teaching geometry Archives - Mathematics for Teaching

Convert a Boring Geometry Problem to Exploratory Version

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The following problem (or proving activity, if you like to call it that) is a typical textbook geometry problem. It is tough and guaranteed to scare the wits out of any Year 9 student.

When I used the given condition to construct the figure using GeoGebra, the only thing I can move is A or B and what it does is simply to reduce or enlarge the circle. Pretty boring. So I thought of making C dynamic. The way to do this is to construct point C along the circle and then construct a perpendicular line to AB. With C moving along the circle, the once static and close task is now a dynamic, exploratory and task.

Your students will observe that for triangle ABC to be an equilateral triangle, CD must be the perpendicular bisector of AB. You can now ask them the problem: Given that CD is the perpendicular bisector of radius AB, prove that ABC is an equilateral triangle, which is what the textbook is asking them to do.

In presenting the problem the way I’ve shown above, you did not only make the problem more interesting (hopefully) and accessible to the majority of the learners (I’m sure most of them can answer the questions), you have also given learners the chance to explore the problem first and be familiar with the situation.

Note that you will be doing a disservice to your students’ geometry life if you will stop at #5 and not give them the opportunity to prove. Proving is what makes mathematics different from other disciplines. It would be a shame if they will go through life only complaining about x and not of proving as well. I’m joking but you know what I mean. You may want to check some of my favorite post about teaching geometry through problem solving: Unpacking mathematics – a geometry example and Problem Solving Involving Quadrilaterals.

Geometry

A problem solving lesson about triangles and circles

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This short lesson was inspired by one of the problems from the blog Five Triangles Mathematics. The author challenges the reader to construct a circle using only a compass and straight edge, through two points X and Y. The centre of the circle must be a point on the line located between the two given points. If you can’t visualise it, click here to see the diagram and try the problem first and then come back if you are interested to see how you might teach this in your class without losing the essence of the problem solving activity.

Here’s my sequence of tasks. Notice that all three tasks involve geometric constructions in increasing complexity, one building on the previous task.

Problem 1

You can use this as context for reviewing the properties of isosceles triangle after the students have come up with at least two solutions.

Problem 2

Solution

This is one of the solution but I suggest you ask students to come up with other ways of constructing the isosceles triangle. The procedure shown involved constructing the perpendicular bisector of CD. F is any point on the perpendicular bisector.

Problem 3

Solution

(Of course I hid some part of the construction to make it a little bit of a challenge. Do you think the location of J is unique?)

In terms of time, this is not actually a short lesson because you need to give students more time to solve the problems. You may also want to read How to scaffold problem solving in geometry. The following book is a good resource for tasks that fosters geometric thinking.

Geogebra

Mathlets – dynamic math applets

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‘An applet is any small application that performs one specific task that runs within the scope of a larger program, often as a plug-in. An applet typically also refers to Java applets, i.e., programs written in theJava programming language that are included in a web page’ -Wikipedia. That settles it. It has nothing to do with Apple and small apples. What about mathlets? Yes, you guess it right that it is an applet about mathematics. Not, it’s not yet in the dictionary. But I find it cute and I intend to use it from now on to describe the math applets I have been creating since I started using GeoGebra to create dynamic worksheets for learning and discovering mathematics and not for demonstrating mathematics. Below is a list of mathlets which I posted in the new website AgIMat which contains resources in science and mathematics teaching.

GeoGebra mathlets are interactive web pages (html file) that consist of a dynamic figure (interactive applet) with corresponding explanations, questions and tasks for students. Students can use the dynamic worksheets both on local computers or via the Internet to work on the given tasks by modifying the dynamic figure.

Geometry

Graphs and Functions

GeoGebra worksheets, Geometry

How to scaffold problem solving in geometry

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Scaffolding is a metaphor for describing a type of facilitating a teacher does to support students’ own making sense of things. It is usually in the form of questions or additional information. In scaffolding learning, we should be careful not to reduce the learning by rote. In the case of problem solving for example, the scaffolds provided should not reduce the problem solving activity into one where students follow procedures disguised as scaffolds. So how much scaffolding should we provide? Where do we stop? Let us consider this problem:

ABCD is a square. E is the midpoint of CD. AE intersects the diagonal BD at F.

List down the polygons formed by segments BD and AE in the square.
How many percent of the area of square ABCD is the area of each of the polygons formed?

Students will have no problem with #1. In #2, I’m sure majority if not all will be able to compare the area of triangles ABD, BCD, AED and quadrilateral ABCE to the area of the square. The tough portion is the area of the other polygons – ABF, AFD, FED, and BCEF.

In a problem solving lesson, it is important to allow the learners to do as much as they can on their own first, and then to intervene and provide assistance only when it is needed. In problems involving geometry, the students difficulty is in visualizing the relationships among shapes. So the scaffolding should be in helping students to visualize the shapes (I actually included #1 as initial help already) but we should never tell the students the relationships among the geometric figures. I created a GeoGebra worksheet to show the possible scaffolding that can be provided so students can answer question #2. Click here to to take you to the GeoGebra worksheet.

Extension of the problem: What if E is 1/4 of its way from C to D? How many percent of the square will be the area of the three triangles and the quadrilateral? How about 1/3? 2/3? Can it be generalized?

Please share with other teachers. I will appreciate feedback so I can improve the activity. Thank you.

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