Category: Geometry

Posted in Elementary School Math, Geometry

Angles are not that easy to see

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Like most numbers, geometric objects such as angles, are abstraction from properties of real objects and quantities. For example, the idea of “two-ness” can be abstracted from real objects such as two apples, two chairs, two goats, etc. It will not take along for a learner to figure out what the idea of two means. Abstracting angles from real objects this way is not as easy as one might think it is.

Look around you and find something that to you looks like an angle. Chances are you would identify corners as forming an angle.  That’s easy because you see two sides meeting at a corner. But doesn’t the door also forms an angle when you open it?  But where is the other side? How about turning the door knob? Doesn’t it form an angle also? But where are the two sides there? It doesn’t even have a corner!

Mitchelmore and White (2000) of Australian Catholic University conducted a study of 2nd, 4th, 6th and 8th grade students understanding and difficulty about angles.  They found that students do not readily incorporate ‘turning’ in their idea of angles. They found that it is the line (or arms) of angle  which are the key to students identifying angles in different physical situation. Their study showed the easiest angles for students to learn are the two-line angles. These are angles in which both arms are visible such as corners of geometrical figures, corners of rooms, blades of a pair of scissors. The second group of angles are the one-line angles. In these angles, only one arm is visible. The other line must be imagined or remembered. Examples are the angles formed by an opening in a door, a hand of a clock and sloping of roofs. The most difficult for the students to identify are the no-line angles in which neither arms of the angles are visible. Examples include the turning wheel and spinning ball.

One can be said to have an understanding of the concept of angle if he/she can recognize all these types of angles in physical objects and is able to see that they all share the same property: they all consist of two linear parts (even if they are not visible) and they cross or meet at a point and that the relative inclination of the two parts has some significance – it defines the sharpness of the corner or the their openness.

So what is the implication of these to teaching? The most obvious is the importance of exposing students to as many different physical situation that can be represented by angles. Starting with the definition an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle and then drawing the angle figure on the board is certainly the most ineffective strategy the teacher  can do to teach students about angles.

 

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

 

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 Geometry

Twelve definitions of a square

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How does mathematics define a math concept?

Definitions of concepts in mathematics are different from definitions of concepts in other discipline or subject area. A definition of a concept in mathematics give a list of properties of that concept. A mathematics object will only be an example of that concept if it fits ALL those requirements, not just most of them. Further, a definition is also stated in a way that the concept being defined belongs to an already ‘well-defined’ concept. On top of this, economy of words and symbols and properties are highly observed.

Does a math concept only have one definition? Of course, not. A concept can be defined in different ways, depending on your knowledge about other math objects. In a study by Zaskin and Leikin, they suggested that the definitions students give about a concept mirrors their knowledge of mathematics. Below are examples of definitions of squares from that research. Do you think they are all legitimate definitions?

What is a square?

A square is

  1. a regular polygon with four sides
  2. a quadrilateral with all the angles and all the sides are equal
  3. a quadrilateral with all the sides equal and an angle of 90 degrees
  4. a rectangle with equal sides
  5. a rectangle with perpendicular diagonals
  6. a rhombus with equal angles
  7. a rhombus with equal diagonals
  8. a parallelogram with equal adjacent angles and equal adjacent sides
  9. a parallelogram with equal and perpendicular diagonals
  10. a quadrilateral having 4 symmetry axes
  11. a quadrilateral symmetric under rotation by 90 degrees
  12. the locus of all the points in a plane for which the sum of the distances from two given perpendicular lines is constant. Click this link to visualize #12.

 

Making (not stating) definitions is a worthwhile assessment task.

Here’s three great references for definitions of mathematical concepts. The first is from no other than Dr. Math (The Math Forum Drexel University). The middle one’s for mom and kids – G is for Google and the third’s a book of definitions for scientists and engineers.

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