Wednesday, April 30, 2014

115: SemiCircles

If purple & green are semi-circles, what do you know about the sum of their areas?

— (@Desmos) April 22, 2014

Tuesday, April 29, 2014

114: Square Root Exponents

We all know that adding/subtracting exponents corresponds to multiplying/dividing the terms, like this:

$x^4 * x^7 = x^{11}$

$\dfrac{x^{14}}{x^9} = x^5$

Then negative exponents logically followed: $x^{-7} = \dfrac{x^2}{x^9}$

Then $\dfrac{x^3}{x^3} = x^0 = 1$ logically followed that. 

Additionally, multiplying/dividing the exponents relates to powers/roots

$ {x^4}^2 = x^{4*2} = x^8$

$\sqrt{x^6} = x^{6/2} = x^3$

So a fractional exponent means a radical, depending on the denominator of the exponent.

So here's my question:

What should we think about $x^{\sqrt{2}}$

How should we interpret that?

Monday, April 28, 2014

113: The Address

How long would it take everyone in your school to recite it?

272 words (or 269, depending on how you count, and which version you use.)

On November 19th Lincoln traveled to Gettysburg to dedicate the new Union Cemetery

The featured speaker was Edward Everett of Massachusetts, a diplomat, clergyman, and celebrated orator. The president had been invited almost as an afterthought to offer a few appropriate remarks. Everett spoke for not quite two hours.
Then Lincoln rose. The local photographer took his time focusing; presumably the president could be counted on to go on for a while ... but he spoke just 269 words.

He started off by reminding his audience that just 87 years had passed since the founding of the nation and then he went on to embolden the Union cause with some of the most enduring words ever spoken.

Lincoln was heading back to his seat before the photographer could open the shutter.

He felt that he had failed, that it was a poor speech, that people didn't like it. It was so brief, less than two minutes, he felt that he had failed. His friend, Ward Lehman, was sitting next to him on the stand when he sat down and there was just a sprinkling of applause.

He said "Lehman, that speech won't scour." That's what you say about the plow in the prairies when the mud won't come off it."
"The cheek of every American must tingle with shame when he reads the flat, silly, dish watery utterances of the man who has to be pointed out to intelligent foreigners as the president of the united states."  Chicago Sun-Times
"Dear Mister President, I should be glad if I could flatter myself that I came as near to the central idea of the occasion in two hours as you did in two minutes." - Edward Everett
President Lincoln delivered the 272 word Gettysburg Address on November 19, 1863 on the battlefield near Gettysburg, Pennsylvania.
"Fourscore and seven years ago our fathers brought forth, on this continent, a new nation, conceived in liberty, and dedicated to the proposition that all men are created equal. Now we are engaged in a great civil war, testing whether that nation, or any nation so conceived, and so dedicated, can long endure. We are met on a great battle-field of that war. We have come to dedicate a portion of that field, as a final resting-place for those who here gave their lives, that that nation might live. It is altogether fitting and proper that we should do this. But, in a larger sense, we cannot dedicate, we cannot consecrate—we cannot hallow—this ground. The brave men, living and dead, who struggled here, have consecrated it far above our poor power to add or detract. The world will little note, nor long remember what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great task remaining before us—that from these honored dead we take increased devotion to that cause for which they here gave the last full measure of devotion—that we here highly resolve that these dead shall not have died in vain—that this nation, under God, shall have a new birth of freedom, and that government of the people, by the people, for the people, shall not perish from the earth."

Sunday, April 27, 2014

112: Calvin, Mind Blown

Can you help Calvin understand this?

Do you know what a record looks like?

Saturday, April 26, 2014

111: Dirt Triangle

Taken from David Cox, changed just a little:

While I was out, I left a few distance/rate/time problems for students to solve.  Upon my return, I was asking students about the problems and many students had similar responses.
S:  "The Dirt Triangle."
Me:  "?"
S:  "Look, Mr. Cox it's like this...
"...You cover up the one you're looking for and if the other two are next to each other, you multiply.  If one is above the other, you divide."

Me:  "Really? That's strange.  I never learned the Dirt Triangle. I learned ... The Turd Triangle

S1: "No, that won't work. That's not what he told us."
S2: "He said it didn't matter how we wrote it."
Me: "So which is it; does one work or are they the same?  Make your case and be ready to defend it."

Friday, April 25, 2014

110: Non-standard Dice

What number of sides are possible for dice?

Can these creations be created in such a way that each side is equally probable?

Some possibilities if you're interested:

Platonic Solids, and 2 decahedra.

1-sided die = moebius strip.

Thursday, April 24, 2014

109: Powers of Two

If the tiles always pop up as 2s, how many moves has this player made?

Wednesday, April 23, 2014

108: How many miles is good enough for a bus?

From the walls of the DC Metro, this ad:

How about that 8,260 miles? No, we can't talk about shoes.


Tuesday, April 22, 2014

Monday, April 21, 2014

106: Sierpinski

This time, YOU ask a question. Put it in the comments.


Go 3D

Sunday, April 20, 2014

105: Linear or Quadratic?

Is the expression  x² + 6x = x² - 8x + 4 "quadratic" or linear?

Thursday, April 17, 2014

102: Watermelon

From Richard Wiseman:

I have a 100 pound watermelon laying out in the sun. 99% of the watermelon’s weight is water. After a few hours 98% of the watermelon’s weight is water. How much water evaporated?

The answer, slightly obscured for the sake of any who want to solve on their own, is 1100102 pounds – but can you explain why?

Wednesday, April 16, 2014

101: Gun Deaths in Florida

You can take it from here ...

Tuesday, April 15, 2014

100: Combinations. One more Twist.

We've seen this a couple of times now.

What if I had a white marker, one that didn't show, but still held a place?
How will that affect things?

Image from Mr. Stadel.

99: More combinations

Blue, green, yellow, and red ... We seem to be all set.

What if I get three black markers out of the drawer?
How will that affect things?

Image from Mr. Stadel.

98: Combinations!

Blue, green, yellow, and red ... We seem to be all set.

Now, what about the black, purple, and orange ones?


Image from Mr. Stadel.

Monday, April 14, 2014

97: n!

It's a straightforward question today. Calculators are allowed for this question because we here do not feel that six-digit multiplication is worth the effort. I mean, we know you COULD do it, but that's what a tool is for, right?
For any confused students, this problem is not suggesting that 851 is special! That's factorial notation, n!, meaning multiply the integer n by all the positive integers less than n.  (Yeah, n must be a positive integer to begin with.)

For example, 6! = 6*5*4*3*2*1 = 720

For completeness, 0! = 1

Thanks to Darren, Right on the Left Coast, for this one.

Sunday, April 13, 2014

96: Flower Petals

What fraction of this is yellow?

95: Circumcenter

A rule from geometry says that three points determine a circle. 

Make a triangle with the three points. The perpendicular bisectors of the three sides will meet at the center of the circle. (Of course, you'll only need two but the third is confirmation.)

The points are A( 20, –18 ); B( 8, 18 ); C( –16, –6)

Graph paper, pencil and straightedge only.

Saturday, April 12, 2014

94: Related Rates

Describe how the two trucks must work together throughout this turn.

Can the rear truck get whipped?

93: Diagonals and Patterns

Step 43?

Each vertex of a n-sided polygon connects to how many other vertices? And count each diagonal only once ...

Thursday, April 10, 2014

91: One-Quarter Times One-Quarter

What is one-quarter times one-quarter?

What if we look at things this way? Do we get the same results if we consider these as overlapping squares or as L-shaped pieces?

Tuesday, April 8, 2014

89: Approximately Three-fifths

Building on the previous few days ... if all vertical lines are equally spaced,

Which is closest to being three-fifths shaded?

88: Truly Bad Graphics

This graph is biased in what way?
How could you change it to bias it the other way?

Sunday, April 6, 2014

86: Three-Fifths

Which represents three-fifths?

Twitter comment:

85: Thirds

Which of these calculations was done correctly?

84: Choices, choices ... Lottery Winnings

You won the Lottery, but not the top prize. Instead, you get to choose which way you get paid your winnings (and we're going to assume "no taxes" because you're awesome). You will be paid for 3 weeks (7 days a week) plus one more day for good measure, so a total of 22 days. You get to choose from these options:
  1. You get $100 for the first day, $200 for the second day, $300 for the third day. Each day you are paid $100 more than the day before.
  2. You get 1 cent for the first day, 2 cents for the second day, 4 cents for the third day. Each day you are paid double what you were paid the day before.
  3. You get 1 dollar for the first day, 1 dollar for the second day, 2 dollars for the third day. Each day you are paid the sum of the two previous days.
Which do you choose?

Friday, April 4, 2014

82: Double Stuffed.

CNN reported last summer on a teacher named Dan Anderson who looked at this weighty issue:

Here's some evidence from his website, in case you can't locate an accurate scale:


10 Original Oreos:
Original Oreos
10 Double Stuf Oreos:
Double Stuf
10 Mega Stuf Oreos:
Mega Stuf
5 Wafers:

Dan says: "I’ll leave the math to you. Or if you’re terrifically lazy, spoilers found here."