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Dear Mitch,

This coming school year I will be teaching tenth grade geometry for the first time since becoming a high school math teacher seven years ago.The principal of our school claims he's confident I'll be a "natural" at it, in part, I think, because he knows I'm interested in the visual arts and have dabbled in painting and have always helped the drama club design and build their stage-sets.  However, he shocked me the other day be emailing me a note saying that even though this is the busiest time of the school year, completing the curriculum, winding down, making finals and filling out progress reports (report cards), he would appreciate it if I could come into another teacher's 10th grade geometry class during my one free prep period and do a "demo lession".  In other words, he would like to see me teach the class -- the entire 44-minute class, and he will be in the back of the room "for moral support" (which really means to assess my lesson!).  I've since learned through the grapevine that there's been phone calls from a parent saying she did not want her child to be a 'guinnea pig' with a "rookie", despite my successful track record teaching algebra and pre-calculus!

So, since I was given the "greenlight" on selecting any topic at all that is included in the textbook and part of the states' mandated curriculum, a colleague told me you published an article several years ago in a teaching journal describing one of your favorite lessons that seems to be the type that brings a 10th-grade geometry concept to life.

Can you please reprint it here or give some kind of summary of it?  I really would appreciate it if you would, especially if you can do it within the next two weeks.

Thank you so much for devising the lessons you've managed to come up with over the past few years and thank you for taking the time to post so many ideas for teachers, students, parents, and anyone else interested in learning math in such clarifying and memorable ways.  (My husband, by the way, who has been a 5th-grade teacher for nine years, truly believes that you're a genius at what you do.) 


Ms. B.

(From one of the hundreds of school districts in Illinois.)


Dear Ms. B.,

As a matter of fact,  I do have some ideas for lessons you should consider.  (I can't pick a 'favorite' geometry lesson, because  the ones that come to mind right now would be tied with about a dozen other lessons, but here's one that immediately strikes me as something so effective and engaging that students from years gone by seem to recall it with such clarity and fondness that it's the one I'll describe.)

First, the MATERIALS:

1. You will need one copy of the the book, A CLOAK FOR THE DREAMER, written by Aileen Friedman and illustrated by Kim Howard.  As I sit here at my laptop and type this, the edition of the book I mention that I see on the bookshelf next to me was published by Scholastic, as part of the series "A MARILYN BURNS BRAINY DAY BOOK."  Unless it is currently being printed by another company, this information should make it fairly easy to find and obtain.

2. You will also need to have a supply of lined, index cards, one for each student.  (I've always used large cards, such as the 5-inch x 7-inch type, but any size should be sufficient for this lesson.)

3. And this one (material #3) should be a given in any classroom in any school -- though, sadly, it often is not:  Each student needs one pencil.  (Actually, now that I recall the details and possible pitfalls of this relatively simple lesson, non-erasible pens or thin markers might be preferrable to pencils; you will understand the reason for this slight adjustment to the usual math class habit of employing pencils rather than pens as we proceed.)

Okay, now, the first thing you will notice when you peruse the book is that it is an illustrated children's book.  It does not even appear to be a math book, and, in a way, it is not; it's basically a story book of the type you might read to a child much younger than a tenth grader, and this is one of the elements which makes the lesson come as a welcome surprise; 10th-graders do not often get opportunities to have books read to them, and, most people, regardless of age or level-of-education, enjoy sitting back and listening to an interesting tale, despite whether they would ever publicly admit to finding the experience to be as big a treat as their rapt attention reveals it to be. 


1) Sketch a giant version of the index card on the chalkboard/dry-erase-board/or-whatever-your-classroom-has for such purposes. Use your sketched version to demonstrate the layout on the students' cards for each step.

2) Distribute the cards and pens and instruct the students to write their name on the top margin.

3) Have the students write the number one (#1) on the left-hand-side of the first line under the top margin of the card.

4) Have the students skip a few lines and write #2 on the left-hand-side of the line that is vertically separated by a few lines from the top line.

5) Have the students skip a few more lines and write #3 on the left-hand-side of the line that is vertically separated from the line they've just labeled #2.

6) Have the students skip another couple of lines or turn their card over and write a #4 on the left-hand side of the card.

7) Have the students place their cards and pens down on their desktops and explain that you are going to read them a story.  Also, inform them that you will pause at a few places in the story in order to ask them one question each time you pause. The students are to make their best guess at an answer and jot down their idea (and the reason for their idea) on the correctly numbered line of their card.  They are to do so on their own (no group work), and they are to do so silently; there will be time later for questions, answers, and explanations, AFTER the entire story is read.

8) Begin reading the story, and do so as animatedly as you would for a younger audience; I have developed a richly textured collection of jokes/one-liners to acompany the text, and the lesson is one of the most commonly cited ones as the 'in-class' experience in which the students felt they learned the most and learned the subject without any memorable challenge.  My personal version of the tale evolved organically over the years as I returned time and time again to the lesson, and would therefore would be unlikely to fit as naturally into the style of another teacher (i.e., whoever is currently reading this).  It is worth reiterating the fact that the topic covered here (the plane geometry of polygons' external angles), is an important and sophisticated geometric concept, and this particular lesson is designed to transform an otherwise abstract and intangible idea into a concept that matters to people in the real world.

O.K.  Now, the story is about a tailor and his three sons, each of whom works for him.  They are given an important assignment to fulfill, and they will be able to make the deadline, as long as each son works independently while the father does the same.

The story's mathematics leads up to -- and then boils down to --sophisticated geometric concepts involving the properties of certain polygons (2-dimensional shapes) to be joined along their edges to form a solid plane, while other polygons require either the inclusion/insertion of other shapes to accomplish the same task of filling a space without leaving any gaps.  In a nutshell, the final concept comes down to whether or not various shapes (such as octagons, squares, hexagons, pentagons, etc.), have exterior angles that have degree measurements which can be combined to form groups of 360 degree arrangements (thereby making a solid piece of cloth without holes or overlaps). 

1) The first pause should occur directly after the tailor and his sons are asked to make cloaks for the important client using separate shapes in a patchwork-quilt-type style.  The students should be asked: Which polygons would work to make a solid cloth without overlaping material? Which polygons would not? And, if the students are capable, they should be asked to explain their reasoning in a sentence or two.

The teacher continues reading the story.

2) The second pause in the reading should be made just before the tailor wakes his sons and tells them he has an idea.  Ask the class to write down what they think the brothers and their father will do to the cloak made of circles in order to turn it into a solid cloak.

3) The third pause should be made after the story is completed. Ask the class to list other polygons that would have worked just as well and have them include the reasons for their choices.  OCTAGONS?  PENTAGONS? ETC.

4) Pause for the fourth time after the students have written down their responses to question number three. Now ask the class to try to come up with a combination of two or more shapes that would work well together to solve the problem and to include WHY this combination would work.

It is usually helpful for this fourth question to remind students of the old-fashioned-but-still-well-known-and-commonly-seen floor pattern in restrooms and bathrooms of apartments built during the early part of the last century, particularly in the New York area; these black and white floors, which are comprised of small white octagons and small black squares, are most comonly seen in apartment buildings in Brooklyn, Queens, the Bronx, and Manhatttan.

Finally, you will notice that at the very end of the book are a few pages of related lessons and explanations to extend the students' practice with the extrenal angles of various polygons, etc.  It is worded for children younger than those in tenth grade, but that is an easy adjustment to make as you present the questions and problems your own way and have the students calculate the portions of 360 degrees in each part of the shapes in question that are to be fitted together.  From that small section, the possibilities for crossing the geometry with algebra and graphing are only limited by your imagination.  It is also a good time to review and build upon the basic operations of addition, subtraction, multiplication and division using large but user-friendly numbers!

And, within a few class periods, a teacher should see if he/she can guide the students toward the magic formula:

((Number of sides of the polygon) - 2 ) x 180 = the total number of degrees within the shape,

and so...

that dividend divided by the number of angles the shape has =

the measure of each 'corner' of a 'regular' polygon. 

Hope this helps.

And Good Luck with your demo lesson!