Tag: computer-based instruction

Posted in Algebra, Geogebra

Teaching with GeoGebra – Investigating coordinates of points

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The most basic mathematics students need to know to understand GeoGebra is the coordinate axes. Must you teach students how to plot points and interpret coordinates of points before they use GeoGebra or the other way around? I think, at the same time. Below is a sample activity on how I think this can be done.  The lesson is about investigating coordinates of points on a Cartesian plane. Its objective is to teach how to use GeoGebra’ s point tool, interpret coordinates of points and make generalizations.

1. Locate the reflections of the points A, B, C, D, E, F, and G if they will be reflected along the y-axis. Use the point button [.A] or the reflect button [.\.] to plot the points.

2. Hover the cursor along the points A to E. These pairs of numbers are called the coordinates of the point. What do you notice about the coodinates of these set of points (A through E)? Will this observation be true to the reflections of A, B, C, D, and E you just plotted?
3. Hover the cursor to the other points. How do the coordinates of the points relate to the values in the x and y axes?
4. In the input bar type P=(5,-2). Before hitting the Enter key, predict the location of the point. Experiment using other coordinates. Use the Move button to drag the grid to see the points you plotted, if they are not visible in the panel.
5. The x and y axes divide the plane into four quadrants. Describe the coordinates of the points located in each quadrant. What about the points along the x -axis and y – axis?

Click here to explore.

Of course, the teacher need to understand a little about GeoGebra first before giving this activity to his/her class.

Posted in Algebra, Geogebra, High school mathematics

Teaching simplifying and adding radicals

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The square root of a number is usually introduced via an activity that involves getting the side of a square with the given area. For example the side of a square with area 25 sq unit is 5 unit because 5 x 5 = 25. To introduce the existence of \sqrt{5}, a square of area 5 sq units is shown. The task is to find the length of its side. The student measures it then square the measure to check if it will equal to 5. Of course it won’t so they will keep on adjusting it. The teacher then introduces the concept of getting the root and the symbol used. This is a little boring.  A more challenging task is to start with this problem: Construct a square which is double the area of a given square. In my post GeoGebra and Mathematics: Squares and Square Roots I described a teaching sequence for introducing the idea of square root using this problem. There are 4 activities in the sequence. The construction below can be an extension of Activity 4. This extension can be used to teach simplifying radicals and addition of radicals. The investigation still uses the regular polygon tool  and introduces the text tool of GeoGebra.  Click links for the tutorial on how to use these tools. You will find the procedure for constructing the figure here.

radicalsThe construction shows the following equivalence:

1. 2\sqrt{5} = \sqrt{5}+\sqrt{5} since EA = EF+FH

2. 4\sqrt{5} = 2\sqrt{5}+2\sqrt{5} since AK = AB+BJ

3. 2\sqrt{10} =\sqrt{10}+\sqrt{10}

4. 4\sqrt{10} = 2\sqrt{10}+2\sqrt{10}

5.7\sqrt{5} = \sqrt{5}+2\sqrt{5}+4\sqrt{5}

6. 2\sqrt{5} = \sqrt{20} because they are both lengths of the sides of square EHBA or poly3 whose area is 20 (see algebra panel)

7. 2\sqrt{10} = \sqrt{40} because they are both lengths of the sides of square AHJI or poly4 whose area is 40.

8. 4\sqrt{5} = \sqrt{80} because they are both sides of square AJLK or poly5 whose area is 80.

Posted in Algebra, Geogebra

GeoGebra, Calculator, and Mathematics

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GeoGebra is a dynamic mathematics software for teaching and learning mathematics. As tool for the teaching part is pretty easy to do. But the learning part, well, that’s always been the one that is problematic, GeoGebra or not GeoGebra.

Studies about integration of technology in teaching and learning have always acknowledged that despite the availability of the technology, teaching and learning tools like GeoGebra is still not widely used in many classes even with the availability of computers for students. If ever, it’s the teacher who uses it and more often, for demonstration and sometimes ‘staged’ discovery of concepts and for visual effects  for all the students to enjoy but not to learn. I discuss my thoughts about it in my first post about GeoGebra and Mathematics.

To date, the calculator is still the undisputed teaching and learning tool in many mathematics classes. And for calculators, I can confidently claim that it is indeed both a teaching and learning tool. Students use it and can use it to investigate mathematical relationships, depending how lucky they are to have a math teacher that makes it possible. I think students use calculator not just because they know how to use it but because they understand the mathematical ideas represented by the keys. Now, if we can do the same for GeoGebra then maybe, just maybe, we can maximize its potential for facilitating mathematics learning.

 

 

Posted in Geogebra

GeoGebra and Learning Mathematics

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GeoGebra is a great software for teaching and learning mathematics. It offers geometry, algebra and calculus tools in one environment, a great support indeed for linking mathematical concepts. On top of that it is free and an open-code software. Click here to download the latest version of the software.

 Is it easy to use? Yes and No. Yes, for math teachers because they know the mathematics and can therefore easily understand the ideas and logic behind the tools. Yes, for students who have been instructed on how to use the tools and understand the mathematics and logic behind it. They can use it in solving problems and for investigating mathematical relationships. But, for the majority of students, especially those who have not learned the basic of graphing, equations, and geometric relationships, the use of GeoGebra is limited to manipulating ready-made GeoGebra applets. (Click here for my posts on solving problems about quadrilaterals or here for introducing function using Geogebra applets.) Well, yes, GeoGebra applets are easy to use but most of the time if you do not know the  mathematics behind the construction or can’t construct it yourself, then the learning of the mathematics may be superficial.

Construction of math models using the software is not accessible for many younger students just starting to learn basic Algebra and Geometry. In order for them to construct a model, they  will have to follow a set procedure (constructed by the teacher) without really understanding why they do what they do. So I thought why not teach GeoGebra tools and mathematics at the same time? This is a challenge I set for myself and I have no idea if it will work or not. I am thinking of doing a research of it later in the year. GeoGebra is free and faithful to mathematics so for countries like us that can’t afford to  buy licensed softwares, we get the same quality teaching tool with Geogebra. I think all students need to know how to learn mathematics with it.