Star Doodles and Factors

Today, we are going to investigate numbers with some doodles. Everyone look at the first side of the worksheet.

What do you think p stands for?

"Points," someone answers.

Not just points, but NUMBER of points. Let's do the first one together.

Choose any point. Then, going clockwise, jump 2 points from your first point. Connect this new point to the last point with a straight line. Go jump another 2 points and join this point to your last point. Keep going until you reach a point already joined up.

What do you get?

"A star."

There is a special name for this star. Does anyone know what this type of star is called?

"A pentagram," says someone.

Excellent. The secret society of Pythagoreans used this star as their secret symbol.

j stand for the NUMBER of points you jump. Write j = 2 under your pentagram. Now try the next group where p = 5 and j = 3.

What do you notice?

Now try the group where p = 6 and j = 2. Stop when you get to a point already connected.

At this point, many students were excited by the stars and completed 2 equilateral triangles to form the Star of David. That is, 2 lots of 3.

What happened?

"We got 2 lots of 3, which makes 6."

Go ahead now and experiment drawing different doodles with different values for p and j.

This is fun. Everyone thinks the stars are cool and many people decide which is their favourite.


Let's look at the group where p = 9. Can anyone predict what value of j we should use so that we connect every point before we get back to the point where we started?

"I think that will happen if we use a value for j that is a non-factor of the p number."

Non-factor; is that a real word? Sounds good to me.

A good conjecture; everyone try it.

I have some balls of wool. Let's gather into circles, pass the wool around and make some of our favourite stars.


This activity led to great discussions and fabulous fun. Terminology included factors (and non-factors), multiples, divisibility and prime numbers.
But the best thing about the lesson was everyone was able to access the work. At the other end of the spectrum, I thought I had understood everything about the lesson, until someone says...

"But what about when p = 10 and j = 4? 4 isn't a factor of 10."

P.S. Thanks to vihart for the inspiration.

Pythagoras' Theorem - From Concrete to Abstract

I have found this the best way to teach Pythagoras' Theorem. The focus of the lesson is actually on showing the students how to go from the concrete to the abstract.
The lesson takes about 100 minutes, a double where I am teaching.
Students have commented on what fun they had during the double.

First, engage the students with Donald Duck in Mathemagic Land.
The first 5 minutes or so talks about the secret sect of Pythagoreans and their contribution to the understanding of music and the golden section.

Don't forget to discuss with your students that the Pythagoreans allowed women in their sect, while women were still not classed as citizens in ancient Greek society. The Pythagoreans were ahead of their times!


Next, introduce the theorem.
It has lasted over 2500 years, although there is evidence that the Babylonian society understood the relationship between the squares of the sides of a right triangle way before the Greeks.
People think that knowing Pythagoras' Theorem makes you smart. Check out the brainless scarecrow in the Wizard of Oz who, when granted a brain, recites a version of Pythagoras' theorem using an isosceles triangle, instead of a right triangle!

Now,

The square on the hypotenuse = the sum of the squares on the other two sides.

Then have your students explore a proof using a great interactive applet available at

http://www.ies.co.jp/math/java/geo/pythasvn/pythasvn.html

Then, give the students a worksheet showing a right triangle and the squares on the three sides. Once they have found a proof using the applet, they cut out the two smaller squares and glue these inside the large square.

Here is the worksheet.

Time to go to the abstract.

Have the students draw a 3cm, 4cm, 5cm triangle as accurately as they can, then draw a square on each side.

Then ask them to find the area of each of these squares.

Now, the abstract connection. Interpret Pythagoras' Theorem and replace the words with the areas of each square. At this point in the lesson, you will here gasps of understanding - the Ah Ha! moments.

Conclusion

I conclude by giving some homework that involves finding the unknown hypotenuse. The worksheet I use is from a set of books called, "Math-O-Magic". The worksheets have corny jokes but the students like doing them. The books are available at http://members.iinet.net.au/~markobri/mom.html

I am sure you will enjoy this lesson. Have fun!