Play an addictive game to learn algebra!

DragonBox title

One of the best ways to learn something is by playing a game. Unfortunately, designing an effective learning game is a lot harder than it looks. So when I read a rave review for DragonBox, I was pretty excited.

DragonBox title page

I tried it out, and this is by far the most complete representation of algebra rules I’ve ever seen on a computer. It rivals the Algebra Models I wrote about before. It makes algebra so easy a five-year-old can learn it. And kids love the game — it’s more popular than Angry Birds on the AppStore.

There are a few relatively minor details I’d change, and I suggest you learn about the balance model of algebra along with playing DragonBox. But if you’ve had trouble with algebra, or if you’re a parent who wants your kids to avoid having trouble with algebra, DragonBox would really help.

Download it for Android, Ipad, Iphone, PC, or Mac. The 12+ version is only available for phones, but there’s a web version available until the game is ported to PC.

Order of operations is all out of order!

Math teachers frequently say that math, more than any other academic subject, is logically sequential. Each topic builds on the previous topics, chapter after chapter.

But that’s not always true. Some topics in math class seem to come out of nowhere, wedged in the middle of some apparently unrelated stuff. Teachers know it’s because they’re considered necessary preparation for future topics, or because they’re considered important even though they’re not directly related to the rest of the semester. Unfortunately, most students have given up asking why.

Possibly the best example of an awkwardly wedged-in topic is Order of Operations. Sure, we all have to agree on a standard set of rules for operator precedence. But there are good reasons for these rules; they’re not just convention.

Unfortunately, the rules look totally arbitrary, because of timing. We teach them as a prerequisite to algebra, so students have no basis for understanding them, so all we can do is fall back on convention. We claim that math is logical, but the way we teach isn’t very convincing!

I propose that we simply skip the section on order of ops. The first time students need to substitute numbers for letters, talk about terms. When you find the area of multiple rooms, first you need to find the area of each room, then you add the areas together. Terms are like rooms: 2x is a term, and -3y is a term, and before adding them together, we have to resolve them. The fact that x is closer to 2 than it is to anything else provides visual reinforcement. Division is usually written as a fraction, so if x is on top of 2, clearly that must be resolved before adding 4. This covers the MDAS of PEMDAS, and for now, I’d leave it at that.

When we get to distributive property, I talk about parentheses being like a picture within a picture, or a story within a story.* If you want to multiply the picture by 3, you have to include the whole picture. So you need to either resolve everything within parentheses before doing anything else with it, or you have to be sure to include everything. Of course, resolving should be preferred whenever possible. I also remind students that their calculator can’t see how long a fraction bar is, so sometimes they need parentheses on the calculator when they don’t on paper.

Finally, when we get to exponents, I explain repeatedly that x, x2, and x3 are like length, area, and volume. You can’t add 5 ft + 3 ft2; it doesn’t make sense. When you substitute numbers, you need to resolve the exponential terms down to just numbers, before you can combine them. Any operator only applies to what’s right beside it,** so 2x3 means 2 cubes or 2xxx, not (2x)(2x)(2x).

In my experience, most students see math as a labyrinth of inscrutable rules. No wonder they hate it! This way of presenting order of operations flows with the rest of algebra — students won’t even think about memorizing because it will just make sense.

This idea developed gradually when I taught a very basic algebra class. I refined my plan in response to two posts by Timon Piccini at Embrace the Drawing Board. Do you see any problems with this? How would you improve on it?

* I just thought of this analogy while writing, and I like it better than my previous explanation.
** Yes, this is also true of addition and subtraction. They only act on the result that’s right beside them, when it’s their turn to act.

The best innovation since the Cartesian coordinate system

I wrote previously about the analogy of an algebra equation being like a pan balance, and recommended a website to use for practicing the concept. But there’s no practical way to do this with three-dimensional objects, because the weight of x would have to change for each equation.

However, there is a way to learn algebra with physical objects — you just have to remember the basic rules without any help from gravity. It’s still much easier than the letters and numbers that have plagued generations of teenagers.

Okay, so what is this system? A set of algebra manipulatives includes little squares to represent ones, two different sizes of narrow rectangles to represent x and y, and larger pieces to represent x2, y2, and xy. The most common way of representing negatives is that the back sides of the pieces are a different color.

What’s so great about them? First, you can use them to solve algebra equations. Rather than a bunch of numbers and letters on paper, you have objects to rearrange. 3x + 6 becomes 3 x pieces and 6 one pieces. These help a student learn algebra just as counters help a first-grader learn addition, because they can do concrete actions to each side of an equation, and see what 3x + 6 = 5x – 4 actually looks like.

Second, and even more exciting to me, is that these manipulatives provide a visual representation of multiplying and factoring polynomials. Rather than trying to remember all the apparently arbitrary rules, a student can see that x times 3 forms a 3x rectangle. The trial-and-error of factoring becomes a simpler matter of rearranging the pieces to make a rectangle, adding pairs of opposites as needed.

It’s algebra without anxiety! Using manipulatives makes the mechanics less error-prone, and reinforces the logical concepts automatically. Over time, students gradually transition to doing algebra exclusively on paper, when they’re ready. These simple plastic pieces are a visual aid as innovative as the Cartesian coordinate system, and I wish more teachers would use them in the classroom.

Now, where can you get some? The best designs are three dimensional: Lab Gear and Algeblocks. Unfortunately both are pricey, and the 3D Lab Gear parts are unavailable as far as I know. The set I use for tutoring is called Algebra Models and is comparatively inexpensive. I recommend the “Cooperative Group Set” for one student — the individual set doesn’t have enough pieces for most algebra problems, unless you use a book designed for the manipulatives. It would be nice to have the “Small Group Set” for even more pieces.

Another option, which may not be much of a cost savings over the Algebra Models, is to cut the pieces out of cardstock. There is also a virtual representation of the manipulatives, but that format is far from ideal.

In the future, I’ll write in more detail about how to use these. Which topic in algebra would you most like me to explain?

A better way to think about solving algebra equations

Most people learn to solve algebra equations in terms of a list of steps for each different type of equation. But really, all those steps are based on a few simple concepts. If you understand those concepts and see an algebra equation within that framework, you can forget about all the different rules and steps that are so overwhelming to beginners.

An algebra equation is like an old-fashioned pan balance used for weighing things. To use a balance, you put an item, say a bag of candy, on one side, and then add weights to the other side until they balance. If the weights are one 5oz and two 1oz, the bag of candy weighs 7oz. In the case of an algebra equation, the equals sign tells you that whatever is on one side weighs the same as whatever is on the other side. Your goal is to figure out what one x “weighs”. You can do whatever you want, like add 5 or multiply by 2, as long as you do it to both sides of the balance so both sides still weigh the same. When you get one x alone on one side of the balance, and only a number on the other side, you’re finished – x “weighs” that many units.

It’s hard to solve algebra equations with an actual balance, because x would have to be a different weight for each equation. However, you can experiment with a virtual balance on the computer. This particular website has some limitations to the size of numbers and what you can do to the equation, but it’s the best program I’ve found for showing a picture of this concept.

So, when solving an equation, you have one goal: get x all by itself. There’s only one rule: whatever you do, you have to do it to both sides. Other than that, you can do whatever you want — whatever you think will help get x by itself. Generally the best strategy is to do the opposite of what’s been done to x. But it’s perfectly acceptable to add, subtract, multiply, or divide by any number, as long as you do it to both sides correctly. You might take the scenic route to a solution, but you’ll get better with practice.

After you practice for awhile and start chafing at the limitations of the virtual balance, there’s another website that will allow you to experiment with steps to solve an equation. There’s no graphical balance, unfortunately, but you can see what happens when you multiply by 3 instead of dividing, or add 5 when 7 would have worked better. You can undo steps and try different ideas, knowing that you won’t be tripped up by an error with negatives when you’re trying to focus on the process.

Of course, with enough practice you’ll be able to solve equations on paper, without any help. And rather than blindly following a list of steps, you’ll be confident that you actually know what you’re doing!