Five Girls Sewing Dresses

Posted on January 25, 2012 by Brett Edwards

Every Wednesday I begin my math classes with a riddle. Most of them are math related and often have a connection to something we are studying or have studied. It is funny how similar they are to word problems but if I call it a “riddle” they instantly consider it a challenge worthy of their excited attention. Nearly all my collected riddles are not my own but come from riddle books or online sources. The riddle for my 7th and 8th graders today was the following:

“If 5 girls can sew 5 dresses in 5 days, how many girls would it take to sew 50 dresses in 50 days?”

I enjoy this riddle because there are number of mathematical ideas I can address regarding this one simple riddle. The solution to this riddle is obviously connected to ratios and proportions and can be an effective teaching moment for those concepts. More importantly though I make a point to emphasize the proper use of basic logic. It is remarkable how many students will answer 50. My response is always, why would I give this “riddle” to you if the answer is 50? This is mental laziness and will cause all kinds of problems if they allow it to find a place in their mental thinking.  The final thing (related to the previous one) I try to emphasize is that often the point of a riddle is to be careful in trusting your intuition. Such riddles are fun because the answer seems so obvious but will contradict your original thought process. The student thinks “5, 5 and 5 in the first scenario so it must be 50, 50 and 50 for the second.” They correctly apply a pattern and they remember that Mr. Edwards is always saying “math is the science of patterns” so 50 must be the answer. Not in riddle land!

For the few out there reading this, if you would like a compilation of my collected riddles I would be more than willing to email you what I have.

Posted in Fun, Riddles | 1 Comment

Hero’s Root Approximation

Posted on January 21, 2012 by Brett Edwards

Hero of Alexandria was a Greek mathematician and engineer living during the 1st century AD.  Among his mathematical contributions was a clever technique of approximating square roots.  He uses an iterative method meaning the process is repeated and each iteration generates a better approximation.  I want to briefly cover this fascinating approach that often brings remarkable accuracy to square root computation after only a few iterations.   Although a bit tedious, the beauty of this method is that it can be done without the aid of a calculator.  I want to first discuss the formula and how to apply it.  Second, I want to discuss the logic behind the method.

The iterative process can be best explained using an example.  Let’s make this interesting by trying to approximate the √243,148.  Read more…

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Why do I have to learn this?

Posted on January 12, 2012 by Brett Edwards

Educators, especially those teaching math, are privileged to here this question often.  I would argue the prevalence rests not in the attitude of our students but rather in the insufficient answers provided on a daily basis by math teachers.  Often, the teacher will say something to the effect of “well if you want to pass this class” or “if you want to go to college.”  Such answers will guarantee that the teacher will hear this question again and again.  The following from Banner & Cannon’s “Elements of Teaching” illustrates a better response to this question:

“A teacher’s confidence in the intrinsic worth of knowledge is fundamental to all instruction. Such deep-rooted belief makes a teacher able to relate knowledge to life, to all human experience. To students’ typical questions, “Why do we have to learn this? What good is such knowledge?” the typical instrumental answers come to mind easily: “Because it’s required by the school board.”  “Because you will do better on your licensing exam.” “Because you’ll need it later when you study economics.”  But the teacher with deep learning answers with conviction and authority more pertinently: “Because acquiring this knowledge is difficult.   Because you will feel triumphant when it no longer confuses you. Because you will enjoy what you can do with it. Because in learning it you may discover new perspectives on life, new ways of thinking.   Because its possession will make you more alive than its alternative, which is ignorance.”

Well said.  I especially appreciate the last line.  “Because its possession will make you more alive than its alternative, which is ignorance.”  All truth is God’s truth and this includes mathematical truths.

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Every Number Has a Story

Posted on January 7, 2012 by Brett Edwards

As the story goes, mathematician G.H. Hardy visited his friend and renowned mathematician Srinivasa Ramanujan in a hospital.  Upon arriving, he informed his colleague that he had taken a taxi numbered 1729 and considered the number very uninteresting.  Ramanujan disagreed saying “No, Hardy! No, Hardy! It is a very interesting number.  It is the smallest number expressible as the sum of two cubes in two different ways.”  Both 13 + 123 and 93 + 103 sum up to 1729.   Ya know… just some trivial math stuff that everyone knows.

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Where is 1 Million?

Posted on January 3, 2012 by Brett Edwards

A general literacy of numbers is lacking in our culture.  A fun exercise I like to do with my students to illustrate this is drawing a large number line on my whiteboard. I go as long as I can. Put 0 on the far left and 1 billion on the far right.   I then ask students to tell me approximately where they think 1 million would be on the number line.

Generally, the students give me answers that show they really don’t know the significant difference between 1 million and 1 billion. Read more…

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Mutilated Chessboard

Posted on December 31, 2011 by Brett Edwards

One distinction all students need to grasp at some point in secondary school is the important difference between mathematical proof and scientific proof.  It is common for the students to think proving things in the math and science class uses the same logical process.  This is not true.  A good way to illustrate this distinction is using the idea of a mutilated chessboard.  First it is necessary to consider a normal chessboard.

If one were asked to take dominoes and cover all the squares of a traditional chessboard, most would see quickly the ease of doing such a task. Read more…

Posted in Fun, Pedagogy | 2 Comments

Constructions and Proofs

Posted on December 27, 2011 by Brett Edwards

There is no mathematical discipline that needs greater reform than geometry.  The Greeks would look at our modern geometry curriculum and would either fall over laughing or have a good ol’ fashioned book burning.  There is much to be said regarding the modern day drift away from Euclid’s Elements but I would like to simply call us back to the two major components of a classical geometry education.

Geometry from 500 years before Christ until the 19th century was centered on constructions and proofs.  Unfortunately, these two things are the very things finding it more and more difficult to find the light of day in the modern geometry class. Read more…

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Forming Math Problems

Posted on December 13, 2011 by Brett Edwards

Central to the discipline of Mathematics is problem solving. Christian math teachers should be trying to developing problem solving skills that students can take with them to every area of their life.  Effective problem solving involves rigor, logic and creativity.  Modern math education is so focused on finding the right answer that it neglects the proper use of logic and the essential role of creativity in the math class.  The current “cookbook” math pedagogy goes something like this.  Provide the student (in the lesson or from a lecture) with a formula or a step by step process to find the answer to a problem.  Do a few examples with the students so that they know exactly how to plug the length of the legs into the “a” and “b” and the hypotenuse into the “c”.  The teacher has communicated clearly the recipe in easy to follow steps and the student simply follows instructions.  This problem solving is shallow and has little application outside of chapter 5, lesson 3, part 2.  Any student that can follow instructions should be able to do well in this environment.

How can we do better?  Albert Einstein said “the formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.”  Math teachers need to do a better job of framing the problems that are tackled in the math classroom.  First let me illustrate a typical problem that doesn’t really challenge our students to be creative problem solvers and how this problem could be changed. Read more…

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A Sunset, Three Perspectives and One God

Posted on November 23, 2011 by Brett Edwards

Let the blogging begin.  I have thought for awhile now that I should start an online effort to help cultivate collaboration among Christian math educators.  I am passionate about revealing the wonders of God’s creation in the math class and enjoy sharing helpful thoughts to other math teachers.  To get this blog started, I wanted to share a few thoughts regarding Christian worldview as applied to mathematics.

Generally, when a person looks at a magnificent sunset, it is received with one of three responses:

  1. “Wow, God is creative!”
  2. “Wow, that’s weird!”
  3. “What sunset?”

The first response recognizes that there is something “other” in the beauty we see.  It admits that such beauty cannot be explained alone with mathematical formulas and equations.  Similarly, these people refuse to believe that love is merely an interaction between the chemicals dopamine, norepinephrine and serotonin.  This paradigm of thinking looks at a beautiful sunset much the same way one looks at a Rembrandt painting.  The assumption behind the eyes in both situations is that this must have come from the hands of a creative being.  G.K. Chesterton said “if the beatification of the world is not a work of nature but a work of art, then it involves an artist.”

Read more…

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