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Posted in Algebra, Geogebra, Geometry

The Pythagorean Theorem Puzzle

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Math puzzles are great activities for enjoying and learning mathematics. The following is an example of Tachiawase. Tachiawase is a popular puzzle in Japan which involve dissecting a geometrical figure into several parts and then recombining them to form another geometrical figure. The puzzle below is credited to Hikodate Nakane (1743). This was one of the puzzles distributed at the booth of Japan Society of Mathematical Education during the ICME 12 in Seoul this year.

Make a shape that is made from two different sizes of squares by dividing them into three parts  then recombine them into one square. [Reformulated version: Make two cuts in the shapes below to make shapes that can be recombined into a bigger square.]

two squares puzzle

Here’s how I figured out the puzzle: I know that it must have something to do with Pythagorean Theorem because it asks to make a bigger square from two smaller ones. But where should I make the cut? I was only able to figure it out after changing the condition of the puzzle to two squares with equal sizes. It reduced the difficulty significantly. This gave me the idea where I could make the cut for the side of the square I will form. The solution to this puzzle also gave me an idea on how to teach the Pythagorean theorem.

I made the following GeoGebra mathlet (a dynamic math applet) based on the solution of the puzzle. I think the two-square math puzzle is a little bit tough to start the lesson so my suggestion is to start the lesson with this mathlet and then give the puzzle later.  As always, the key to any lesson are the questions you ask. For the applet below, here’s my proposed sequence of questions:

  1. What are the areas of each of the square in the figure? Show at least two ways of finding the area.
  2. How are their areas related? Drag F to find out if your conjecture works for any size of the squares.
  3. Can you think of other ways of proving the relationships between the three squares without using the measures of the sides?
  4. If the two smaller squares BEDN and GFNH have sides p and q, how will you express the area of the biggest square LEJG in terms of the area of the smaller ones?
  5. Express the length of the sides LEJG in terms of the sides of BEDN and GFNH.

[iframe https://math4teaching.com/wp-content/uploads/2012/08/Pythagorean.html 550 450]

After this lesson on Pythagorean relation you can give the puzzle. Once they have the correct pieces, ask the students to move the pieces using transformation in the least possible moves. They should be able to do this in three moves using rotation. Click here to download the applet. Note: If you don’t see the applet, enable java in your browser

Use the comment sections to share your ideas for teaching the Pythagorean relation. If you like this post, share it to your network. Thank you.

Posted in Elementary School Math, Geometry

A triangle is a fish

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Why is it that students find it easier to calculate the area of triangle ABC but will have difficulty calculating the area of triangle DEF? Middle school students even believe that it’s impossible to find the area of DEF because the triangle has no base and height!right triangles

That knowing the invariant properties that makes a triangle a triangle (or any geometrical shape for that matter), is not an easy concept to learn is illustrated by this conversation I had with my 4-year old niece who proudly announced she can name any shape. The teacher in me has to assess.

Thinking about how a four-year old could possibly think of these meaning of the shapes made me ask: If four-year olds are capable of thinking this way then why do we think that there are students who can’t do math or doubt the idea that algebra is for all? 

Posted in Teaching mathematics

What is variation theory of learning?

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variation theoryVariation theory of learning was developed by Ference Marton of the University of Gothenburg. One of its basic tenets is that learning is always directed at something – the object of learning (phenomenon, object, skills, or certain aspects of reality) and that learning must result in a qualitative change in the way of seeing this “something” (Ling & Marton, 2011). Variation theory sees learning as the ability to discern different features or aspects of what is being learned. It postulates that the conception one forms about something or how something is understood is related to the aspects of the object one notices and focuses on.

Here’s an example: In linear equations you want your students to learn that a linear equation in one unknown can only have one root while an equation with two unknowns can have infinitely many roots.  You also want them to learn that in an equation of one unknown, the root is represented by x only while in equation with two unknowns, the root is represented by an ordered pair of x and y. It is also important that students will see that while both roots can be represented by a point, the root of the equation in one unknown can be plotted in a number line or one-dimensional axis while the root of the equation in two unknowns are plotted in two-dimensional coordinate axes. Will the students discern these particular differences between the roots of the two types of equation in the natural course of teaching linear equations or should you so design the lesson so that students will focus on these differences? Variation theory tells you, yes, you should.

At the World Association of Lesson Studies (WALS) conference in HongKong in 2010 most of the lesson studies presented were informed by variation theory. The teachers reported that students achievement showed significant increases in the post test. Everybody seemed to be happy about it. I think it is not only because of its effect on achievement but it also gave the teachers a framework for structuring their lesson particularly on the design and sequencing of tasks. This sounds very simple but it is actually challenging. The challenge is in identifying the critical feature for a particular object of learning – what is it they need to vary and what needs to remain invariant in the students experiences. Variation theory asserts that change in conception can occur by highlighting critical elements of the object of learning and creating variation in these while all other elements are held constant.

Variation theory directs the teacher to focus on the critical aspect of the object of learning (a math concept, for example), identify differing level of conceptions, and from each of these conceptions identify the critical elements (core ideas) which needed to be varied and those that will remain invariant. In mathematics, these invariants are usually the properties of the concept. In the case of the angles for example, in order for students to have a ‘full’ understanding of this concept they needed to experience it in different forms – the two-line angles, the one-line angles, and the no-line angles.

teaching angles
‘Types’ of Angles

What they need to learn (abstract) from these is that 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. Given these, the teacher now has to design the lesson/ tasks that will provide the necessary variation of learning experiences. You can read my post Angles aren’t that Easy to See for further explanation about understanding angles. Check also my post on how to select and sequence examples to see how variation theory is useful for thinking about examples.

Teachers must always remember however that “even if they aware of the need for the appropriate pattern of variation and invariance, quite a bit of ingenuity may be required to bring it about. Providing the necessary conditions for learning does not guarantee that learning will take place. It is the students’ experience of the conditions that matters. Some students will learn even though the necessary conditions are not provided in class. This may be because such conditions were available in the students’ past, and some students are able to recall these experiences to provide a contrast with what they experience in class. But, as teachers we should not leave learning to happen by chance, and we should strive to provide the necessary conditions to the extent that we are able” (Ling & Marton, 2011). I think we should also remember that the way the learners are engaged is a big factor in learning. You may have addressed the critical feature through examples with appropriate pattern of variation but if this was done by telling, learning may still be limited and superficial.

Another useful guide for effecting learning is creating cognitive conflict. Click Using cognitive conflict to teach solving inequalities to see a sample lesson.

Posted in Geometry

Guest Post: Real World Uses of Geometry

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“When am I going to use this?” This question has been asked in almost every geometry class at one point or another. Many students introduced to advanced mathematics, such as geometry and trigonometry, will deem it worthless. This could not be further from the truth. There are many real world uses for geometry and many careers that require a functional knowledge of it to be successful. If you are currently studying geometry and finding it difficult, consider hiring a professional geometry tutor to assist you in your studies. Also reach out to your teachers and other students to ensure you leave your geometry class with a solid understanding. Using geometry is an essential skill to master.

Every Day Uses of Geometry

Geometry is used throughout many areas of daily life, even if it is not required by your career. John Oprea, author of the book Geometry in the Real World, discusses how many areas of life can benefit from the use of geometry. Below are a few examples of uses of geometry almost everyone will encounter throughout their lives.

  • Lawn Care – When you purchase fertilizer or grass seeds, you may notice that the bags are listed with a square foot measurement. To properly purchase the correct amount of seed or fertilizer, you will need to determine the square footage of your lawn. Without doing some quick geometry you may not purchase enough, or waste money purchasing too much.
  • Purchasing Items – Have you ever moved into a new residence and had the task of filling it with furniture and appliances? Even the seemingly simple task of determining the best use of your area can benefit from basic geometry. How much area will the recliner occupy? Is there room for the seven sectional couches?  Geometric calculations can answer these questions. Purchasing certain appliances also requires geometry. Freezers and refrigerators list their storage capacity in cubic feet. By understanding what a cubic foot means, and calculating an estimate of your household storage needs, you can purchase an appliance that will accurately address your requirements.
  • Household Repairs – A variety of different household repairs can benefit greatly from running some quick geometric equations. Repairing your roof will require you to determine the square footage so you can purchase the appropriate amount of shingles. Any sort of repair involving carpentry will require geometry. You must ensure the corners are perfectly square and the walls are plumb. Geometry will help you determine the design of a new project and how much material you will need.
Careers That Require Geometry

Hundreds of careers require an expert level understanding of geometry in order to be successful. David Eppstein, author of Geometry in Action and writer for the University of California Irvine, states the below careers are heavily involved with geometry.

  • Architecture – From the Pyramids in Egypt to the skyscrapers of New York, geometry is the building block of architecture. Before the ground is broke and foundation is laid, an architect will draft a complete model of the new building. The architect’s primary focus when designing a building is using geometry to create a safe structure. Every angle, and the length of every side, is carefully calculated in accordance with geometric principles to create a structure that can safely withstand the elements and any other hazards it may encounter.
  • Computer Graphics Artist – This modern field of artistry and design merges almost every aspect of geometry in a computer simulation to create a variety of graphics. Cutting edge software allows graphic artists to create visually compelling and aesthetically pleasing graphics that are used in video games, movies and presentations. While the computer is able to handle a lot of the behind the scenes math, a solid understanding of geometry is required to be able to construct the complex models artists create.
  • Video Engineering – How do projectors create a crisp compelling image that fills up the screen? How do directors determine which lens to use for their ideal shots? They employ video engineers to solve these problems. Using their mathematic prowess, they are able to calculate which lens will create the optimal field of view the director is requesting. They also determine the perfect location to setup the projector and the best angle to produce a clear and crisp image on the screen.
Geometry: Well worth Learning

The examples above are only a small sampling of the uses of geometry. It is used every day in a variety of careers and tasks. Applying yourself fully to your geometric studies can prepare you for your ideal career and help solve many problems you may encounter throughout life.
About the Author:

Andrew Boyd is a writer who has enjoyed geometry since he was introduced at an early age. As a hobby carpenter, he uses geometry on a daily basis and loves showing others why it’s such a worthwhile field of study.

Recommended readings:
Fostering Geometric Thinking: A Guide for Teachers, Grades 5-10
Understanding Geometry for a Changing World: NCTM’s 71st Yearbook