# Algebra Pinball (a Fun Way to Master Algebra I) & More

[Second of three talks as a Featured Speaker at CAMT 2015; updated August 2022.]

One hundred (of 164+) lessons in Carol's online sequenced Algebra I curriculum have timed exercises that challenge students to master important foundational skills—from adding signed numbers to factoring trinomials. With the click of a button, students get information on number of problems attempted, number correct, and average seconds per correct problem.

Add a bit of teacher coercion (it's homework) or incentive (extra credit), and you have an easy-to-grade way to improve basic skills—Algebra Pinball! Turn it into a class (or school) competition for even more fun. This talk also explores other fun classroom techniques and digressions.

## 60 Minutes, 60 Morsels

This is designed as a self-guided talk. Just click to open each morsel and then follow the suggested links. Feel free to email me with any questions or comments. Enjoy!

1. Welcome, everyone! I'm really excited to be here.
2. I'm very thankful to the conference organizers for inviting me. This is my second talk—they told me to make my three talks independent (not consecutive), so I'll be repeating some introductory material from my first talk.
3. If you went to my first talk, I certainly don't want you to be bored! So, here's something for you to do: (If you're a good multi-tasker, you can do the puzzle and listen to me at the same time!) Just read it and follow the directions!
1. I'll present 60 morsels in 60 minutes.
2. All my talks are online, so there's no need to take notes. If you're interested in (say) morsels 27 and 53, just jot those numbers next to this talk in your program guide.
1. My waaayyy too long name is Dr. Carol JVF Burns.
2. If you Google (just) Carol Burns, you won't get me! (I'm not the Australian actress.)
3. Here's my online vita.
I have a Doctor of Arts in Mathematics, which is a doctoral-level degree designed for effective teaching.
4. I've taught mathematics for about three decades at both the university and high school levels.
1. Raise your hand (or make some noise) if you have any association with each place I've taught
(if you taught or went there yourself, or if you know someone who teaches or goes there):
1. University of Massachusetts (Amherst): Bachelor of Science degree Magna Cum Laude in Civil Engineering
2. University of Oklahoma (Norman): Master of Arts in Mathematics
3. Idaho State University (Pocatello): Doctor of Arts in Mathematics
4. Miss Hall's School (private all-girl's day and boarding school, Pittsfield, Massachusetts): Algebra Pinball got started here, and was a BIG hit in the school.
5. Lenox Memorial High School (public high school, Lenox, Massachusetts): I was Chair of the Mathematics Deparment for a year, before I left to pursue math-on-the-web fulltime.
6. Northern Arizona University (Flagstaff): I moved closer to my daughter, who is pursuing her doctorate in neuro-linguistics at the University of Arizona. Then, I met my husband Ray through dancing, and moved to Tucson.
2. Scroll down to reveal the answer!

one from the same site.

(If you already heard the next couple morsels in my first talk, feel free to TRY to stop your foot from changing direction!)
1. You can type ‘Carol Burns math’ or ‘math cat burns’ in most any search engine.
2. Take the ‘One Mathematical Cat, Please!’ entry to get to my homepage.
3. You can type ‘Algebra Pinball’ in most any search engine.
1. Everything I offer online is completely free and immediately available. No logins. No downloads. No pop-ups.
2. I have 400+ sequenced lessons from basic arithmetic to calculus. About 100 of them have timing features—these form the basis for Algebra Pinball. We'll take a quick look at quite a few of them during this talk.
3. Every lesson is the result of decades of refining and tweaking to maximize understanding. I begin creating each lesson with notes that I've used for many, many years in my teaching.
3. To do just about anything in math, you need strong algebra skills. Students want loads of practice, with answers. Some like a bit of competition to spice things up. Teachers want students to understand ideas, not just memorize particular problem types; quizzes and worksheets at the touch of a button would be great, too. Put it all together, and you’ve got Algebra Pinball.

Algebra Pinball is a collection of timed online exercises—part of an entire online Algebra I course titled ‘One Mathematical Cat, Please!’ Students are challenged to improve their times on basic algebra skills. Teachers can create individual, class, or all-school competitions. There are many ways to proceed, but they all lead to the same place—mastery of important math skills.

4. I remember watching my students at Miss Hall's School trying to beat a really fast time. Their fingers were flying!! It made me think of someone using a pinball machine—and thus was born the name ‘Algebra Pinball’.
5. Let's try a first Algebra Pinball exercise, all together. We'll use a technique that I like to call a ‘choral response’ (choral, as in a singing group—a chorus). Here are the rules:

• I'll read the question aloud, followed by a few seconds of silence for thinking time.
• I'll start moving my hands in some weird way, to clue you in that you'll be responding soon.
• At the instant my hands HIT the top of my head, everyone calls out their answer at once.

Let's practice:

• Say ‘ONE’ at the appropriate time.
• Say ‘TWO’ at the appropriate time.
• Say ‘THREE’ at the appropriate time.

Now, let's try a real question. For this question, you'll ANSWER YES or NO:

If you Google ‘Carol Burns’, will you get me? (Do a choral response.)

6. There are several benefits to the Choral Response technique:
• It can get shy students to participate, without the anxiety of being called on individually.
• You (the teacher) listen to the responses. It's pretty easy to tell when you're getting a mixture of answers, that will require some clarification.
• You (the teacher) get some arm exercise!
• Students are usually amused by their teacher's weird movements.
• On the Algebra Pinball page, search for ‘Deciding if a number is a whole number, an integer, etc.’. (This is the fifth Algebra Pinball exercise in the series.) Click on this exercise. The link always opens in a new page.
• Quickly scroll through the page and review: real numbers, positive, negative, nonnegative, whole numbers, integers; numbers have lots of different names (Is $\,-\frac{18}{6}\,$ an integer? Yes!)
• Now let's do a few exercises, with the Choral Response, just to warm up.

Remember:
I'll read the question aloud.
Brief thinking silence.
Weird arm movements.
When my hands HIT the top of my head, you'll all answer together.
Close the window when you're done warming up.
7. Now, let's time ouselves on Deciding if a number is a whole number, an integer, etc.   Jump to the exercises. To time yourself, do the following:

• Click in the ‘Want to time yourself?’ box and type your name (do it).
• When you're ready, click the  ‘Start timing’  button. Then, press  ENTER  to get your first problem. (Do a few.)
• After each problem, you're updated on number correct and average speed per correct problem.
• Click the  ‘End timing’  button when done, and take a look at the timing summary sheet. Close it when you're done.
8. Now, look up the best time at Algebra Pinball. Masha Jones holds the best time, at 1.4 seconds per correct problem. If you want, try it at home, and see how close you can come to this fastest time!
9. A few of my Algebra Pinball exercises were created after I left Miss Hall's School. These typically have my own time listed in the Algebra Pinball chart (which should be easy to beat).

For example, go to the Algebra Pinball chart and search for ‘mental math’ (follow the link). This particular lesson has ‘guided exercises’ so students can practice some basic mental math strategies. Try a few!

• First guided exercise: no carrying
• Second guided exercise: turn it into a simpler problem and then adjust

The timed exercise at the bottom gives mixed practice. Try a few!

10. Another one of my newer exercises was inspired by a teacher, Jo Johansen, from Rutgers University. She wrote me a series of emails discussing ideas and language that she found really helpful to students—I liked them so much I got her permission to share them with the world.

Take a look at Relatively Prime Numbers and Related Concepts. The first three sketches show the ways that two numbers can relate to each other:

• ‘Strangers’ share no common prime factors;
here, the least common multiple (lcm) is the product.
• For ‘live-ins’, one number contains all the prime factors of the other; here, the lcm is the bigger number.
• It's the ‘overlapping’ case that requires some work in finding the lcm.

Jump to the exercises and try a few! (We'll time ourselves, and use a Choral Response.) Then, check out the fastest Algebra Pinball time—this one should be easier than others to beat!

11. Take a look at the top of the Algebra Pinball chart. Here are the requirements that I used for recording times in this chart:

• Do at least $\,20\,$ problems.
• The number correct must be at least $\,90\%\,$ of the number attempted. This prevents people from just clicking quickly until they get the easier problem(s).
• I had an ‘Algebra Pinball’ box on my desk. Students who beat a fastest time would print out the Algebra Pinball timing summary sheet and put it in this box. No more than one sheet per person per day.

Of course, you could choose different requirements for your own competitions.

Let's look at a bunch of exercises, and use them to point out a variety of features:

1. Many students have become too calculator-dependent; they need lots of practice with things like $\,3 + (-5)\,$ or $\,7 - (-1)\,$. I call numbers with an optional minus sign ‘signed numbers’. There are separate exercises for just addition and just subtraction, and then a mixed one.

Here's the subtraction exercise:   Subtraction of Signed Numbers Quickly scroll through the lesson, then try a few! I'll use a one-hand choral response.

Lots of my exercises (like this one) include variables—so an additional benefit of Algebra Pinball is that students improve their typing skills.

2. My lessons are carefully sequenced so that prerequisite skills are in place before presenting a new concept.

For example, when averaging two numbers, like $$\displaystyle\,\frac{4+(-9)}{2} = -2.5\,,$$ students must already be comfortable with adding signed numbers, dividing by $\,2\,$, and reporting in decimal form.

If you take the lessons in order, all these skills have been thoroughly practiced before getting to Average of Two Signed Numbers. Quickly scroll through the lesson, then note the instructions just above the exercise Here, students must report their answers in decimal form to have them counted as correct. Always take a look at the instructions—sometimes I allow different names for answers, sometimes not.

3. In the same Average of Two Signed Numbers lesson, scroll down past the timed exercise to the Concept Questions Exercise. Try a few! Many of my lessons have two different exercises:

• a timed exercise to develop particular skills (the Algebra Pinball exercise)
• another exercise to reinforce important concepts

Now, click for a worksheet. As with all my worksheets, they're different every time you click. (Close the worksheet, then click again.)

Worksheets are guaranteed to give you all different problem types. That is, if there are ten different kinds of problems (as there are here), then you'll get one of each type in a ten-problem worksheet.

4. Go to Interval and List Notation. I have several exercises involving sets and set notation.

In interval notation, parentheses are used for endpoints that are NOT included. Brackets (sometimes called ‘square brackets’) are used for endpoints that ARE included.

Here's a memory device I use: dirt collects in corners! Brackets have corners, and they correspond to filled-in dots (filled with dirt)!

I believe it's more fun to learn from someone you ‘know’. So, users have access to Fun Facts about me!

5. Go to Solving Simple Sentences by Inspection. Throughout my curriculum, I encourage students to think, not just memorize or manipulate symbols.

This exercise requires students to think about when an equation or inequality is true—long before they've learned standard solution techniques.

The goal is to get students thinking about truth—the property of being true or false. Some sentences have a unique solution; others have infinitely many (you'll choose just one).

Choral response should be fun with this one, because there might be LOTS of different answers! Try a few!

6. Go to Using Mathematical Conventions. One thing unique about my curriculum is that I put a heavy emphasis on the language of mathematics—students are taught how to teach themselves mathematics while they're learning it .

Consequently, some of the exercises are pretty non-traditional. Here, I want students to start getting comfortable with some conventions regarding the names of variables:

• Numbers are usually named with lowercase letters.
• Sets are usually names with uppercase letters.
• A variable with universal set $\,\Bbb R\,$ (or, any interval of real numbers) is most likely to be named with a letter near the end of the alphabet, particularly $\,t\,$, $\,x\,$, or $\,y\,$.
• A variable with universal set $\,\Bbb Z\,$ (or, any subset of the integers) is most likely to be named with a lowercase letter near the middle of the alphabet; particularly $\,i\,$, $\,j\,$, $\,k\,$, $\,m\,$, or $\,n\,$.
Try a few!
7. Go to Practice With the Form $\,a\cdot\frac{b}{c}$. How many times have you heard students lament that their answer doesn't match the one in the back of the book—and they know they did it right!? Often, they just need a simple reminder—that numbers have lots of different names!

In this exercise, students have to decide if two fractional expressions are always equal or not. I want students to be comfortable recognizing fractions in many common forms.

Pause at the top of the lesson and look at some of the many ways the fraction $\displaystyle\,a\cdot\frac{b}{c}\,$ might be presented. Then, try a few!

(By the way, my third talk focuses on the language of mathematics.)

8. Go to Divisibility Equivalences. In my curriculum, I talk a lot about ‘equivalence’. Two mathematical sentences are equivalent when they always have the same truth values—they're true at the same time, and false at the same time. Equivalent sentences can be used interchangeably.

Just as ‘numbers have lots of different names’, it is also true (via equivalence) that ‘sentences have lots of different names’. Students need lots of practice recognizing equivalent sentences.

Notice that $\,2\,$ goes into $\,6\,$ evenly. How else might we commonly say that?

• $\,6\,$ is divisible by $\,2\,$
• $\,6\,$ is a multiple of $\,2\,$
• $\,2\,$ is a factor of $\,6\,$

Jump to the exercises, and try a few!

9. Many of my exercises have dynamic graphics, made possible by JSXGraph. (Open the link, scroll down a bit, and play with the ‘packing’ graphic on this page, as an example of what JSXGraph can do.)

• It's free!
• It works really well.
• The source code is compact so it doesn't slow down page loads.
• It works in all major browsers and platforms.

In Solving Equations of the Form $\,xy = 0\,$, students readily see that the zeros of the function are the solutions of the equation. Try a few!

10. Let's take a look at Classifying Units as Length, Time, Volume, Weight/Mass. My HTML web pages typically give an ‘in-a-nutshell’ lesson. Want a more traditional-looking textbook?

On the lesson page, search for ‘full text’ and follow the link. Scroll through the pdf and notice the fun, alternative page numbers.

There are ‘active reading’ exercises scattered throughout the lesson, with answers at the end. If desired, you can file-print this pdf file.

11. Let's take a look at Multi-Step Exponent Law Practice. Different levels of challenge are often provided for critical skills. For learning the exponent laws, I have exercises that test each law individually, then all mixed up (but still using only one exponent law), and finally this mixed exercise for mastery.

Generate a worksheet, so you can see the variety of problems.

12. Do a couple exercises from each of these, to see the skill level build:

13. Go to Simplifying Expressions Like $\,-a(3b-2c-d)\,$. Extra spaces don't usually make a difference when keying in answers. Sometimes, order does matter, though!

If you know you're giving a correct answer, and it keeps saying ‘Sorry’, then be sure to read the instructions above the exercise!

For students, this kind of ‘persnickety-ness’ is not necessarily bad, since it reinforces a correct, efficient approach.

Try a few exercises. Put in some extra spaces. Get one ‘wrong’ by putting the variables ‘backwards’.

14. Try some exercises from each of the lessons below, to see what's available for FOIL practice (and more).

Take a look at the end of the concept discussion, just above the exercises. There, you often find additional instructions about how to key in answers.

15. Go to Solving More Complicated Linear Equations with Integer Coefficients. Take a look at the top, just before the concept discussion. There, you're often pointed to prerequisite (or simpler) related exercises.

Also (in the exercises) notice how you can hide the graph (it's a toggle). Click quickly until you get a simple equation that's easy to solve mentally. Here, answers can be given in fractional or (exact) decimal form.

Suggestions for Using Algebra Pinball In and Out of the Classroom:
1. Do you have students seeking a better grade on a returned quiz or exam? Scan through their assessment, and locate skill weaknesses that contributed to their mistakes (e.g., faulty arithmetic with signed numbers, or adding fractions incorrectly).

Guide them to appropriate Algebra Pinball exercise(s) to strengthen the skills that are hindering their progress. Give them an opportunity to earn back points by passing in timing summary sheet(s) that indicate mastery.

2. Want an easy-to-grade homework assignment? Have students pass in an Algebra Pinball timing summary sheet with a required time. (They might—for example—take a snapshot of their timing information and show it to you.)

3. [August 2022 update: My updated lessons don't open a separate window for the timing information. Students can just take screenshots of successively better timing info summaries.]

To save paper, have students print the timing summary sheet to a PDF file and save on the desktop. Keep saving it ‘over itself’ as faster times are achieved. Only print out a hardcopy with the required (or best) time!

4. LISTEN-THINK-SHARE: This is good for exercises which have single, well-determined answers. The teacher has the responsibility to listen carefully, and see if mostly correct answers are being called out.

Project an Algebra Pinball exercise:

• LISTEN: Teacher clicks for a question, and reads it aloud. Students listen.
• THINK: Silence while students think about the answer.
• SHARE: Use the Choral Response. On signal, all students call out their answer at once.
5. RANDOM GROUPINGS: (For when you need to divide students into groups—and, this is great when students need to wiggle!)

• Ask students to (quickly and efficiently) order themselves around the edge of the room. Use creative/fun criteria: # of magnets on home refrigerator; length of right thumb; last digit of cell phone # (Fun project: create a class poster with lots of good grouping ideas.)
• Want two big groups? Have them count off. Then, separate into even/odd; or, divide total by 2.
• Want four groups? Count off 1-2-3-4 (repeat as needed); 1 goes here; 2 goes here, and so on.
• Use lots of different ordering/choosing methods, to keep students guessing!
6. TWO LINE PRACTICE: This is good for exercises which might be correctly answered in several different ways.

• Use random groupings (above) to divide the entire class into two lines.
• Teacher clicks for a question, and reads it aloud.
• First student in line must answer the question. They can ask the next person in line (and only this person) for help.
• Student rotates to end of line after their question.
• If question is missed, other team gets a chance (collaborative team effort, but first person gives the answer).
• Take turns, switching from team to team; teacher keeps score.
• If you're using the Quick Quiz method (morsel 49), each person on the winning team might earn $\,1\,$ QQ point! (Some incentive makes the game more fun and increases student focus.)
7. Let's try ‘Two Line Practice’ with Identifying Common Factors. First row, please stand up! Order yourselves by number of siblings (you can interpret this however you like): zero goes here—go!! When done, check the ordering: go down the line, and state many siblings you have.

8. Please count off: one-two-one-two (and so on). Ones line up over here; twos over there.
• We'll skim through the examples so you know how to answer.
• I'll click for a question, and read it aloud.
• First student in line answers. Pretend you don't know the answer—consult with the next person in line!
9. Over to the next team! Go back and forth a few times.

10. Get one wrong, on purpose. The same question then moves to the other team: collaborative effort, but the first person in line gives the answer.

Thanks, everyone! You can sit down now!

11. Periodically, students want extra credit to boost their grade. Algebra Pinball can assure that such extra credit is truly earned by a motivated student, and that they benefit mathematically in the process.

Pick an important skill and an average time per correct problem that is a stretch to the individual. Offer extra credit for passing in timing summary info showing they achieved that time.

12. Remind students that a ‘pop quiz’ rewards those who are focusing. You can use a pop quiz to replace a poor quiz or homework grade (if it helps). That way, it can only help a student—not hurt.

At a moment's notice, you can project a worksheet (questions only). Read each question; have students write in their answers.

Here's a form you can print: Web Exercise Practice Sheet. You might want to keep a pile of these forms in your desk drawer.

Exchange papers to grade. Have graders write in correct answers (as needed).

13. Why do Algebra Pinball? (Review and elaborate!)

• It's a GREAT review of math skills.
• It's FUN. (Some people even say it's addictive.)
• Students love the bonus points they (might) earn by setting a new record.
• Students love to see their name on the web.
• It's cool to see Algebra Pinball come up when someone (like a college admissions person!) searches for your name on the web.
• Teachers love getting students, friends and family doing math.
• You can have an Algebra Pinball all for yourself.
• You can have an Algebra Pinball for a family competition.
• You can have a different Algebra Pinball for each of your math classes.
• You can have an Algebra Pinball for the entire school.
• You can just keep a paper/pencil copy of Algebra Pinball (the low-tech version!) and update scores as they come in.
• Your Department Chair/Principal/Head of School will love your initiative in promoting mathematics.
14. Each student can maintain their own personalized ‘Algebra Pinball’ chart using this form: Individual Algebra Pinball.

• File-save this page as (say):   algebraPinball.htm   The ‘htm’ extension is required, so it will be recognized by your computer as an HTML document.
• Personalize the file: Open it with a text editor (e.g., Notepad on a PC, or TextEdit on a MAC). The file is richly commented to tell you how to make changes. Be sure to save any changes.

You'll learn a bit of HTML (Hyper Text Markup Language) while you're learning math!

15. Many teachers have their own web pages. You can customize a copy of the Individual Algebra Pinball form for each of your classes, or for the entire school.

For example, change the column heading to:   ‘Best Time at Miss Hall's School’.   You'll probably want to let people know how often you'll update the document with faster scores: e.g., every Friday afternoon.

Here are two classroom techniques that I would never teach without:

1. I've used the ‘Quick Quiz’ technique for decades. One of my most common evaluation comments over the years? ‘LOVE THE QUICK QUIZZES!’

• A pile of scrap paper sits by the door; students take a sheet as they enter. I use one-side-blank 8.5" x 11" paper, cut in half each direction. (Make sure there's no sensitive material on the printed side!)
• As the ‘bell rings’, a Quick Quiz (QQ) goes on the front board. One minute max—often it's only about 15 seconds.
• Establish an efficient pass-in-paper method (e.g., each row to a specified side; they pass forward).
• Quick Quizzes are not returned—they're graded and discarded. Bonus: they can be used for attendance!
• There are absolutely no QQ make-ups.
• Quizzes are graded either with a ‘$\,1\,$’ (one point) or an ‘ N ’ (Not completely correct). Points accumulated are bonus points.
• The QQ gets students to class on time, and they love the tangible daily benefit for having studied.
2. Each student gets a ‘Grade Sheet’ that they maintain throughout the course. It's simple math: their points versus total points possible.

Whenever an assessment is returned, students look to the side board to update their Grade Sheets. They come to appreciate how a good effort raises their grade; as the course progresses and more points are amassed, they understand the increased difficulty in making a change.

This simple ownership and clarity of one's grade has been an outstanding motivational tool.

More fun in and out of the classroom:

1. Anti-glaze is a really fast get-the-blood-flowing game. It gets students focused when class energy has waned.

Needed: $1$ Koosh ball

• Each student thinks of a number between $\,1\,$ and $\,x\,$.
• [Teacher counts off, from $\,1\,$ to $\,x\,$.]
• When you hear your number, stand up.
• Throw the Koosh to someone else who is standing.
• After you've thrown the Koosh, sit down.

Let's try it! (I just happen to have a Koosh ball with me!)

2. For many years at Miss Hall's School, we maintained an extremely popular school-wide PuZzLeR board:

• Not just math! Word puzzles, riddles, optical illusions, more.
• Math teachers took turns coming up with the puzzles.
• A new puzzle every week. (Solutions from prior week were posted when the new puzzle went up. Sometimes we posted several different solutions.)
• [optional] Weekly random drawing from participants: They got a homemade batch of brownies!
3. If you haven't ever seen the Proverbidioms posters by T.E. Breitenbach, then you're in for a treat!

I laminated the sayings and put tiny pieces of velcro near the graphical depiction of each proverb/idiom. If students finished a task early (or in off-class hours), they could choose a slip, find the associated picture, and stick it on.

It was a big hit! I think anything that gives students practice being attentive to detail is beneficial, and this is really fun.

4. I mentioned Number Gossip in my first talk, and it's worth repeating here. This is one of my favorite sites!

Type in a number, and it tells you all kinds of cool things about it. My family members and friends are frequently subjected to fun facts on birthdays and anniversaries. The results make fun cake decorations.

For example, at age $\,56\,$ I was (among other things):

• abundant:
The sum of all the proper divisors of $\,56\,$ is more than $\,56\,$; $$1 + 2 + 28 + 4 + 14 + 7 + 8 > 56$$
• a lazy caterer:
$\,56\,$ is the greatest number of pieces a pizza can be cut into, with $\,10\,$ straight slices
• odious:
There are an odd number of ones in the binary representation of $\,56\,$:   $56 = 111000_2$
• practical:
Every number less than $\,56\,$ is a sum of distinct divisors of $\,56\,$. For example: $$55 = 1 + 4 + 8 + 14 + 28$$
5. As I mention in my Teaching Philosophy, I believe it's more fun to learn from someone you know.

I let students get glimpses of who I am as a human being. When a story problem evokes a memory, I may share it with the class. They're likely to find out that I do some of my best thinking on long walks, and that I believe in moderation in all things (even moderation in moderation).

It's my responsibility to keep digressions short—but when class energy is low, these little fun facts can again restore it to full speed.

So, here's one of my funniest (scariest?) teaching moments. I regularly lower myself on bent knees to write near the bottom of the board. One day, upon doing so, I heard a very loud ‘rrrriiiippppp’ and felt the back seam of my pants open up. I immediately turned so my back was against the board, walked with my back against the wall to the nearby door, zipped into my office, tied a sweater around my waist, and returned to finish the discussion!

6. This is a fun story that I cherish.

One day, while teaching at Miss Hall's School, a frantic-looking student knocked on my door and said ‘Ms. Walters wondered if you could come with me. We're having a math emergency.’

Before I was even fully in the room, I figured out what the problem was. (It had arisen from that nasty fact that the trigonometric functions aren't one-to-one, so they don't have ‘true’ inverses.) Years later, I still had students reminding me of that day when—in a matter of seconds—I had solved their math emergency!

7. I was writing this morsel close to ‘Super Pi Day’ (3/14/15), so it made me recall an activity that my students really enjoyed.

It comes from this Mudd Math Fun Fact: the Buffon Needle Problem.

Many classroom floors are tiled with one foot tiles.

• Bring in a bunch of thin pieces of wood that are one foot long.
• Focus attention on one direction of lines on the floor (and ignore the lines going the other direction).
• Tell students that (calculus can be used to show that) the probability that a dropped stick will cross a line is $\displaystyle\,\frac{2}{\pi}\,$.

Thus, if $\displaystyle\,\frac{\overbrace{\text{# crossings}}^{:=C}}{\underbrace{\text{# drops}}_{:=N}}\,$ is the fraction of times a stick crosses the line, we should have $\displaystyle\,\frac{C}{N}\approx \frac{2}{\pi}\,$ for a large number of drops.

• We can use this to approximate $\,\pi\,$:   $\displaystyle\pi\approx \frac{2N}{C}\,$.
• Drop sticks!
• After about every ten drops, have students go up to the board, record their drops, and update the cumulative amounts:
• In the chart, $\,N\,$ is the cumulative number of drops.
• In the chart, $\,C\,$ is the cumulative number of crossings.
• Hmmm ... is that last column approaching $\,\pi\,$?
 # of drops $N$ # of crossings $C$ $\displaystyle\frac{2N}{C}$ $10$ $10$ $3$ $3$ $\displaystyle\frac{2\cdot 10}{3}\approx 6.7$ $10$ $20$ $8$ $11$ $\displaystyle\frac{2\cdot 20}{8}\approx 5.0$ $10$ $30$ $6$ $17$ $\displaystyle\frac{2\cdot 30}{17}\approx 3.5$

Closing up:

1. Please don't hesitate to email me with questions/comments on any of my talks or anything on my web site. I love to hear from my users!

Over the years, comments from users have made my site better for everybody.

2. My last talk will also consist of $\,60\,$ morsels.

You'll find out where the name ‘One Mathematical Cat’ comes from. I hope to give you an entirely new perspective on:

• simplifying expressions (numbers have lots of different names)
• solving equations/inequalities (sentences have lots of different names)