false. that is, 3|x + 5| = 3x + 15 is, I believe, a false statement, right? (I hate formal logic.)

July 18th, 2020 at 7:41 pm

Yep! But |3x+15| = 3|x+5| is true.

July 19th, 2020 at 1:03 am

I should have been clearer. a|b + c| != |ab + ac|. It’s not always true.

So if you have |-2x+4|, you’d have to factor out the -2 within the absolute value |-2(x-2)| and then that becomes 2|x-2| which, if you distribute it back over is |2x-4|. But -2|x-2| wouldn’t yield that result. So I’d rather teach them how to factor out a common term from a function but not distribute it, since the there are only a few times it kind of works, like if the term is positive and the function is absolute value.

July 17th, 2020 at 4:44 am

Hi! A second comment regarding the Swedish guy’s review: what has been your own experience teaching with colleagues from other countries, notably the former USSR?

In general, they seem to know math very well, at a level above even other American math teachers. This is anecdotal, of course.

However, I do wonder if the Swedish guy was right in claiming that integrating math early on can help with a broader degree of understanding of algebra? Because the Soviet curriculum integrated math, including algebra, at a much slower pace at much earlier grades, which might make it a lot more accessible to a lot more people than dumping all that abstraction on kids in grade 9.

July 17th, 2020 at 6:43 pm

Integrating math has always, but always, failed in America. Not just in popularity, but also as demonstrated by learning and test scores. We don’t stop teaching kids math in high school, while Europe tracks.