Category: Geogebra

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.