```
https://t.co/WQB5vGf8PB #mathchat
```

— Desmos.com (@Desmos) April 22, 2014
180 Days of Ideas for Discussion in Math Class. (as of 9July2014, we're in overtime!)

## Wednesday, April 30, 2014

### 115: SemiCircles

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

## 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.

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

How should we interpret that?

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

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

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

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

## Sunday, April 27, 2014

## 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

Me: "?"

Me: "Really? That's strange. I never learned the

S

S

Me: "So which is it; does one work or are they the same? Make your case and be ready to defend it."

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**S

_{1}: "No, that won't work. That's not what he told us."S

_{2}: "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?

```
Q that perplexed my class: can you have a 3-sided die? 4-sided? 5-sided? 2-sided? #MTBoS @MathCurmudgeon -- we are now creating dice.
```

— Hunter Patton (@professorpatton) April 16, 2014

Some possibilities if you're interested:

Platonic Solids, and 2 decahedra. |

1-sided die = moebius strip.

## Thursday, April 24, 2014

## 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.

Source:

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

Source:

```
@MathCurmudgeon math argument?
Via @JohnAllenPaulos
```

— Megan Schmidt (@Veganmathbeagle) February 21, 2014

## Tuesday, April 22, 2014

### 107: Road Cycling.

Rates, anyone?

```
#soretomorrow There's gotta be some math questions here. #wcydwt #caedchat #mathchat pic.twitter.com/wW1RCXMSOK
```

— Jedidiah Butler (@MathButler) March 18, 2014

## Monday, April 21, 2014

## Sunday, April 20, 2014

### 105: Linear or Quadratic?

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

```
Would you consider x² + 6x = x² - 8x + 4 "quadratic" or, because the x² terms can be subtracted from both sides, linear? #mathchat
```

— David Wees (@davidwees)

## Saturday, April 19, 2014

## Friday, April 18, 2014

## 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 110010

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 110010

_{2}pounds – but can you explain why?## Wednesday, April 16, 2014

### 101: Gun Deaths in Florida

You can take it from here ...

```
So wrong. MT @aatishb: How to lie w/ data visualization http://t.co/CjaCfQkwBa 1 of the most misleading figures ever! pic.twitter.com/6ircSgZ3R1
```

— Frank Noschese (@fnoschese) April 14, 2014

## 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.

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.

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?

And BROWN!

Image from Mr. Stadel.

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

And BROWN!

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,

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.

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?

```
What fraction is yellow? How did you see it? pic.twitter.com/Y7OYAnkIgb
```

— Learning Maths (@LearningMaths) April 13, 2014

### 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?

```
Thinking... Thinking... #wcydwt RT @eeportal_com: Dispatching wind turbine blade to the site! Amazing!
```

— Geoff Krall (@emergentmath) March 28, 2014

### 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 ...

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

## Friday, April 11, 2014

## 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?

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?

## Wednesday, April 9, 2014

## 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?

How could you change it to bias it the other way?

```
A quick Fox Graphic for you
```

— Christopher (@Trianglemancsd) March 31, 2014

## Monday, April 7, 2014

## Sunday, April 6, 2014

### 86: Three-Fifths

Which represents three-fifths?

Left

Right

Neither

Both

Twitter comment:

Left

Right

Neither

Both

Twitter comment:

```
34/50 = 17/25 shaded not 3/5 pic.twitter.com/dFtXC2dC7E
```

— Tim Buckton (@mrbuckton4maths) April 7, 2014

### 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:

- 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.
- 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.
- 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.

## Saturday, April 5, 2014

## 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 Double Stuf Oreos:

10 Mega Stuf Oreos:

5 Wafers:

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

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

# Evidence:

10 Original Oreos:10 Double Stuf Oreos:

10 Mega Stuf Oreos:

5 Wafers:

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

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