Posted in Geometry

Unpacking mathematics – a geometry example

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

Pentagon to Quadrilateral Puzzle

Puzzles involving cutting shapes and forming them into different shapes helps reinforce the idea that area do not necessarily change with change in shape. It is also a good activity for developing visualisation skill and spatial ability.

The puzzle below is from one of the leaflets at the booth of Japan Society of Mathematical Education last ICME 12 in Seoul, Korea. The original puzzle is suited for Grade 4. The instruction was to cut the pentagon along the dotted lines and then form them into the shapes shown. The shapes shown in the leaflet is a parallelogram, a rectangle, an isosceles trapezoid, and a general trapezoid. I modified the puzzle for students in the higher level. I have indicated the measure of the two angles just in case you want your students to justify that the pieces really form into quadrilaterals. This is one way to assess your students knowledge of the properties of these parallelograms, trapezoid and trapezoids as they justify each shape formed.

pentagon puzzle

Here are two solutions – rectangle and isosceles trapezoid. Form the other two shapes.

trapezium and rectangle

Posted in Algebra

Visual representations of the difference of two squares

Students’ understanding of mathematics is a function of the quality and quantity of the connections of a concept with other concepts. As I always say in this blog, ‘To understand is to make connections’.

There are many ways  of helping students make connections. One of these is through activities involving multiple representations. Here is a lesson you can use for teaching the difference of two squares, x^2-y^2.

Activity: Ask the class to cut off a square from the corner of a square piece of paper. If this is given in the elementary grades, you can use papers with grid. If you give it to Grade 7 or 8 students you can use x for the side of the big square and y for the side of the smaller square. Challenge the class to find different ways of calculating the area of the remaining piece. Below are two possible solutions

Solution 1 – Dissect into two rectangles

 

Solution 2 – Dissect into two congruent trapezoids to form a rectangle

 

Extend the problem by giving them a square paper with a square hole in the middle and ask them to represent the area of the remaining piece, in symbols and geometrically.

Solution 1 – Dissect into four congruent trapezoids to form a parallelogram

 

Solution 2 – Dissect into 4 congruent rectangles to form a bigger rectangle

These two problems about the difference of two squares will not only help students connect algebra and geometry concepts. It also develop their visualization skills.

This is a problem solving activity. It’s important to give your students time to think. Simply using this to illustrate the factors of the difference of two squares will be depriving students to engage in thinking. They may find it a little difficult to represent the dimensions of the shapes but I’m sure they can dissect the shapes. Trust me.

Posted in Geometry

Twelve definitions of a square

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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