Beautiful moments in the HS math curriculum (part 1 of n)
It’s not hard to find people railing against the current math curriculum these days, and I’m not here to say that it’s perfect. There are many changes I would like to see implemented, but in my experience those changes are slow to come, if they ever do. So in the meantime, I think it’s important to look for the magical moments in our current curriculum to help make mathematics more meaningful, interesting, and engaging for our students.
Solve for x:
(x-2)(x+3)=0
I thought this was supposed to be about beautiful moments in the math curriculum, why are we looking at a factored quadratic? Because this is one of the most unappreciated moments in HS mathematics. Where I teach, students learn how to solve for x in an equation like this in 9th grade. Now, I’m not in the room when they learn it, but I can tell you that by the time they reach me what they know is to make a “T-chart”, set both “sides” equal to zero, and then solve for x.
I’m sure there is some attention paid to the logic behind what’s going on here, but it quickly becomes procedural in nature, and even my double accelerated students struggle to give any reason to justify this behavior.
But there’s actually something quite beautiful happening here! Think about what’s really going on. We may not know the value of x, but we know it’s a number. That means (x+2) and (x -3) are also just numbers. Now let’s play a game. Imagine I’m thinking of two numbers. I want you to guess them, and I give you one clue: when you multiply them, the product is zero. What can you tell me about my numbers?
One of them has to be zero! The only way to get zero is to multiply by zero (nice chance to review multiplicative identity).
So, when we do this “T-chart” thing, what we’re really saying is that the only way the product of the two numbers can be zero, is if at least one of those numbers is zero. That’s why I never use a “T-chart.”
Instead, I solve the equation by acknowledging the logical flow, where the initial equation leads to two possible outcomes (which also relates back to the inclusive OR vs exclusive OR).
This small difference is important for two reasons. First, it doesn’t naturally reduce to a procedure as easily as a “T-chart” does. Second, and more important, it provides an opportunity to revisit the logic with each new solution, and help reinforce understanding over procedure.
Focusing on the logic opens up the door to understanding other related problems as well. For example, when learning to solve quadratic equations of the form x² -x -2 = 4, what are students taught to do?
Step 1, get it equal to zero.
Step 2, factor & solve.
Why? In my experience you’d have to ask a lot of students to get an answer besides “That’s what my teacher told us to do.”
Well, what if we don’t get it equal to zero and instead just factor?
We get (x-2)(x+1)=4. It’s important to point out that there’s nothing wrong with this, it just isn’t particularly helpful. How many times have you seen a student go from here to the “T-chart” step and write x -2=4 and x+1=4? This is clearly wrong, but if we understand the logic of the previous example we are well equipped to see why. Imagine we play the same game as before, where I have two numbers in mind, but this time I give you the clue that the product is 4. What do you know about the numbers?
Literally nothing, aside from that their product is 4.
Perhaps the numbers are 1 and 4, but they could also both be 2. It gets worse when we realize that fractions are fair game: maybe 8 and .5, or 16 and .25, or… wait a second, what about sqrt(8) and sqrt(2)!? Obviously, there are infinitely many solutions, and as my analysis professor would have said, “This is a highly non-trivial result.”
The realization that having two numbers multiply to a constant is useless unless that constant is zero, is a powerful, and important moment in a students’ Algebra journey.
The reason we get that quadratic equal to zero before factoring, is so that we can leverage that power to our advantage. When students truly understand that, they won’t need to remember what steps to use to solve a quadratic, they will just be using the power of a product equal to zero!
With understanding also comes the ability to solve similar, but different problems without requiring new sets of rules or procedures.
For example, students often stumble when first confronted with equations such as x(x-2)=0. Once a student understands the power of a product equal to zero this is a trivial exercise. Even though it looks different, it’s still just two numbers whose product is zero, which means at least one of them must be zero. What about something really scary looking like:
(x-1)(x+2)(x-3)(x-7)=0
Most Algebra students balk when first presented with something like this, because it doesn’t fit the rules or procedure they’ve learned previously. A student who understands what we’ve been discussing here can easily see that the only way for that to be true is if at least one of the terms in the equation is zero. That means that (x-1)=0, or (x+2)=0, or (x-3)=0, or (x-7)=0.
This also leads nicely into an understanding of why the x-values always take on the opposite sign of the constant in the parenthesis with the x (a pattern students often pick up on, without really understanding why). The only way for (x-1) to be equal to zero, is if x is the additive inverse of -1, namely +1 (yeah that’s right, we’ve managed to talk about the multiplicative identity and additive inverse just by thinking deeply about solving a quadratic).
I hope you agree that there’s a lot of worthwhile and interesting mathematics going on in this simple HS Algebra topic. Every time we enter a classroom of students, we have a choice to teach a procedure or to teach for understanding. Sometimes, there are rich opportunities for understanding even within apparently simply and rudimentary skills such as solving a quadratic equation. Whenever we pull back the veil of procedure, and provide students the opportunity to make sense of the mathematics we’re learning, we give them the chance to really experience the math and make it their own. If we look for them, there are many opportunities to do this; often hiding in plain sight, and often obscured by routines and procedures which have become so automatic, we no longer question them.
Where do you find beauty in the HS curriculum? Comment below and share one of your favorite math moments.
