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.

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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!

We Got A Problem #4: Chips!

Three problems about chips in a bag…

  1. A bag contains four green chips and three white chips.  You win if you can pull all the green chips before pulling all the chips of a different color.  Find the probability of winning the game.
  2. Now two red chips are added to the bag.  The rules are the same.  What is the probability of winning this game?
  3. On “The Price Is Right”, the “3 Strikes” game is played with five white chips and one red chip.  White chips are not replaced, but the red chip is.  What is the probability of pulling all five white chips before pulling the red chip for the third time?  (In the real game, after pulling a white chip the player must make a decision that might return the white chip back to the bag, so the probability of winning the full game is significantly lower.)

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 #3: ABACAB

The Genesis song “Abacab” was named after the original order of its sections: A-B-A-C-A-B.  Poetry rhyme schemes use the same convention: the rhyme scheme for a limerick is AABBA.

How many rhyme schemes are there for six-line poems?  ABACAB is one of them.  So are AAAAAA and ABCDEF.

(The fine print: if a new line doesn’t rhyme with any previous line, it must use the next available letter in the alphabet.  BAAB is wrong: it should have been ABBA. ABACAF is wrong too.  The first letter is always A.)

We’d love for readers to be able to explore this problem, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to approach the problem, 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 #2: The current time is… 120 degrees

The hour, minute, and second hands of a continuously-moving clock all point the same direction at 12 noon.  But when are they equally spaced around the clock face?

In attacking this problem, Kathy from Pittsburgh pointed out that at 12:20:40, the hands almost form three 120-degree angles.  But they don’t, because the hour hand has moved a little passed 12 on its way to 1, and the minute hand has moved a little too.

So, when do the three clock hands form three 120-degree angles?

We’d love for readers to be able to explore this problem, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to approach the problem, 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 #1: 6/5/11

Today is June 5, 2011, or 06-05-11 in shorthand (05-06-11 in many countries).

Interestingly, 06 + 05 = 11.  How many more times in this century will it happen that the month number plus the day number equal the last two digits of the year number?

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