Trivial Pursuit

 

By Rebecca L. Adcock

 

            Student teaching is over but our class has been assigned to observe in middle schools over a three county area. It gives us exposure to students younger than the ones most of us taught during our student teaching stints in senior high schools. It has been an interesting experience. I am at a Title I school and my sixth-graders come to class better prepared than the high school students. The younger kids actually bring paper, pencils, and their textbooks to class! I will admit, though, that they really love the noisy, disruptive electric pencil sharpener.

J.E. Richards Middle School

 

            The students attend a class that is called ‘CQI’. That stands for Continuous Quality Improvement. Each student attends a CQI class for the subject that they struggle with the most. Since my mentor teacher, Tammy Thompson, has taught in an alternative school, she is experienced in teaching struggling learners. In fact, she will have some involvement in writing curriculum for next year’s CQI  and she hopes to drum up some support for an idea she is passionate about. From her studies and her experiences, she has concluded that the hour spent in CGI is most beneficial to the student  if it is spent previewing the concepts and vocabulary the student will soon encounter in his/her regular mathematics class.. The struggling student, having already seen some of the material, is more prepared to grasp the concepts, keep up with the rest of the class and as a result feels more successful. The class is currently used to follow up on past class work

         Now that you’ve “got the picture”,  I want to describe a lesson that the CQI class worked on and some of the questions that I had that arose as a result and some of the ideas the lesson invoked.

 

               

Tammy Thompson and class.                                                                    Some of our “free spirits”!

 

            The students were still grappling with the concepts of area and perimeter of polygons so Ms Thompson assigned the students to create as many polygons as they could imagine, using twelve one-inch plastic squares. They could create both regular and irregular polygons (and some were highly irregular!) and they had to document their shapes on graph paper and ‘calculate’ the perimeter of each. Since each tile had an area of one, they knew that the area of each shape would be 12 square inches. Here’s one student’s paper:

 

 

 

 

And another student’s…

           

 

 

Although it may not have been apparent to the students, I noticed that given a polygon created by 12 squares, the perimeter was always an even number. The students may not have noticed the pattern because they were still having trouble computing the perimeters. I saw a lot of promise in how this lesson could be expanded and turned into a classroom investigation.

 

Here’s how it could go…

 

One of the questions that may arise would be the definition of a polygon. In sixth grade, the students are exposed to regular polygons but the shapes they were creating did not look “regular”. Some of the shapes above would work very well into the following definitions:

Š      A polygon is a closed plane figure whose sides are line segments that intersect only at the endpoint.

Š      A convex polygon is a polygon in which any segment connecting any two vertices lies inside of the polygon.

Š      A concave polygon is a polygon in which at least one segment connecting two vertices lies outside of the polygon.

 

         Once the students have created a number of shapes, ask them to speculate on anything interesting that they have noticed. For example, a student may have noticed that the area is an even number, and all the perimeters on his shapes are also even numbers. If the area was odd, would the perimeter be odd? The students would draw more shapes, this time using 11 squares…

 

 

Once again the perimeters are all even. So where would the investigation go from here? Perhaps half of the class would continue to investigate shapes composed of a different number of squares, while the other half uses a different shape, like a triangle, that has an odd-number of sides, rather than the even-number of sides. The problem with using an equilateral triangle with a side-length of 1 is that the area of that triangle is not equal to 1. So our project changes from comparing area to perimeters to comparing the number of figures to the perimeter. This could lead to an introduction of the Pythagorean Theorem, which would be needed to calculate the area of the triangle. More investigations…

 

 

So using an even number of odd-sided figures would create an even perimeter. What if an odd number of odd-sided figures was used?

 

 

Finally an odd perimeter! It worked for 11 triangles and worked again for 7 triangles.

 

 

So we finally end up with a conjecture. It appears that a figure created from an odd number of odd-sided shapes will have a perimeter that is an odd number. An even number of odd-sided shapes, an odd number of even-sided shapes, and an even number of even-sided shapes all create figures with a perimeter that is an even number.

“So what?”, the students ask. Well, I don’t really know the significance so let’s look at the number of shared sides in each figure and see it any patterns appear. We can put all of our numbers into a spreadsheet and see if we can discover a pattern.

 

Sided Shape

Number of  Shapes Used in Figures

Perimeter

Number  of Common Sides

3

7

7

7

3

7

9

6

3

11

9

12

3

11

11

10

3

11

11

11

3

11

13

13

3

12

10

13

3

12

14

11

4

11

14

15

4

11

16

14

4

11

18

13

4

11

20

12

4

11

22

11

4

11

24

10

4

12

16

14

4

12

16

15

4

12

16

16

4

12

18

13

4

12

18

14

4

12

18

15

4

12

18

16

4

12

20

11

4

12

20

14

4

12

22

13

4

12

22

21

4

12

24

12

4

12

26

11

 

 

 

 

 

I can detect no patterns, but does this mean that the exercise was a waste of time? Not in my opinion! Here is a list of concepts and skills that could be taught through a lesson like this, depending on what tools the teacher chooses:

Š      Definition of a polygon.

Š      Examples of convex and concave polygons.

Š      Collecting (creating) data.

Š      Detecting patterns.

Š      How to construct an argument to support an idea.

Š      Constructing a table with data in Excel or Word.

Š      Using the Pythagorean Theorem.

Š      Using Geometer’s Sketch Pad to construct a square and triangle and using Transform and Reflect.

 

I enjoyed the potential of this lesson. The students were really engrossed in creating as many different shapes as they could and that interest could have been channeled into a longer lesson, complete with u-turns and pitfalls as the investigation developed. In addition, it could have opened the door to introducing the students to Geometer’s Sketch Pad and other software. Once that door was open, I could imagine a second lesson that would involve the centers of triangles and Euler’s line. Where this lesson turned out to be a trivial pursuit that failed to develop anything of mathematical significance, the second lesson could end with the “discovery” of a renowned geometric property.

 

 

Click on the square to see how I constructed one in GSP for the examples above.

 

 

Click on the triangle to see how I constructed one in GSP for the examples above.

 

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