Category: GeoGebra worksheets

Posted in GeoGebra worksheets, Geometry

The house of quadrilaterals

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In Investigating an Ordering of Quadrilaterals published in ZDM, Gunter Graumann shared a good activity for developing students mathematical thinking. The activity is about ordering quadrilaterals based on its characteristics. He gave the following list of different aspects of quadrilaterals as possible basis for investigation.

  1. Sides with equal length (two neighbouring or two opposite or three or four sides)
  2. Sum of the length of two sides are equal (two neighbouring or two opposite sides)
  3. Parallel sides (one pair of opposite or two pairs of opposite sides)
  4. Angles with equal measure (one pair or two pairs of neighbouring or opposite angles, three angles or four angles)
  5. Special angle measures (90° – perhaps 60° and 120° with one, two, three or four angles)
  6. Special sum of angle measures (two neighbouring or opposite angles lead to 180°)
  7. Diagonals with equal length
  8. Orthogonal diagonals (diagonals at right angles)
  9. One diagonal bisects the other one or each diagonal bisects the other one,
  10. Symmetry (one, two or four axis’ of symmetry where an axis connects two vertices or two side-midpoints, one or three rotation symmetry, one or two axis’ of sloping symmetry). With a sloping-symmetry there exists a reflection – notabsolutely necessary orthogonal to the axis – which maps the quadrilateral onto itself. For such a sloping reflection the connection of one point and its picture is bisected by the axis and all connections lines point-picture are parallel to each other.

The house of quadrilaterals based on analysis of the different characteristics of its diagonals is shown below. Knowledge of these comes in handy in problem solving.

House of quadrilaterals based on diagonals

Read my post Problem Solving with Quadrilaterals. You will like it.:-)

Posted in Algebra, GeoGebra worksheets, Math Lessons

Teaching maximum area problem with GeoGebra

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Remember that classic maximum area problem? Here’s a version of it: Pam wishes to fence off a rectangular vegetable garden in her backyard. She has 18 meters of  fencing wire which she will use to fence three sides of the garden with the existing fence forming the fourth side. Calculate the maximum area she can enclose.

This problem is usually given as an application problem and is solved algebraically. For example if x is one of the two equal sides to be fenced then the area is the function f(x) = x (18-2x). The maximum area can be found by graphing or by inspection. If students have done a bit of calculus already then they can use the first derivate to solve the problem. But with free technology such as GeoGebra, there should be no excuse not to make the teaching of this topic less abstract especially for Year 9 or 10 students. It need not be at the end of the chapter on quadratic but as an introductory lesson for quadratics. Here’s a GeoGebra applet I made which can be used to teach this topic more visually and conceptually. Below is the image of the applet. I did not embed the applet here because it takes a while to load. Click maximum_area_problem worksheet to explore.

Here’s my suggested teaching approach using this applet. Students need to be given a bit of time exploring it before asking them the following questions:

  1. Pam wishes to fence off a rectangular vegetable garden in her backyard. She found fencing wires stored in their garage which she will use to fence three sides of the garden with the existing fence forming the fourth side. How long is Pam’s fencing wire? What are some of the sizes of gardens Pam can have with the fencing wires?
  2. If you were Pam, what garden size will you choose? Why?
  3. What do the coordinates of P represent? How about the path of P, what information can we get from it?
  4. As the length of BD changes so does the length of the other two sides. What equation will describe the relationship between the length of BD and EF? between BD and DE? between BD and area BDEF.
  5. What equation of function will run through the path of P? Type it in the input bar to check.
  6. What does the tip of the graph tell you about the area of the garden?

Feel free to use the comments sections for other questions and suggestions for teaching this topic. How to teach the derivative function without really trying is a good sequel to this lesson. More lessons in Math Lessons in Mathematics for Teaching.

Posted in Algebra, Geogebra, GeoGebra worksheets

Teaching mathematics with GeoGebra

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GeoGebra constructions are great and it’s fun ‘watching’ them especially if you know the mathematics they are demonstrating. If you don’t and most students don’t then we have a bit of a problem. Even if the applet demonstrates the mathematics to students I don’t think there’ll be much learning there. No one learns mathematics by watching. We know that ‘mathematics is not a spectator sport’. You have to play the game. In Geogebra  and Mathematics I proposed that if GeoGebra is to help students in learning mathematics with meaning and understanding, then students should know how to use it. But these GeoGebra tools will be most useful only to students if they know the mathematics behind the tools and why they work and behave like that.  And so we teach students the mathematics first? Where’s the fun in that?

I believe (Translation: I’ve yet to do a study if my theory works) that it is possible to learn mathematics and the tools of GeoGebra at the same time .  I will be sharing in this blog GeoGebra activities where students learn to use GeoGebra as they learn mathematics. The main objective is of course to learn mathematics. The learning of GeoGebra is secondary. I will start with the most basic of mathematics and the most basic of the tools in GeoGebra: points, lines, and the coordinates system.

The lesson includes four GeoGebra activities:

Activity 1 – What are coordinates of points? Read the introduction about coordinates system here.

Activity 2 – What are the coordinates of points under reflection in x and y axes?

Activity 3 – How to describe sets of points algebraically Part 1?

Activity 4 – How to describe sets of points algebraically Part 2? (under construction)

Posted in 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.

  1. List down the polygons formed by segments BD and AE in the square.
  2. 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.

More Geometry Problems:

  1. The Humongous Book of Geometry Problems: Translated for People Who Don’t Speak Math
  2. Challenging Problems in Geometry