503: Circles

This is a straightforward question. I'd like to make all of you students into teachers for a minute ... Let's create a test question !

• Do we have to specify angle AOB?
• Is there a better way to say something without actually saying it?
• What other instructions and given information could we provide that would lead to the same answer? 
• What is the best question here?

414: Special Right Triangles

#mathsTLP Really must share this... Pythagoras meets surds - what's so special about these triangles? Higher GCSE pic.twitter.com/0v8vB6dEW4
— James Pearce (@MathsPadJames) March 22, 2015

384: Inscribed Squares

In terms of "this inscribed in that", what do we have here?
Describe the squares and their sizes compared to the circle.

1. If you were to calculate distances (radii, side lengths) or areas, what given information would you like to start with and what would you like to solve for?

• What "givens -- answer" path would be easiest?
• What "givens -- answer" path would be difficult?
• Which would you choose to give to your fellow students as a collaborative question?

N.B. The original question is slightly different. You can see it below.


Source.
Original image, click to enlarge.

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?

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?

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?