Evaluating vs Solving

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 x² + y²=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.

 

Teaching Transformations

One of the most important new concepts in algebra 2 and beyond is the notion of transformation. That is, given the function f(x), we  can change any function’s position and growth by using the same instructions, much like giving directions from a map.

I’ve just introduced functions at this point in the calendar, so I’ve designed this activity to reinforce f(x) as a rule, that once a mapping is created, the mapping holds for all subsequent calls.

So just create a random table, one that’s simpler than anything I’d do in class. (One of the incredibly irritating things about blogging is that it’s insanely time-consuming to create images for publication that take next to no time at all to do on  a smartboard, but I never think of capturing images while on a smartboard.)

x	f(x)
-3	2
-1	5
1	6
3	3
5	-1

That looks like this:

So then I ask if this is f(x), what would f(x+2) look like? Someone brave will always say “Two to the right”.

At that point, I always say “This is a totally logical guess and one of the most annoying things in math from this point on is that your guess is wrong.” (I originally developed the concept of a parabola as the product of two lines as another way of explaining this confusing relationship. Confusing to normal people. Mathies think it makes sense, but they’re weird.)

I add a column to the table. “We start with x. Then we add 2. Then we make the function call. Note the function call comes after the addition of the value. This is important. Now, we have three columns, but we are starting with our x and that’s still our input value. We graph it against the outer column, the output value for f(x+2).”

x	x+2	f(x+2)
-3	-1	5
-1	1	6
1	3	3
3	5	-1

I’ll ask how we can bring the -3 back in, and after some mulling, they’ll suggest that I add -5 to the table. So I add:

-5

-3

2

to the bottom. But I’ve been plotting points all along, so the kids can see it’s not going as expected.

“Yes, indeed. I’ll be teaching this concept in many ways over the next few months, and I ask you to start wrapping your head around this now. We have many ways of envisioning this. When working with points as opposed to an entire function, it might be helpful to think of it this way: Suppose I’m standing at -3, and I want to add two. This has the effect of me reaching to the right on the number line and pulling the output value back to me–to the left, as it were.”

I go through this several times. Whether or not students remember everything I teach, I always want them to remember that at the time, they understood the concept.

“So if standing on -3 and reaching ahead is addition and move the whole function to the left, how would I move the whole function to the right?”

If I don’t get a ready chorus of “subtract?” I know that I need to try one more addition example, but I usually get a good response.

“Exactly. So let’s try that.”

x	x-2	f(x-2)
-3	-5	NS
-1	-3	2
1	-1	5
3	1	6
5	3	3
7	5	-1

One year, I had a doubter who noticed that I’d made up these numbers. How did we know it’d work on any numbers? I told him I’d show him more later, but for now, imagine if I had a table like this:

x	f(x)
1	1
2	2
3	3

etc.

Then I told him, “Now, imagine I put decimal values in there, fractions, whatever. Imagine that no matter how I change the x, the new value has an entry in the table and thus an output. So imagine I added 50. There’d be a value 50 ahead that I could reach forward or backwards.”

“In fact, we’ll eventually do all this with equations that are functions, instead of randomly generated points. But I start with points so you won’t forget that it works with any series of values that I can commit math on. Which isn’t all functions, of course, but that’s another story.”

“But if adding makes it go left and right, how do we make a function go up and down? Discuss that among yourselves for a minute or so.”

Sometimes a student will see that we’ve been changing x so far. Otherwise I’ll point it out.

“The function call itself is key to understanding this. If you change the value before you make the function call, then you are changing the input to the function. Simpler: you’re changing x before you call the function. But once the value comes out of the function, that is, once it’s no longer the input, it’s the….” I always wait for the class to chime in again–are they paying attention?

“Output!”

“Right. But output is no longer x. Output is”

“f of x!”

At this point, I call on a mid-level student. “So, Sanjana, up to now, we’ve been changing x before making the call to the function. See how the new column is in the middle? What could I do differently?”

And I wait until someone suggests making the column on the right, after the f(x).

x	f(x)	f(x)- 3
-3	2	-1
-1	5	2
1	6	3
3	3	0
5	-1	-3

I’m giving a skeletal version of this. Often the kids have whiteboards and are calculating all this along with me. I’ll give some quick learning checks in terms of moving to the right and left, up and down.

The primary learning objective for is to grasp the meaning of horizontal and vertical translations–soon to be known as h and k. But as an introduction, I define them in terms of function notation.

 

We usually end this activity by combining vertical and horizontal shifts.

What would f(x-2)+ 3 look like? Well, you’d need another column.

x	x-2	f(x-2)	f(x-2)+1
-3	-5	NS
-1	-3	2	5
1	-1	5	8
3	1	6	9
5	3	3	7
7	5	-1	2

I connect them this time just to show that one point is in both the original and the transformation.

Ultimately, this goes to transforming functions, not points. That’s the next unit, transforming parent functions. I have a colleague who teaches transformations entirely by points. I start down that path (not from his example, just because that’s how this works), but the purpose of transformations, pedagogically speaking, is for students to understand that entire equations can be changed at the unit level, without replotting points. At the same time, I want the students to know that the process begins at the point level.

Over time, the students start to understand what I often call inside and outside, or before and after. Changes to the input value affect the x, or the horizontal because they occur before the function is called. Changes to the output value affect the y, or the vertical, because they occur after the function is called. Introducing this on a point by point basis creates a memory for that.

At best, this lesson functions as more than just a graphing exercise, something to introduce vertical and horizontal shift. It should ideally give students an understanding of the algebra behind it. Later on, when they are asked to solve equations like:

Find f(a) = 32 for f(x)=3(x-2)²+5

Weaker students have trouble with understanding order of operations, and a memory of “inside” and “outside” the function can be helpful.

If I were writing algebra 1 curriculum, I’d throw out quadratics, introduce a few parent functions, and teach them function notation and simple transformations. It’s a complicated topic that they’ll see all the way through precalc, at least.

I’ll discuss stretch and its complexities in another post.

The Product of Two Lines

I can’t remember when I realized that quadratics with real zeros were the product of two lines. It may have been this introductory assessment that started me thinking hey, that’s cool, the line goes through the zero. And hey, even cooler, the other one will, too.

And for the first time, I began to understand that “factor” is possible to explain visually as well as algebraically.

Take, for example, f(x)=(x+3) and g(x)=(x-5). Graph the lines and mark the x-and y-intercepts:

Can’t you see the outlines of the parabola? This is a great visual cue for many students.

By this time, I’ve introduced function addition. From there, I just point out that if we can add the outputs of linear functions, we can multiply them.

We can just multiply the y-intercepts together first. One’s positive and one’s negative, so the y-intercept will be [wait for the response. This activity is designed specifically to get low ability kids thinking about what they can see, right in front of their eyes. So make the strugglers see it. Wait until they see it.]

Then onto the x-intercepts, where the output of one of the lines is zero. And zero multiplied by anything is zero.

Again, I always stop around here and make them see it. All lines have an x-intercept. If you’re multiplying two lines together, each line has an x-intercept. So the product of two different lines will have two different x-intercepts–unless one line is a multiple of the other (eg. x+3 and 2x+6). Each of those x-intercepts will multiply with the other output and result in a zero.

So take a minute before we go on, I always say, and think about what that means. Two different lines will have two different x-intercepts, which mean that their product will always have two points at which the product is zero.

This doesn’t mean that all parabolas have two zeros, I usually say at this point, because some if not all the kids see where this lesson is going. But the product of two different lines will always have two different zeros.

Then we look at the two lines and think about general areas and multiplication properties. On the left, both the lines are in negative territory, and a negative times a negative is a positive. Then, the line x+3 “hits” the x-axis and zero at -3, and from that zer on, the output values are positive. So from x=-3 to the zero for x-5, one of the lines has a positive output and one has a negative. I usually move an image from Desmos to my smartboard to mark all this up:

The purpose, again, is to get kids to understand that a quadratic shape isn’t just some random thing. Thinking of it as  a product of two lines allows them to realize the action is predictable, following rules of math they already know.

Then we go back to Desmos and plot points that are products of the two lines.

Bam! There’s the turnaround point, I say. What’s that called, in a parabola? and wait for “vertex”.

When I first introduced this idea, we’d do one or two product examples on the board and then they’d complete this worksheet:

The kids  plot the lines, mark the zeros and y-intercept based on the linear values, then find the outputs of the two individual lines and plot points, looking for the “turnaround”.

After a day or so of that, I’d talk about a parabola, which is sometimes, but not always, the product of two lines. Introduce the key points, etc. I think this would be perfect for algebra one. You could then move on to the parabolas that are the product of one line (a square) or the parabolas that don’t cross the x-intercept at all. Hey, how’s that work ?What kinds of lines are those? and so on.

That’s the basic approach as I developed it two or three years ago. Today, I would use it as just as describe above, but in algebra one, not algebra two. As written,I can’t use it anymore for my algebra two class, and therein lies a tale that validates what I first wrote three years ago, that by “dumbing things down”, I can slowly increase the breadth and depth of the curriculum while still keeping it accessible for all students.

These days, my class starts with a functions unit, covering function definition, notation, transformations, and basic parent functions (line, parabola, radical, reciprocal, absolute value).

So now, the “product of two lines” is no longer a new shape, but a familiar one. At this point, all the kids are at least somewhat familiar with f(x)=a(x-h)²+k, so even if they’ve forgotten the factored form of the quadratic, they recognize the parabola. And even better, they know how to describe it!

So when the shape emerges, the students can describe the parabola in vertex form. Up to now, a parabola has been the parent function f(x)=x² transformed by vertical and horizontal shifts and stretches. They know, then, that the product of f(x)=x+3 and g(x)=x-5 can also be described as h(x)=(x-1)²-16.

Since they already know that a parabola’s points are mirrored around a line of symmetry, most of them quickly connect this knowledge and realize that the line of symmetry will always be smack dab in between the two lines, and that they just need to find the line visually, plug it into the two lines, and that’s the vertex. (something like this).

For most of the kids, therefore, the explanatory worksheet above isn’t necessary. They’re ready to start graphing parabolas in factored form. Some students struggle with the connection, though, and I have this as a backup.

This opens up the whole topic into a series of questions so natural that even the most determined don’t give a damn student will be willing to temporarily engage in mulling them over.

For example, it’s an easy thing to transform a parabola to have no x-intercepts. But clearly, such a parabola can’t be the product of two lines. Hmm. Hold that thought.

Or I return to the idea of a factor or factoring, the process of converting from a sum to a product. If two lines are multiplied together, then each line is a factor of the quadratic. Does that mean that a quadratic with no zeros has no factors? Or is there some other way of looking at it? This will all be useful memories and connections when we move onto factoring, quadratic formula, and complex numbers.

Later, I can ask interested students to sketch (not graph) y=x(x-7)(x+4) and now they see it as a case of multiplying three lines together, where it’s going to be negative, positive, what the y-intercept will be, and so on.

At some point, I mention that we’re working exclusively with lines that have a slope of positive one, and that changing the slope will complicate (but not alter) the math. Although I’m not a big fan of horizontal stretch outside trigonometry, so I always tell the kids to factor out x’s coefficient.

But recently, I’ve realized that the applications go far beyond polynomials, which is why I’m modifying my functions unit yet again. Consider these equations:

and realize that they can all be conceived as as “committing a function on a line”. In each case, graphing the line and then performing the function on each output value will result in the correct graph–and, more importantly, provide a link to key values of the resulting graph simply by considering the line.

Then there’s the real reason I developed this concept: it really helps kids get the zeros right. Any math teacher has been driven bonkers by the flipping zeros problem.

That is, a kid looks at y=(x+3)(x-5) and says the zeros are at 3 and -5. I understand this perfectly. In one sense, it’s entirely logical. But logical or not, it’s wrong. I have gone through approximately the EIGHT HUNDRED BILLION ways of explaining factors vs. zeros, and a depressing chunk of kids still screw it up.

But understanding the factors as lines gives the students a visual check. They will, naturally, forget to use it. But when I come across them getting it backwards, I can say “graph the lines” instead of “OH FOR GOD’S SAKE HOW MANY TIMES DO I HAVE TO TELL YOU!” which makes me feel better but understandably fills them with apprehension.