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.