GeoGebra activities Archives - Page 2 of 3

Constructing polygons with equal area

Posted on by

The power of GeoGebra lies not only for demonstrating a concept but more so for creating a situation where students are made to think, solve problems, and reason mathematically. Here is a sample lesson on how this can be done. The lesson involves the concept of area of triangles and constructing parallel lines applied to problems involving preserving areas of polygons. The most ideal situation is for students to explore on the applets individually or in small group. If this is not possible and the students have no computers, the applets can be projected. The teacher can then call a students or two to explore the applets. The idea is to stimulate students’ thinking to think of explaining why the transformations of the shapes are possible. The students should have a triangle rulers or straight edge with them. The lesson will not be complete if the students cannot devise a procedure for transforming a triangle into other polygonal shapes with the area preserved.

The following applet may be used in the introductory activity to teach about triangles with equal areas. Click here to explore. The applet shows that all triangles with equal base and have equal heights or altitude have equal areas. That’s pretty obvious of course.

The next applet is more challenging but uses the same principle as the first.
Depending on the previous knowledge of your students you can give this second applet right away without showing the first applet on triangles. Click here or the worksheet below explore and answer the problem.

Extension problems using the same principle about area of triangles.

Given a triangle, construct the following polygons equal in area to the given triangle using triangle rulers and pencil only: a) parallelogram; b)trapezoid; c) hexagon. (There’s nothing that should prevent you from using GeoGebra to construct them.)
The Rosales and Ronda families are not very happy with their piece of land because of the narrow corners. Help this family to draw a new boundary line without changing the land area of each family.

I will deprive you of the fun if I will show the answers here right away, won’t I?

I will appreciate feedback on this lesson.

Geogebra, Geometry

Problem on proving perpendicular segments

Posted on by

This problem is a model created to solve the problem posed in the lesson Collapsible.

In the figure CF = FB = FE. If C is moved along CB, describe the paths of F and E. Explain or prove that they are so.

This problem can be explored using GeoGebra applet. Click this link to explore before you read on.

perpendicular segments

One way to prove that FC is a straight line and perpendicular to AC is to show that FC is a part of a right triangle. To do this to let x be the measure of FCB. Because FCB is an isosceles triangle, FBC and CFB is (180-2x). This implies that EFB is 180-2x being supplementary to CFB thus CFB must be 2x. Triangle EFB is an isosceles triangle so FBC must be (180-2x)/2. Adding CFB and FBC we have x+ (180-2x)/2 which simplifies to 90. Thus, EB is perpendicular to CB.

The path of F of course is circular with FB as radius.

Algebra, Geogebra

Teaching with GeoGebra – Investigating coordinates of points

Posted on by

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.

Algebra, Geogebra, High school mathematics

Teaching simplifying and adding radicals

Posted on by

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.