Category: Algebra

Posted in Algebra, Geogebra, High school mathematics

Embedding the idea of functions in geometry lessons

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GeoGebra is a great tool to promote a way of thinking and reasoning about shapes. It provides an environment where students can observe and describe the relationships within and among geometric shapes, analyze what changes and what stays the same when shapes are transformed, and make generalizations.

When shapes or objects are transformed or moved, their properties such as location, length, angles, perimeters, and area changes. These properties are quantifiable and may vary with each other. It is therefore possible to design a lesson with GeoGebra which can be used to teach geometry concepts and the concepts of variables and functions. Noticing varying quantities is a pre-requisite skill towards understanding function and using it to model real life situations. Noticing varying quantities is as important as pattern recognition. Below is an example of such activity. I created this worksheet to model the movement of the structure of a collapsible chair which I describe in this Collapsible  chair model.

Show angle CFB then move C. Express angle CFB in terms of ?, the measure of FCB. Show the next angle EFB then move C. Express EFB in terms of ?. Do the same for angle FBG.
[iframe https://math4teaching.com/wp-content/uploads/2011/07/locus_and_function.html 700 400]
Because CFB depends on FCB, the measure of CFB is a function of ?. That is f(?) = 180-2?. Note that the triangle formed is isosceles. Likewise, the measure of angle EFB is a function of ?. We can write this as g(?) = 2?. Let h be the function that defines the relationship between FCB and FBG. So, h(?)=180-?. Of course you would want the students to graph the function. Don’t forget to talk about domain and range. You may also ask students to find a function that relates f and g.

For the geometry use of this worksheet, read the post Problems about Perpendicular Segments. Note that you can also use this to help the students learn about exterior angle theorem.

Posted in Algebra, Geogebra

Solving algebra problems – which one should be x?

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Every now and then I get an e-mail from a friend’s son asking for help in algebra problems. When it’s about solving word problems, the email will start with “How about just telling me which one is the x  and I’ll figure out the rest”. The follow-up email will open with “Done it. Thanks. All I need is the equation and I can solve the problem”. The third and final e-mail will be “Cool”. Of course I let this happen only when I’m very busy. Most times I try to explain to him how to represent the problem and set-up an equation. Here’s our latest exchange.

Josh: What is the measure of an angle if twice its supplement is 30 degrees wider than five times its complement? All I need is to know which one’s  the x.

Me: How about sending me a drawing of the angle with its complement and supplement?

Josh: Is this ok?

Me: Great. Let me use your drawing to make a dynamic version using GeoGebra. Explore the applet below by dragging the point in the slider. What do you notice about the values of the angles? Which angle depend on which angle for its measure? If one of the measure of one of the angles is represented by x, how will you represent the other angles? (Click here for the procedure of embedding applet]
[iframe https://math4teaching.com/wp-content/uploads/2011/07/angle_pairs1.html 650 435]

Josh: They are all changing. The blue angle depends on the green angle. Their sum is 90 degrees. The red angle also depends on the green angle.  Their sum is 180 degrees. The measure of the red angle also depends on blue angle.

Me: Excellent. Which of the three angles should be your x so that you can represent the others in terms of x also? Show it in the drawing.

Josh: I guess the green one should be x. The blue should be 90-x and the red angle should be 180-x.

Me: Good. The problem says that twice the measure of the supplement is 30 degrees wider than five times the complement. Which symbol >, <, or = goes to the blank and why, to describe the relationship between the representations of twice the supplement and five times the complement:

2(180-x) _____ 5(90-x)

Josh: > because it is 30 degrees more.

Me: Good. Now, what will you do so that they balance, that is make them equal?  Remember that  2(180-x) is “bigger” by 30 degrees? What would the equation look like?

Josh: I can take away 30 degrees from 180-x. My equation would be (180-x) -30 = 5(90-x)?

Me: Is that the only way of making them equal?

Josh: Of course I can add 30 to 5(90-x). I will have 180-x = 5(90-x)+30.

Me: You said  you can do the rest. Try it using both equations and tell me the value of your x and the measures of the three angles.

Josh: x = 40. That’s the angle. It’s complement measures 50 degrees and its supplement is 140 degrees. They’re the same for both equations.

Me: Does it makes sense? Do you think it satisfies the condition set in the problem?

Josh: 2(140) = 280. 5(50) = 250. 280 is 30 degrees wider than an angle of 250 degrees. Cool.

Me: What if you make A’DC your x? Do you think you will get the same answer?

No reply. I guess I’ll have to wait till the teacher give another homework to get another e-mail from him.

I don’t know if the questions I asked Josh will work with other students. Try it yourself. Please share or send this post to your co-teachers. Thanks. I will appreciate feedback.

Problem solving is the heart of mathematics yet it is one of the least emphasized activity. Solving problems are usually relegated at the end of the textbooks and chapters.

Posted in Algebra, What is mathematics

Love and Tensor Algebra

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Come,  let us hasten to a higher plane

Where dyads tread the fairy fields of Venn,

Their indices bedecked from one to n

Commingled in an endless Markov chain!

Come, every frustrum longs to be a cone

And every vector dreams of matrices.

Hark to the gentle gradient of the breeze:

It whispers of a more ergodic zone.

In Riemann, Hilbert or in Banach space

Let superscripts and subscripts go their ways.

Our asymptotes no longer out of phase,

We shall encounter, counting, face to face.

I’ll grant thee random access to my heart,

Thou’lt tell me all the constants of thy love;

And so we two shall all love’s lemmas prove,

And in our bound partition never part.

For what did Cauchy know, or Christoffel,

Or Fourier, or any Bools or Euler,

Wielding their compasses, their pens and rulers,

Of thy supernal sinusoidal spell?

Cancel me not – for what then shall remain?

Abscissas some mantissas, modules, modes,

A root or two, a torus and a node:

The inverse of my verse, a null domain.

Ellipse of bliss, converge, O lips divine!

the product o four scalars is defines!

Cyberiad draws nigh, and the skew mind

Cuts capers like a happy haversine.

I see the eigenvalue in thine eye,

I hear the tender tensor in thy sigh.

Bernoulli would have been content to die,

Had he but known such a2 cos 2 phi!

Love and Tensor Algebra is from the book The Cyberiadwritten by  Stanislaw Lem. Stanis?aw Lem was a Polish writer of science fiction, philosophy and satire. The Cyberiad is one of his best work.

The Cyberiad (Polish: Cyberiada) is a series of short stories. The Polish version was first published in 1965, with an English translation appearing in 1974. The main protagonists of the series are Trurl and Klapaucius, the “constructors”. The vast majority of characters are either robots, or intelligent machines. The stories focus on problems of the individual and society, as well as on the vain search for human happiness through technological means. The poem Love and Tensor Algebra found its way to this blog because of its mathematical flavor. And I love reciting it.