]]>RAY: This puzzler is from a fellow named Jerry Olsen. Ed and his two sons, Biff and Skip, have been hired to paint the floor of a merry-go-round. They want to make sure they measure the floor area exactly, because they don’t want to buy any extra paint. The carousel, of course, is a circle. Here’s the catch: In the middle of the carousel is a smaller circle, which contains all the machinery for the carousel.

TOM: An annulus, in other words.

RAY: Exactly. It’s a ring we’re dealing with. Ed tells Biff, “We need to know the area of the carousel, including the area of the big outer circle that we’re going to paint and the area of the inner circle where there’s nothing but the machinery. “Once we have the areas of both circles we can subtract the inner circle from the other circle and we’ll know how much paint we need.”

Biff goes to the carousel and says to himself, “I can’t do this. All the machinery is in the middle. I can’t get to the center to measure the diameter.” He thinks, “I’ll cheat. The old man will never know!” Biff measures a straight line from one edge of the carousel to the other edge, not going through the center.

TOM: In other words he’s going to make what’s called a chord of the big circle.

RAY: Right. Any line that goes from one edge of the circle to the other that isn’t a diameter is a chord. As luck would have it, the tape measure touches the inner circle, or in geometric terms, is tangent to the inner circle at one point.

Biff returns to his dad and says, “I couldn’t do what you wanted me to do. I got this measurement and it’s 70 feet.” The old man administers a swift dope slap. He says, “How the heck are we going to figure this out. We don’t know either diameter.”

The other brother Skip says “I think I can figure it out. ”

Can he or can’t he?

The Introduction to Computer Programming Class had a Skype Q and A with Jason from Dark Sky (Dark Sky is an iOS weather app, check it out, it’s fantastic. They also make the great weather site forecast.io.) Super nice guy, I just emailed and asked, and it was scheduled a day later. Great experience for these kids.

Here’s some student responses to the prompt: “**What blew your socks off? What’d advice/stories/information was surprising?**”

- When how he told that if you really want to learn something. you need to be able to do it on your own time
- I think it is motivational that someone who is successful had a hard time and still does sometimes and still does what he wants to do.
- I thought it was really interesting when he talked about “reverse engineering” video games, and that was how he learned trigonometry. But now that I think about it, it isn’t surprising that it was easier for him to learn something difficult while immersing in something he was passionate about.
- The most surprising fact was that he wrote 45,000 lines of code to make the app originally, and then he modified it to do more, but only required 8,000 lines of code. I also really liked how he encouraged people to go on their own and explore other programs by themselves.
- What I guess what surprised me the most was how he compared computer programming to dance or singing or art. Going into this class I perceived computer programming as a very technical and systematic subject…that everything is by the book. While this may be true… talking to the developer brought to my attention that computer programming can be largely reliant on self discovery and self error.
- I was shocked when Jay told us how long it took to create his app and how many lines of code it required (45,000).

Really cool.

Steps (from wikipedia):

- Take any four-digit number, using at least two different digits. (Leading zeros are allowed.)
- Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.
- Subtract the smaller number from the bigger number.
- Go back to step 2.

Here’s an example (also from wikipedia):

For example, choose 3524:

5432 – 2345 = 3087

8730 – 0378 = 8352

8532 – 2358 = 6174

Fun right? Also fun to program, here’s the python code that tries every number from 1000 to 10000, and counts how many steps it takes to get to 6174 and puts the results in a csv file:

def largest(nstr): if (len(nstr) == 0): return "" elif (len(nstr)==4) and (nstr[0] == nstr[1] == nstr[2] == nstr[3]): return "7641" digit = -1 index = 0 for i in range(0,len(nstr)): if (digit < int(nstr[i])): digit = int(nstr[i]) index = i return str(digit) + largest(nstr[0:index]+nstr[index+1:len(nstr)]) def kaprekarSteps(n): count = 0 nstr = str(n) while (n != 6174): l = int(largest(str(nstr))) lstring = str(l) s = int(lstring[::-1]) n = l - s nstr = str(n) while (len(nstr) < 4): nstr = "0" + nstr count += 1 return count f = open('kaprekar.csv','w') c = 1000 while (c < 10000): f.write(str(c)+ "," + str(kaprekarSteps(c)) + "\n") c += 1 f.close()]]>

Darker pixel = longer Euclidean algorithm to find GCD(x,y). Here x & y range from 1 to 500. So many questions… pic.twitter.com/k8N6jiGxSd

— Matt Enlow (@CmonMattTHINK) September 24, 2014

I saw this toot by Matt yesterday morning and I loved the visual of the color based on the number of steps of Euclid’s GCD algorithm. The algorithm is pretty straightforward and it’s a nice example for either using recursion or using a loop. The coloring is fun to mess with too. The coding went quickly for me because I already had the code for breaking a 1D pixel array (why processing??) into x and y coordinates.

There’s two variations:

Here’s the link for the GCD steps version.

And here’s the link for the GCD version where the closer the GCD is to the minimum of x and y, the more white the pixel is.

Here’s the recursive algorithm:

int euclidGCD(int x, int y) { if (y == 0) { return x; } else if (x >= y && y > 0) { return euclidGCD(y,(x%y)); } else { return euclidGCD(y,x); } }

Here’s the loop algorithm to count the number of steps:

int euclidGCDsteps(int x, int y) { int t; int steps = 0; if (x >= y) { while (y != 0) { t = y; y = x % y; x += t; steps++; } } else { return euclidGCDsteps(y,x); } return steps; }]]>

- You want the students to find your subject interesting.
- You want the students to investigate your subject in their own time, like an adult would do.
- You want the students to continue to develop their presentation skills.
- You want the students to pique the interest of the other students in the class in your subject.
- You want to expose the students in your class to a diverse set of ideas, opinions, and experiences.

When I (briefly) attended Twitter Math Camp last year, I really enjoyed the My Favorite section of the conference. This is a section where teachers came forward and presented *something* that they liked that they thought the other teachers would be interested in. These included activities, strategies, etc. Pretty much whatever the presenter thought the other teachers would enjoy. They had about 5-10 minutes to present.

To co-opt this **My Favorite** structure to my classroom, I asked every student in all of my classes to present one thing that they found interesting about math.

Here’s the document that I gave the students:

Great! I loved it, and I (think) the kids loved it. We had so many different and interesting topics. Here’s a quick subset of the topics that were covered in three of the classes:

Fibonacci and Bartok

Piles

Friendship Paradox

Volleyball winning odds

Pi Cupcakes

Vortex Math

Birthday Paradox

Chaos Theory

Hairy Ball Theorem

3D Polyhedra

Pythagorean Thm

Riot Theory

Monty Hall Paradox

Pappus of Alexandria Thm

Cryptarithms

Golf Handicap

Quad Midpoints make Parallelogram

Kaprekar’s Constant

Number Trick

4 color theorem

Diving Scoring

Brouwers fixed point theorem

P vs NP

Mole Train Woot Woot

Sound and Sine

Fourier Transforms

Photography

4D

Golden Ratio

Font Layout and Yearbook

Binary Numbers

Fermat’s Last Theorem

Rule of 72

Symmetry

Mobius Strip

Paradoxes

Nines

P vs NP

Euler

Golden Ratio and Phi

Mandelbrot Fractal

Reuleaux Polygons

Graham’s Number

Pascal’s Triangle

Tesselations

Schwarz lantern

Monty Hall

Math (and Physics) of Bowling

What’s faster, going up or going down?

Only recognize about 2/3rds of them? Me too. It was awesome seeing the different mathematics that was discussed in our classes. What was the percent of topics that lined up with any of the final exams? I’d say under 5%.

Do you have 2-5 min to spare 30 or so times per class per year with very little burden of prep on the teacher? I would bet that you do.

]]>Thoughts?

**Left image**

Length: 905 pixels

Width (between front feet): 275 pixels

1 inch: 90 pixels

**Right image**

Height: 150 pixels

1 inch: 115 pixels

**Left image**

Length: 1014 pixels

Width (between front feet): 292 pixels

1 inch: 60 pixels

**Right image**

Height: 200 pixels

1 inch: 98 pixels

Original Length: 10.06″

Original Width: 3.06″

Original Height: 1.30″

After Length: 16.90″ (increase of 68%)

After Width: 4.87″ (increase of 59%)

After Height: 2.04″ (increase of 57%)

Liars!

Volume of box that would just contain original lizard: 10.06 * 3.06 * 1.30 = 40.02 cubic inches

Volume of box that would just contain soaked lizard: 16.90 * 4.87 * 2.04 = 167.90 cubic inches (* increase of 319%*)

Liars!

Thoughts? Did I mess up?

- How big would it be if the volume *did* increase by 600%?
- How big would the lizard be if it grew 1000%? 10,000%?
- How big would the lizard have grown if it just barely fit in *your* classroom?
- …

"'M&M's Mega' To Have Three Times More Chocolate." Oh brother here we go again. http://t.co/zrBE5tZ3Al /cc @dandersod

— Dan Meyer (@ddmeyer) May 16, 2014

What a great Calculus activity (hint: rotational volumes).

Cross sections with an exacto:

What shape fits best? Click on image to get directly to the desmos interactive.

Pretty good fit.

Here’s the data for each type. I used a caliper to measure the dimensions.

To account for the candy shell you can measure pixels in an image manipulation program. If you’d like to follow along at home, the M&M Mega has a thickness of 404 pixels, and the chocolate only (no shell) has a thickness of 333 pixels. The regular M&M has a thickness of 251 pixels, and a chocolate thickness of 210 pixels.

You have enough information to do some damage, go ahead and see if the M&M’s Mega truly have **3x** the chocolate of the regular M&M’s.

M&M Mega volume of 1,423 mm^3.

Regular M&M volume of 474.8 mm^3.

Fun. Great activity for calculus.

*Phew, no fox news.*

]]>

Careful SBG, the all seeing eye of pershauron is trained on you: http://t.co/S6QAJz1p86 c/o @mpershan

— Dan Anderson (@dandersod) May 19, 2014

In response to the Dark Lord Pershauron, I’ve decided to see what feedback you all have for my current SBG setup (v 5.0).

The course is Pre-Calculus honors. There are about 40ish standards for the year. About once a week students take a test on several standards (2-4). The first time they see the standard there is no grade; this is an opportunity for the students to see where they are. I started off giving them feedback but moved to self-feedback only: when finished, they go to a feedback station (@fnoschese style), grab a colored pen, look at the key and give themselves feedback. Sometimes they work in their groups to get the feedback and I put the key up on edmodo (all the keys end up on edmodo). They keep these feedback quizzes and put them in their folder.

The second time they see a standard it is for a grade, normally a week or two after we’ve taught the topic in class (grade out of 5: 5, 4.5, 4, 3, or 2). The grade goes in the books as is. They get three reassessment opportunities per term (out of 10-15 total standards per term). They see their graded quiz, but they don’t get to keep the graded assessment.

Things I need to get better at for next year:

Providing feedback myself to the student instead of just leaving it up to them. I like they they take responsibility in knowing what they know and don’t know, but I don’t think they are always able to give themselves the directed feedback that a teacher could give. I get a general idea of where the class is by informally going around and looking at quizzes post feedback, but I don’t have the whole picture. This informal feedback has changed the class, sometimes its obvious that they’ve understood a topic and we can move on, other times it’s clear that we need to take a different look at this material before moving on.

Thoughts? Comments? Let me have it. I see this as shot peening for my grading system. Thanks in advance!

]]>]]>How did the magistrate recommend that the two bedouins divide the money?