When am I ever going to use this: Proportions

A few months ago, I started taking an online drawing class because I decided I needed to be able to illustrate my work. With my faithful 5.5- by 8-inch sketchbook in hand, I started working my way through the beginner lessons.

One of them recently required me to construct a proportionate drawing space that could be 4 by 6, 6 by 9, or 8 by 12. Since I like to draw in the top half of my page and reflect on my work at the bottom, turning my book to achieve even the smallest of these dimensions was out of the question.

But I noticed that 4/6, 6/9, and 8/12 are equivalent ratios and they’re all equivalent to 2/3, which fit in the top half of my sketchbook page. I created the drawing space and completed the lesson.

My roommate, who’s studying to be an animator, deals with proportions and scales in her work all the time, especially when she’s laying our her design space or planning out a design. Math has long been her enemy, but she faces it on a near daily basis to make sure her incredible work comes out proportional. It’s a detail she often says can’t be ignored or overlooked.

Think about your day. Think about your parents’ jobs. Look for situations where understanding how to handle proportions would make the problem much easier to solve. You’ll find math isn’t as pointless as you think!

Finding the familiar

Right now, I’m trying to get my students to understand that math has patterns to it. If you can see the pattern, then your chances for successfully solving the equation increase.

For example, take this expression:

x³ + 7x² + 12x

Too many of my students look at this and panic. They don’t know where to start.

But if they looked closely at the expression, they might notice something. They might notice every term has at least one x, and that means an x can be factored out of every term, leaving us with:

x(x² + 7x + 12)

Look at what’s in between the parentheses. That’s a quadratic expression, and most students encountering the original expression have learned how to factor quadratic expressions. Once they factor the expression, it looks like this:

x(x + 3)(x + 4)

The once frightening expression is factored with a pair of simple steps.

If you’re struggling with a challenging math problem, look for a pattern. Look for anything familiar. Often, you can simplify the problem to something you recognize, which leaves you with something you can solve.

Punnett Squares in Factoring

Thinking about lattice multiplication got me to thinking about factoring quadratics. I don’t know which currently adopted curriculum does it, but one teaches factoring quadratic expressions by using a Punnett Square.

You remember Punnett Squares, right? They’re an integral part of understanding genetic inheritance in high school biology. Four squares form a larger square. The traits of one parent along the top, the traits of the other along the left side. The resulting possibility for combining the top trait for that column and the left trait for that row go into the square, and you’re left with every possible trait combination for these two parents’ children.

Algebra teachers who employ it to teach factoring use it in reverse to a certain degree. The x² term is placed in the upper left square. The constant term is placed in the lower right corner. The student can then write an x above the left column and to the left of the top row, because x times x equals x².

But then the student is confronted with the same problem they would be if they were being taught to simply factor the equation into (x + a)(x + b) form: what numbers go in the other two boxes?

The student is often then encouraged to “guess and check” until they find a pair of numbers that adds to the middle term and multiplies to the last.

If the student is going to end up at the exact same place they would if they were factoring in a more traditional way, what is the point of asking them to draw the Punnett Square? Most of the students I’ve encountered who were taught this particular factoring method don’t actually understand what to do when they’ve finally found the two numbers, and they can’t make the connection between what they write along the edges of the square and the roots of the equation they were factoring.

Why can’t we just teach them a simple, organized method for finding roots? Why does it need to be made more complicated with fancy squares?

Just teach the math, thanks!

Every time I see “multiplication math tricks” like the lattice method or this line trick, it really boils my blood.

The problem with tricks like these is that they aren’t fool-proof, nor are they scalable. Don’t believe me? Ask a student who has only been taught to multiply using the lattice method to multiply two decimals. He’ll either ask for a calculator or throw the paper back in your face. Neither is good for the student’s self-esteem.

The reason why we teach algorithms is so the student has something scalable that can be adapted to the type of number he’s working with. Even if it’s a bit daunting to the student at first, with practice they get the hang of it. When they run into problems multiplying fractions or decimals down the road, you can then tell them how the rules change (and those are pretty slight changes) and they’ll be far more successful.

Tricks are an easy out, but they don’t do the student any favors. It’s far better, both for you and the student, to just teach the algorithm from the start and then give the student ample opportunities to practice the algorithm.

Sounds like my students

Let’s hope this year proves my students, and math students everywhere, very wrong!