Algebra is a fundamental topic in mathematics and it can be taught by using some applications like GeoGebra.
Let’s look at GeoGebra application and some questions and try to understand how we can construct algebraically thinking.
Q6. Examining patterns and describing them in words occurs early in children’s school experiences, often as early as Pre-K. How can teachers build on patterning activities to foster students’ algebraic thinking?
As a math teacher candidate, I would ask my students to create a letter and a fruit shape to choose. Therefore, I put these letters in one of the two baskets, and the fruits in the other, to make them establish algebraic relationships between them by adding or subtracting letters or fruits. Hence, we can construct and represent sequences in numerical way and geometrical way.
Open a new GeoGebra document with a Spreadsheet view:
Q8. Describe in your own words what each of the steps used to create the sequence and cumulative sequence means.
Sequence can be defined as, when somebody goes to different stores to get number of books with a progress. After n stores s/he can calculate the number of books by using the number of books that this person got previous store.
For example, from this table we can say that, someone has 5 books. S/he will progress by buying 7 more books than twice the number of books s/he bought from the previous store s/he went to. Then s/he goes to stores and s/he takes from each store 7 more than twice the number of books she has. From this point s/he gets [(2 x 5) + 7] which makes 17 books from 1st store. Similarly, s/he has [(2 x 17) + 7] which makes 41 books from 2nd store. This process will go and we can call it sequence.
Cumulative sequence can be defined as, when somebody goes to different stores to get number of books with a progress, then after n stores s/he can calculate the number of total books that s/he had. While calculating this, she will buy 7 more than twice the number of books s/he bought from the previous store from the new store.
For example, from this table s/he gets [5 + ((2 x 5)) + 7)] which makes 22 books from 1st store. Similarly, s/he has [22 + ((2 x 17) + 7)] which makes 63 books from 2nd store. So the total number of books that s/he got from the 1st and 2nd store is 63.
Geometric representations of convergent cumulative sequences:
Q26. Construct the midpoints of each of the three sides of the triangle and connect them by segments. The figure you have constructed separates your equilateral triangle into smaller triangles. Examine your construction. How many smaller triangles have you constructed? What kind of triangles have you constructed? What conjectures can you make about them?
First of all, I choose a equilateral triangle with side length is 4. After that I find midpoints of each side then I reach another equilateral triangle by piecing together the midpoints. I did this implementation 3 times. So, I have 3 equilateral triangles. Therefore, I go to infinity by finding midpoints and reunion these points to create new equilateral triangle. However, at one step I will reach a point not equilateral triangle. At that point, my sequence will be finished.
Q30. What do you notice about the consecutive ratios of the areas of the triangles? Be sure to drag a vertex of T0 to change the sizes of the triangles. Describe and justify your observations in your own words.
New tringle divides the old triangle that is drawn into each triangle formed by joining the middle points into 4 equal parts. So, the ratio of the new triangle to the old triangle is ¼. When we come to the area, I can say that since one side length is reduced to half, the similarity ratio of the new side to old side becomes ½. Moreover, the area ratio would be ¼ as it would be the square of the similarity ratio.
Q31. What do you notice about the cumulative sequence and its relationship to the ratios of the areas? Describe your observations in your own words.
If we know the length of one side of an equilateral triangle given to us, we can calculate the length of the side in the new triangle formed by the combination of the midpoint of each side. We can find the area of triangles from the formula for calculating the area of an equilateral triangle with one side known. At the end we will come up to ¼ the ratios of areas. for Different way of making analyze for the ratios for the areas, we can see from the example, 4 times the new triangle gives our old triangle. As a result, the new triangle becomes ¼ of the old triangle.
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