Kiss those Math Headaches GOODBYE!

Archive for April, 2008

How to Use Color to Explain Algebra Concepts

Lately I’ve been having great success using COLOR to explain algebraic concepts, as strange as that may seem.

I’ve found that is that adding color to equations helps students see important relationships among the terms, for it lets them visually group related terms.

This is impossible to explain without an example, right? So here’s one to show you how it’s done.

GOAL: Give students a way of understanding the following exponent rule, usually learned in Algebra II:

In explaining this rule, I first show students a common sense but critically important idea:

GIVEN: Two terms are equal to each other, and each term has two parts.

CONCLUSION: IF two of the parts are equal (blue), you can CONCLUDE that the other parts are also equal (pink).

Here’s a visual WITH COLOR that gets this idea across:

Next I help students recall an early math concept that uses this principle. In this case, the following example gives them a nice way of seeing this concept with color:

The conclusion, of course, is that x + 4 = 9, meaning that x = 5.

Notice that I consistently use the same colors to indicate the same ideas: blue, to show the parts that we know are equal; pink, to indicate the parts that we conclude are equal.

Finally I use this same color scheme to show the logic behind the Algebra II exponent rule, like this:

The conclusion, of course, is that THEREFORE x = y.

I get a lot of quiet “Now I Get Its” when I show this to Algebra II students.

The “Aha” looks that I get remind me that while this concept is obvious to me and you, many students don’t grasp it fully.

In any case, I wanted to share this because I’ve found that there’s great potential in using color to explain aspects of algebra. I hope that you find this useful, and I encourage you to use color yourself.

Just for the record, the rule I explain in this section is a very useful rule of exponents. To show you its use, I am including the example of a problem that requires its use, below:

How to Multiply Two Teen Numbers — FAST

Ever wondered if there’s a quick way to multiply two numbers in the teens, like:
14 x 17, or 18 x 16?

Turns out, there is. Stick around a minute, and you’ll learn it. Let’s try it for:
18 x 16.

The trick involves doing two operations with the numbers in the ones place of the teen numbers. The numbers in the ones place are 8 (in the ones place of 18) and 6 (in the ones place of 16).

First, ADD the two digits in the ones place.
8 + 6 = 14.

Take that sum, 14, and add it to 10:   14 + 10 = 24.

Tack a zero on to the end. 24 becomes 240. (Keep that number in mind.)

Next MULTIPLY the two digits in the ones place:
8 x 6 = 48.

Now just add this product, 48, to the 240.
240 + 48  = 288.

That’s the answer. This may seem a little tricky and a little weird, at first, but it gets easy after a few times. Trust me …

O.K., fine, don’t trust me. But just try it one more time, with 14 x 17, and see for yourself.

4 + 7 = 11. 11 + 10 = 21.

21 becomes 210.

Then 4 x 7 = 28, and 210 + 28 = 238.

That’s all there is to it.

Now try these:

a) 13 x 16
b) 12 x 17
c) 14 x 19
d) 12 x 19
e) 13 x 14
f) 17 x 18
g) 19 x 17
h) 15 x 19
j) 16 x 17
k) 18 x 19


a) 13 x 16 = 208
b) 12 x 17 = 204
c) 14 x 19 = 266
d) 12 x 19 = 228
e) 13 x 14 = 182
f) 17 x 18 = 306
g) 19 x 17 = 323
h) 15 x 19 = 285
j) 16 x 17 = 272
k) 18 x 19 = 342

April Problem of the Month


What provides a better fit? A square peg in a round hole, or a round peg in a square hole? How well does each peg fit in its respective hole?

To provide a complete answer, express the “fit” of each plug as a percentage. To find
this percentage, divide the area of the peg by the area of the hole.

Saying that one of these creates a “better fit” means that the peg fills up a greater percentage

To get credit, you must not only provide the correct.
answer, you must also show your work.

Send answers to:
Make the subject POTM.

How to Best Learn the Multiplication Facts


Is there a particular order that is most beneficial to learning times tables? For example, I have heard that learning 1, 5, 10, 11, 2, 4, 8, 12, 3, 6, 9 is a good order. What say you?


There are many “schools of thought” on what makes the best order for teaching the times tables.

I encourage everyone out there who has a favorite order to share it with us in the comments. And feel free to explain why this is your favorite order.

Here is my preferred order — and a few notes on why:

1, 2, 4, 10, 5, 11, 3, 6, 9, 8, 12, 7

IMPORTANT NOTE: I teach the facts for the multiplicands 1 – 6 first. Only after
students have those down do I advance to the multiplicands of 7 – 12.

i.e.: When teaching the x 4 facts the first round through, teach 1 x 4 through 6 x 4. On the second round through, teach the 7s through 12s, i.e.: 7 x 4 through 12 x 4.

Using this staggered approach helps students avoid getting befuddled by the size of numbers they encounter early on. This approach builds success and confidence early on.

A few notes on my order:

1st) I teach x 2 second. That’s because you can teach multiplying by 2 as simply “doubling” the number, a concept almost all kids understand. The “doubling” approach gives teachers a good way to talk if you’re going to use manipulatives to reinforce the concept.

In suggesting that you use manipulatives, I mean that you may want to use counters such as tiddly winks, paper clips, pennies, etc. To demonstrate a fact like 2 x 7, you would first lay out a row of 7 counters, then underneath them put out another row of 7 counters, and you would have the student count them all up.

2nd) I put x 4 third because you can teach multiplying by 4 as doubling a number TWICE. e.g.,: 4 x 3: 3 doubled is 6. And 6 doubled is 12, so 4 x 3 = 12.

Again, it would be a good idea to use manipulative counters to demonstrate this concept. Here you would lay out four rows of three counters, to show the fact that 4 x 3 = 12

3rd) I teach x 5 right after x 10 because multiplying by 5 gives you a number that’s exactly half of what you get when you multiply by 10.

When teaching the 5s tables, use the trick that multiplying by 5 is the same as multiplying by 10 and cutting in half. (And, for even multiples of 5, first cutting the number in half, then tacking on a 0 at the end.)

e.g.: for 4 x 5: Half of 4 is 2. Tack on a 0, and you get 20 4 x 5 = 20 [yes, it is that easy!]

4th) You can teach multiplying by 8 in several ways, depending on what works for your children. One way is to see that multiplying by 8 is doubling a number three times. e.g., for 6 x 8: 6 doubled is 12. 12 doubled is 24. 24 doubled is 48. So 6 x 8 = 48

Another strategy for 8: Multiply by 10, then take away 2 x the number. e.g., for 6 x 8: 10 x 6 = 60, and 2 x 6 = 12. 60 – 12 = 48, so 6 x 8 = 48

5th) Multiplying by 12 offers you a great chance to introduce the distributive property for multiplication. That’s just fancy language for saying that when you multiply, say, 6 x 12, you can look at it like this: Since 12 = 10 + 2, 6 x 12 can be viewed as: (6 x 10) + (6 x 2) = 60 + 12 = 72

Most children can get used to this quickly, if they are taught it EARLY ENOUGH.

Exploring Addition Facts


I have been asked to help my grandson memorize the addition tables. If I ask him what is 8 + 5, I watch him doing it in his head. He gets the right answer but takes a long time, which may hurt him later when he is introduced to multiplication. Do you have any suggestions?


Great question.

Without knowing your grandson, it’s impossible to know what he is doing when he takes some time to figure out addition facts.

But I’d suggest is that you just ask him — in a happy and curious way — what thought steps he is going through.

You can tell him that you’ve heard that students do mental math in different ways, and that you’d be interested to know how he is doing it. You might also want to reassure him that there is no “wrong” way to do math in your head.

[Be aware: some children have trouble verbalizing what they’re “doing” when they do math mentally. If your grandson has trouble telling you, you might prompt him by asking if he’s using either of the strategies I describe below.]

It may be that your grandson is trying to retrieve a memorized fact, but it’s more likely that he is using some kind of mental operation to arrive at the answer.

For example, it may be that he is “counting up” 5 from 8, to get to the answer. (If so, it would be commendable that he can count up 5 in his head — without using his fingers. Not all students can do this.) It’s also possible that he is taking 5 from the 8, and giving it to the first 5 to make 10, and then tacking on the extra 3, to get to 13 (an advanced strategy).

The main point, though, is that you can’t know till you ask him.

And the other point is that it’s often fascinating to open up a dialogue like this with kids, to find out how they do mental math.

Once you get the dialogue underway, I’d suggest that you just follow wherever it leads. For example, if your grandson is using the second strategy I mentioned (making 10 and adding on), ask him if he can extend the process a bit, and do problems like 18 + 5, 28 + 5,

If, on the other hand, he is “counting up” 5 from the 8, see if he can use the second method, too.

Essentially, what you have here is a great opportunity to find out how your grandson does addition, and to explore the operation with him. And have fun doing it.

Please feel free to write back if you do open up this kind of dialogue. I’d be curious to know what happens.

And to get to the heart of your question, I would say: Yes, you do want your grandson to develop speed or “fluency,” as teachers like to say. But when he is first learning the facts, it’s critical that he think about the operation, not just memorize facts.

To help him gain speed, I would suggest that you use flash cards or fact worksheets (just google math addition worksheets, and you’ll find loads of them).

And to help him develop a range of good strategies to help him learn the facts with understanding,
I suggest the Facts That Last series by Creative Publications.

Good Question

Isn’t it great when kids ask good questions?

(Rhetorical question, that, of course.)

I got a good question today, about factoring.

I was showing this student how to factor by taking out the GCG, and he asks me, “So what’s the difference between factoring and dividing?”

You see, we had been using dividing when factoring the GCF. For example, to factor an expression like 4x + 16, we divided both terms by 4 after seeing the 4 is the GCF. So in this boy’s mind, factoring seemed akin to dividing.

What I liked about the question is that it made me think … and clarify something.

I realized that when you factor, you do divide, but you do more than divide.

Essentially, when you factor, you use division to make rename an expression. 

In the example I gave, you equate 4x + 16 with its factored form, 4(x + 4)

When you divide, on the other hand, you are just doing a small piece of this.

You divide, for example, when you ask:  4x divided by 4 = what? Answer: x

You use that answer to lay out the factored version, but dividing is only a step.

So hooray for good questions and congratulations to those who ask and recognize them.  Good questions make the act of teaching come alive.

(Less) Curiouser and Curiouser

If curiosity killed the cat, the cat took one too many tests

Today I tutored two girls … with two radically different levels of curiosity

The first girl, an eleventh grader, taking Algebra 2. The second, a fifth grader, working on elementary school math.

The older girl was studying logarithms. I started to try to tell her about the mysteries of the number “e.” I find this interesting. I got nothing but yawns.

The younger girl, a fifth grader, was studying elementary-level geometry. She needed help on measuring angles. I showed her that, and then she wanted to learn more. I showed her how to do constructions. She loved it, and even made her very first circle with a compass on the spot. I showed her how tesselations work. She loved that, and created her own M.C. Escher look-a-like.

What struck me after this girl left is the question: “What a difference?” And then the follow-up … how much of the difference in curiosity is due to years in school?

Answer:  a LOT!

All I can really say is that it’s sad how the focus on tests, and passing tests, and studying for tests, can really dampen a kid’s curiosity. It’s just sad, when you consider how many fun math topics there are, that so many kids get turned off.