We Got A Problem #8: Super Ultra Mega Man!

The new toy craze is Mega Men, where kids buy a Mega Man in a box. The box won’t tell you which one it is, and kids open it up when they get home in hopes of getting one they don’t already have. (Sadly, eBay is not an option, since the only cool Mega Men are the ones still in their original packaging.)

There are ten toys in all, each equally likely when you buy a box. If you collect all ten, you can make Super Ultra Mega Man!

On average, how many Mega Men will you have to buy for your kid before he can finally collect them all?

(Stolen from PCMI 2007 Day 1 Problem #11: http://mathforum.org/pcmi/hstp/sum2007/morning/bowen/day1handout.pdf)

We Got A Problem #7: Drawin’ on the Multiplication Table

Draw a rectangle on top of the multiplication table.  The numbers in opposite corners of the rectangle form pairs.

What can you say about the products of the two numbers in each pair?  Explain.

What can you say about the sums of the two numbers in each pair?  Explain.  Ooh, that’s a good one.

See if you can find anything similar in the addition table.  Or, drop a comment with your favorite pattern from either table.

Wow. Just wow.

A colleague directed me to this material called “Memorize in Minutes”.  While it’s not completely devoid of math content, it … well, really, the last 80 pages should speak for themselves.

http://sessums.mysdhc.org/teacher/3922casey/Class_Homework/Memorize%20Times%20Tables%20in%20Minutes.pdf

Just to clarify, I do not think this is a very good way to present multiplication!

Mathematics of Game Shows, Plus!

Thanks to everyone who came out at 8 am on Thursday morning for “Mathematics of Game Shows, Plus”, the second installment of everyone’s favorite game-show-math-analysis talk.

The slides are here.  In particular, read about “Shell Game”.  The next post here will be an analysis and proof of the general case for that game.  Oh, and if you haven’t seen it yet, here are the slides from last year’s talk, with different games.

It was terrific seeing so many people at NCTM, new and old friends.  Best wishes for the rest of the school year.

Semi-Mathy Games

Here are some silly semi-mathy things I’ve done with my students either as bonus problems or fillers.  I cannot vouch for their pedagogical value.

• +1 or +3.  A bonus question appeared at the end of a test.  ”Write Free +1 and you’ll get a bonus point on the test, no questions asked.  Or write Go For +3 and if two-thirds of the class does it, all who do will get 3 bonus points.  Saying anything out loud voids the bonus for everyone.”  I was pretty surprised that in most classes, not enough students went for the +3.  One student wrote Free +3 and got nothing…
• Choose A or B.  In class, students were given this choice: “Write A and you’ll get a point for everyone in the class who writes A.  Write B and you’ll get twice as much as anyone who writes A.”  After a long debate, most students picked B.  Far fewer students picked B when the choices were revealed publicly instead of privately.  (Social commentary followed…)
• The probability you’ll get this question right.  Still one of my favorite bonus problems of all time, this came from my wife: “If train A leaves Denver at 50 miles per hour, and train B leaves Chicago at 60 miles per hour, what is the probability that you’ll get this question right?”  Students who wrote 1 or 100% were right, students who wrote 0 were wrong, and anything in between was resolved with a coin or random number generator.  (There’s a fun discussion over whether a student who writes 0 can be right by being wrong… but if they’re right then they’re wrong…)
• The dollar auction.  This one is evil.  See http://en.wikipedia.org/wiki/Dollar_auction for a full description.  In one class, a student successfully “escaped” by outbidding the other player significantly.   In another class, a third student jumped into the auction midstream allowing another to escape.  If I were still teaching, I would repeat the process using the bidding systems by Beezid and other “penny auction” sites (http://en.wikipedia.org/wiki/Penny_auction).  Hopefully by exploring these auction styles with students they will avoid being scammed for real money later in life.

What else?

 

We Got A Problem #6: Useless Rulers

By the way, there’s an interesting problem kicking around Dan Meyer’s blog.  But now for something completely different…

Take a ruler or any stick.  Mark it off into three equal parts.  Then, perhaps in a different color, mark it off again into five equal parts.  This divides the ruler into how many total parts?

(The mathematician’s version: Consider a number line from 0 to 1.  Mark all multiples of 1/3 and 1/5.  How many intervals are produced?)

Start over.  On another ruler, mark it off into four equal parts, then again into ten equal parts.  How many total parts this time?

On another ruler, mark it off into twelfths and thirtieths.  (This is close to what happens with inches and centimeters, but not quite.)  How many parts now?

If you mark off the ruler into M equal parts, then again into N equal parts, how many parts are there in total?

Can you think of ways to extend either this problem or the rectangle-diagonal problem?

We’d love for readers to be able to explore these problems, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to think about them, successful or not. If you’d like to provide a full solution, do so with a pingback to your own blog!

We Got A Problem #5: Tiled Pools

Lots of math books have problems about tiled pools, and they always seem to have a tiled border.  Area and perimeter, and all that.  (Never mind that the tiled border isn’t the perimeter, because of the corners…)

Picture a pool (okay, a rectangle) made of white square blocks, surrounded by a border of black square blocks.  Is it possible to use exactly the same number of white and black square blocks?  If so, find all the ways.  If not, prove it can’t be done.

Bonus points for creating an interesting 30-second video to motivate this exact question and no others!

More bonus points: think of some other questions that extend this problem, and let everyone know.  There are lots of interesting ones!

(Thanks to Matt Chedister and the PROMYS for Teachers crew for bringing up this interesting problem.)

We’d love for readers to be able to explore these problems, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to think about them, successful or not. If you’d like to provide a full solution, do so with a pingback to your own blog!