Category: Geometry

Posted in Algebra, Geogebra, Geometry

The Pythagorean Theorem Puzzle

Posted on by

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

Posted on by

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 Geometry

Guest Post: Real World Uses of Geometry

Posted on by

“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

Posted in GeoGebra worksheets, Geometry

The house of quadrilaterals

Posted on by

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.:-)