266: Lending $50

If I were to give you $50 we'd have the same amount. If you were to give me $50 I'd have 9 times as much as you. How much do we each have?

Source.

265: Nix The Tricks

Dear Students:

• Which of the following "tricks" was helpful to you in learning and understanding what was going on?
• Which ones just got you confused when the problem changed form slightly?
• Did you ever confuse $\dfrac{3}{5}x\dfrac{1}{5}$ with $\dfrac{3}{5} = \dfrac{x}{5}$ with $\dfrac{3}{5} + \dfrac{1}{4}$?

Which "tricks" should teachers stop using?

264: Add Operators

Using only the symbols +, -, *, and ÷ as replacement for the blanks, find the greatest value of the expression given.

1 ___ 2 ___ 3 ___ 4 ___ 5 ___ 6 ___ 7 ___ 8

What would be the greatest value if you could also use parentheses?

How do you know this is the largest value?


source: Algebra Problems, Dale Seymour.

263: Make Equal with Parentheses

Insert parentheses into each of the following expressions so that each will have the same value.

4 + 3 * 7 - 4

2 * 3 + 3 * 5

2 * 5 - ½ * 10 * 9

3² ÷ ⅓ + 3 * 6

Algebra Problems, Dale Seymour.

262: Fractions by hand

You don't need a calculator for this one either. Why not?

Source: UVM, 2011

261: Logarithm by hand

Why don't we need a calculator for this problem?

Source: UVM, 2011

260: Annual Factorial

Why don't we need a calculator for this?

Source: UVM, 2011

259: Another Sets Puzzle

There are 40 students in the Travel Club. They discovered that 17 members have visited Mexico, 28 have visited Canada, 10 have been to England, 12 have visited both Mexico and Canada, 3 have been only to England and 4 have been only to Mexico. Some club members have not been to any of the three foreign countries and an equal number have been to all three countries. How many students have been to all three countries?

Source: UVM, 2011

258: Roots of an equation.

There's nothing RealWorld about this question.

The two roots of the quadratic equation x² - 85x + c = 0 are prime numbers. What is the value of c?

What part of that sentence is the most important - that couldn't be left out and still have a solution.?

257: Two Triangles

We have this "hard" SAT question:

In the figure above, if AB = 4, BC = 24, and AD = 26, then CD = ?

How could we redraw this to make it simpler to see and understand the solution?

256: Nets - Rectangular Prism

Which of these could be a net for a rectangular prism (a box)?
If it cannot be folded into a box, which "face" would need to change and in what way?

255: Oscar Winners

Below are are four lists:
Best Actor, Best Actress, Best Director, Best Supporting Actress.

Do you expect to see any significant differences in their ages?

Do you expect to see any changes over the years?

Why are there different numbers of entries?

254: Football Puzzle

From Sam Loyd's Puzzle Encyclopedia:

We lived way back in the country and used to order our ball by mail, according to sizes advertised in a sporting house catalog, which advised patrons to "give the exact number of inches required," and that is where the problem comes in. We were told to give the size in inches, but we did not know whether it meant the number of square inches of rubber on the surface or the number of cubic inches of wind contained within  the ball so we combined  the two principles and ordered a ball which should contain just as many cubic inches of wind as it had superficial inches of surface!

How many of our puzzlists can guess the diameter of the ball which was ordered?

253: Half of the blocks have paint

I liked this puzzle, so I included it.

A rectangular wooden block (not necessarily a cube) is painted on the outside and then divided into one‐unit cubes. It turns out that exactly half of the cubes have paint on them. What were the dimensions of the block before it was painted?

(many answers)

So, where do we start?
What information is really being supplied here?



source.

252: Circle and Quarter-circles

What do you notice about the blue figure and the yellow one?

What could we do to find perimeter?
How about area?

251: Semicircles 1

Let's talk about perimeter. What do you notice about the perimeter of the big semicircle alone and all of the smaller ones together?

What will happen if we changed this so that all the small semicircles were really, really, really small. Wouldn't that essentially be a straight line?

250: The Shadow Knows

The man is walking down the sidewalk in the evening, at a constant speed. As he walks away from the streetlight, he casts a shadow in front of him.

If a second man were to try and keep up with the very front tip of his shadow, would he be walking at a constant speed, accelerating or decelerating, or what?

How do you know your answer is correct?

249: Create some data

Can you create a set of data (more than 12 numbers, please) with a mean of 17 and a median of -17 and a mode of 0?

248: Raising one out of two.

Can you create a data set of more than 5 numbers so that when you add one additional number to the set, you can raise the median but not the mean?

247: SAT Number theory

This is an actual SAT question, from later in a section so it is one that a lot of test-takers got wrong because it was a hard question, was worded strangely, had too many undefined letters, or was near the end of a section and the test-takers simply ran out of time.

What are some of the difficulties you think most students would have with it? What are some things we could do to rewrite the question so it's more clear?

246: Square and Circle

Two opposite vertices of a square lie on a circle. Sure as sunrises, someone is going to ask about area.


What information do we need?

What would be the easiest scenario?

Is there more than one easy scenario?

source.

245: Multiplication

What are some different ways we might find the greater product here?

35 * 29  OR  39 * 25

Can you do this algebraically?
Graphically?

source: Fawn Nguyen @fawnpnguyen

244: Percent Sugar

Do we need any more information to find which has lower percentage of sugar?

243: Powers and Squares

The process seems to be the interesting thing here. How would you begin to work on this?

In how many ways can you write 2ⁿ as a difference of two squares?

source:

In how many ways can you write 2^n as a difference of two squares?
— James Tanton (@jamestanton)

242: Trapezoid Puzzle

figure not drawn to scale.

An isosceles trapezoid has the following properties:

Leg = 30
Diagonal = 82
The difference of the bases is 48

Find the area of the figure.

241: Mode not Median

Given the same numbers as we've been working with for a couple of days, along with an extra number X:

X, 12783, 12784, 12785, 12790, 12792, 12795, 12799, 12801 

What values can X take on so that it is the mode but not the median?

240: Median not Average

Given the same numbers as we've been working with for a couple of days, along with an extra number X:

X, 12783, 12784, 12785, 12790, 12792, 12795, 12799, 12801 

What values can X take on so that it is the median but not the average?

239: Outliers

Given the same numbers as we've been working with for a couple of days, along with an extra number X:

X, 12783, 12784, 12785, 12795, 12792, 12790, 12799, 12801 

How small (or large) do you feel that value of x should be to be considered an outlier?

and of course, Ben Shabad, full-time graduate student and part-time cartoon-drawing person.

238: Skew

Given the same numbers as yesterday:

12783, 12784, 12785, 12790, 12792, 12795, 12799, 12811 

What characteristics can we look at to tell whether this data is skewed?

237: Average

Given the numbers:

12783, 12784, 12785, 12795, 12792, 12790, 12799, 12801 

There are a lot of tricks and shortcuts, and of course, the pencil and paper add-em-up method of finding the average of a set of numbers.
What's your best way to get the average of those eight numbers?