524: Is a HotDog a sandwich?

Does the topology matter? If I were to completely separate the two halves, would it make a difference?

False Negatives and Cancer.

In the voiceover, Cologuard claims it finds 92% of colon cancers. The ad also provides a generously large and easy legible banner stating that false positives and false negative results can occur. 

The next banner gives more detail.

If your grandparent gets a "Yes" response, what is the probability that they do have colon cancer? If they get a "No", what are the chances they don't have it?

https://www.ispot.tv/ad/op0r/cologuard-finding-things

If you've never paid much attention to advertisements before, this is pretty amazing. Usually the fine print is illegible even on a 62" ultra hi-definition tv paused at just the right moment.

So ... thank you Cologuard for being honest and up-front.

522: Where to start with factoring

You are asked to find the roots and factors of the following polynomial function:
f(x) = x⁴ – 2x³ – 18x² + 6x + 45

By the rational root theorem, possible rational roots are
± 1, 3, 5, 9, 15, or 45.

In order to minimize your effort, you know that you should begin with the possibility that is most likely to be a root. Which one is most likely?

521: When is an outlier still an outlier?

In a set of data, can the outlier also be the mode or does it stop being an outlier?

520: Dan or Mike? Who's Better at Doodle Jump?

Every year I give this graphic comparing the Doodle Jump stats for @ddmeyer and his friend Mike.


Give a couple of reasons why we could consider Dan the better player.
Give a couple of reasons why we could consider Mike the better player.

Measures of Central Tendency.

519: Concavity

I have a question that will help me understand the fundamental definition of concavity. If we have y=x^4, can we say that is concave up from (-inf, +inf) or do we need to say (-inf,0)U(0,+inf)?

518: Interpret your results.

Say you had data that told you that
The proportion of all US public school students whose families are low-income:
In 1989, less than ⅓
In 2013, more than ½

Interpret this in a positive way.
Interpret this in a negative way.

Proportion of all US public school students whose families are low-income: In 1989, <1/3. In 2013, >1/2.

— Alfie Kohn (@alfiekohn) December 21, 2016

517: What is the function?

Algebra 2:
Write a function for this graph ...

Estimation:
What are some possible values for a,b,c,d?
If I told you it also went through (0,54), what might the leading coefficient be?

I would not use the phrase "in factored form", if I were using this in a review or summative assessment since students should be selecting the form that's most appropriate.  I might use it if we are in the middle of learning about functions for the first time and we hadn't really made the case for the utility of various forms of the equation. 

source:

How would your Ss write a function in factored form for this graph? Nice #MP7 #CthenC problem from Bossé #ncctm16 pic.twitter.com/DoUZUzNR7L

— Jennifer Wilson (@jwilson828) October 28, 2016

516: Data

Is money a discrete variable or a continuous one?

515: Pascal Proof

An interesting Question ... is an algebraic method the only way to do this?

Prove that anywhere on Pascal's triangle, products of numbers in yellow and in orange are equal.
Via @HopeHana pic.twitter.com/zgs3d0wenJ

— ⋁△ ᴹᴬᵀᴴcafé ☕ (@Five_Triangles) August 18, 2016

514: Probability Game

Temporary return! A quick probability question ...

Player A's score is determined by taking the highest of 3 dice.
Player B's is determined by taking second-highest of 8.
Who wins more games?

Player A's score is determined by taking the highest of 3 dice.
Player B's is determined by taking second-highest of 8.
Who wins more games?

— Ben Orlin (@benorlin) June 16, 2016

513: Compound Area

Can you come up with other, better, divisions?

source.

512: Factory Ratios 3

We started with this:

In a factory, the ratio of men to women is 2:3. The ratio of right-handed men to left-handed men is 7:3. The ratio of right-handed women to left-handed women is 11:1. What fraction of the factory workforce is right-handed?
And then we extended with: What is the fewest number of employees possible in this building?

If the ugly-sweater party was held in that factory with the minimum number of workers possible and the  ratio of red to green ugly sweaters was 7:13, how likely is it that a left-handed man wore a red ugly sweater?

Source:

New GCSE Maths question: Ratio and proportion is the buzz thing! How are your students going to tackle it? pic.twitter.com/RyPsmc9z4m

— m4ths.com (@m4thsdotcom) December 16, 2015

511: Factory Ratios 2

We started with this:

In a factory, the ratio of men to women is 2:3.
The ratio of right-handed men to left-handed men is 7:3
The ratio of right-handed women to left-handed women is 11:1
What fraction of the factory workforce is right-handed?

But let's extend things a bit.

What is the fewest number of employees possible in this building?

Source:

New GCSE Maths question: Ratio and proportion is the buzz thing! How are your students going to tackle it? pic.twitter.com/RyPsmc9z4m

— m4ths.com (@m4thsdotcom) December 16, 2015

510: Factory Ratios 1

In a factory, the ratio of men to women is 2:3.
The ratio of right-handed men to left-handed men is 7:3
The ratio of right-handed women to left-handed women is 11:1

What fraction of the factory workforce is right-handed?


Source:

New GCSE Maths question: Ratio and proportion is the buzz thing! How are your students going to tackle it? pic.twitter.com/RyPsmc9z4m

— m4ths.com (@m4thsdotcom) December 16, 2015

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