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?

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?

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

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?

355: Millionaire

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

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?

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

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?

351: Balance 6

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

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?

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?

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?

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?

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?

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.

Is this die loaded?
What is the probability he will get a 6 on the next roll?

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?

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?

342: Algebret

That takes care of four letters.

twenty-two to go ....

341: Sensible or Not? Study Types

Defend your answer! Here's your claim ... what's your warrant?
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.

340: Giving Raises

A small company wants to give raises to their 5 employees. They have $10,000 available to distribute. Imagine that you are in charge of deciding how the raises should be determined.

1. What are some variables you should consider?
2. Describe mathematically as many different methods you can think of to distribute the raises. (We came up with nine; can you beat that?)
3. What information will you need to compute those raises according to your various methods?
4. Which of your methods do you feel is the most fair?

source: illustrativemathematics.org

Use an atbash cipher decoder: http://rumkin.com/tools/cipher/atbash.php

1. zm vjfzo znlfmg, gdl gslfhzmw vzxs.
2. vjfzo kvixvmgztv yzhvw lm hzozib.
3. mvklgrhn: 1p, 1p, 1p, 1p, Ldmvi'h hlm: 6p.
4. R'n lmv lu gsv vnkolbvvh: 0p, 0p, 0p, 0p, nv: 10p
5. kilwfxgrergb tlzoh: 1p, 1p, 2p, 2p, 4p
6. olggvib, zoo li mlgsrmt.
7. olggvib, 0p, 1p, 2p, 3p, 4p.
8. zokszyvgrxzo liwvi, 0p, 1p, 2p, 3p, 4p.
9. vnkolbvv'h xsrow'h gvhg hxlivh rm gsv olxzo hxsllo ... ru gsviv'h ml rnkilevnvmg, ml ylmfh.

334: Which values of x do we choose? Absolutes

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?

$y = \lvert x \rvert$

$y = \lvert x \rvert - x$

$y = 2\lvert x+2 \rvert$

$y = \dfrac{1}{2}\lvert x-3 \rvert$

What are your favorite examples of this?

333: Which values of x do we choose? Radicals

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?

$y = \sqrt{x+2}$

$y = \sqrt{1-x}$

$y = \sqrt[3]{2x+1}$

What are your favorite examples of this?

332: Which values of x do we choose? Rational functions

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?

$y = \dfrac{4}{(x+2)(x+11)}$

$y = \dfrac{1}{x^2+1}$

$y = \dfrac{1}{x^-1}$

$ y = \dfrac{x+2}{x^2-4}$

$y=\dfrac{x^2-1}{x^2+1}$

What are your favorite examples of this?

331: Which values of x do we choose? Trigonometry.

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?

$y = sin(3\theta)$

$y = \Big{sin(\theta)}^2$

$y = sin^{-1}(\theta)$

$r = 3sin(3\theta)$

What are your favorite examples of this?

330: Which values of x do we choose? Linear Functions

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?

$y = \dfrac{1}{3}x + 7$

$y = \dfrac{2}{7}(x + 4)$

$y = \dfrac{x + 2}{5}$

What are your favorite examples of this?

329: Which values of x do we choose? Quadratics

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?

$y = (x+2)(x-4)$

$y = -(x+1)(x+5)$

$y = x^2 + 6x + 9$

$y = \dfrac{1}{4}(x+5)^2$

What are your favorite examples of this?

328: Which values of x do we choose? Conics

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?

$x^2+y^2=25$ 

$\dfrac{(x-1)^2}{9}+\dfrac{(y+2)^2}{16} = 1$

$\dfrac{(x-4)^2}{9}-\dfrac{(y-4)^2}{16} = 1$

What are your favorite examples of this?

327: Sensible or Not? Sampling

Defend your answer! Here's your claim ... what's your warrant?
Give an example or a situation to bolster your position.

1. For a high school project, I conducted a census to determine the average rate that teenagers charge for babysitting. 
2. Even though the study used a convenience sample, the results may be meaningful. 
3. The study must have been biased, because it concluded that the 75% of Americans are more than 6 feet tall. 
4. We obtained a simple random sample of milk-producing cows in Jefferson County by drawing the names of 50 dairy farms from a hat and asking the owners of those farms to select three cows for us to study. 
5. A good strategy for stratified sampling involves first using simple random sampling to choose 500 people, then randomly dividing them into ten groups of 50 to represent the 10 strata. 
6. Although the study was conducted with a representative sample and careful analysis, the conclusions still reflect the researcher's anti-death penalty bias. 

ch1.2

326: ASN - Quadrilaterals

source

an image from there:

325: ASN - Another Algebra Set

Always True, Sometimes True, Never True?

Once again ... The why is the important part.

source, with lesson plans and materials.

324: ASN - Algebra

Always True, Sometimes True, or Never True?

Cut out the cards and have students sort them into one of the three categories.
In each case, students need to say WHY its always true or NEVER true, or give an example and a counterexample for when it is SOMETIMES true.

source: lesson plan and handouts.

323: Sensible or Not? Study structure.

Defend your answer! Here's your claim ... what's your warrant?
Give an example or a situation to bolster your position.

1. When the IRS decided to determine how many people were cheating on their taxes, they did a study in which the sample consisted of every adult in the United States.
2. My professor conducted a study in which he was unable to measure any sample statistics, but he succeeded in determining the population parameters with a very small margin of error.
3. A poll conducted two weeks before the election found that Smith would get 70% of the vote, with a margin of error of 3%, but he ended up losing anyway.
4. The goal of a new startup company is to compete with Nielsen Media Research in compiling television ratings. They intend to succeed by providing data with a larger margin of error than Nielsen's while charging television stations the same price for their service.
5. The goal of the research is to learn about depression among people who have suffered through a family tragedy, so the population of interest is everyone who has been sick in the past month.
6. We know for certain that a majority of Americans support the President's position on this issue because an opinion poll found support from 65% of Americans, with a margin of error of 5%.

ch1.1

322: Sensible or Not? distribution

Distribution Statements: Sensible or Not?
Defend your answer! Here's your claim ... what's your warrant?

1. Because this data has two modes, it cannot be symmetrical. 
2. This distribution is left-skewed because it has outliers to the left. 
3. Josh works at a veterinary clinic and weighs the dogs. He claims there is less variation in the weights of 10 Rottweilers than in the weights of 10 dogs of different breeds. 
4. Josh combines the weights of 10 toy poodles, 10 Rottweilers and 10 St. Bernards into one big group. He says the distribution has one mode. 
5. Jean concludes that the mean of her symmetric distribution is greater than the median. 

ch4.2

321: Sensible or Not? central measures

Measures of Central Tendency Statements: Sensible or Not?
Defend your answer! Here's your claim ... what's your warrant?
Give an example or a situation to bolster your position 

1. A data set should be discarded if the mean exceeds the mode. 
2. A student with an average of 65 computes her new average after earning a 70 on the last exam. Her new average is 72. 
3. Observing that the mean weight of a group of patients is 154 pounds and the median weight is 145 pounds, the doctor concludes that there must be an outlier on the heavy side. 
4. Noting that there are three modes in his data set, Rob assumes there was an error in his data gathering. 
5. The two means in the data lie at 102 and 201
6. The two medians in the data set lie at 23 and 28. 
7. The two modes in the data set lie at 45 and 100. 

ch4.1

320: Sensible or Not? percent

Percent Statements: Sensible or Not?
Defend your answer! Here's your claim ... what's your warrant?

1. The percentage of households with more than four children decreased by 100,000 households
2. Brent makes 120% less than Bill each month 
3. Ann is 10% taller than Brenda, so Brenda is 10% shorter than Ann
4. Fifty percent of the people in the room are men and 50% of the people in the room are single. Therefore, 25% of the people in the room are single men.
5. Pete’s prices are 10% more than Paul’s prices, so Pete’s prices are 110% of Paul’s prices.
6. The interest rate at the bank increased by 100% 

ch2.3

319: Sensible or Not? error

Error Statements: Sensible or Not?
Defend your answer! Here's your claim ... what's your warrant?

1. There are 138,232 species of butterflies and moths on the Earth. 
2. The measurement taken by an electronic timer must be more accurate than that taken by stopwatch. 
3. The relative error that a microbiologist makes in measuring a cell must be less than the relative error that an astronomer makes in measuring the width of a galaxy, because cells are smaller than galaxies. 
4. The bank teller claims that his errors are random even though they are always to his advantage. 
5. The 6 billionth person on Earth was born on October 12th 1999, in Bosnia 
6. I would rather be shortchanged by $1 than by 1% 

ch2.2

318: Exponential Function

Is this enough information to find the equation?

317: Church

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

316: Intersections

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

315: Pizza Maker

 

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

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.

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.

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

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

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!

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:

Edge of a cube has length 2, P is the midpoint of an edge. Show that the greatest distance from P to a vertex is 3.
- David Marain (@dmarain) October 28, 2014

308: Congruent Lines?

Can two lines of infinite length be considered congruent?

Student question that sparked a good debate: are all lines congruent? Not segments, but whole lines. Thoughts?
— Justin (@JustinAion) October 20, 2014

307: Expected Value vs Psychology

Which would you press ... and why?

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?

Plot of log(g(n)) versus n @jamestanton g(n) = smallest integer such that g(n)*n! is a perfect square. pic.twitter.com/tUvhiR0LjI
— Republic of Math (@republicofmath) October 18, 2014

305: Space and the Atmosphere

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

304: Speed Reading

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

Reading at 500 words a minute? Maybe it's all in the delivery. — Ed Southall (@solvemymaths) October 18, 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"

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?

Where should we start this shape substitution puzzle?

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

301: Systems of a Sort 7

What's the best way to proceed here?

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:

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?

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?