Kiss those Math Headaches GOODBYE!

Archive for July, 2008


Here’s the problem —

Al’s Discount Clothing store is trying to entice customers with a

special “20-40” Percent Sale.


Here’s how it works. For all sale items, the Al’s

first takes off 20%. Then the store takes an

additional 40% off the sale price.




a)  How much is the discount, under the terms

of this sale, for a bundle of clothes with retail

value of $250?


b)  How much would the discount be if Al’s had

just run a straight 60% discount sale?


c)  Why do you think that Al’s is running the

sale in this way?


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Make the subject:  POTM

Making Sense of Negative Signs

If you’ve ever seen students struggle with expressions that have a negative sign in front of parenthses (I sure do!), expressions like this:  – (8x – 5)

then this blog entry is for you!

I’ve developed a new way to help students get this concept right — and to remember the concept so they continue to get it right, week after week. I’ll explain the basic approach in this entry, then give more details in the next entry.

The big problem with the typical textbook presentation for this concept is that it is filled with math gobbledygook. I’ve found that you can cut out the gobbledygook and instead relate this kind of problem to everyday life. Once students learn it this way, they’ll never forget it!


MAIN IDEA: encourage students to think of the negative sign as the as the everyday word, OPPOSITE.

So start out by asking simple questions, but writing them in a pseudo-math format.


Write:  opp (black)

while you ask:  What is the opposite of black?

When students give the answer, what you write now looks like this:

opp (black)

= white


Continue with other examples, like this:

opp (tall)

= short


opp (down)

= up


Once students get the basic idea, expand on the idea by telling students they can take the opposite of two concepts, not just one. Show this by writing expressions like:

opp (white, short)

Students should answer:

opp (white, short)

= black, tall


opp (left, slow)

= right, fast


Once students master this idea, extend the lesson to NUMBERS, first by pointing out that all numbers (except 0) and MONOMIALS have opposites. e.g., that the opposite of + 3 is – 3; the opposite of – 5 is + 5, the opposite of – 8x is + 8x, the opposite of 6ab is – 6ab, etc.

Now challenge students to do problems like this:

opp ( + 5x, – 7 )

They should get:

opp ( + 5x, – 7 )

=     – 5x, + 7


Then simply explain that in math, we express the idea of opposite by using a “–” sign in front of the (    ), and that we write the terms inside (    )  without commas.

Give students this problem:

– ( 7x + 4 )

See if they can get the answer, which should look like this:

–  ( + 7x – 4 )

= – 7x + 4


I hear a lot of: “Oh, that’s easy,” when I explain it this way. I think that’s because opposites is a concept students know “cold.” Plus, using the opposites concept connects the algebra to non-math concepts, and students often find that refreshing. I’m sure you’ve noticed that, too.

Then give students a set of problems, like these:

1)  – ( – 7a – 4 )

2)  – ( – 3x + 7 )

3)  – ( + 8y – 3 )

4) – ( 4p – 6a + 12 )

5)  – ( 9x + 4y – 3 )

Have fun with this lesson. If you think of any ways to extend it, or if you find any tricks that make it work especially well, feel free to share them by sending them to:


Well, my obsession this summer turns out to be knives.

But don’t get any wrong ideas. When I say knives, I mean knives for whittling and wood carving. That’s the true obsession. But knives themselves are a pretty close second. I am a guy after all.

In getting obsessed with knives, it’s impossible not to get obsessed with that critical aspect of knives: sharpness.

If you’re a guy, you already understand how this could be an obsession. If you’re not — and you don’t get it — just do a Google search for “sharpening knives.” I mean, there’s almost as much stuff on sharpness as there is on, say, wrinkle cream. Well, at least one-tenth as much!

So you’re probably wondering by now why a guy like myself, who’s really into math, would be writing about knives and sharpness. 

The reason has to do with that concept: sharpness. As I’ve been poring over websites about sharpness and then buying and sharpening my own knives for hours on end, to the point where there’s hardly any knife even left, I’ve had some time to think about the sharpness of knives and the sharpness of minds. Those two words rhyme, so take a second look:  knives and minds.

As I’ve been honing and polishing away, I’ve been wondering how much use algebra is for most kids, and I’ve come to the conclusion that for most of us, algebra itself is not really of that much “use.” I mean, c’mon: how many of us — other than engineers — use algebra on a daily basis. Even I don’t! But, what I’ve also realized is that algebra is kind of like a sharpening stone, and our minds are like the knives.

By doing math like algebra, geometry, trig, analytic geometry, and calculus, we are sharpening our minds.

So even though we don’t use these math topics every day, we make our minds sharper and sharper for any occasion when we use numbers.

It might happen when we (when we’re not sharpening our knives) are planning out our garden. We can use math to figure out how much space our rows of carrots will take up when we follow the directions and plant the rows 1.5 inches apart. Or it might happen when we are buying a car and we’re trying to follow the interest and the payment info. 

If we have sharp minds, we can do this stuff with relative ease. We can cut right through it, almost with a joy of power. If we have dull minds, we might just “guess-timate” or leave it up to the car financial guy. Whoa! What a mistake, huh!

So keep on sharpening those minds … and knives. It will pay off, even if you don’t do this very kind of math today or even tomorrow. Numbers are all around. There are all kinds of opportunities to put those sharp minds to good use.

Place Value Metaphor

During the summer I get to tutor a lot of elementary age students, remediating them on the basics.

Almost invariably I find that these students are confused about PLACE VALUE, and considering how critical this concept is to all of math, I decided to write this post.

Whenever I have the least suspicion that a student might be confused about place value, I check with a simple test.

I have them write down the number 22, then I ask them if they can tell me the difference between the two 2s. Often they cannot.

Tutoring a girl this past week I came up with a way of understanding place value that really resonated with the student. I want to share it because you may be able to use it, or a modification of it, with your students. First it’s important to know that this student’s mom teaches ballet, and the girl dances at her mom’s studio. 

I asked the girl if she has ever been to a ballet performance, and of course she said yes.

Then I drew a quick diagram of the stage and first few audience rows. I pointed to two seats, one in the front row, another seat several rows back. I asked her if the two seats would cost the same amount. This girl knew that the close seat costs more money because it is closer to the action on stage.

Then I used that idea to explain place value. I showed this girl that just as seats can be more or less valuable because of where they are, so too digits can be more or less value based on where they are in a number. 

She got this idea very quickly, and now she understands place value.

For children with different interests, use whatever makes sense. For example if you’re teaching a boy who loves baseball, make the rows of seats those at a baseball game, and so on.