## Kiss those Math Headaches GOODBYE!

### Let STUDENTS make the math Problems, for a change

If you want students to look at you like you’re crazy — and have fun because you know you’re doing a good thing — try this.

Tell students it’s their turn to make up a math problem.

Math Meeting Board and Lesson (Photo credit: Old Shoe Woman

Yes, they’ll give you that look like, what are you talking about? But it’s o.k. Persist. Not only that … tell them to make up a word problem just like one in the textbook or on the worksheet. And tell them to make it relevant to their own lives.

For example, if you’re doing problems on rate, time and distance, suggest that students make up a skateboarding problem. One of my students came up with this:

You want to skate over to Ted&Tom’s (a local hangout), and you need to get there by 2:15 pm. If you’re 3 miles away and you leave at 1:30, going 4 mph, will you get there in time? [Answer:  You’ll get there right on time, not a minute too soon or too late.]

See how easy it is? Not really hard.

Or, let’s say that you’re doing ratio problems. Suggest that students do a problem on price comparisons. Another one of my tutees came up with this:

Lip gloss is on sale, 4 tubes for \$7. At that rate, can you buy 12 tubes if you have exactly \$20? [Answer:  No, since you won’t get the special if you have only \$6 for the last set of lip gloss tubes.]

The benefits for students are many.

1)  Students start to see that math problems are “all around them.” i.e., They start to see math in their everyday situations. And they start to realize that they can actually use the math you’ve been teaching them to figure out  real-life problems.

2)  By developing their own problems, students grasp the concepts in the problems more deeply. In the same way that we teachers learn by teaching, students learn by making (and solving) their own problems.

3)  Making problems is a creative activity, and once students see they can pull their problems from real life, they start to enjoy the activity. And because this involves creativity, this exercise engages the “creative types” who often feel like math does not “speak to them.”

4)  If you take the activity one step further, you can help students build their critical thinking skills. The one step further is: require that students get a whole number answer for their problem. This requirement forces students to think about how the numbers in the problem affect the value of the answer. And when they need to fine-tune those problem numbers to get out a particular kind of numerical result (like a whole number answer), they learn about the “innards” of the problem. They learn how the problem works more deeply than they would if they only were solving a problem someone else gave them.

5)  If you make the solving process cooperative, you can add even more fun to the process. By this I am suggesting that after students make the problems, they give them to other students to solve them. This way two students can exchange problems. I’ve seen students really get into this. They start making problems harder until they are just at the level that makes their partner “sweat.” But they enjoy this process, and it helps them get to know each other. I’ve found that this is a good way to get some fun socializing into a math class.

One last nice thing:  I’ve found that students cannot actually make up problems if they don’t know how to solve the problems. That means that this exercise tells you, the teacher, which of your students do understand the problem. And if they don’t get it, you can help them get it by helping them make the problem. It’s a nice, indirect way to teach.

So give it a try in your class or teaching situation, whatever that may be. I have a hunch you’ll find it as helpful and enjoyable as I have found it to be.

### “Hacks” for Slaying Proportions, Part 1: the Amazing Horizontal Canceling Trick

Proportions can seem intimidating, but they’re actually one of the easiest types of word problems to master. In this series I’ll offer a number of tips that help you conquer algebraic proportion problems.

But first, a cool shortcut you can use whenever you’re facing down an algebraic proportion …

In working with proportions, I’m amazed that few students know how a canceling process that would help them find the solution more quickly and efficiently.

So I want to share the trick, for all who’ve never seen it.

Of course, given a problem like:  6/x  =  24/32,

most of us know that we can cancel vertically with the two numbers in the fraction on the right, to get:

6/x  =  3/4

Then we just cross-multiply to get:

3x  =  24, and see that x  =  8.

In other words, we know we can cancel vertically given a proportion, just as we can cancel vertically with any fraction.

What many people don’t know though, is that there’s another way we can cancel when solving proportions — horizontally!

— What? you say.

Horizontally, I say. And no, I’m not joshing.

For example,  given the proportion:  7/4  =  21/x

you can cancel horizontally with the two numbers in the numerator:  the 7 and the 21. These reduce to 1 and 3.

The proportion then becomes:

1/4  =  3/x  [I’m really not kidding.]

Cross-multiplying, you get the answer in one quick step:   x = 12.

What’s really convenient is that you can also cancel both vertically and horizontally in the same problem. For example, in

6/x  =  42/28,

you could first cancel horizontally, to get:

1/x  = 7/28

Then you can cancel vertically, to get:

1/x  =  1/4

Cross-multiplying, you get the answer in just a step:  x = 4

I find that when students cancel before cross-multiplying, they’re more apt to get the right answer, and to get less frustrated, for the numbers they deal with remain small.

For example, in the last problem, if the student had not canceled at all, he would have a cross-multiplication mess of:

6 x 28 = 42x

That sort of problem just opens up the door to arithmetic mistakes. But canceling before cross-multiplying shuts that door since it makes the numbers smaller and easier to manage.

So now you get a chance to practice horizontal cancelling!

First use horizontal cancelling to get the answer to these
proportions. Those who’d like an added challenge might like to try them in their head:

a)   x/12  =  3/4

b)  3/7  =  x/35

c)   z/48  =  7/12

d)  y/56  =  7/8

Now go really wild! Use both horizontal and vertical canceling to make quick work of these proportions:

e)  x/9  =  16/36

f)   x/22  =  30/66

g)  32/56  =  y/14

h)  13/q  =  65/35

And here are the answers to all of these problems:

a)  x  =  9

b)  x  =  15

c)   z  =  28

d)   y  =  49

e)   x  =  4

f)   x  =  10

g)   y  =  8

h)  q  =  7

### “Simple” equations? Not always so simple.

Have you ever wondered how students can make mistakes with equations that appear extremely simple to solve, equations as basic as: 9 – x = 11, or: – 9 + x = 11?

To me this was rather baffling until I started to see these equations through students’ eyes, thanks to some tutorees who told me why they were making some mistakes here.

Through this experience, I’ve come to realize that I should be careful before I label any equations “simple equations.”

Here’s a case in point, which I’ll call Example #1:

9 – x = 11

If you ask students how the 9 in this problem is connected to the left side of this equation, a distressingly LARGE NUMBER will say that the 9 is connected by SUBTRACTION. If you just think about it, it’s a very understandable mistake. There’s a big glaring negative sign to the the right of the 9. When students look at that negative sign, many think it tells them that the 9 is connected by SUBTRACTION. Using this analysis, and armed with the knowledge that you do “the opposite operation,” these students will blithely ADD 9 to both sides of the equation, but they’ve just gotten off on the wrong track, getting: 18 – x = 20.

Similarly, given an equation as “simple” as this, Example #2:

– 9 + x = 11

many students will tell you that the – 9 is connected by addition. Why? Because of the big glaring + sign to the right of the – 9. . Reasoning thus, these students will gleefully do the opposite operation and subtract 9 from both sides of the equation, landing in an equally sticky puddle of wrong thinking, getting: – 18 + x = 2

How can we help students avoid these common algebraic pitfalls? Here’s what I’ve found helps.

Tell students that if a number is connected to a variable by an ADDITION or SUBTRACTION sign, you have to look to the LEFT of the number — not to the right — to determine how it is connected to that side of the equation.

To see how this works, let’s look back at Example #1: 9 – x = 11

A student who understands the correct process knows she must look to the LEFT of the 9 to see how it is connected to this side of the equation. Since there’s NO VISIBLE SIGN to the LEFT of the 9, that means that there’s an invisible + sign, so the problem can be re-written, for sake of clarity as: + 9 – x = 11

Now the student is getting somewhere. Using the idea of looking to the LEFT, she looks to the left of the 9 and sees this + sign. This tells her that the 9 is connected by ADDITION, not by subtraction. Following the rule of inverse operations, she now knows to SUBTRACT 9 from both sides of the equation, to get: – x = 2

From there it’s just a matter of multiplying both sides by (– 1) to get the variable alone in its positive form, obtaining the answer: x = – 2

Similarly, Example 2 can be solved using the Look to the LEFT rule of thumb.

This problem is: – 9 + x = 11

The student must ignore the + sign to the right of the 9, and instead look to the LEFT of the 9, finding a – sign. That – sign tells the student that the 9 is connected to the left side of the equation by SUBTRACTION. Using the rule of inverse operations, he knows to add 9 to both sides of the equation, getting: x = 20 And that is the final answer.

As I think about this area of confusion, I think it occurs because we teachers sometimes tell kids that they must “get rid of the number that is connected to the variable.” I know I’ve been guilty of talking this way (but I am trying to reform myself).

When a number is multiplying or dividing a variable, it is perfectly appropriate to tell kids to “get rid of the number next to the variable.” For example, in: 3x = 21, students need to get rid of the 3 which is connected to x through multiplication. Likewise, in x/5 = 6, students must “get rid” of the 5 that is connected through division.

But my recent algebra epiphany is that when numbers are connected through addition or subtraction, as in these problems:

10 – x = 23

x + 7 = – 5

– 8 + x = 13

– 2 – x = 8

we need to be more subtle in our instructions. We must tell students to Look to the LEFT of the number on the same side of the variable. And then tell students to let the sign they see to the LEFT guide them as to how to get rid of the number term that is in the way.

Just a little self-confession that I thought I’d pass along.

Good news: I’ve already started using this “Look to the Left” approach, and it is clearing up lots of confusion for students.

Happy teaching!

— Josh