43: What is a Root?

We all know that $\sqrt{4*4} = 4$ and $\sqrt{3*3} = 3$. We know that $\sqrt{12.4*12.4} =12.4$.

What's $\sqrt{-4*-4} = ?$

Which is it ... or both?

$\sqrt{4} = 2$ or $\sqrt{4} = -2$ or $\sqrt{4} = \pm 2$ ???

From Gabriel Rosenberg via email:

Which is it ... or both?

$\sqrt{4} = 2$ or $\sqrt{4} = -2$ or $\sqrt{4} = \pm 2$ ???

42: How many numbers are there?

When I ask "How many numbers are there?", I get the usual "An infinite number!"

• If there are the same number of positive numbers and negative numbers, is the number of positive numbers (∞) - the number of negative numbers (also ∞) = 0?

• What infinite group of numbers is one member greater than the set of positive integers?

• Are there more rational numbers or irrational numbers?

• The distance between 0 and 1 is equal to 1, right? If I went halfway from 0 to 1 (i.e., I moved to the 0.5 mark) and then moved halfway from there to the end, and then I moved halfway from where I was to the end, can I ever reach the 1.0 mark?

41: Numbers at the edge of reason

From Gabriel Rosenberg via email:

True or False?

• ∞ is a number.
• ∞ can be negative.
• ∞ - ∞ = 0
• 0 * ∞ = 0
• ∞ / ∞ = 1 

40: Parallelism

From Gabriel Rosenberg via email:

True or False?

A line is parallel to itself.   

39: zeroes AND exponents.

Jim Hays via email:

If anything to the power of 0 is 1, and 0 to any power is 0, then what is $0^0$?

38: 0.9999999999999 ... = 1?

Jim Hays et. al. reminded me of this old classic from 8th grade math:

Consider $\dfrac{9}{10}+\dfrac{9}{100}+\dfrac{9}{1000}+\dfrac{9}{10000}+ ...$, which can also be written as $0.\overline{9}$ or $0.999999 ...$?

Is that less than 1, or equal to 1?


Naturally, you'll need to show them this method [from Constance Mueller (and several others) via email]:

If N = 0.9999999 ... and 10N = 9.99999 ...


 10N=   9.99999...
 -  N =   0.99999...
--------------------------
  9N = 9
   N = 1

37: Line CG is perpendicular to line EB

I like this one: angle CAB is right, the quadrilaterals are squares. 
— Max Ray (@maxmathforum)

36: What is a circle?

Taken from Chris Lusto ‏@Lustomatical

1. In your groups, answer the question, "What is a circle?"
2. Absolutely no book-looking or Googling.  If all goes well, you will be frustrated.  Your peers will frustrate you.  I will frustrate you.  Don't rob anybody else of this beautiful struggle.  If your definition includes the word locus, you are automatically disqualified from further participation.
3. Each group will have one representative present your definition to the class.  No clarification.  No on-the-fly editing.  No examples.  No pantomime.  Your definition will include, and be limited to, English words in some kind of semantically meaningful order.  Introduce variables at your own risk.
4. If you're going to refer to some other mathematical object (and I suspect you will), make sure it's not an object whose definition requires the concept of circle in the first place.  (Ancillary benefit: you will be one of the approximately .01% of the population who learns what "begging the question" actually means.)
5. Once a group presents a definition, here is your new job: construct a figure that meets the given definition precisely, but is not a circle.  Pick nits.  You are a counterexample machine.  A bonus of my undying respect for the most ridiculous non-circle of the day.
6. When you find a counterexample, make a note of the loophole you exploited.  What is non-circley about your figure?

35: Even-ness

Can something be more even than even?

Uber-even?
Arch-even?

Is 6 less even than 4? It has only one even factor while 4 has two.

@MathCurmudgeon, Is 6 less even than 4?
— Max Ray (@maxmathforum) January 30, 2014

34: Bi-weekly Financing

From Kevin Shonk (via email)

Is it fair for car dealerships to advertise ‘biweekly’ lease prices (“It’s only $88 bi-weekly!”)

Should basic personal finance courses be mandatory in high school?

I'll add this: For a home mortgage, is it better to pay $1000 per month or $500 every two weeks?

33: Congratulations! It's a child!

All students with exactly 1 sibling, please stand.

If your sibling is of opposite sex, stay standing. Otherwise, sit down.

Do you predict half of them will stay standing? More than half? Less than half?

Fun: Ask all your students with exactly 1 sibling to stand. Ask each if their sibling is of opposite sex. If you're right they stay standing
— Max Ray (@maxmathforum)

@maxmathforum Do you predict half of them will stay standing? More than half? Less than half? #matharguments
— Max Ray (@maxmathforum)

32: Roll, roll, roll the two dice.

When rolling 2 dice are there 36, 11, or 42 outcomes in the sample space?

Or is it: 2,3,4,5,6,7,8,9,10,11,12 ... thus 11?

Or is it:

Does it matter if the dice are the same color?
Does having different colored dice change the probabilities?

31: Is this rational?

Can complex numbers be categorized into rational and irrational, or is it only the real numbers that get divided that way? What do you think about this idea?

Must irrational numbers be real?

If you think so, how do you reconcile the various definitions of irrational?
If you don’t think so, why do we seem to perpetuate this idea with students that irrationals are composed entirely in the real number system...perhaps not by stating that directly, but by using representations such as the ones below?

This next is an extra credit project for a college teacher prep program ... these students obviously don't know their subject all that well and this "teacher" is no better. "Hands On Math: Burn The Textbooks, Shred The Worksheets, Teach Math." is the blog motto.

This is incorrect?

Are the visual organizers getting in the way of the understanding?

Source:

@MathCurmudgeon Here's a possible math argument: Are Complex Numbers Irrational? are-complex-numbers-irrational
— Matt Enlow (@CmonMattTHINK) January 30, 2014

30: Holy Moley Exponents

In the comments on Day 7b, More Exponents, Liz, on January 29th, said:

Wow, answer on Wolframalpha was pretty surprising!
Is it true for all numbers a and b, when a < b, then $a^b > b^a$?

Well? What do you say, Internet?

29: Mars ! Who's with me?

Actually, I'm not, but a close friend of mine is on that shortlist.

How likely is that?

28: Trigonometric Identity?

Express the value of s as a rational number in lowest terms where

$ s = & sin^2(10^{\circ}) + sin^2(20^{\circ}) + sin^2(30^{\circ}) +\\ &sin^2(40^{\circ}) + sin^2(50^{\circ}) + sin^2(60^{\circ}) +\\ &sin^2(70^{\circ}) + sin^2(80^{\circ}) + sin^2(90^{\circ}) $?


And NO Calculator allowed.
UVM 2004-20

27: Batting Average

A major league baseball player can have a batting average of .333 on the first day of the season, but not a .334 average.

When can he have a .334 average?

26: The Units Digit

Let's play "Spot the Pattern" !

What is the units digit of $3^{3^{3^{3}}}$ ?

25: Fraction Magic

Find x and y so that $\dfrac{1}{x} + \dfrac{1}{y} = 6$.

24: Arithmetic Challenge

Choose the path through the grid that you feel will gather the most points. You start out at the bottom with 1 point.

While moving from the starting point to the goal, calculate your score by using the "+", "x", and "-" symbols along the way. Calculate at each step - order of operations is turned OFF for this puzzle.

For example: N-N-N-N-N-N-E-E-E-E-N would earn 54 points. Post your maximum in the comments!

You can cross your own path, but you can't take the same route twice, or return along a path you've already taken.

Copyright(c) 2003 Ryosuke Ito

23: Graphicacy and the Science Fair

Wandered around Science Fair, found this ...

Here's the data.

	Trial 1	Trial 2	Trial 3	Trial 4
Hot Water	30	25	25	25
Cold Water	600	720	720	540
Warm Water	350	360	420	360

The basic experiment was to have water of differing temperatures and see how long it takes for a bag of pop rocks to fully dissolve. Discuss this experiment.

Graphicacy is the ability to create a visual (usually in graph form) that communicates well with the reader. Generally, if it takes the reader more than a few seconds to figure out what's going on, it's a bad diagram or graph.

1. Is this the best type of graph for this data?
2. How is it communicating badly: what normal assumptions and expectations does this graph contradict?
3. What errors did the creator make?
4. What would you do to improve this graph?

22: The Five Digit Number

I am thinking of a 5-digit number. Well, I'm not thinking of it right now, but just go with it, okay?

If you put a "1" at the end of the number, you get a 6-digit number that is three times as big as the 6-digit number you'd get if you put a "1" at the beginning of the number.

In other words, the number _ _ _ _ _ 1 is three times as big as 1 _ _ _ _ _

What is the 5-digit number?

21: Change this 9-Clock to 8s

Or sevens, or sixes, or whatever strikes your fancy.

20: Cranberry Pyramid

How many cans in the pyramid?

Is this the most stable stacking arrangement?

19: It's fifty-fifty, right?

True or False

If it could happen or not happen, then the odds must be 50-50.

From Hunter Patton @professorpatton,

What's the probability you make a free throw? -- 50/50? 

Story via The Bad Astronomer and Mark C. Chu-Carroll, but I supplied the question:

This guy sued in a Honolulu court to stop the Large Hadron Collider. If he could either win or lose, his lawsuit has a 50-50 chance of going his way -- either he'll win or he won't, right?

Watch it all or forward to Walter Wagner at about the 2:15 mark to see the statement in the interview.

The Daily Show With Jon Stewart	M - Th 11p / 10c
Large Hadron Collider
thedailyshow.com

18: Breaking twenty-five

In this "Tiny Math Games" post by Dan Meyer, there's an idea from [Malcolm Swan]

Pick a number. Say 25. Now break it up into as many pieces as you want. 10, 10, and 5, maybe. Or 2 and 23. Twenty-five ones would work. Now multiply all those pieces together.


What's the biggest product you can make?

What's your strategy?

Will it always work?

Does it work for fractions?

Is there another set of numbers you could use that isn't explicitly against the stated rules?

17: Pi versus Tau

Here's Vi Hart:

What do you think?

What changes to math would occur as a result of switching from using pi to using tau?

• Which will be good/useful/more efficient?
• Which be bad/annoying/less efficient?

Does she make a good enough case for us to switch in this class or should we continue to use pi?

(at 2:30, "The way of mathematics is to make stuff up and see what happens." I love that statement.)

16: Exponents and powers of ten.

Following on an earlier question from Day 7 ...

If I were to tell you that $2^{100} - 100^2 = 1,267,650,600,228,229,401,496,703,195,376$,
can you tell me what would change if I hadn't subtracted $100^2$ ?

What is $2^{100}$?

How do you know?

Just for the record, what -illion is that?

15: Error Analysis - Can You Be A Teacher, Too?

In a fascinating bit of testimony before a Michigan Senate Hearing, this slide was presented. The presenter (Dr. Ball) asked the legislators to identify where and how the 4th-grade solvers made their mistakes. 

David Wees ‏@davidwees asks:

"What is the mistake?"
"What is the thinking that led to this mistake?"

I saw both for (b) and (c) but the train of thought in (a) escaped me completely. What do you think happened in (a)?

What would you tell the students in each case?

If you'd like to watch, the video is about 5 minutes long.