Started off prep this morning with this question:

Hate how the book just shoves this formula conversion at the students without thought of explanation. pic.twitter.com/P3YfVLjSBD

— Dan Anderson (@dandersod) February 29, 2016

And ended up getting an answer:

@dandersod Took some fancy, non-intuitive, algebra to prove a specific case. pic.twitter.com/Nr6u2J0CSe

— Dan Anderson (@dandersod) February 29, 2016

eventually…

@dandersod ugh, error in last line. The sum of squares should be divided by 4.

— Dan Anderson (@dandersod) February 29, 2016

Anyway, I had the kids expand the sigma notation and I gently guided them towards an unknown goal. They did well, all considering. Instead of it being some formula from a teacher or a book, they have a bit more of a mental grip on the different representations and formula. Here is one group’s scratch work.

I also was able to find this error. A lot! Surprising to me, but useful to know that they have some work to go on sigma notation.

Thanks for the help David Griswold and Jason Merrill!

@dandersod Here's the proof

Useful for proving you need n-1 for samp. var. to be unbiased estimator of var. pic.twitter.com/wbY6re7X5F

— David Griswold (@DavidGriswoldHH) February 29, 2016

@dandersod The easy way to do this is to expand the square *inside the sum*, and notice that the cross term simplifies to -2*μ^2.

— Jason Merrill (@shapeoperator) February 29, 2016

@dandersod Yeah, that's the extra context that is critical. (∑ᵢ μ²)/n = μ² might not be immediately obvious either.

— Jason Merrill (@shapeoperator) February 29, 2016

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