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1 S t r o n a 1 Author: Anna Rybak Subject: GeoGebra program will help you... Introduction In this article the program GeoGebra will be presented as a measure in teaching mathematics in a secondary school. Generally, the demonstrated material is a suggestion for a student. While solving particular problems the main focus is on the student. It is a student who has to be active, build constructions, measure, carry out researches, make conclusions and put forward hypotheses. Due to purpose of the article and the type of a recipient I decided to present statistics with screenshots rather than with links to ready to use applets of GeoGebra program. Students must build constructions on their own instead of using already built constructions. Demonstrated constructions are not complicated similarly as operating the program. GeoGebra is a free program. It can be downloaded from the website: and installed on a computer or simply operated on the Internet. The program is in my opinion the best program for researches carried out by a student and constructions of mathematical knowledge done by students on their own (however, with directions and help provided by a teacher). Program GeoGebra is very popular among teachers all around the world. The international society of Geogebra users has already been created; Polish users are also members of this group. There are remote courses of operating and using the GeoGebra program, as well as the exchange of users experience. There are didactic materials sources created concerning the usage of the program. I strongly recommend the website Problem Bisector of a triangle A bisector of the inner angle in a triangle divides the opposite side of a triangle into two sections. Is there any relation between the lengths of these sections and lengths of remaining sides of a triangle? Construct and examine Below you can see a constructed triangle and a bisector of one of its angles. You can study the construction analyzing objects in Algebra View. To easily set up a hypothesis concerning the relation between the proper sections in a triangle begin the considerations from an isosceles triangle and a bisector of an angle next to the apex between the triangle s arms. Notice that in the demonstrated triangle the sections AC and AB are arms, whereas BC is a base. Thanks to measurements done (and to the knowledge of an isosceles triangle) it is easy to state that BD and DC are equal, as well as AB and AC.

2 S t r o n a 2 Fig.1. An isosceles triangle and the bisector of an angle Is it the same in any other triangle? In GeoGebra program you can construct a triangle, do the required measurements and then move the chosen apex and study other triangles. The measurement and calculation results are updated automatically. In the screenshot below you can see a triangle formed by moving apex of the original triangle. As can be noticed, the previously considered intersections are not equal any more. What is, then, the relation between the lengths? Notice that in an isosceles triangle the ratio of intersections lengths into which the bisector divided the base and the ratio of the lengths of the remaining sides equaled 1. Perhaps in the case of a triangle with various sides lengths there is a proportion of considered pairs of intersections maintained? Examine the case.

3 S t r o n a 3 Fig.2. A triangle with sides of various lengths and a bisector of an angle Notice that in Algebra View the following objects appeared: i = e/f oraz j = c/b. You will create them easily, completing given quotients in fields Enter (Wprowadź) and accepting each of them. A precise description of each object in Algebra View will be obtained by setting in the upper menu the path Options/Description (Opcje/Opis) in Algerba View/Definition (Widok Algebry/Definicja). The introduced quotients enable you to study proportion of distinguished sections. To read the values of the quotients set in the upper menu the path: Options/Description (Opcje/Opis) in Algebra View/Value (Widok Algebry/Wartość). You can read (from the below demonstrated screenshot): i = j =

4 S t r o n a 4 Attention: If you calculate quotients i and j with the help of a calculator and you will obtain an approximation with e.g. ten decimal places, you will easily notice that at the seventh decimal place there is a difference of numbers in both quotients. Such a situation is the result of calculations with rounding which is done on a computer. The mistakes of rounding multiply. Thus, calculations carried out by a computer are always burdened with a kind of inaccuracies. Therefore, formal thinking occurs in relation to a general case. Fig.3. the ratio of sides lengths Below you can see another triangle. Analyze the issue in relation to this triangle.

5 S t r o n a 5 Fig.4. the ratio of sections lengths in another case Moving the triangle s apex examine the issue for other triangles. Remember that such an observation (the examination of measurement and calculation results in a few cases) does not constitute formal evidence, but it helps to set up a hypothesis. Formal evidence is based on mathematical thinking. Hypothesis A bisector of an inner angle in a triangle divides an opposite side into two sections whose lengths ratio equals the ratio of lengths belonging to adjacent sides of this triangle. Problem Diagonal in trapezoid In a trapezoid diagonals have been marked. Four triangles have been created. Are there any relations between them? Revise needed knowledge Examining relations between two triangles consider their congruence or similarity. The triangles are congruent if their corresponding angles have the same measures, and corresponding sides have the

6 S t r o n a 6 same lengths. The triangles are similar in scale k (where k>0) if their corresponding angles have the same measurements and corresponding sides are proportional. Construct and examine Below you can see a constructed trapezoid. You can study the construction analyzing objects in Algebra View (Widok Algebry). Fig. 5. Trapezoid construction Even not a thorough observation of the created triangles proves that in this case triangles will not be congruent. Perhaps there are any similar triangles among those constructed? Pay attention to the triangles: DEC and ABE. Their angles next to apex E are formed by two intersecting straights, so they should be equal. Angles next to remaining apexes have been created as a result of intersection of two parallel straights by the third straight. Therefore, there should also be angles of the same measurements. Check, measuring them properly:

7 S t r o n a 7 Fig.6. Angles measuring Indeed, in triangles DEC and ABE there are pairs of angles which are equal respectively. Are the proper sides proportional? Measure and examine the ratio of lengths of proper sides. Notice that in Algebra View (Widok Algebry) the objects appeared: g = BE / ED and h = AE / EC. You will create them easily writing down the given quotients (without vertical lines) in the field Enter (Wprowadź) and accepting each of them. The precise description of each object in Algebra View (Widok Algebry) can be obtained by setting in the upper menu the path Options/Description (Opcje/Opis) in Algebra View/Definition (Widok Algebry/Definicja).

8 S t r o n a 8 Fig.7. Examining the proportion of sides To read the value of mentioned quotients it is necessary to set up in the upper menu the following path: Options/Description (Opcje/Opis) in Algebra View/Values (Widok Algebry/Wartość). Read (looking at the screenshot presented below): g = h = That means that in the examined trapezoid corresponding sides in triangles DEC and ABE are proportional:

9 S t r o n a 9 Fig.8. Values in Algebra View (Widok Algebry) Is it the same in any trapezoid? In GeoGebra program you can construct a trapezoid, do the proper measurements, and then move a chosen apex to study other trapezoids. The measurement and calculation results are updated automatically. In the screenshot below you can see a trapezoid created by moving apexes of an original trapezoid. As it can be easily read, here also in triangles: DEC and ABE corresponding angles are equal and corresponding sides are proportional:

S t r o n a 10 Fig.9. Studying angles and sides of other trapezoid examples Examine the case creating other trapezoids by the manner of manipulation.

10 S t r o n a 10 Fig.9. Studying angles and sides of other trapezoid examples Examine the case creating other trapezoids by the manner of manipulation. Remember that such researches (observations of measurement and calculation results for a couple of cases) do not constitute formal evidence but it helps to set up a hypothesis. Formal hypothesis is based on mathematical thinking. Hypothesis Triangles DEC and ABE are similar. Construct and examine on your own whether triangles AED and BCE are similar. Problem Solve the following problem on your own. The centre of area in a triangle The centre of area in a triangle is a point of intersection of its medians. Does this point divide medians in any particular manner? Construct and examine

11 S t r o n a 11 Look at the construction presented below and analyze measurement results. Then, make your own constructions and do the measurements with various accuracies (option Options/Rounding) and make conclusions. Remember that such research (the observation of measurement and calculation results for a couple of cases) does not constitute formal evidence but it helps to set up a hypothesis. Your conclusion will be a hypothesis. To verify your hypothesis (confirm or refute) you must think mathematically while considering a general case. Fig.10. Medians of sides in a triangle Summary The experience coming from lessons taught at middle schools and secondary schools inclines that students do well operating Geogebra program from the very beginning of starting it on the computer. In fact, presenting them problems to be solved and inspiring them to research work concerning properties of geometric figures on a plane influences understanding mathematics and attitude to learning it in a positive manner. Bibliography 1 Skiba R., Winkowska-Nowak K. (red.), GeoGebra. Wprowadzanie innowacji edukacyjnej, Wydawnictwo Naukowe UMK, Toruń 2011, ISBN: Skiba R., Winkowska-Nowak K., Pobiega E. (red.), GeoGebra. Innowacja

12 S t r o n a 12 edukacyjna kontynuacja, SEDNO Wydawnictwo Akademickie, 2013, ISBN Supplements Suggestions of a few additional problems to solve using Geogebra program: Problem 1. Does a section joining the middles of two triangle s sides have any specific features? Problem 2. Does a section which is parallel to the triangle s side have any specific features? Problem 3. Can you determine parallelism or perpendicularity of two straight lines knowing coefficients in their equations? Problem 4. Is there any relation between lengths of the altitude going from the apex of a right angle in a right triangle and of a hypotenuse?

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