One of the fun aspects of learning while having access to a computer center is the exploration style of learning that is possible accurately with Geogebra on the computer. To learn about shapes like triangles, angles, circles, etc through measurements. Looking or patterns and making observations and deriving some sort of a generalization made for an excellent exercise.
Here are some of the things that worked well us:
1) Using geogebra as a drawing tool in practical geometry to draw various equilateral triangles. A line segment is drawn with A as center passing through B. Then a circle is drawn with center of A passing through B and then a circle with center at B drawn through A. The meeting point of the two circles is equidistant from both A and B for the same lenght thus giving the equilateral triangle. The children can then measure the angles and distances. As they create and measure many more triangles using different lengths for a side they realized that all equilateral triangles have the same internal angle of 60 degrees.
2) A triangle with three sides can be drawn with a line of one of the lengths and then circles with radius of the other two sides. The point of intersection of the circles gives the third point of the triangle.
Picking up three random sides of a triangle in a class brings up the triangle inequality, when children are unable to find a point of intersection and naturally moves one to the realization that two sides have to be greater than a third to make a triangle. I encourage the children to try using the first side as any of the other sides and see what happens e.g. 1 cm, 2 cm, 4 cm has the following three ways of looking at it. Interestingly, I only thought of the inequality as the third figure, but some of the children seemed to find one of the other two more sensible.
Here are some of the things that worked well us:
1) Using geogebra as a drawing tool in practical geometry to draw various equilateral triangles. A line segment is drawn with A as center passing through B. Then a circle is drawn with center of A passing through B and then a circle with center at B drawn through A. The meeting point of the two circles is equidistant from both A and B for the same lenght thus giving the equilateral triangle. The children can then measure the angles and distances. As they create and measure many more triangles using different lengths for a side they realized that all equilateral triangles have the same internal angle of 60 degrees.
Picking up three random sides of a triangle in a class brings up the triangle inequality, when children are unable to find a point of intersection and naturally moves one to the realization that two sides have to be greater than a third to make a triangle. I encourage the children to try using the first side as any of the other sides and see what happens e.g. 1 cm, 2 cm, 4 cm has the following three ways of looking at it. Interestingly, I only thought of the inequality as the third figure, but some of the children seemed to find one of the other two more sensible.
3) Some other observations about triangles also came about by this process
- Isosceles triangle has two angles the same
- Sum of the angles was always 180' no matter what the triangle
- The largest side is opposite the largest angle.
Then we did some games using a combination of these e.g. would the central angle of a triangle 4, 5, 5 be less or more than 60'.
The children were starting to get a feel that sides and angles are not independent quantities.
4) Circles - circumference in relation to circle diameter, area relation to the area of a square with one side as radius. These were fun exercises that I had attempted by getting the kids to do physical measurements last year. It took quite some time and due to measurement inaccuracies (especially when measuring small objects like ear rings) could throw the kids off. The ratio of pi (that they presently know as 3.14) came out like magic as geogebra could be used to measure these quantities of interest no matter how small or how large the circle was. I skipped the calculations at this point and stuck to creating a spreadsheet of the circles that children drew.
The children were each able to document 10 circles or more that we were unable to last year. The number has really stuck with them as an assessment recently (with circumference given and radius/diameter to be calculated) showed with over 75% of the class guessing these right.
5) Algebra
The notion of what lines like x+y=10, x+y=20 look like coupled with stories like you and I share 10 chocolates, if I get one more you should get one less (negative slope) had done. Again an assessment later indicated that most children were able to get this.
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