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.
June 2nd, 2017 at 12:43 am
Just asking questions, not passing judgment, so don’t yell at me!
You teach factoring quadratics using the area model, yes? Would be curious to hear your thoughts on the connection of that to this. (I know that you said that this helps kids who flip the zeros after factoring.) How close are factoring and graphing in your typical sequence of topics? You do this before factoring?
How long do you have kids “hold that thought” on parabolas that don’t have linear factors?
When you say “most of them quickly connect this knowledge and realize that the line of symmetry will always be smack dab in between the two lines,” do you mean the x-coordinate of the vertex will be between the two zeros? In the graph linked after the quote, I don’t see how the line of symmetry is “between” the two factor lines.
I’m honestly just asking.
June 2nd, 2017 at 3:49 am
The line of symmetry is the x-coordinate of the vertex, isn’t it? I wrote this late in the evening, so I’ll check to be sure the right graphs are in place.
I’ve always taught graphing before factoring.
June 3rd, 2017 at 5:21 am
I just noticed the middle part of your question, sorry. Quite a bit. It takes a while until we get to that part, towards the end of the unit.
June 8th, 2017 at 12:01 am
Yes. I was thrown off more by “between the two lines,” but I think you meant what I thought you meant. Thanks.
Cool method. I like the visual emphasis. Another possibility, at the cost of the connection to factoring and zeros, is to think of y = ax^2 + bx + c as the sum of a parabola (ax^2) and a line (bx + c).
June 8th, 2017 at 1:43 am
I’ve already got that one written up: Sum of a Parabola and a Line. It’s a much cleaner way of teaching completing the square for lower ability kids.
June 8th, 2017 at 12:07 am
You could go a step further and talk about it as a sum of a quadratic term, a direct variation function (bx) and a constant function (c). That’s one way I have introduced linear equations, so showing that connection has been helpful at times with some students.
June 3rd, 2017 at 3:36 am
“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.
…the product of two different lines will always have two different zeros.”
I apologize if this is misguided or too trivial, but taken literally, the latter quoted statement seems false. Two different lines may have the same x-intercept.
E.g., the parabola f(x)=(x-2)(2x-4) is the product of two different lines, and has only one zero.
June 3rd, 2017 at 4:46 am
The only time the lines will have the same zeros is if both terms have a common factor, as in your example.
I’d call it trivial, since this is clearly an intro lesson and it was a throwaway comment. But I’ll add a note to that effect.
June 3rd, 2017 at 1:24 pm
A nice extension to this line of thinking is noticing what happens when you have two intersecting lines. I’ll add more later, but have been meaning to follow up since I saw your post. I saw someone mention this idea of parabolas as products of lines a few years back and really explored it for a bit. So many great little nuggets to unwrap. Also, see if you can find a relationship between the lines and the focus and directrix. It’s a nice thing to explore with a class that’s become familiar and comfortable with parabolas already.
June 10th, 2017 at 6:43 pm
If students actually understand that lesson then you must be doing something very differently in lectures than you are in your written explanation and posted diagrams, which, as with other lessons you have posted, were incomprehensible to me despite always scoring in the top percent in standardized math tests (far better in verbal) and pursuing independent study and application of math and physics for decades. Only by ignoring your explanations and most of the notations on the graphs could I understand what you were talking about. Perhaps in the classroom quick feedback lets you know when you’re not being understood, but you consistently complain that your students are dim for not understanding what you are saying, and no doubt many of them would have trouble understanding even the best explanation, but it seems quite likely that a lot of their difficulties come from your explanations being incomprehensible to almost anyone, even the extremely gifted.
June 10th, 2017 at 7:54 pm
I’m always cheered up that these nasty comments that betray total ignorance and dishonesty (I complain my students don’t know what I’m talking about?) are about the posts that get the most praise from math teachers.
In any event, this isn’t a description of a lesson, but of a concept. Which you might understand if your verbal skills were in any way equal to your whizbang math skills.
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