509: Circles 2

Several days ago, I posted a slightly different question about tangential circles and the spaces in between. Compare this question, from the 2004 SAT Practice Test to that question from Emma Bell.

• Which question seems harder? What are the difficult aspects of each?
• Do we have to specify the angle APB?
• How are these two questions different in terms of the knowledge they require for solving?
• Is this question made harder by the "how many times" part? Does that phrasing make the question unnatural?

source: ETS, 2004

508: Shorter Path

What car will have the shorter path?
How could you tell for certain?

source: Justin Aion

507: Orderly Probability

Scenario 1: Which winning number group is more likely to occur?

1-2-3-4-5-6  OR 4-8-15-16-23-42

Scenario 2: Which winning number group is more likely to occur if the numbers are drawn in any order and THEN put into ascending order by the presenter?

1-2-3-4-5-6  OR 4-8-15-16-23-42

In which of the above two scenarios is getting the winning numbers more likely?

source: Jeff Suzuki

506: The Conical Tank

The last of the three related-rate geogebra problems from Kate Nowak.  It's the related rate problem from calculus: the conical tank being filled with water.

Adjust the slider and... wait, what is changing and how?

For every click of the slider:
Is the depth increasing at a constant rate?
Is the radius increasing at a constant rate?
Is the volume increasing at a constant rate?
 How can you tell?

• Where or how, in the RealWorld^tm, could we see the constant increase in volume?
• Where or how, in the RealWorld^tm, could we see the constant increase in radius, or depth?

If you want to play with the animation, Conical Tank Problem. source: @k8nowak

505: The Balloon Problem

We've all seen this problem, but many of our students haven't.  It's the related rate problem from calculus: the balloon being filled with air.

There are two questions being demonstrated here.
(1) "If the volume increases at a constant rate, what is happening to the radius?" and
(2) "If the radius increases at a constant rate, what is happening to the volume?"

The first question is to figure out which situation is modeled in red and which in blue.
Then we can ask:

• Does the radius increase at a constant speed in both models? How can you tell?
• Does the volume increase at a constant speed in both models? How can you tell?
• Where or how, in the RealWorld^tm, could we see the constant increase in volume?
• Where or how, in the RealWorld^tm, could we see the constant increase in radius?

If you want to play with the animation, Balloon Problem. source: @k8nowak

504: The Ladder Problem

We've all seen this problem, but many of our students haven't.

It's the related rate problem from calculus: the ladder sliding down the wall.

The "official" question?

How fast is the ladder's top sliding down the wall if the bottom is being pulled out at a rate of 1 ft/sec?

We can ask a few questions of kids at any level, though, based on the given that the bottom of the ladder is being pulled to the left at 1 foot per sec.

• Does the top drop at a constant speed?
• Does the top drop a distance equal to the horizontal movement?
• When is the speed of the top greater than 1, less than 1, and equal to 1?
• If this is a 25 foot ladder, with the bottom 7 feet out from the base of the wall, and the top drops 4 feet ... how far out does the bottom of the ladder have to go?

If you want to play with the animation, Ladder Problem. source: @k8nowak

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?

502: Powerful Question

It's not included in the PEMDAS Order of Operations ...

Should $a^{b^c} = ({a^b})^c$ or should it be $a^{b^c} = a^{(b^c)}$ ??

Does $3^{2^0}$ equal 1 or 3?
 
Let's just consider easy numbers {1, 2, 3, 4} so we can explore. What's the probability that the two methods arrive at the same answer?

For the record,  $a^{b^c} = a^{(b^c)}$ is the accepted order of operations here.