140: Topology Bagels - Two Rings

Can you cut your bagel so that you get two linked rings?

For Sunday brunch, cut your bagel so the two halves are linked together like a chain.
— Steven Strogatz (@stevenstrogatz) May 25, 2014

139: Largest Integer, part 2

After yesterday have to ask ... Is there a largest odd number that is not the sum of three composite odd numbers?
— James Tanton (@jamestanton) May 26, 2014

138: Largest Integer

What is the largest even integer that cannot be written as the sum of two odd composite numbers?

For example, 42 = 21 + 21, so it is not a candidate. 22 is a candidate because no pair of 9, 15, or 21 can equal 22.

As with many of Mr. Tanton's puzzles, there's a way to know that you are absolutely correct. Can you find the number and the explanation?

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
— James Tanton (@jamestanton) May 25, 2014

137: McNuggets

McNuggets come in boxes of 3, 6, or 20. You can get 9 by buying a 3-piece and a 6-piece, but you can't get 13.  You can get 21 by buying seven 3-piece boxes, but not 22. You can get 23 with a 20-piece and a 3-piece, etc.

What is the largest number of McNuggets you CAN'T get with some combination of 3-piece, 6-piece or 20-piece boxes?

136: Jeopardy!

Assuming that you get every question correct, and bet it all on the Daily Double, and the Daily Double happens at the most opportune time, and the Daily double is located in the absolute best place, how much can you win in the first round of Jeopardy?

And what is the best place for the Daily Double?
And the best time to get the Daily Double?

The prizes are twice as big when you get to the Double Jeopardy Round:

So let's keep this going ... how much can you theoretically win to this point?

In Final Jeopardy, you can bet it all ("Let it Ride", as gamblers say).

135: Raising the Average

In a charter school of 400 students, the average score on a standardized test at the beginning of the year was 82%.

By the end of the first few months, 92 students couldn't take it anymore: the 9-hour days of which 8.5 hours were spent staring at computer screens, the required Saturday classes and the constant badgering over being quiet, walking dully in a single file, drill, drill, drill, drill, and drill, all seemingly in an effort to get them to drop out.

The 92 students, who were getting Ds and Fs, had a collective average of 60, which is why they were targeted.

In late October, Jay Matthews of the Washington Post interviews the principal for a Thanksgiving-themed piece in the paper and asked about the gains and improvements that the wonderful charter-school had been able to accomplish.

The principal did not mention that the students had not taken the test again or that the rise was simply because of the elimination of what he considered inferior students.  Instead, the principal said proudly, "Our program is amazing. We have been able to raise their scores by _________ in just two months. If the City gives over some more public schools to our control, we'll be able to do more of this."

How much did scores rise because of the expulsion/removal of the 92 lower-scoring students?

134: Savings?

#unitratechat RT @hpatt27: Interesting how they delivered the numbers to support their message. #WCYDWT #mtbos 
— Christopher (@Trianglemancsd) May 19, 2014

133: Concentric Circles

So Sayeth Mr Honner:

I wonder if it’s possible to find the angle between my sight-line and the central axis of the cylinder by comparing the centers of the various circles?

I've got a hunch the relative size of the circles counts, too, but I haven't thought deeply about this yet.

132: Cubes

If you've ever played with a Rubic's Cube, you know it's possible to divide a cube into 27 smaller cubes. Eight and sixty-four is pretty obvious, too.

Can you show how to divide a cube into other numbers of sub-cubes, numbers that aren't perfect cubic numbers?

Like 15? or 20?

Possible to divide a cube into 1, 8, and 27 cubes. Show how to also divide a cube into 20 or 15 sub cubes. Any other nmbrs under 27?
— James Tanton (@jamestanton) May 11, 2014

Because you don't have enough awesome in your life, I present watermelon cubes.

131: Traveling Salesman

from MathGIFs:

This is a visualization of the Traveling Salesman problem.

"One of the most famous problems of math and computer science is the Traveling Salesman Problem. Given a list of n cities, what is the shortest tour that visits each city exactly once and returns back to the starting city? One of the reasons why this problem is so famous is that it is easy to understand the problem and incredibly hard to solve the problem. The animation above illustrates the steps a computer algorithm used to arrive at what we think is the shortest distance - 12,930 miles - needed to visit all 48 continental US capitals. (We used Google maps to use actual roads and highways when computing the distances between cities.) The best way to view the animation is to watch only one part of the US at a time, like the northeast. Then you can see how the web of paths works itself out into an efficient route."

Let's simplify things a bit ...

Find the shortest route through the capitals of the six New England states.

130: Catching a Pass

So ... is this possible? Or is it a well-done fake?

Who has physics/math students that can analyze this: is this real? Can you catch your own pass? https://t.co/JWYzoNPS0t #mathchat #anyqs
— Max Ray (@maxmathforum) April 28, 2014

129: Helpers

If it takes me 60 minutes to cook dinner by myself, and 90 minutes if one of my kids is "helping" me, how long would it take her to do it herself?
— Matt Enlow (@CmonMattTHINK) May 10, 2014

Q15 - Negatives

Question 15: Negative times a negative is ... ?

Q14 - Point Nines

Question 14: Point Nines

Q13 - Axioms

Question 13: Axioms.

Q12 - Undefined Terms

Question 12: Undefined Terms in Geometry.

Q11: Precise vs Accurate

Question 11: Precision and Accuracy.

Q10: Conventions

Question 10: What is the difference between a convention (such as PEMDAS) and a law (such as the distributive property)?

Q9: Imaginary Numbers

Question 9: Why were imaginary numbers invented?

Q8 - Conversions

A simple question today.

Question 8: Create the equations that a website or a calculator might use.

Q7 - The Mean and Hidden Information

Question 7: Most teachers use the mean to assign a grade, but this measure hides information. What information is hidden and what, if anything, should we do about it?

Q6 - Catering the Party

The form is embedded after a jump. View this post directly to see it.

Question 6: Catering the Party.

How many tables will be needed?

Q5 - Repeated Addition

Question 5: Multiplication is Just repeated Addition.
Explain why this statement is false, giving examples.

Q4 - Ordering Numbers

Question 4: Organized list of numbers
Place these numbers in order from largest to smallest and predict the errors that some of your classmates might make.

Q3 - Invert and Multiply

Question 3: Invert and Multiply
You are told to “invert and multiply” to solve division problems with fractions. To the best of your understanding, explain why this works and when it works; include any situations for which this instruction doesn't work.

Q2 - Equivalence

The form is embedded after a jump. View this post directly to see it. We think it would be best if you point your students to this form and have them record their thoughts on each question BEFORE any class discussion, perhaps as a homework assignment the evening before. If they wish to change their thoughts afterwards, we would ask that they submit a new entry.

Question 1: Equivalence
"Solving problems typically requires finding equivalent statements that simplify the problem." Explain.

Q1 - Divide By Zero

Question 1:  "You can't divide by zero". Explain why not.
To the best of your understanding, explain why you can't divide by zero, even though you can multiply by zero.

128: Which is bigger?

We'll just let this one speak for itself ...

Quick, which infinite sum is greater?

1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 ...,
or
1/2 + 1/4 + 2/8 + 3/16 + 5/32 + 8/64 + 13/128 ...

— Matt Enlow (@CmonMattTHINK) April 29, 2014

127: Consecutive Sums Are Squares

A Challenge from StandupMath:

Can you arrange the integers 1 to 17 so that each adjacent pair adds to a square number? 

@standupmaths: arrange integers 1 to 17 so that each adjacent pair adds to a square number #mathspuzzle”
— Christopher Rath (@Rathematician) September 24, 2013

126: Teachers - Balanced or Not?

Building on yesterday's ideas, we present this thought-bomb:

FiveThirtyEight had a graph on baseball teams, comparing the abilities of the players with a type of Winning percentage metric:

 

Teams to the right consisted of a few great players mixed with some average and below-average players - a big spread of abilities.

Teams to the left didn't have any superstars, but they had above-average players.

Which group of faculty would you think would make for a better school? The one with some GREAT teachers and some LOUSY teachers, or the one with only GOOD teachers?

Which school would you rather be enrolled in?

125: Average Shmaverage

Assume a classroom where every grade has equal weight. Here are two sets of grades:
Student A (ave=80.0): 100, 100, 100,  0,  100
Student B (ave=80.2):  81,  80,   80,  80,  80

Questions:

1. Who has better mastery? 
2. Who will be better prepared for the next course?
3. Which would you rather be?

Another set of grades, in sequence:
Student A: 25, 25, 50, 100, 100
Student B: 100, 100, 50, 25, 25

Same questions:

1. Who has better mastery? 
2. Who will be better prepared for the next course?
3. Which would you rather be?

Examples from Step 1: Try Something.

124: Dividing a Number by UmberN

Can you find a number such that,
if you move the leading digit to the units place of a number, i.e. 72 becomes 27 or 381 becomes 813, the original number is a factor or multiple of the new number? Oh, and not all digits are identical.

1. Can you find a two digit number that will work in this way?
2. If you can't, was it because it can't happen?
3. In his tweet, James Tanton claims that there are no three or four digit digit numbers that will work. He doesn't make a statement like that unless he has found some way to show it NEVER works. Can you find that rule?

There is no three-digit number abc divisible by bca. (Move first digit to end) No 4-digit number abcd divisible by bcda. Five? Six?
— James Tanton (@jamestanton) April 26, 2014

123: Binary Names

@MathCurmudgeon @crstn85 Great math question here: how many of these questions would you need to get right before you could ID the teacher?
— Bowen Kerins (@bowenkerins) April 25, 2014

122: Packing the Crates

There are ten boxes weighing 15, 13, 11, 10, 9, 8, 4, 2, 2, and 1 kgs. They are to be packed into three crates, 25 kg each.

How many different ways can this be done?

New problem of the week just added to http://t.co/VbtHG8Lui3 #ukedchat
— Graham Andre' (@grahamandre) March 30, 2014

121: Train Revisited

What about the Sequel?

Check out AWESOME "how many ways can something be arranged?" from @afs8math http://t.co/P5dBwtDHau & http://t.co/qPR7Vqw6SU cc @mr_stadel
— Fawn Nguyen (@fawnpnguyen) April 15, 2014

120: Permutations - Arranging the Train

Stop the video before he starts giving out answers.

Well?

Check out AWESOME "how many ways can something be arranged?" from @afs8math http://t.co/P5dBwtDHau & http://t.co/qPR7Vqw6SU cc @mr_stadel
— Fawn Nguyen (@fawnpnguyen) April 15, 2014

119: Area of Triangles

From the Amazingly Amazing Fake Martin Gardner Twitter Account:

Which has the larger area?

• A triangle with sides of 3, 4, and 5.
• A triangle with sides of 300, 400, and 700.

Challenge

• Change one dimension of the 300, 400, 700 triangle to give the two triangles the same area.

"Which has the larger area? A triangle with sides of 3, 4, 5, or a triangle with sides of 300, 400, and 700?" p17 Riddles of the Sphinx
— Martin Gardner tweet (@WWMGT) April 27, 2014

118: Questions for Sale

Got this email today:

I teach high school math ... is that going to be enough?
At $99 for a school-wide license, is it worth it?

117: Fibonacci divided by Twos

1/4 + 1/8 + 2/16 + 3/32 + 5/64 + 8/128 + 13/256 + 21/512 + 34/1024 + ...

1/4 + 1/8 + 2/16 + 3/32 + 5/64 + 8/128 + 13/256 + 21/512 + 34/1024 + ... = ?
— Matt Enlow (@CmonMattTHINK) April 16, 2014

116: Dating

It's a well-known formula ... your age divided by two, and add seven. If your date is any younger than that, it's just creepy.

By this calculation, what is the youngest you are allowed to go on a date?