Division is repeated subtraction

Multiplication is repeated addition. My math methodologies teacher taught us how to show students that multiplication is a short-cut for adding the same thing over and over. The curriculum I’m currently teaching has an entire lesson devoted to teaching children that multiplication is a short-cut for repetitive addition problems.

Logically, it follows that division is repeated subtraction. It’s a short-cut for subtracting the same thing over and over.

So why do so many students resort to addition to solve a division problem? Is it because we don’t help them make that connection, or do they just fall back to what’s familiar?

Translating percent problems into solvable equations

It’s usually pretty easy to see when you have a percent problem on your hands. They generally come in one of three varieties:

• x is what percent of y?
• What is z percent of y?
• x is what percent of z?

When you understand how x, y, and z fit together, solving percent problems becomes a snap. I’m going to show you two different ways to tackle each type of question, one using proportions, and one using a straightforward equation. Both work, so pick the one that makes more sense for you.

1. x is what percent of y?

=     OR    x = ? * y

2. What is z percent of y?

=     OR    ? = z * y

3. x is z percent of what?

=     OR    x = z * ?

It’s really just two simple formulas that can be adapted to whichever situation you’re in. Just insert the numbers you know, and solve for the one you’re missing. Beware, though. If you choose the multiplication formula, make sure to convert the percent into a decimal!

Why do I have to learn algebra?

Because there’s more to algebra than just the math.

Algebra is really about learning how to apply math to real world situations. It’s about setting up the right equation to give you the answer you need when you need it. It’s about modeling data to help you make good decisions. It’s about making sure you’re getting a fair deal.

In short, algebra teaches you to think critically. It teaches you to look at what you have, and to use what you have efficiently to find what you need. It teaches you to look at patterns, translate them, and apply them. It teaches you to make connections, to apply what you learned in one situation to a seemingly random situation.

It also lays the groundwork for you to be successful in geometry and any later math classes you choose to take, thereby allowing you to graduate and move on to successfully complete your math requirements in college. And those critical thinking skills come in handy regardless of your discipline. Being able to extract the right information and work with it will get you pretty far through your career.

So, the next time you complain about how you’re never going to use what you’re learning in algebra class, remember that even if you don’t use everything, you are still coming away with valuable life skills.

Why labels are good

I thought I had covered somewhere in the past, but it would appear that I’ve only shared this bit of wisdom with my students.

Labels are important. Labels are things like units of measurement, what you write along axes on a graph, or materials you’re counting. Labels are what provide the context for your response. They explain clearly what is going on. I often tell my students that the difference between an answer with the correct label and an answer with an incorrect or missing label is the difference between a building and a pile of rubble.

It’s true if you think about it. Imagine you’re trying to install a pool in someone’s backyard. The homeowner tells you they want a rectangular pool that’s 5 yards by 3 yards. As you’re doing your calculations for the materials, you lose that, and suddenly you’ve built a lovely 5-foot by 3-foot wading pond and really upset the owner in the process. By your calculations, your numbers were reasonable for the space. You just forgot to include the unit.

And if you look at a graph where there aren’t any labels, then it’s really hard to interpret the data. You have nothing to base it on. Dig out your science or history book, find a graph, and cover up anything written next to the axes and the graph’s title. What does this graph tell you? Does it make sense, or is it just some weird graph? Now, remove the covers and look at the graph again. With the labels, you can now find answers with this graph, allowing the graph to do its job.

Even word problem answers have labels. If you’re trying to find how many books fit a shelf, a number alone is not a sufficient answer. Did you mean that number of books? Did you misread the question and mean that many shelves? It’s also a bad idea to get the right number and put the wrong label next to it. Always reread the question to make sure your answer is reasonable and your label is correct.

When you’re taking an open-ended test (one that requires you to show your work or explain your answer), forgetting to include labels can cost you points. There’s nothing worse than doing the math perfectly and failing a test because you weren’t careful with your labels.

Take an extra moment when you think you’ve finished a problem. Ask yourself if your answer is reasonable, and then ask yourself what label you need to include. It’s a good habit to develop.

Moving a student from confusion to understanding

I work in a tutoring center, often helping students struggling with the initial algebra skills. There is one student in the center who has been working on solving for a variable on both sides of the equation for a couple of months now. Four or five of us have tried to help him, but he was just stuck.

Actually, that’s not fair. He has actually shown considerable growth. When he first started bringing in worksheets filled with these types of problems, he’d ask for a graphing calculator. The first couple of teachers didn’t question him; they just handed him the calculator. They never noticed he was quickly discarding the calculator in frustration.

Then, he came to sit with a teacher who sits near me and asked me for a graphing calculator. I have a pretty strict policy with the calculators. I only hand one over if it’s the right type of calculator, and if it’s a skill or time limit that could actually benefit from one. So, I asked the student what he was working on, and he showed me. I asked him to show me what problem he was stuck on, and he said, “Well, I haven’t started yet because I need the graphing calculator.”

Somehow, he was convinced that if he typed the equation exactly as it was on the worksheet, the graphing calculator would give him the answer. I sent him back to the other teacher, sans calculator, and asked the other teacher to actually show the student the process for solving those equations.

About a month later, the student was sitting with me, still working on solving for a variable on both sides. He was no longer asking for a calculator (unless he needed to solve something taxing like 48 / 6), but he was doing something else odd that none of the other tutors had noticed.

Given a problem like:

2x + 3 = 4x – 7

he was dutifully grouping the like terms:

2x + 3 = 4x – 7
4x – 2x = 2x
3 + 7 = 10

and then he was stuck. He had no idea what to do next, and he was positive his teacher said he wasn’t supposed to set them equal to each other (as I so foolishly suggested).

But I persisted and finally got him to try setting his two answers equal to each other. Lo and behold, he had an answer.

It’s been a couple of weeks, but the student is finally allowing us to show him the actual process for solving these problems, and he’s picking it up quite well. Occasionally, he’ll regress back to that method above, but generally, he does it our way…and he understands it.

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.

What’s the point of homework?

The school year is slowly beginning, and my students are starting to bring in their homework. It seems like the ideal time to remind everyone what homework’s purpose is in this life.

For those not familiar with me, I work in a K-12 learning center. I spend a lot of my time with the advanced math students, many of whom bring in their homework either for me to check over or to explain.

I know what you’re thinking. They’re teenagers. They probably didn’t pay attention in class and that’s why they don’t understand. In some cases, that’s true and the student admits it, but we’ve had a startling trend over the past couple of years (as the integrated math program has invaded the younger grades, coincidentally) where math teachers are assigning middle school and high school students homework for the skill they’re studying the next day. In a frightening number of cases, the teacher then fails to either check the homework or teach the skill.

Naturally, I have some very frustrated kids who just want to understand how to do the math being shoved on them.

The point of homework is not to replace instruction. That’s not what the school district is paying for. That’s not what parents want. Amazingly, that’s not what students want, either. Sure, you have the slackers, but eventually even the slackers want to be able to pass the class and not fail the next one.

The point of homework is to give students an opportunity to practice the skill you taught them. It’s a chance to make sure they understand it before you give them a harder skill built upon this skill. It’s a chance to make sure you taught the skill to them in a way the majority of the class understands.

If you’re expecting your students to teach themselves, perhaps it’s time you reconsidered your career choice, if only for the sake of the students.