Making Perfect Circles by Using Coffee Filters

When I teach angles or the properties of circles, I find that most children have difficulty cutting out a true circle (even with a blackline).  I have resorted to purchasing cheap coffee filters (not the cone shaped ones) and ironing them flat. You can iron several filters at one time, and once they are ironed, they form excellent ready-made circles. Here are some of the ways you can teach angles using these circles.

Writing Formulas on the Coffee Filter Circle

1. Introduce the fact that each and every circle contains 360 degrees.
2. Have the students fold their coffee filter in half. Discuss that this is a straight angle. Ask, “How many degrees does it contain if it is one-half of a circle?” (180 degrees)
3. Have the students fold the coffee filter one more time, into fourths. Talk about this angle being called a right angle and that it contains 90 degrees. Ask, "What fractional part of a circle is this?"
4. Have the students use this fourth of a circle to locate places in the classroom where it will fit (e.g. the corner of their desk, a corner of a book, a corner of the board).
5. Explain that these corners are right angles and without right angles, we would live in a crooked world. Nothing would be straight!
6. With older students, have them write the parts of the circle and the formulas needed for solving problems about circles on the coffee filter circle.

Linking Math and Literature for Older Students

Read Sir Cumference and the First Round Table (A Math Adventure) by Cindy Neuschwander. This is a story about a clever knight of King Arthur’s named Sir Cumference. By using ideas offered by the knight’s wife, Lady Di of Ameter, and his son, Radius, King Arthur finds the perfect shape for his table. Basic geometric vocabulary involving circles (circumference, radius, and diameter) is introduced.

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Want more hands-on ideas for teaching angles? Check out Angles: Hands-On Geometry Activities.

Celebrate March 14th (Pi Day) Using Crossword Puzzles to Review Math Vocabulary

March 14 is Pi Day because March is the third month, and with 14 as the day, we get the first three digits of pi - 3.14! On Pi Day, nerds, geeks, and mildly interested geometry students alike come together and wear pi-themed clothing, read pi-themed books and watch pi-themed movies, all the while eating pi-themed pie. 

Pi is an irrational number that approximately equals 3.14. It is the number you get if you divide the circumference of any circle by its diameter, and it's the same for all circles, no matter their size. You can estimate pi for yourself by taking some circular things like the tops of jars or round plates and measuring their diameter and their circumference. Then divide the circumference by the diameter, You should get an answer something like 3.14. It should be the same every time (unless you measured wrong).  In other words, π is the number of times a circle’s diameter will fit around its circumference

Actually, 3.14 is only approximately equal to pi. That's because pi is an irrational number. That means that when you write pi as a decimal it goes on forever and ever, never ending. (It is infinite.) Also, no number pattern ever repeats itself.

Usually in math, we write pi with the Greek letter π, which is the letter "p" in Greek. You pronounce it "pie", like the pie you eat for dessert. It is called pi because π is the first letter of the Greek word "perimetros" or perimeter.  What is interesting is that in the Greek alphabet, π (piwas) is the sixteenth letter; likewise, in the English alphabet, the letter "p" is also the sixteenth letter.

But hold your horses!  Whether you like it or not, pi is everywhere. Here are a few more places it has popped up:

1. The main character in the award-winning novel (and 2012 film) Life of Pi nicknames himself after π. 
2. A circular room in the Palais de la Découverte science museum in Paris is called the pi room. The room has 707 digits of pi inscribed on its wall. (The value of pi has now been calculated to more than two trillion digits.)
3. In an episode of Star Trek: The Original Series, Spock commands an evil computer to compute π to the last digit which it cannot do because, as Spock explains, “The value of pi is a transcendental figure without resolution.”
4. Pi is the secret code in Alfred Hitchcock’s Torn Curtain and in The Net starring Sandra Bullock.

Here is more arbitrary information related to pi that I found interesting.

1. If you were to print one billion decimal values of pi in an ordinary font, it would stretch from New York City to Kansas (where I live). 

2. 

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   3.14 backwards looks like PIE. 
3. "I prefer pi" is a palindrome. (It reads the same backwards as forwards)
4. Albert Einstein was born on Pi Day (March 14) in 1879.

All this information about pi and circles can be found in a Pi Day Crossword. It includes two different math crossword puzzles about Pi Day and features 20 words that have to do with pi or circles. One crossword includes a word bank which makes it easier to solve while the more challenging one does not. Even though the same vocabulary is used for each crossword, each grid is laid out differently. Answers keys for both puzzles are included.

By the way, notice my "handle" of Scipi.  The Sci is for science (what my husband teaches) and the pi is for π because I teach math.

Using Math Humor in Geometry

 I 've been using Pinterest (as well as Tailwind) for as long as I can remember, and I love it. Not only do I post many resources and teaching ideas there, but I learn so-o-o much. For example, I learned how to pack one suitcase with enough stuff for a week. (My husband is thrilled with this one.) I also learned that when you fry bacon, to make a small cup out of aluminum foil; pour the bacon grease into it; let the grease harden; then close up the aluminum cup and toss it into the trash. That is one I use all of the time!

On my Pinterest account I have a board entitled Humor - We Need It! I post many math cartoons or humorous sayings there. My favorite subject to teach my college remedial math students is geometry, and I have plenty of corny jokes that I intersperse into my lessons. Here's one.

What did the little acorn say when it grew up? Gee- I'm - A - Tree! (Geometry)

Or about this one?

What did the Pirate say when his parrot flew away? Polly-Gone (Polygon)

Here are some other geometry funnies from Pinterest.

Try placing a riddle or cartoon in the middle of a test.  I often do, and I know exactly where the students are by their laughs.  It helps them to relax and maybe get rid of those mathphobic tendencies.  I hope these math cartoons brought a smile to your face.  Have a great week of teaching!

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You might also like Geometry Parodies, a math handout that includes 20 unusual definitions of geometry terms. Each definition is a play on words or a parody. Twenty-six geometric terms that are possible answers are listed in a word bank, but not all of the words are used in the matching exercise. An answer key is included.

In Math, what is the Difference Between Drill and Practice?

When I was a kid, one of the things I dreaded most was going to the dentist. Even though we were poor, my Mom took my brother and me every six months for a check-up.  Unfortunately, we didn’t have fluoridated water or toothpaste that enhanced our breath, made our teeth whiter, or prevented cavities.  I remember sitting in the waiting room hearing the drill buzzing, humming, and droning while the patient whined or moaned.  Needless to say, I did not find it a pleasant experience.

I am troubled that, as math teachers, we have carried over this idea of drill into the classroom. Math has become a “drill and kill” activity instead of a “drill and thrill” endeavor.  Because of timed tests or practicing math the same way over and over, many students whine and moan when it is math time.  So how can we get students to those “necessary” skills without continually resorting to monotonous drill?

First we must understand the difference between drill and practice.  In math drill refers to repetitive, non-problematic exercises which are designed to improve skills (memorizing basic math facts) or procedures the student already has acquired. It provides:

1)   Increased proficiency with one strategy to a predetermined level of mastery. To be important to learners, the skills built through drill must become the building blocks for more meaningful learning. Used in small doses, drill can be effective and valuable.

2)   A focus on a singular procedure executed the same way as opposed to understanding.  (i.e. lots of similar problems on many worksheets)  I have often wondered why some math teachers assign more than 15 homework problems.  For the student who understands the process, they only need 10-15 problems to demonstrate that.  For students who have no idea what they are doing, they get to practice incorrectly more than 15 times!

Unfortunately, drill also provides:

 3)  A false appearance of understanding.  Because a student can add 50 problems in one minute does not mean s/he understands the idea of grouping  sets.

 4)  A rule orientated view of math.  There is only one way to work a problem, and the reason why is not important!  (Just invert and multiply but never ask the reason why.)

5)   A fear, avoidance, and a general dislike of mathematics. A constant use of math drills often leaves students uninterested.

On the other hand, practice is a series of different problem-based tasks or experiences, learned over numerous class periods, each addressing the same basic ideas. (ex. different ways to multiply)  It provides:

1)   Increased opportunity to develop concepts and make connections to other mathematical ideas.  (i.e. A fraction is a decimal is a percent is a ratio.)

2)   A focus on providing and developing alternative strategies.  My philosophy, which hangs in my classroom, is: “It is better to solve one problem five ways than to solve five problems the same way.”  (George Polya)

3)   A variety of ways to review a math concept.  (ex. games, crosswords, puzzles, group work)

4)   A chance for all students to understand math and to ask why. (Why do we invert and multiply when dividing fractions?) 

5)   An opportunity for all students to participate and explain how they arrived at the answer. Some may draw a picture, others may rely on a number line, or a few may use manipulatives. Good practice provides feedback to the students, and explains ways to get the correct answer.

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Let’s look at it this way. A good baseball coach may have his players swing again and again in the batting cage. This drill will help, but by itself it will not make a strong baseball player whereas practicing hitting a ball with a pitcher requires reacting to the different pitches with thought, flexibility, and skill.

I am of the opinion that drill should not be omitted from the math classroom altogether.  Basic math skills should be automatic because being fluent in the basics makes advanced math easier to grasp.  There is a place for drill; however, its use should be kept to situations where the teacher is certain that is the most appropriate form of instruction.  Even though practice is essential, for math it isn't enough. If understanding doesn't come, practice and drill will only leave a student with disjointed skills. If we want to produce strong mathematicians, we must focus on the BIG conceptual ideas through practice in problem-based lessons. We must present ideas in as many forms as we can so that students will go beyond rote drill to insight.

If you are interested in sharing this with your staff, colleagues or parents, check out the EDITABLE power point entitled: Drill vs. Practice.