## Wednesday, December 31, 2014

### 359: Lines and Diagonals

There's sequences all over the place. What are they?

Does your sequence predict the 20th case?

And the 11th and 12th?

## Tuesday, December 30, 2014

### 358: Division and Remainders

Is that diagram correct?

What if the question were 13 ÷ 5 = 2 R3? Could we diagram it in the same way?

## Monday, December 29, 2014

### 357: Box Office Receipts for The Interview.

"Sony doesn't say" .... but do we have enough information to tell anyway?
‘The Interview’ Brings In $15 Million on Web LOS ANGELES — “The Interview” generated roughly$15 million in online sales and rentals during its first four days of availability, Sony Pictures said on Sunday.

Sony did not say how much of that total represented $6 digital rentals versus$15 sales. The studio said there were about two million transactions over all.

source. @ddmeyer

## Sunday, December 28, 2014

### 356: Partridge in a Number Tree

You know the song. "And a Partridge in a Pear tree." What patterns of numbers can we find here?

 The first partridge.

If you look at the total gifts each day, what sequence of numbers is this?

 Four Calling Birds, calling out numbers ...
How many different ways are there to find the total number of gifts given over the twelve days?

## Saturday, December 27, 2014

### 355: Millionaire

Would you guess? $250,000 if correct and$100,000 if she refuses to try for it.

## Friday, December 26, 2014

### 354: How to Teach Division

So, students. You've had a chance to weigh in on addition and multiplication.

What is the best way to do division?

## Thursday, December 25, 2014

### 353: How to Teach Multiplication

There are several ways to teach multiplication. Many people seem to feel that students know best how they learn, so I'm asking students to weigh in on this particular issue.

Which method is best? Is there a difference between what we should be doing with elementary students and with high school students? With how much and with what do students need to graduate high school and enter the RealWorld?

Hindu Lattice Method Grid Method

Standard Algorithm

Japanese Sticks

## Wednesday, December 24, 2014

### 352: How to Teach Addition

Everyone seems to have an opinion and now, students, we're asking for yours.

Scott Macleod says, "We now live within multidirectional conversation spaces in which 12-year-olds can reach audiences at scales that previously were reserved for major media companies, large corporations, and governments. We all now can have a voice. We all now can be publishers. We all now can find each other’s thoughts and ideas and can share, cooperate, collaborate, and take collective action."

So how should addition, and by extension subtraction, be taught?The standard algorithm or by piecewise addition?

Weigh in on the "Letter to Jack".

How would you teach these two problems?

## Tuesday, December 23, 2014

### 351: Balance 6

Weights 1lb through 6lbs.
Where should we start?

## Monday, December 22, 2014

### 350: Primes

This little puzzle, via @mathmovesu, asks for three consecutive prime numbers.

Is the guess and check method the best way to go here?

Which prime numbers are candidates and which ones can we safely ignore?

## Sunday, December 21, 2014

### 349: Hole-in-One Insurance

If the average golfer is able to get a hole-in-one once in approximately 3000 rounds of golf (18 holes apiece), then what is the probability of any one of 100 average golfers getting a hole-in-one on the 5th hole during the weekend golf tournament?

What's the best way to find this out if you're the insurance company that will write this policy?

## Saturday, December 20, 2014

### 348: Homer's Pythagorean proposition

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

Wait, didn't Fermat say this was impossible?
What's a three-second way to tell that this equation is false?

## Friday, December 19, 2014

### 347: Combinatorics

Consider eight objects. We will choose them one at a time, two at a time, three at a time, and so on.

Which of these will result in identical numbers of ways?
Why?

## Thursday, December 18, 2014

### 346: Casting the Play

The cast of a school play that requires 4 girls and 3 boys is to be selected from 7 eligible girls and 9 eligible boys.

• Will it be a different calculation if the boys are willing to play girls' parts, as in Shakespeare's time? If so, how will it be different?

## Wednesday, December 17, 2014

### 345: Fair or Foul?

Sullivan bought a die at the magic shop. He rolls it 155 times and gets the following results:
• ONE: twenty-eight times
• TWO: twenty times
• THREE: fifteen times
• FOUR: thirty-one times
• FIVE: thirty-two times
• SIX: twenty-nine times.
What is the probability he will get a 6 on the next roll?

## Tuesday, December 16, 2014

### 344: Monty Hall

Once upon a time, the world's smartest person (Marilyn vos Savant, IQ: 228) received a question for her newspaper column …
Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors? Craig. F. Whitaker, Columbia, MD

Marilyn's answer was surprising to many people. What do you think?

## Wednesday, December 10, 2014

### 343: Forces and Friction

Your teenage son has a fast car.

He knows that friction is determined by the weight of the car over the wheels, the "normal" force. He also knows that additional weight means that the car can't accelerate as fast, but he's also having problems with the rear tires spinning out.  He's convinced that having Fat Eddie sit in the back will help his quarter-mile time.

Will the extra weight help him or hurt him?

## Tuesday, December 9, 2014

### 342: Algebret

That takes care of four letters.
twenty-two to go ....

## Monday, December 8, 2014

### 341: Sensible or Not? Study Types

Give an example or a situation to bolster your position
1. The Department of Education conducted an observational study to determine the average salary of high school teachers in each of the United States
2. A paint company conducted a double-blind experiment to determine which of two types of exterior paint was more resistant to rain.
3. The lab conducted an experiment to determine whether the throat culture was positive for strep.
4. In a study of medications designed to slow the rate of balding in men, a placebo group had better results than the control group.
5. A meta-analysis was conducted to determine the population of New Zealand.
6. A case-control experiment was used to determine the average family size in Utah.

ch2.2

## Friday, November 21, 2014

### 318: Exponential Function

Is this enough information to find the equation?

## Thursday, November 20, 2014

### 317: Church

How many different math concepts can you use this church as an example for?

## Wednesday, November 19, 2014

### 316: Intersections

Which one is better?
Which one is easier to solve?

## Tuesday, November 18, 2014

### 315: Pizza Maker

Talk about the rate at which pizza sauce is being pumped out through the tube ...

## Monday, November 17, 2014

### 314: Stats Starter 2

Following on from yesterday's question, we have a puzzle from the same source.

Which of those central tendency statistics are necessary in order to find the missing numbers? (Necessary meaning that you can't find a particular number without it.) In order words, do we really need all five statistics?

Source.

## Sunday, November 16, 2014

### 313: Stats Starter 1c

Here is that same list of numbers.

You create a new problem this time ...
Can you create a solvable problem with just two hints?
Or do we need three?

Original source.

## Saturday, November 15, 2014

### 312: Stats Starter 1b

Here is a list of numbers.

You create the problem ...
What are some different statistics that you could give to a classmate (other than the ones below) yet still keep this a solvable problem, with the same answers as the original problem?

Original source.

Original problem:
mean = 76; range = 32; IQR = 21

## Friday, November 14, 2014

### 311: Stats Starter 1a

Here is a list of numbers.

What information would you NEED in order to determine the missing numbers?

Source.

If you want to solve the original problem yourself, you can go there and look for the rest of the problem and a discussion on finding the answers, but here is the set of numbers provided:
mean = 76; range = 32; IQR = 21

## Thursday, November 13, 2014

### 310: Missing Area

What things do we know?
What lines do we need to construct?
What unknowns do we need to plop a variable on?

Is this a problem best given to a Geometry class, an Algebra class, or Pre-Calculus class?

Source, For the Nguyen!

## Wednesday, November 12, 2014

### 309: Rational Cube Routes

Imagine a cube, 2 inches on a side ...

Actually, don't bother, I'll put one over there on the right side. ==>>

Now imagine if you were sitting on a vertex. How far is it (straight line distance) to the other vertices?

How many of those paths would be rational number distances?

What if you were on the midpoint of an edge and considering the paths to the vertices again. How many of those paths would have rational lengths distances?

And, no, I won't apologize for the pun. Pfft!

source:

## Tuesday, November 11, 2014

### 308: Congruent Lines?

Can two lines of infinite length be considered congruent?

## Monday, November 10, 2014

### 307: Expected Value vs Psychology

Which would you press ... and why?

## Sunday, November 9, 2014

### 306: Factorials and Perfect Squares

Let's examine the function g(n):

g(n) = smallest integer such that g(n)*n! is a perfect square.

How should we go about finding if there's a pattern in that?

## Saturday, November 8, 2014

### 305: Space and the Atmosphere

What kind of approach might give us the thickness of the atmosphere in the video?

## Friday, November 7, 2014

What's a good problem you can make based on this video?

## Thursday, November 6, 2014

### 303: Reworking Pythagoras

Pythagorean theorem:  a² + b² = c².

“In a right triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two legs.”

Here is the problem: does the figure whose areas we compare, drawn on the triangle's legs, have to be square?

Can there be other shapes – triangles, rhombuses, regular pentagons, etc. – that make the Pythagorean Theorem more generally true?

Source: Grant Wiggins, "The Problem of So-Called Problems - unpublished paper 2013"

## Wednesday, November 5, 2014

### 332: Which values of x do we choose? Polynomials

For the following functions, think about "How to graph like a math teacher." Math teachers want to sketch graphs quickly and efficiently and choose values of x that work "nicely" in the equation and  generate integer values of y, thus making it easier to graph.
• Which points are best found by inspection?
• Which points are best found by substitution?
• Which points are best found by symmetry?

$y = (x-2)^2(x+2)$

$y = (x-1)^3$

$y = x^3-8$

$y=x^3+3x^2$

$y=(x-3)^2(x+3)^2$

What are your favorite examples of this?

### 302: Systems of a Sort 8

Where should we start this shape substitution puzzle?

Source: Mimi! (I Hope This Old Train Breaks Down...)

## Tuesday, November 4, 2014

### 301: Systems of a Sort 7

What's the best way to proceed here?

## Monday, November 3, 2014

### 300: Systems of a Sort 6

Your teacher (me) is thinking about buying a new car. Presently, the cost of gas is $3.60 per gallon and he knows that he is going to commute to work each day and drive errands on the weekend ... about 300 miles per week. He wonders whether or not to buy a Corolla or a Prius. Currently, he drives a Ford Ranger pickup truck, which is paid off but is starting to have some expensive repair bills. If he averages those costs, it's about$200 per month.

Option 1: The Corolla costs $16,800 (about$400/month) and gets roughly 36 mpg.
Option 2: The Prius costs $24,200 (about$570/month) and gets roughly 49 mpg.
Option 3: The Ranger is paid off but has repairs (about \$200/month) and gets 21 mpg.
What should he do?

Prius:

Corolla:

## Sunday, November 2, 2014

### 299: Systems of a Sort 5

Is there an easy way to tell if those lines will have one solution, no solution, or an infinite number of solutions?

## Saturday, November 1, 2014

### 298: Systems of a Sort 4

This system of equations has a peculiar characteristic ... I think it is easier to solve it by analytical means than by using Desmos or a TI-84.

Do you agree?

What about it makes a graphical solution difficult?