Tag: geometry lessons

Posted in Geometry

Unpacking mathematics – a geometry example

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Engineers, mathematicians, and mathematics teachers all deal with mathematics but it is only the math teacher who talks about math to non-mathspeakers and initiate them to ‘mathspeak’. To do this, the math teachers should be able to ‘unpack’ for the students the mathematics that mathematicians for years have been so busy ‘packing’ (generalising  and abstracting) so that these learners will learn to do the basics of packing by themselves. This is in fact the real job description of a mathematics teacher. I won’t comment about the remuneration as this is not this blog is about. I thought it would be best for me to continue sharing about the ways we can unpack some of the important ideas in mathematics as this is the mission of this blog. Just in case you haven’t read the blog description, this blog is not about making mathematics easy because math is not so stop telling your students that it is because that makes you a big liar. What we should try to do as math teacher is to make math make sense because it does. This means that your lesson should be organised and orchestrated in a way that shows math does makes sense by making your lesson coherent and the concepts connected.

Today I was observing a group of teachers working on a math problem and then examining sample students solutions. The problem is shown below:

congruent triangles

The teachers were in agreement that there is no way that their own students will be able to make the proof even if they know how to prove congruent triangles and know the properties of a parallelogram. They will not think of making the connection between the concepts involved. I thought their concerns are legitimate but I thought the problem is so beautiful (even if the way it is presented is enough to scare the wits out of the learners) that it would be a shame not to give the learners the chance to solve this problem. So what’s my solution to this dilemma? Don’t give that problem right away. You need to unpack it for the learners. How? To prove that AFCE is a parallelogram, learners need to know at least one condition for what makes it a parallelogram. To be able to do that they need to know how to prove triangle congruence hence they need to be revised on it. To be able to see the necessity of triangle congruence in proving the above problem, learners need to see the triangles as part of the parallelogram. So how should the lesson proceed?

Below is an applet I developed that teachers can use to initiate their learners in the business of making proofs where they apply their knowledge of proving triangles and properties of quadrilaterals, specifically to solving problems similar to the above problem.  Explore the applet below. Note the order of the task. You start with Task 1 where the point in the slider is positioned at the left endpoint. Task 2 should have the point positioned at the right end point. You can have several questions in this task. Task 3 should have the point between the endpoints of the slider. Of course you can also present this using static figures but the power of using a dynamic one like the geogebra applet below not only will make it interesting but the learners sees how the tasks are related.

Task 1

  1. What do the markings in the diagram tell you about the figure ABCD? What kind of shape is ABCD? Tell us how you know.
  2. Do you think the two triangles formed by the diagonals are congruent? Can you prove your claim?

Task 2 – Which pairs of triangles are congruent? Prove your claims

Task 3 – What can you say about the shape of AFCE? Prove your claim.

[iframe https://math4teaching.com/wp-content/uploads/2013/05/Parallelogram_Problem.html 550 500]

Here’s the link to the applet  Parallelogram Problem

Note that Task 3 has about 4 different solutions corresponding to the properties of a parallelogram. I will show it in my future post.

More of this type: Convert a Boring Geometry Problem to Exploratory Version

Posted in Geometry

Teaching triangle congruence

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In this post I propose a way of teaching the concept of triangle congruence. Like most of the lessons I share in this blog, the teaching strategy for this lesson is  Teaching through Problem Solving. In a TtPS lesson, the lesson starts with a situation that students will problematize. The problems either have many correct answers or have multiple solutions and can always be solved by previously learned concepts and skills. Problems like these help students to make connections among the concepts they already know and the new concept that they will be learning in the present lesson. The ensuing discourse among students and between teacher and students during the discussions of the different solutions and answers trains students to reason and communicate mathematically and thereby help them to appreciate the power of mathematics as a language and a way of thinking. In mathematics, language is precise and concise.

Here’s the sequence of my proposed lesson:

1. Setting the Problem:

Myra draw a triangle in a 1-cm grid paper. Without showing the triangle, she challenged her friends to draw exactly the same triangle with these properties:  QR is 4 cm long. The perpendicular line from P to QR is 3 cm. 

Pose this question: Can you draw Myra’s triangle?

Give students enough time to think. When each of them already have at least one triangle, encourage the class to discuss their solutions with their seat mates. Challenge the class to draw as many triangles satisfying the properties Myra gave.

2. Processing of solutions: Ask volunteers to show their solutions on the board. Questions for discussion: (1) Which of these satisfy the information that Myra gave? (2) What is the same among all the correct answers? [They all have the same area]. Possible solutions are shown below.

triangle congruence


3. Introducing the idea of congruence: Question:  If we are going to cut-out all the triangles, which of them can be made to coincide or would fit exactly? [When done, introduce the word congruence then give the definition.]

Tell the class that Myra only drew one triangle. Show the class Myra’s drawing. Question: In order to draw a triangle congruent to Myra’s triangle, what conditions or properties of the triangle Myra should have told us?

Myra’s triangle

Possible answers:

  1. QR is 4 cm long. The perpendicular line PQ  is 3 cm.
  2. QR is 4 cm long. PQ is 3 cm and forms a right angle with QR.
  3. PQR is a right triangle with right angle at Q. QR is 4 cm and PQ is 3 cm.

4. Extending the problem solving activity: Which of the following sets of conditions will always give triangles congruent to each other?

  1. In triangle ABC, AB and BC are each 5 cm long.
  2. ABC is a right triangle. Two of its shorter sides have lengths of 4 cm and 5 cm.

I would appreciate feedback so I can improve the lesson. You feedback will inform the sequel to this lesson.. Thank you.

Posted in Geogebra, Geometry

Constructing polygons with equal area

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The power of GeoGebra lies not only for demonstrating a concept but more so for creating a situation where students are made to think, solve problems, and reason mathematically. Here is a sample lesson on how this can be done. The lesson involves the concept of area of triangles and constructing parallel lines applied to problems involving preserving areas of polygons. The most ideal situation is for students to explore on the applets individually or in small group. If this is not possible and the students have no computers, the applets can be projected. The teacher can then call a students or two to explore the applets. The idea is to stimulate students’ thinking to think of explaining why the transformations of the shapes are possible. The students should have a triangle rulers or straight edge with them. The lesson will not be complete if the students cannot devise a procedure for transforming a triangle into other polygonal shapes with the area preserved.

The following applet may be used in the introductory activity to teach about triangles with equal areas. Click here to explore. The applet shows that all triangles with equal base and have equal heights or altitude have equal areas. That’s pretty obvious of course.

The next applet is more challenging but uses the same principle as the first.
Depending on the previous knowledge of your students you can give this second applet right away without showing the first applet on triangles. Click here or the worksheet below explore and answer the problem.

Extension problems using the same principle about area of triangles.

triangle rulers
  1. Given a triangle, construct the following polygons equal in area to the given triangle using triangle rulers and pencil only:  a) parallelogram; b)trapezoid; c) hexagon. (There’s nothing that should prevent you from using GeoGebra to construct them.)
  2. The Rosales and Ronda families are not very happy with their piece of land because of the narrow corners. Help this family to draw a new boundary line without changing the land area of each family.

I will deprive you of the fun if I will show the answers here right away, won’t I?

I will appreciate feedback on this lesson.