Showing posts from March, 2006

Crazy 4 Math contest

You might feel that your kid/students will never be able to take part in any math contest, but this one is different. It is for everybody, and you don't have to solve any math problems at all.

Crazy 4 Math contest simply asks you to...

"Describe how you use math in any activity you love to do - a sport, game, craft, hobby or anything else. Send in a description of the activity and how you use math. You can also include a drawing or diagram. Class entries are also accepted. All participants will receive an MP3 of Googol Power's new song "Crazy 4 Math" when you enter."

So this might be a nice way to motivate a student, plus underline the fact, like I've mentioned before, that math is part of our life - not some obscure 'no-one needs it' type of thing.

I encourage you to check some of the last year's entries at

Teachers: you can submit a class entry too.

Refreshing algebra books from Dr. Math®

I've just finished writing a review on Dr. Math&reg algebra books by The Math Forum. There are two: one for pre-algebra and one for algebra 1 students.

They are NOT algebra textbooks because they don't have any exercises. Rather, they're supplemental.

BUT they're very nice AND fairly inexpensive (around $9-10 new, and maybe $6-7 used). Also they weren't awfully long; both were under 200 pages with clear layout. I really think you should take a look at these if you or your student has trouble with algebra!

Dr. Math's books are written in a very easy-reading and friendly tone. They are compiled from question-answers that real students have asked the Dr. Math® service at The Math Forum over the years. It's like reading letters, in a sense, but they deal with math questions. And these letters or explanations are not reading like a textbook but more like a real teacher talking.

One thing I liked was that the books included some 'philosophical' question…

Challenging / open-ended math problems

I just read a very inspirational article Teenager or Tyke, Students Learn Best by Tackling Challenging Math (PDF) (html).

It tells about two teachers who frequently employ open-ended problem-solving sessions in their teaching - and the students (almost all) like it well and are very motivated.

In math education, OPEN-ENDED problem usually means it doesn't have a specific step-by-step solution. You can solve it in many different ways. Or, it may have more than one solution.

The problems these teachers use are often from real life, and not quick to solve. Instead it can take some time and struggling to get anywhere. (Hey, that's how problems in real life often are, too!)

But, struggling can be valuable. One of the teachers featured in the article, Heidi Ewer, says:

"Struggling helps them see this as an investment of their own time and energy. It makes them more willing to learn," Ewer says. "Struggling to solve problems requires students to use their intuitive skill…

Two Choices - story

I got this in an email... so I'm "forwarding" it to all of you. A touching story about choices in all of our lives.

What would you do? You make the choice! Don't look for a punch line; there isn't one! Read it anyway. My question to all of you is: Would you have made the same choice?

At a fundraising dinner for a school that serves learning disabled children, the father of one of the students delivered a speech that would never be forgotten by all who attended. After extolling the school and its dedicated staff, he offered a question:


The audience was stilled by the query.

The father continued. "I believe, that when a child like Shay, physically and mentally handicapped comes into the world, an opp…

Prove that irrational*non-zero rational number is equal to an irrational number

How can you prove that irrational*non-zero rational number is equal to an irrational number?

I suspect this is a university student asking me this.

One important step in how to make proofs is to understand clearly what you're asked to prove. It often involves statements that are true for ALL numbers in a certain set (such as all real numbers, all rational numbers etc.).

This one does not include the word "all" nor the word "any", but it still is of that type. I can reword it like this:

Prove that any irrational number multiplied by any non-zero rational number is equal to some irrational number.

Or, this way:

Prove that for all irrational numbers x and for all rational numbers y excluding 0 the following is true: xy is an irrational number.

This question basically ASKS for indirect proof.

In indirect proof, we assume the OPPOSITE is true, and show that would lead to a contradiction.

We are supposed to prove that if you take ANY irrational number times ANY non-zero rati…

Carnival of Homeschooling

This week's Carnival of Homeschooling is up at PHAT Mommy. Again, lots of interesting stuff to read about homeschooling.

Review of Time4Learning is an online curriculum covering not only math but language arts, science, and social studies.

I've done a review on their mathematics curriculum (their content is provided by Compasslearning Odyssey).

It's quite motivating, because kids get to do these animated lessons and on lower grades they can aftewards go to "playground". My child sure likes it a lot!

Read my review and see some pics at

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Strategies to solve simple math equations

What are the strategies for solving simple equations?

I got this question in the mail just today.

I assume the person means LINEAR equations - those where you only have one variable (usually x), and that x is not raised to second or third or any other power, nor is it in the denominator or under square root sign or anything. Just x's multiplied by numbers and numbers by themselves, such as:

2x - 14 = 9x + 5


1/3x - 3 = 2 - 1/2x


2(5x - 4) = 3 + 5(-x + 1)

Here are the strategies for solving these:
* You get rid of paretheses using distributive property
* You may multiply both sides of the equation by the same number
* You may divide both sides of the equation by the same number
* You may add the same number to both sides of the equation
* You may subtract the same number from both sides of the equation

You might think, "Which one of those will I use, and in which order?"

That depends. There is no clear cut-n-dried answer.

Whatever you do, you try to transform your equation towards th…

Addition facts with 6 and 8

Sums with 8

Well today's post will not contain any deep mathematical insights... Just two pics of the things we're doing at our house. We're studying simple addition facts. I just keep writing problems in her notebook. Have a happy weekend!

Sums with 6

Word problem situations in elementary grades

I still want to continue on the topic of word problems - not because it's my 'soap box' subject but because I feel parents and students can use lots of help with them.

If you didn't catch the earlier post on how your typical math books subtily teach kids NOT to think carefully with word problems, read it here.

The easy 'routine' word problems in early grades usually require just one operation to solve. I would recommend studying a bunch of such word problems WITHOUT calculating the answers but only thinking and finding which operation is needed to solve each problem.

After you do that enough times, the student should start associating the types of situations with the appropriate operations:

Total is divided into so many parts/containers, each part having same amount.

This is the multiplication/division situation:
(number of parts) × (amount in each) = total

If you know how many parts and how much in each, MULTIPLY.If you know the total and the number of parts, DIVIDE.…

Pi Day

I almost forgot...! Today is Pi Day! See, it's 3-14 (March 14) and pi ≈ 3.14.

To celebrate, check some pi day activities.

The problem with word problems 2

For the purpose of this post, we could divide word problems to three different categories:
1) routine word problems
2) non-routine word problems
3) algebra word problems

Actually you could divide algebra word problems to routine and non-routine as well, but I want to now talk about word problems kids encounter in school before algebra - in grades 1-8 usually.

J.D. Fisher suggested in the comments section of my previous post on word problems that kids are encouraged to think linearly, step-by-step. Then, when the word problems they encounter don't anymore follow any step-by-step recipe, they are lost. You might want to go back and read that.

Don't typical math book lessons kind of follow this recipe:

Explanation and examples.
Numerical exercises.
A few word problems.

In other words, the word problems are usually in the end of the lesson. (That might make solving them a rush.)

Then, have you ever noticed... If the lesson is about topic X, then the …

Logarithms in a nutshell

Someone asked me recently to make a post about logarithms. So here goes. I already answered the person in an email but I thought I could include some interesting history tidbits here, too.

Logarithms are simply the opposite operation of exponentation.

For example, from 23 = 8 we get log28 = 3, and we read it "base 2 logarithm of 8 equals 3".

So it's not difficult: if you understand how exponents work, logarithms have the same numbers, just in a little different places.

Just as in exponentiation, a logarithm has a base (2 in the above example). Remember that in 53, 5 is called the base and 3 is the exponent.

Other examples:
53 = 125 and log5125 = 3.
104 = 10,000 and log1010,000 = 4.
2x = 345 and log2345 = x.

As you can see from the last example above, you can use logarithms to solve equations where the x is the exponent:
4x = 1001
x = log41001.

Then you'd get the value of x from a calculator.

However, the base of the logarithm can be anything and most calculators only have two bu…

Example of failed problem solving

My intention for today's post was to solve a problem, and write down my thinking process, so it would be an example of problem solving process.

But I failed to solve the problem!

I went to this site first but the problems there looked routine, plus I didn't want to choose a calculus problem but keep it somewhat simple.

Then I went to and looked at their one-star problems... they seemed kind of easy, I was able to "see" the solution immediately. I wanted a problem that I personally could not solve just by reading it.

I finally settled on a two-star geometry problem.

It has a right triangle and lines drawn in it, and you're supposed to prove that a certain angle is equal to another angle.

The details of this are not important so I'm not going to post here all that I typed... (I tried to type in my thinking process while solving it). I quickly saw some right triangles there, and that instead of proving the angles equal, I could try prove a certain …

The problem with story problems

I was trying to think today why it is that so many kids feel that word problems or story problems are difficult.

It didn't make sense to me initially. See,
Most kids just love stories.Usually kids love words, too, based on the fact they use them a lot.And problems - I can't imagine that kids don't like word problems just because they need find an answer to something. Most of us even adults get fascinated by puzzles, for example.
So what is the problem with word problems?

It surely can't start on 1st grade. You know, someone tells you a story problem such as: There are five ducks on the lake and three on the shore. How many ducks are there total? And often the math book has a nice picture there to accompany it. Surely kids don't think that as being difficult.

My child has gotten to like "subtraction stories" pretty well - just simple situations where someones or some things go away. She has even made up some herself.

Could it be that they don't underst…

Addition fact flashcards

This morning, my baby found a set of addition flashcards I had bought at some yardsale, and scattered them on the floor.

While picking them up, I was lamenting in my mind the fact how they are not usable with my other daughter. The set has 32 cards so obviously it doesn't cover all basic addition facts. It seems to have just random ones, such as 7 + 6 = __ or 8 + 4 = __ or 2 + __ = 2 or 1 + __ = 5 all mixed together.

But what I'd want for right now is those cards only where the sum is five:
0 + __ = 5 and 5 + __ = 5
1 + __ = 5 and 4 + __ = 5
2 + __ = 5 and 3 + __ = 5

That's the approach I have used in my ebook Addition 1: to practice the facts in small groups, grouping them by the sum.

First you could study those facts where the sum is 4, then those with sum 5, and so on.

That would make more of a logical approach to all this fact practicing. You know, everybody understands that you're supposed to memorize multiplication tables by the tables and not by 'all at once' or…

10th Carnival of Homeschooling

10th Carnival of Homeschooling is online at PalmTree Bundit. Anne is celebrating spring!

What to do when math gets dry

Have you ever felt like your child needed something different for math than whatever you were currently doing? Like your math program didn't work anymore: your child didn't learn, or was bored? What to do?

Here a little while back I received a question along those lines. We exchanged a few emails and I got permission to post her situation here on this blog. I think it is a good example of what many times happens in home school.

"We have 4 children; boy-12, girl-8, girl-5 and boy-3. With oldest we have used the gamut of curricula for math and eventually settled out with Abeka for the past 3 years.

However, the material seemed to not sufficiently explain new material, but run old material in the ground until the grinding sensation was numbing. That in itself seems to be contradictory, but my son was really not getting much better at the basics, mainly I think because he was bored with the repetition.

I narrowed his assignments down to every other one and if I found he missed a…

Tilings and shapes on the way

Today we went to town, for most of the day. At one point I noticed I was walking on hexagonal tiles, and I noted that to my daughter.

From then on, we just kept looking around at store floors, walls, the sidewalk to find various tiling patterns. Most of them were either simple squares, or were using rectangular tiles, but with rectangle-shape tiles people had made all sorts of different designs or patterns. In some, the tiling combined two different shapes.

She especially enjoyed seeing the 'artistic' ones (not true mathematical tilings) where people just had put all sorts of broken tile pieces in a cement.

I guess that was our math lesson of the day. Made the long walking around town go quicker for her, for sure!

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Using mental math vs. paper and pencil

The example of decimal division I found recently made me think some more about how we teach our students the use of mental math.

For example, how would you want your (current or future) high schooler to find the answer to 5 × 24 or maybe 6 × 71 and such like?

a) I wish he'll use pencil and paper, put 5 under 24 and use the algorithm.
b) I wish he will go 5 × 20 and 5 × 4 in his head and add those.
c) I wish he grabs the calculator.
d) I wish he asks me.

Also think WHICH of the four options is the most efficient 'tool' for finding the answer? After all, just like a carpenter, a good problem solver knows which tools are available and which one is the best to use for a particular task.

Now let's back up in time when the student is in third grade. How are we encouraging third graders to do the same problem? As you may realize, kids usually spend lots of time mastering the multiplication algorithm (the pencil & paper method) and go thru tons of practice problems similar to my…