Category: Geogebra

Posted in Algebra, Geogebra

Making connections: Square of a sum

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One of the ways to connect concepts in mathematics is to make use of the same representation to teach mathematics. Let me take for an example the visual representation of the square of a sum, (a+b)^2 = a^2+2ab+b^2 . This concept is usually ‘concretized’ using algebra tiles. However, if you have facility for computer technology I would recommend using the GeoGebra applet below: [iframe https://math4teaching.com/wp-content/uploads/2011/09/square_of_a_sum.html 650 550]

Suggested tasks:

1. You can show the grid (click view to do that) to teach Grade 3 pupils about area. You can change the dimensions of the rectangles and squares by dragging D.

2. For older students you can give this task (don’t show grid but you can show lengths then drag D): The square AGHC is dissected into rectangles and squares.If the sides of square AEFD is 2 units and that of square DIJC is 3 units,

a) calculate the area of the other rectangles and square?

b) write two numerical expressions representing two ways of getting the area of the big square.

4. Having worked with numerical expressions, students will be ready to work with variables: If AD is x and DC is y, find two expressions for the area of the square AGHC. This of course leads to the identity (x+y)^2 = x^2 + 2xy + y^2 . The popular FOIL method should only come after this activity. For the record, I’m not a fan of this method.

5. This representation can also be used to teach how to calculate expressions such as 0.75 x 0.75 +0.25 x 0.75 x 2 + 0.25 x 0.25. This is one way to help students appreciate an application of the square of a binomial.

6. You can also use this figure to teach the idea of function: If you drag D along AC, how will a change in the distance of AD affect the area of the rectangles and squares?

a) Create a table comparing the area of the quadrilaterals as the side AD increases from 0 to 5 units.

b) Do as in a) but this time compare the perimeters.

c) Express the area/perimeter of each quadrilateral as a function of the length of AD.

I will write about square of a difference in the next post.

Posted in Algebra, Geogebra, GeoGebra worksheets

Teaching mathematics with GeoGebra

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GeoGebra constructions are great and it’s fun ‘watching’ them especially if you know the mathematics they are demonstrating. If you don’t and most students don’t then we have a bit of a problem. Even if the applet demonstrates the mathematics to students I don’t think there’ll be much learning there. No one learns mathematics by watching. We know that ‘mathematics is not a spectator sport’. You have to play the game. In Geogebra  and Mathematics I proposed that if GeoGebra is to help students in learning mathematics with meaning and understanding, then students should know how to use it. But these GeoGebra tools will be most useful only to students if they know the mathematics behind the tools and why they work and behave like that.  And so we teach students the mathematics first? Where’s the fun in that?

I believe (Translation: I’ve yet to do a study if my theory works) that it is possible to learn mathematics and the tools of GeoGebra at the same time .  I will be sharing in this blog GeoGebra activities where students learn to use GeoGebra as they learn mathematics. The main objective is of course to learn mathematics. The learning of GeoGebra is secondary. I will start with the most basic of mathematics and the most basic of the tools in GeoGebra: points, lines, and the coordinates system.

The lesson includes four GeoGebra activities:

Activity 1 – What are coordinates of points? Read the introduction about coordinates system here.

Activity 2 – What are the coordinates of points under reflection in x and y axes?

Activity 3 – How to describe sets of points algebraically Part 1?

Activity 4 – How to describe sets of points algebraically Part 2? (under construction)

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.