Engaging students: Rational and irrational numbers

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Emma Sivado. Her topic, from Algebra: rational and irrational numbers.

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D.1: What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The famous story on the first discovery of irrational numbers is one of violence. We all know the Pythagorean theorem, a2+b2=c2 , but what happens if we have a right triangle with height 1 and base 1? The hypotenuse becomes √2. So, √2, what’s the big deal? Well this is where we turn to history for the answer. Hippassus was an ancient greek philosopher who belonged to the Pythagorean school of thought. Now the Pythagorean’s had a saying, “All is number.” What do we think this means? What Pythagoras meant was that everything in the universe had a numerical attribute. For example, one is the number of reason, five is the number of marriage. So one day when Hippassus was playing with the length of the diagonal of the unit square, or the hypotenuse of a right triangle with base 1 and height 1, he discovered the number √2. Hippassus tried to write √2 as a fraction, or rational number, and found it to be impossible. Therefore, √2 is what we call an irrational number. Well this is where the history turns violent. There are numerous stories to explain the death of Hippassus, but all of them point to his ultimate cause of death being the discovery of these irrational numbers. Irrational numbers were so against Pythagoras and the Pythagorean school of thought that they had this man killed!




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B.1: How can this topic be used in your students’ future courses in mathematics and science?

I believe that the irrational number would be a great place to introduce a simple proof. Students will have to do proofs in multiple math classes in the future and to give them an example with an interesting story might be a good place to start. For example, after telling the story of the discovery of irrational numbers ask the students how Hippassus might have proven that this was true; possibly his dying words. Then give them an outline or fill in the black of the proof that √2 is irrational. This example I found on homeschoolmath.net is given in good language and gives good explanations of why everything is done in the order it is:

Let’s suppose √2 is a rational number. Then we can write it √2  = a/b where a, b are whole numbers, b not zero.

We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction.

From the equality √2  = a/b it follows that 2 = a2/b2,  or  a2 = 2 · b2.  So the square of a is an even number since it is two times something.

From this we know that a itself is also an even number. Why? Because it can’t be odd; if a itself was odd, then a · a would be odd too. Odd number times odd number is always odd.

Okay, if a itself is an even number, then a is 2 times some other whole number. In symbols, a = 2k where k is this other number. We don’t need to know what k is; it won’t matter. Soon comes the contradiction.

If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get:

2 = (2k)2/b2
2 = 4k2/b2
2*b2 = 4k2
b2 = 2k2

This means that b2 is even, from which follows again that b itself is even. And that is a contradiction!!!

WHY is that a contradiction? Because we started the whole process assuming that a/b was simplified to lowest terms, and now it turns out that a and b both would be even. We ended at a contradiction; thus our original assumption (that √2 is rational) is not correct. Therefore √2 is rational.


Obviously this would have to be presented slowly, but I believe that the students could do this and understand it.




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I would begin by showing the movie clip from Life of Pi when Pi is reciting all the digits of Pi that he knows, or another video of someone reciting a ridiculous number of digits of pi. Then I would ask the students how many digits of Pi there are? When no one could tell me an exact answer I would introduce the irrational number and explain how the decimals will go on forever because this number cannot be written as a fraction like a rational number. At the end of class you could show the kids the Princeton University Pi Day celebration complete with Einstein look alike contests, and pi reciting competitions to win $314.15!






One thought on “Engaging students: Rational and irrational numbers

  1. “If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get:”
    It is slightly simpler to substitute 2k for a in the next equation, a^2 = 2 · b^2.

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