Posted in Elementary School Math, Geometry

A triangle is a fish

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

“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 Algebra, GeoGebra worksheets, Math Lessons

Teaching maximum area problem with GeoGebra

Remember that classic maximum area problem? Here’s a version of it: Pam wishes to fence off a rectangular vegetable garden in her backyard. She has 18 meters of  fencing wire which she will use to fence three sides of the garden with the existing fence forming the fourth side. Calculate the maximum area she can enclose.

This problem is usually given as an application problem and is solved algebraically. For example if x is one of the two equal sides to be fenced then the area is the function f(x) = x (18-2x). The maximum area can be found by graphing or by inspection. If students have done a bit of calculus already then they can use the first derivate to solve the problem. But with free technology such as GeoGebra, there should be no excuse not to make the teaching of this topic less abstract especially for Year 9 or 10 students. It need not be at the end of the chapter on quadratic but as an introductory lesson for quadratics. Here’s a GeoGebra applet I made which can be used to teach this topic more visually and conceptually. Below is the image of the applet. I did not embed the applet here because it takes a while to load. Click maximum_area_problem worksheet to explore.

Here’s my suggested teaching approach using this applet. Students need to be given a bit of time exploring it before asking them the following questions:

  1. Pam wishes to fence off a rectangular vegetable garden in her backyard. She found fencing wires stored in their garage which she will use to fence three sides of the garden with the existing fence forming the fourth side. How long is Pam’s fencing wire? What are some of the sizes of gardens Pam can have with the fencing wires?
  2. If you were Pam, what garden size will you choose? Why?
  3. What do the coordinates of P represent? How about the path of P, what information can we get from it?
  4. As the length of BD changes so does the length of the other two sides. What equation will describe the relationship between the length of BD and EF? between BD and DE? between BD and area BDEF.
  5. What equation of function will run through the path of P? Type it in the input bar to check.
  6. What does the tip of the graph tell you about the area of the garden?

Feel free to use the comments sections for other questions and suggestions for teaching this topic. How to teach the derivative function without really trying is a good sequel to this lesson. More lessons in Math Lessons in Mathematics for Teaching.