424: Point-point-slope

Challenges

1. Use four different digits (from 2-9) to create two points which determine a line with the greatest possible slope.
2. Use four different digits (from 2-9) to create two points which determine a line with the least possible slope.
3. Use four different digits (from 2-9) to create two points which determine a line with a slope as close to zero as possible.

source.

What does "greatest" slope mean?
What does "least" slope mean ... most negative?

423: If only

There's a neat game you can play with the English language:

Place the word "only" anywhere in the sentence. How does it change the meaning?

Only she told him that she loved him.
She only told him that she loved him.
She told only him that she loved him.
She told him only that she loved him.
She told him that only she loved him.
She told him that she only loved him.
She told him that she loved only him.
She told him that she loved him only.

What would it look like if we were trying to create a math problem that worked the same way, except "only" is replaced by the equals sign?

422: Inscibed Semicircle, part three

Last question with this visual:

How could you draw the inscribed semicircle (area = π) so that the rectangle is of maximum size?

A semicircle is inscribed in a rectangle. If the area of the semicircle is π, area of rect=? [4] #SATPrep #mathchat #Geometry
— David Marain (@dmarain) March 26, 2015

421: Inscribed Semicircle, part two

This problem was posed on Twitter the other day.

A semicircle is inscribed in a rectangle. If the area of the semicircle is π, what's the area of the rectangle? [4]

My question yesterday was ... how could you draw the inscribed semicircle in a way that gives a rectangle of a very different (and larger) area?

My question today is, given the arrangement below, what points did I choose that had integer coordinates? I chose a larger semicircle - for convenience - how big was it?

A semicircle is inscribed in a rectangle. If the area of the semicircle is π, area of rect=? [4] #SATPrep #mathchat #Geometry
— David Marain (@dmarain) March 26, 2015

420: Inscribed Semicircle

This problem was posed on Twitter the other day.

A semicircle is inscribed in a rectangle. If the area of the semicircle is π, what's the area of the rectangle? [4]

My question is ... how could you draw the inscribed semicircle in a way that gives a rectangle of a very different (and larger) area?

A semicircle is inscribed in a rectangle. If the area of the semicircle is π, area of rect=? [4] #SATPrep #mathchat #Geometry
— David Marain (@dmarain) March 26, 2015

419: The Value of Why

Can you find values for y that will make each answer true?
Generalize the rules here.

418: Which is larger?

Which is true?

Left side is larger than Right side

Right Side si larger than Left Side.

The two sides are equal.

Not enough information?

How can we deal with this WITHOUT a calculator?