An updated version:

And a video zooming in ... 2 * 10

^{275}

180 Days of Ideas for Discussion in Math Class. (as of 9July2014, we're in overtime!)

Because it's Sunday, here's another:

So ... what do you think? How likely is it that you'll get wet if you walk outside today?

So ... what do you think? How likely is it that you'll get wet if you walk outside today?

```
Son (age 6) says "there's a 240% chance of rain" and I say ____? #tmwyk
```

— Geoff Krall (@emergentmath) March 26, 2014

Which do you think works better?

"Decimals are a subset of fractions." OR "Fractions are a subset of decimals."

```
Fractions are a subset of decimals. Always true, sometimes true, or never?
```

— Professor Yaffle (@adamcreen)

Say What?

```
Found this online today. Ack! It's not hard to explain w/real math, people. Why confuse kids? #NixTheTricks @crstn85 pic.twitter.com/EqEHWQeKCx
```

— Denise Gaskins (@letsplaymath) March 13, 2014

2² ends in a 4

12² ends with 44

Find a square that ends with 444.

Find another that ends with 4444.

12² ends with 44

Find a square that ends with 444.

Find another that ends with 4444.

```
2^2 ends in a 4.
12^2 ends with 44.
Find a square that ends with 444.
One that ends with 4444?
```

— James Tanton (@jamestanton) March 13, 2014

FiveThirtyEight had this graph
and the comment. "It looks to me as if Mr. Cain had been on a positive
trajectory before, perhaps having moved up to about 28 percent of the
Republican vote."

Is that blue trendline reasonable?

Some background information. From Wikipedia, "Cain ran as a Washington outsider and became a front-runner in the race in the fall of 2011. However, Cain's support plummeted after several women alleged that he had engaged in sexual harassment or, in one case, a 13-year extramarital affair. Cain and his wife unequivocally said the accusations were false, but Cain, citing the toll the allegations had taken on his family and his political support, suspended his campaign on December 3, 2011."

Is that blue trendline reasonable?

Some background information. From Wikipedia, "Cain ran as a Washington outsider and became a front-runner in the race in the fall of 2011. However, Cain's support plummeted after several women alleged that he had engaged in sexual harassment or, in one case, a 13-year extramarital affair. Cain and his wife unequivocally said the accusations were false, but Cain, citing the toll the allegations had taken on his family and his political support, suspended his campaign on December 3, 2011."

Another Basketball Bracket Question:

Which is easier ... a completely correct bracket or a completely wrong one?

Which is easier ... a completely correct bracket or a completely wrong one?

```
Question for you all: how much easier is it to get a completely wrong bracket vs a completely correct bracket?
```

— Dan Anderson (@dandersod) March 23, 2014

1, 3, 21, 55 only triangular Fibonacci numbers.

Any tetrahedral beyond 1?

```
1,144
only square Fib nmbrs. 1,3,21,55 only triangular Fib nmbrs. 2 only
oblong Fib nmbr. 1,8 only cubes. (Weird!) Any tetrahedral beyond 1?
```

— James Tanton (@jamestanton) January 21, 2014

I lied. It's not a Pythagorean triangle. It's TWO Pythagorean triangles that make up a Heronian triangle.

What other Pythagorean Triangle combinations make a triangle like this?

What other Pythagorean Triangle combinations make a triangle like this?

From Gabriel Rosenberg via email

True or False?

True or False?

If my stock increases in value by 28% one year, and then decreased in value by 2% the following year, then on average it increased by 13% per year.

If I powerwalk to school at a constant speed of 4 miles per hour, and then saunter home at a constant speed of 2 miles per hour, then I averaged 3 miles per hour for my trip.

3 William Carey, on a Dan Meyer post:

What's the strategy?

We play Prime fish with the sixth and seventh graders. You need a special deck of cards, but it’s an easy deck to make (edited, - ed.):

Ten cards with a "2" on them (or two fish)Each player draws four cards. They multiply their hand together, and announce only the product (!) to the group. They then play go-fish.

Ten cards with a "3" on them (or three squid)

Eight cards with a "5" on them (or five eels)

Five cards with a "7" on them (or seven sea-slugs)

Two cards with a "11" on them (or eleven shrimp)

It’s fun to watch the kids debate whether there’s a strategy to the game. It’s more fun to watch them work out the strategy once they decide that there is a strategy.

What's the strategy?

What are the only square Fibonacci numbers?

How can we show that these are the only ones?

How can we show that these are the only ones?

```
1,144 only square Fib nmbrs. 1,3,21,55 only triangular Fib nmbrs. 2 only oblong Fib nmbr. 1,8 only cubes. (Weird!) Any tetrahedral beyond 1?
```

— James Tanton (@jamestanton) January 21, 2014

Dan might not even remember this ...

How many different ways can you compare these two to decide that Dan is better?

How many different ways can you compare these two to decide that Mike is actually better?

Posted years ago by @ddmeyer

I think my pre-algebra came up with seven different ways to decide.

How many different ways can you compare these two to decide that Dan is better?

How many different ways can you compare these two to decide that Mike is actually better?

Posted years ago by @ddmeyer

I think my pre-algebra came up with seven different ways to decide.

There are 68 teams in the U.S. College Basketball Tournament. The bottom eight play an extra round; the winners get the bottom seeds in the main tournament. The main bracket has 64 teams in a single elimination tournament.

In the main bracket, how many games will be played, total?

If you just flipped a coin, what is the probability that you'd win them all?

Warren Buffet has offered $1 Billion Dollars to anyone who does pick all the games. What are the chances he will lose 1/60th of his money?

If you'd like to join the fun. with the rest of the Mathtwitterblogosphere.

In the main bracket, how many games will be played, total?

If you just flipped a coin, what is the probability that you'd win them all?

Warren Buffet has offered $1 Billion Dollars to anyone who does pick all the games. What are the chances he will lose 1/60th of his money?

If you'd like to join the fun. with the rest of the Mathtwitterblogosphere.

You can tell by the font this is a textbook. You can tell by my tone that this is an error.

Fix that Graph ... Fix That Question.

Well, recite it, actually ...

3.14159265 3589793238 4626433832 7950288419 7169399375 1058209749

4459230781 6406286208 9986280348 2534211706 7982148086 5132823066

4709384460 9550582231 7253594081 2848111745 0284102701 9385211055

5964462294 8954930381 9644288109 7566593344 6128475648 2337867831

6527120190 9145648566 9234603486 1045432664 8213393607 2602491412

7372458700 6606315588 1748815209 2096282925 4091715364 3678925903

6001133053 0548820466 5213841469 5194151160 9433057270 3657595919

5309218611 7381932611 7931051185 4807446237 9962749567 3518857527

2489122793 8183011949 1298336733 6244065664 3086021394 9463952247

3719070217 9860943702 7705392171 7629317675 2384674818 4676694051

3200056812 7145263560 8277857713 4275778960 9173637178 7214684409

0122495343 0146549585 3710507922 7968925892 3542019956 1121290219

6086403441 8159813629 7747713099 6051870721 1349999998 3729780499

5105973173 2816096318 5950244594 5534690830 2642522308 2533446850

3526193118 8171010003 1378387528 8658753320 8381420617 1776691473

0359825349 0428755468 7311595628 6388235378 7593751957 7818577805

3217122680 6613001927 8766111959 0921642019 8938095257 2010654858

6327886593 6153381827 9682303019 5203530185 2968995773 6225994138

9124972177 5283479131 5155748572 4245415069 5950829533 1168617278

5588907509 8381754637 4649393192 5506040092 7701671139 0098488240

1285836160 3563707660 1047101819 4295559619 8946767837 4494482553

7977472684 7104047534 6462080466 8425906949 1293313677 0289891521

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6391419927 2604269922 7967823547 8163600934 1721641219 9245863150

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2654252786 2551818417 5746728909 7777279380 0081647060 0161452491

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6843852332 3907394143 3345477624 1686251898 3569485562 0992192221

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9608412848 8626945604 2419652850 2221066118 6306744278 6220391949

4504712371 3786960956 3643719172 8746776465 7573962413 8908658326

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0349625245 1749399651 4314298091 9065925093 7221696461 5157098583

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9185592524 5953959431 0499725246 8084598727 3644695848 6538367362

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2168299894 8722658804 8575640142 7047755513 2379641451 5237462343

6454285844 4795265867 8210511413 5473573952 3113427166 1021359695

3623144295 2484937187 1101457654 0359027993 4403742007 3105785390

6219838744 7808478489 6833214457 1386875194 3506430218 4531910484

8100537061 4680674919 2781911979 3995206141 9663428754 4406437451

2371819217 9998391015 9195618146 7514269123 9748940907 1864942319

6156794520 8095146550 2252316038 8193014209 3762137855 9566389377

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0260541466 5925201497 4428507325 1866600213 2434088190 7104863317

3464965145 3905796268 5610055081 0665879699 8163574736 3840525714

5910289706 4140110971 2062804390 3975951567 7157700420 3378699360

0723055876 3176359421 8731251471 2053292819 1826186125 8673215791

9841484882 9164470609 5752706957 2209175671 1672291098 1690915280

1735067127 4858322287 1835209353 9657251210 8357915136 9882091444

2100675103 3467110314 1267111369 9086585163 9831501970 1651511685

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6163718027 0981994309 9244889575 7128289059 2323326097 2997120844

3357326548 9382391193 2597463667 3058360414 2813883032 0382490375

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1976211011 0044929321 5160842444 8596376698 3895228684 7831235526

5821314495 7685726243 3441893039 6864262434 1077322697 8028073189

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8537634668 2065310989 6526918620 5647693125 7058635662 0185581007

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3580893306 5757406795 4571637752 5420211495 5761581400 2501262285

9413021647 1550979259 2309907965 4737612551 7656751357 5178296664

5477917450 1129961489 0304639947 1329621073 4043751895 7359614589

0193897131 1179042978 2856475032 0319869151 4028708085 9904801094

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4306332175 1829798662 2371721591 6077166925 4748738986 6549494501

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6080998886 8741326047 2156951623 9658645730 2163159819 3195167353

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3540370141 6314965897 9409243237 8969070697 7942236250 8221688957

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8151262720 3734314653 1977774160 3199066554 1876397929 3344195215

4134189948 5444734567 3831624993 4191318148 0927777103 8638773431

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3231666365 2861932668 6336062735 6763035447 7628035045 0777235547

1058595487 0279081435 6240145171 8062464362 6794561275 3181340783

3033625423 2783944975 3824372058 3531147711 9926063813 3467768796

9597030983 3913077109 8704085913 3746414428 2277263465 9470474587

4459230781 6406286208 9986280348 2534211706 7982148086 5132823066

4709384460 9550582231 7253594081 2848111745 0284102701 9385211055

5964462294 8954930381 9644288109 7566593344 6128475648 2337867831

6527120190 9145648566 9234603486 1045432664 8213393607 2602491412

7372458700 6606315588 1748815209 2096282925 4091715364 3678925903

6001133053 0548820466 5213841469 5194151160 9433057270 3657595919

5309218611 7381932611 7931051185 4807446237 9962749567 3518857527

2489122793 8183011949 1298336733 6244065664 3086021394 9463952247

3719070217 9860943702 7705392171 7629317675 2384674818 4676694051

3200056812 7145263560 8277857713 4275778960 9173637178 7214684409

0122495343 0146549585 3710507922 7968925892 3542019956 1121290219

6086403441 8159813629 7747713099 6051870721 1349999998 3729780499

5105973173 2816096318 5950244594 5534690830 2642522308 2533446850

3526193118 8171010003 1378387528 8658753320 8381420617 1776691473

0359825349 0428755468 7311595628 6388235378 7593751957 7818577805

3217122680 6613001927 8766111959 0921642019 8938095257 2010654858

6327886593 6153381827 9682303019 5203530185 2968995773 6225994138

9124972177 5283479131 5155748572 4245415069 5950829533 1168617278

5588907509 8381754637 4649393192 5506040092 7701671139 0098488240

1285836160 3563707660 1047101819 4295559619 8946767837 4494482553

7977472684 7104047534 6462080466 8425906949 1293313677 0289891521

0475216205 6966024058 0381501935 1125338243 0035587640 2474964732

6391419927 2604269922 7967823547 8163600934 1721641219 9245863150

3028618297 4555706749 8385054945 8858692699 5690927210 7975093029

5532116534 4987202755 9602364806 6549911988 1834797753 5663698074

2654252786 2551818417 5746728909 7777279380 0081647060 0161452491

9217321721 4772350141 4419735685 4816136115 7352552133 4757418494

6843852332 3907394143 3345477624 1686251898 3569485562 0992192221

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8382796797 6681454100 9538837863 6095068006 4225125205 1173929848

9608412848 8626945604 2419652850 2221066118 6306744278 6220391949

4504712371 3786960956 3643719172 8746776465 7573962413 8908658326

4599581339 0478027590 0994657640 7895126946 8398352595 7098258226

2052248940 7726719478 2684826014 7699090264 0136394437 4553050682

0349625245 1749399651 4314298091 9065925093 7221696461 5157098583

8741059788 5959772975 4989301617 5392846813 8268683868 9427741559

9185592524 5953959431 0499725246 8084598727 3644695848 6538367362

2262609912 4608051243 8843904512 4413654976 2780797715 6914359977

0012961608 9441694868 5558484063 5342207222 5828488648 1584560285

0601684273 9452267467 6788952521 3852254995 4666727823 9864565961

1635488623 0577456498 0355936345 6817432411 2515076069 4794510965

9609402522 8879710893 1456691368 6722874894 0560101503 3086179286

8092087476 0917824938 5890097149 0967598526 1365549781 8931297848

2168299894 8722658804 8575640142 7047755513 2379641451 5237462343

6454285844 4795265867 8210511413 5473573952 3113427166 1021359695

3623144295 2484937187 1101457654 0359027993 4403742007 3105785390

6219838744 7808478489 6833214457 1386875194 3506430218 4531910484

8100537061 4680674919 2781911979 3995206141 9663428754 4406437451

2371819217 9998391015 9195618146 7514269123 9748940907 1864942319

6156794520 8095146550 2252316038 8193014209 3762137855 9566389377

8708303906 9792077346 7221825625 9966150142 1503068038 4477345492

0260541466 5925201497 4428507325 1866600213 2434088190 7104863317

3464965145 3905796268 5610055081 0665879699 8163574736 3840525714

5910289706 4140110971 2062804390 3975951567 7157700420 3378699360

0723055876 3176359421 8731251471 2053292819 1826186125 8673215791

9841484882 9164470609 5752706957 2209175671 1672291098 1690915280

1735067127 4858322287 1835209353 9657251210 8357915136 9882091444

2100675103 3467110314 1267111369 9086585163 9831501970 1651511685

1714376576 1835155650 8849099898 5998238734 5528331635 5076479185

3589322618 5489632132 9330898570 6420467525 9070915481 4165498594

6163718027 0981994309 9244889575 7128289059 2323326097 2997120844

3357326548 9382391193 2597463667 3058360414 2813883032 0382490375

8985243744 1702913276 5618093773 4440307074 6921120191 3020330380

1976211011 0044929321 5160842444 8596376698 3895228684 7831235526

5821314495 7685726243 3441893039 6864262434 1077322697 8028073189

1544110104 4682325271 6201052652 2721116603 9666557309 2547110557

8537634668 2065310989 6526918620 5647693125 7058635662 0185581007

2936065987 6486117910 4533488503 4611365768 6753249441 6680396265

7978771855 6084552965 4126654085 3061434443 1858676975 1456614068

0070023787 7659134401 7127494704 2056223053 8994561314 0711270004

0785473326 9939081454 6646458807 9727082668 3063432858 7856983052

3580893306 5757406795 4571637752 5420211495 5761581400 2501262285

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5477917450 1129961489 0304639947 1329621073 4043751895 7359614589

0193897131 1179042978 2856475032 0319869151 4028708085 9904801094

1214722131 7947647772 6224142548 5454033215 7185306142 2881375850

4306332175 1829798662 2371721591 6077166925 4748738986 6549494501

1465406284 3366393790 0397692656 7214638530 6736096571 2091807638

3271664162 7488880078 6925602902 2847210403 1721186082 0419000422

9661711963 7792133757 5114959501 5660496318 6294726547 3642523081

7703675159 0673502350 7283540567 0403867435 1362222477 1589150495

3098444893 3309634087 8076932599 3978054193 4144737744 1842631298

6080998886 8741326047 2156951623 9658645730 2163159819 3195167353

8129741677 2947867242 2924654366 8009806769 2823828068 9964004824

3540370141 6314965897 9409243237 8969070697 7942236250 8221688957

3837986230 0159377647 1651228935 7860158816 1755782973 5233446042

8151262720 3734314653 1977774160 3199066554 1876397929 3344195215

4134189948 5444734567 3831624993 4191318148 0927777103 8638773431

7720754565 4532207770 9212019051 6609628049 0926360197 5988281613

3231666365 2861932668 6336062735 6763035447 7628035045 0777235547

1058595487 0279081435 6240145171 8062464362 6794561275 3181340783

3033625423 2783944975 3824372058 3531147711 9926063813 3467768796

9597030983 3913077109 8704085913 3746414428 2277263465 9470474587

From Gabriel Rosenberg via email:

If Ï€ is a non-repeating and non-terminating value, is it a number?

If Ï€ is a non-repeating and non-terminating value, is it a number?

$3+\dfrac{4}{2*3*4}-\dfrac{4}{4*5*6}+\dfrac{4}{6*7*8}-\dfrac{4}{8*9*10}+ ....$

Feel free to steal the code. I converted to javascript from the Original python code by @rjallain.

I don't care if it's a RealWorld question. It's still fascinating.

Exactly when are the two areas equal?

From @ddmeyer

Exactly when are the two areas equal?

From @ddmeyer

Where is the steepest part of this bowl?

What is that slope?

What is the angle in degrees?

(Click to embiggen)

What is that slope?

What is the angle in degrees?

(Click to embiggen)

Go full size:

The 7.6-mile-long Mt. Washington Road is lined with trees, drop-offs, and winds on the way to the 6,288-foot summit. Travis Pastrana in a 2011 Subaru Impreza WRX STI rally car, with an officially timed run of 6 minutes 20.47 seconds.

There are questions all over the place.

The 7.6-mile-long Mt. Washington Road is lined with trees, drop-offs, and winds on the way to the 6,288-foot summit. Travis Pastrana in a 2011 Subaru Impreza WRX STI rally car, with an officially timed run of 6 minutes 20.47 seconds.

There are questions all over the place.

- Average speed
- Average slope of the road.
- What do the calls mean?
- Add more in the comments.

If you've worked with the Pythagorean Theorem, you've come across some integer solutions.

3-4-5 and 5-12-13 come to mind, especially if you've studied for the SAT; 8-15-17 is another good one.

How many can you find?

Which one is your favorite?

Is there a way to find as many as you want?

(For those who haven't seen this trivia snippet)

pick distinct positive integer values for u and v: perhaps 3 and 4.

The three sides are 2*u*v, u² + v², and |u² - v²|, thus 24, 25, 7

The hypotenuse is simply the longest of the three sides, but won't be generated by the same expression all the time.

3-4-5 and 5-12-13 come to mind, especially if you've studied for the SAT; 8-15-17 is another good one.

How many can you find?

Which one is your favorite?

Is there a way to find as many as you want?

(For those who haven't seen this trivia snippet)

pick distinct positive integer values for u and v: perhaps 3 and 4.

The three sides are 2*u*v, u² + v², and |u² - v²|, thus 24, 25, 7

The hypotenuse is simply the longest of the three sides, but won't be generated by the same expression all the time.

Why is dividing by a fraction equivalent to multiplying by the reciprocal ...
Here's one explanation, from my other blog:
We "invert and multiply", "multiply by the reciprocal" or insist on using the fraction key because we can't remember or were never really taught the reasons or the algorithm. Is there a simple explanation for the method we old farts memorized years ago in third or fourth grade? Why does it work?

Let's start with a problem: $\frac{3}{4} \div \frac{5}{6}$ and change to a compound fraction: $\dfrac{\frac{3}{4}}{\frac{5}{6}}$

Now what? Dividing by a fraction is confusing, but dividing by 1 is obvious. So we turn $\frac{5}{6}$ into unity by multiplying by its reciprocal. Of course, you can't just multiply part of our problem by $\frac{6}{5}$ without changing its value, so we multiply by 1: $\dfrac{\frac{6}{5}}{\frac{6}{5}}$

All in one image: $\dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \rightarrow \dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \cdot \dfrac{\dfrac{6}{5}}{\dfrac{6}{5}} \rightarrow \dfrac{\dfrac{3}{4} \cdot \dfrac{6}{5}}{\dfrac{1}{1}} \rightarrow \dfrac{3}{4} \cdot \dfrac{6}{5} \rightarrow \dfrac{18}{20} \rightarrow \dfrac{9}{10}$

Divide by one. Seems simple to me.

Let's start with a problem: $\frac{3}{4} \div \frac{5}{6}$ and change to a compound fraction: $\dfrac{\frac{3}{4}}{\frac{5}{6}}$

Now what? Dividing by a fraction is confusing, but dividing by 1 is obvious. So we turn $\frac{5}{6}$ into unity by multiplying by its reciprocal. Of course, you can't just multiply part of our problem by $\frac{6}{5}$ without changing its value, so we multiply by 1: $\dfrac{\frac{6}{5}}{\frac{6}{5}}$

All in one image: $\dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \rightarrow \dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \cdot \dfrac{\dfrac{6}{5}}{\dfrac{6}{5}} \rightarrow \dfrac{\dfrac{3}{4} \cdot \dfrac{6}{5}}{\dfrac{1}{1}} \rightarrow \dfrac{3}{4} \cdot \dfrac{6}{5} \rightarrow \dfrac{18}{20} \rightarrow \dfrac{9}{10}$

Divide by one. Seems simple to me.

"A perfect fit with no overlapping"

Take the squares as defined below and fit them into one big rectangle with no gaps or spaces between the squares or in the corners.

The squares have sides of the following lengths 2, 5, 7, 9, 16, 25, 28, 33, & 36.

Take the squares as defined below and fit them into one big rectangle with no gaps or spaces between the squares or in the corners.

The squares have sides of the following lengths 2, 5, 7, 9, 16, 25, 28, 33, & 36.

Pizza ... makes you hungry doesn't it?

What definition did you have to "adjust" in order to get David's answer of 55?

Do you agree that this is a fair interpretation of the problem?

source:

What if you had ten toppings available. How many different two-topping pizza variations can you make ...

What definition did you have to "adjust" in order to get David's answer of 55?

Do you agree that this is a fair interpretation of the problem?

source:

```
10 pizza toppings
Choose any 2 - different or same;
How many choices?
```

10×9÷2=45?

10×10÷2=50?

10×11÷2=55?

10C1+10C2?

Ans: 55 Why?

— David Marain (@dmarain)

Why are coordinate axes perpendicular?

Do they have to be?

What if there are more than two dimensions?

Do they have to be?

What if there are more than two dimensions?

```
My question of the day: what's so great about axes being perpendicular? #math #mathchat #CoordinateSystemchat
```

— Patrick Honner (@MrHonner) February 28, 2014

What questions does this image bring up?

What part of the dinosaur is that?

How big was it?

What measurement system are we using?

Explain WHY this works for adding fractions.

Three very similar-looking, but very different, ideas:

Three very similar-looking, but very different, ideas:

- The Butterfly, aka., "Cross-Adding".
- Cross-canceling.
- Cross-multiplying.

From Gabriel Rosenberg via email:

True or False?

To see why this is tagged Pre-Calculus and Complex Numbers, please see the comments.

True or False?

Two-thirds and four-sixths are the same number.

To see why this is tagged Pre-Calculus and Complex Numbers, please see the comments.

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