Posing interesting questions that make students want to think deeply

[Dr_Nate]

I regularly start my classes with an opener that I hope might 'spark something' in the minds of my high school physics students. This is usually the Astronomy Picture of the Day or some interesting science news article. Sometimes it is more mathematical.

On the math side, so far, I've thrown up on a slide (i) the 'proof' that $a=2a$, (ii) $\cos(x)-x=0, ~x=?$, and (iii) $\lim_{n \rightarrow \infty} {\cos(x) =?}$. They loved them.

I'm looking for more math or physics teasers like these. I'd appreciate any suggestions.

 

[WWGD]

Have you considered the recently-trending $0.9999...=1$ or the continuation of the Riemann Zeta :$1+2+...= \frac{-1}{12}$*? If you cover Stats, D Montgomery's Prob&Stats for Engineers has a list of "Mind-expanding exercises" at the end of most chapters.

* Where this is not equality in the usual sense, before everyone starts piling on me.

 

[PeroK]

I regularly start my classes with an opener that I hope might 'spark something' in the minds of my high school physics students. This is usually the Astronomy Picture of the Day or some interesting science news article. Sometimes it is more mathematical.

On the math side, so far, I've thrown up on a slide (i) the 'proof' that $a=2a$, (ii) $\cos(x)-x=0, ~x=?$, and (iii) $\lim_{n \rightarrow \infty} {\cos(x) =?}$. They loved them.

I'm looking for more math or physics teasers like these. I'd appreciate any suggestions.

Just some ideas:

Hilbert hotel

Countability of the rationals.

The ubiquity of $\pi$:

$\sum \frac{1}{n^2} = \frac{\pi^2}{6}$

$\int_{-\infty}^{+\infty} e^{-x^2} dx = \sqrt{\pi}$

Check out 3blue1brown YouTube channel.

Also, this guys channel. Especially:

 

[vela]

There are these classics:
$$-1 = i^2 = \sqrt{-1} \sqrt{-1} = \sqrt{-1\cdot -1} = \sqrt{1} = 1$$
and
\begin{align*}
a &> 3 \\
3a &> 9 \\
3a-a^2 &> 9 - a^2 \\
(3-a)a &> (3-a)(3+a) \\
a &> 3+a \\
0 &> 3
\end{align*}

 

[berkeman]

I'm looking for more math or physics teasers like these. I'd appreciate any suggestions.

I used to buy math puzzle calendars every year. My Google-foo is failing me right now, but they had a math puzzle for every day whose answer was the date. The challenge was to figure out how to work the problem to get the right answer (the proverbial "show your work").

Now I'm inspired to purchase one for 2020 to post outside my cubicle at work. It's a great conversation starter...

 

[Dr_Nate]

There are these classics:...

Yes, these are exactly the flavour of things I want to show them.

 

[Dr_Nate]

Just some ideas:...

Originally, I thought that these may be too involved for what I want to do. But, I can see how I could show these and many of the students would mull them over at night. They may not be successful, but, hey, just thinking about it may enlighten them.

 

[berkeman]

Now I'm inspired to purchase one for 2020 to post outside my cubicle at work. It's a great conversation starter...

Just bought this one with my Amazon account. Hopefully it's the same series that I used to get every year.

 

[Dr_Nate]

I thought of a good question today, which also happens to be a classic.

Why does a mirror flip the image left and right but not up and down?

 

[PeroK]

I thought of a good question today, which also happens to be a classic.

Why does a mirror flip the image left and right but not up and down?

Spoiler

I was explaining this to someone recently. Here's my take.

Part (a)

Take some solid cut-out letters (*) and arrange them left to right, so you can read them normally. Hold the letters up in front of you while you are in front of a mirror. Look at the image in the mirror. You can read the letters in front of you AND in the mirror image.

Now, turn the letters round, so they are right to left. Hold them up. Now, they are the wrong way round in front of you AND in the mirror image

Conclusion: the mirror does not flip right and left or up and down.

(*) I actually did this with a single sheet of newspaper where you could see through to read the headline front either side.

Part (b)

What the mirror does is change the direction in which the front of an object is pointing, relative to the room. Where front is taken to be what is facing the mirror.

If we take an object and decide that the side facing away from us is the front, then we interpret what is on our right as also the right-hand side of the object. But, if we decide that the side facing towards us is the front, then we also redfine our definition of right and left for the object. Now, the side on our left is defined to be the right hand side of the object. Note that we do not redefined up and down!

Hence, because the mirror image is pointing in a different direction, we redefine left and right for that image, but not up and down.

Part (c)

To confirm part (b), take a 3D object and mark each face according to the direction it is pointing relative to the room. Label the floor, ceiling and walls of the room as A-F. And label the faces of the object accordingly. Assume the mirror is at A and the opposite wall is B.

The mirror image of the object has the same orientation relative to the room for C-F: i.e. up, down, left and right. But, the orientation is different for A-B. Therefore, the mirror flips front and back, but not any other faces.

 

[Dr_Nate]

I was explaining this to someone recently. Here's my take.

That could be a great way for students to explore to get to the solution. Thanks for that!

 

[256bits]

I thought of a good question today, which also happens to be a classic.

Why does a mirror flip the image left and right but not up and down?

That always is a mind boggler. Even when you know the answer you have to do a re-think the next time around.
Does the mirror actually flip left and right though?

Get two people facing each other.
One persons right is on the other persons left; and one person left is one the other persons right.
For the two people to face each other, one of the two had to do a 180 degree rotation if they had both been facing in the same direction to begin with ( in which case the person behind the other would see the others back ). Put a frame between the two people to simulate a mirror.

Now place the two people side by side facing the same direction, perhaps east.
Have them hold their two hands nearest each other, which would be one persons left with the other person right hand. Have them raise their holded hands, which would nearer to each other. Have them raise their other hand in unison. Compare that with what you would see if you were standing sideways in front of a mirror. In this case it doesn't appear to be a just a flipping of left/right hands ( you raise your left and the mirror raises right ), but something to do with near and far. After all, if you face to the east, so does the mirror image. Where is the rotation here?

Very interesting for your students and it will drive them bonkers.

Some related reading, ( of course Wiki )
https://en.wikipedia.org/wiki/Reflection_symmetry
https://en.wikipedia.org/wiki/Mirror_image

After writing this, I forgot the answer to the mirror flip.
Had to go back and read Perok again.

 

[PeroK]

That always is a mind boggler. Even when you know the answer you have to do a re-think the next time around.
Does the mirror actually flip left and right though?

If you lie on your side facing the mirror and touch your head, then the image in the mirror is touching its head. If the mirror physically flipped left and right, then the image in the mirror would have its head where its feet are!

 

[256bits]

If you lie on your side facing the mirror and touch your head, then the image in the mirror is touching its head. If the mirror physically flipped left and right, then the image in the mirror would have its head where its feet are!

I had to stop and think if there was a symmetry that worked that way, which would entail a rotation about an axis perpendicular to the mirror.

If you are lying on your left side, the person in the mirror is lying on his.her left side.
Or paint one hand blue, the other red. No matter what orientation you are at, and no matter which hand you move, the person in the mirror moves the same colored hand, except that their hands are painted in the opposing color of your painted hands. If your right is red and left is blue, the mirror person has their left painted red and their right hand painted blue.
So its not really a 180 degree rotation either.
In my example it is.
In your example it is not.

 

[PeroK]

If you are lying on your left side, the person in the mirror is lying on his.her left side.

That should be "his/her right side", surely?

In any case, it's clear that the mirror is doing the same in the horizontal and vertical directions. It's important to understand that. Many people get stuck at the assumption that something is happening horizontally that is not happening vertically.

 

[256bits]

That should be "his/her right side", surely?

In any case, it's clear that the mirror is doing the same in the horizontal and vertical directions. It's important to understand that. Many people get stuck at the assumption that something is happening horizontally that is not happening vertically.

Yes thanks for the correction, which is what I meant to type.

 

[gmax137]

Calculator exercise
$1782^{12} + 1841^{12} = 1922^{12}$

 

[Vanadium 50]

Calculator exercise

Is that supposed to be wrong? (Because it is)

 

[gmax137]

it checks out on my HP11. but we know it's wrong. that's the thing to talk about. plus, how do you know it is wrong? you don't have to go thru Andrew wiles' paper to prove it. plus plus, it is almost right. that's kind of interesting, too.

 

[Vanadium 50]

Well, I saw even + odd = even, which was my first hint. I didn't even think about Fermat/Wiles.

If I calculate (1782¹² + 1841¹²)^1/12 I do in fact get 1922. But if I calculate 1782¹² + 1841¹² - 1922¹², I do not get zero. So if you want the "right wrong answer" on your calculator, you need to ask it the right question.

By the way, the left hand side ends in 7 and the right hand side ends in 6.

 

[Dr_Nate]

Calculator exercise
$1782^{12} + 1841^{12} = 1922^{12}$

My Nonlinear ODE's prof liked to call these numerical monsters. I vaguely recall that he used to show which calculators were not worth their salt because they quickly accumulated truncation errors and the like.

 

[MidgetDwarf]

The proof of the approximated numerical value of pi using exhaustion?

 

[Dr_Nate]

The proof of the approximated numerical value of pi using exhaustion?

I don't know that one.

 

[gmax137]

BTW, wolframalpha.com will calculate huge numbers, presumably with the necessary precision. It shows
$1922^{12} = 2541210259314801410819278649643651567616$
forty digits. And
$1922^{12} - 1782^{12} - 1841^{12} = 700212234530608691501223040959$
thirty digits.

Sorry, I realize I'm off on a tangent here. I thought I could create an excel spreadsheet to calculate values digit by digit but it turns out to be rather messy with all of the carrying.

 

[DEvens]

For brain-teasers and physics chestnuts, one of my favorites is this.

A 100 kg cart has a spring that let's it fling a 1 kg mass at 100 m/s. So the cart starts at rest, it obviously must finish at 1 m/s in the other direction. So it gains kinetic energy of 1/2 m v^2, just 50 Joules.

Now let the cart start from 1 m/s, and it flings the mass out the back. Now it must clearly go from 1 m/s to 2 m/s. This is necessary, or else the cart would "know" it had started from a moving frame. Relativity (whether Einstein relativity or Galileo relativity ) requires it finish at 2 m/s. But it started with 50 Joules of kinetic energy. (See the first part.) And it finishes with 200 Joules. So it gains 150 Joules.

How can the same spring let it gain 50 Joules in the first case, but 150 Joules in the second case. What happened to conservation of energy?

Spoiler: The key to making this difficult is...

Try not to mention the 1 kg mass after describing the first case. Since it goes at 100 m/s in the first case, it has to go at 99 m/s in the second case. And when you work out the kinetic energy it gains (don't forget it starts at 1 m/s in the second case) it, of course, balances out.

After they work through this case, make them do it for two arbitrary masses with an arbitrary amount of kinetic energy. Then, when they are starting to feel strong, make them do it all in special relativity.

 

[Dr_Nate]

After they finish mechanics with all the standard problems in it, I like to show them the attached photo and ask them how much force I need to pull down with to make the mass go up.

 

[WWGD]

Well, I saw even + odd = even, which was my first hint. I didn't even think about Fermat/Wiles.

If I calculate (1782¹² + 1841¹²)^1/12 I do in fact get 1922. But if I calculate 1782¹² + 1841¹² - 1922¹², I do not get zero. So if you want the "right wrong answer" on your calculator, you need to ask it the right question.

By the way, the left hand side ends in 7 and the right hand side ends in 6.

I guess the example you brought up is from the Simpson's , right?

 

[gmax137]

I guess the example you brought up is from the Simpson's , right?

I'm not sure what that means.