Correlation Coefficients - A Graphic Organizer for Your Students

Just a simple graphic that might bring clarity to the idea of correlation coefficients. A.4A

Quadratic Transformations - Tying it together

A.7C offers a new twist on an old idea.

Parabolas still move up and down, but they also move left and right now.

They still get narrower and wider, but we also describe those changes as compressions and stretches. And while the old standards only included vertical stretches and compressions, the new standards discuss horizontal stretches and compressions.

Distinguishing between horizontal compressions and vertical stretches is tricky. Take a look at this example:

Horizontal Compression: f(x) = (2x)^2  

Vertical Stretch: f(x) = 2x^2

Both of these transformations will result in a graph that is narrower than the parent function. An important difference to notice is that the narrowing effect of the horizontal compression will be stronger than that of the vertical stretch since the "a" coefficient of the compression is squared.

One more point to keep in mind: a horizontal compression of f(x) = (2x)^2 is equivalent to a vertical stretch of f(x) = 4x^2! The effect on the graph of the parabola will be exactly the same; we can only distinguish between the two when we look at the value and placement of the coefficient in the equation.

Fun with images in Google Slides

So I just learned this new trick that you can use to make your images in Google Slides look even better. Watch this one-minute tutorial to learn how to mask images.

Before

After

You could mask the images using circles...

Or alternating trapezoids...

Or thought bubbles...

Choose from dozens of different shapes!

Have fun!

Danger averted! Is it dangerous to swim after watching a Nicholas Cage film?

Texas is adding a few new standards to the algebra I EOC test this year, three of which deal with correlations between real-world sets of data.

• A.4A - Calculate the correlation coefficient using technology
• A.4B - Compare and contrast association and causation in real-world problems
• A.4C - Write linear functions the provide a reasonable fit to data to make predictions

My favorite is A.4B because it's so great for discussion. For example, the number of films with Nicholas Cage correlates closely with the number of people drowned by falling into a pool.

Image from http://www.tylervigen.com/spurious-correlations 

So should we close down swimming pools during years when Nicholas Cage makes a lot of movies?

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Understanding causation is crucial because it guides our decisions and policies. Stats with Cats identifies 6 key purposes for studying causation.

"For example, if you can figure out what causes a condition or event, you can:

• Promote the relationship to reap benefits, such as between agricultural methods and crop production or pharmaceuticals and recovery from illnesses.

• Prevent the cause to avoid harmful consequences, such as airline crashes and manufacturing defects.

• Prepare for unavoidable harmful consequences, such as natural disasters, like floods.

• Prosecute the perpetrator of the cause, as in law, or lay blame, as in politics.

• Pontificate about what might happen in the future if the same relationship occurs, such as in economics.

• Probe for knowledge based on nothing more than curiosity, such as how cats purr."

But since correlation doesn't prove causation, how can we tell whether two variables are correlated due to a third factor or due to a genuine causal relationship?

LearnAndTeachStatstics lists 9 criteria identified in Chance Encounters by Wild and Saber.

1. Strong relationship: For example illness is four times as likely among people exposed to a possible cause as it is for those who are not exposed.
2. Strong research design
3. Temporal relationship: The cause must precede the effect.
4. Dose-response relationship: Higher exposure leads to a higher proportion of people affected.
5. Reversible association: Removal of the cause reduces the incidence of the effect.
6. Consistency: Multiple studies in different locations producing similar effects
7. Biological plausibility: there is a supportable biological mechanism
8. Coherence with known facts.

Is there a strong relationship between Nicholas Cage and drowning deaths? Did we conduct an experiment with strong research design? Would removing Nicholas Cage from the film industry keep us safer? Importantly, is our observation coherent with known facts?

Check out this collection of Spurious Correlations from Tyler Vigen. But preview the charts first before sharing them with your students, since a few are for too mature for school.