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Karen Campe @karencampe.bsky.social · 14h

Ask @mnmmath.bsky.social she knows!
{I won't be at nctm}

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Karen Campe @karencampe.bsky.social · 16h

Oh dear...

a south park cartoon shows a woman and a boy and the caption he seems to get worse every day

ALT: a south park cartoon shows a woman and a boy and the caption he seems to get worse every day

media.tenor.com

Karen Campe @karencampe.bsky.social · 1d

I used 3 ideas:
-Right triangle with hypotenuse = 10
-Geometric mean** of the 2 parts of diameter
-the question asks for area, not the dimensions separately

**thanks to @howiehua.bsky.social I've had the different means on my mind.
In marked diagram --> y is geometric mean of w and x.

Semicircle and rectangle, with base of rectangle along diameter of semicircle, and top edge of rectangle tangent to semicircle.

Left side of rectangle divides diameter of semicircle into two parts, labeled w and x.
Let y be the part of the left side of rectangle from bottom corner up to semicircle intersection.

Then y is geometric mean of w and x:
w/y = y/x

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Karen Campe @karencampe.bsky.social · 1d

Why thank you... 😊

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Karen Campe @karencampe.bsky.social · 1d

Cool!!
My usual approach (w/out technology) is rational root theorem and use the remainder theorem to help P(root) = 0 so can do some easy tests with 1, -1, 2 etc.

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Karen Campe @karencampe.bsky.social · 1d

Yes, that's typical phrasing in US.

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Karen Campe @karencampe.bsky.social · 1d

🤣

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Karen Campe @karencampe.bsky.social · 1d

And here's the full post "One Big Rule to Rule Them All"

#iTeachMath #MathsToday #UKMathsChat

karendcampe.wordpress.com/2023/10/13/o...

One “Big Rule” To Rule Them All

We were talking online1 recently about parallel lines that are intersected by a transversal, and that there are multiple ways a student could justify that two particular angles were congruent. Sinc…

karendcampe.wordpress.com

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Karen Campe @karencampe.bsky.social · 1d

The marked angle is half the sum of the intercepted arcs, so (1/2)•(50+110) --> (1/2)•160 = 80°

This is one of 7 angle-arc cases that can be boiled down to "one big rule"

#iTeachMath #MathsToday #UKMathsChat

Big Rule: Identify the location of the angle’s vertex, which gives the relationship of the angle measure to the intercepted arc(s):

At center: angle = arc
On circle: angle = (1/2)•arc
Inside circle: angle = (1/2)•(arc 1 + arc 2)
Outside circle: angle = (1/2)•(arc 1 – arc 2)

Notes: This condenses 7 rules into 4 rules. Use these varied situations to practice circle vocabulary. In class, I would pantomime an “intercepted arc” with my two arms, as if I were a football player making an interception.

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Karen Campe @karencampe.bsky.social · 1d

Nicely done, David.
At first glance, cubic not easily factored.
Did you use rational root theorem or a graphing calculator to help finding that factorization... or other ideas??
Would love to hear more about your thinking.
#iTeachMath #ProblemSolving

Karen Campe @karencampe.bsky.social · 1d

Yes the harmonic mean is good for average rates, such as rate of work or rate of speed.
I am waiting for Howie's next video explainer to see if it is also the "average" of divisors or something like that...

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Karen Campe @karencampe.bsky.social · 1d

Dislike... hang in there.

Karen Campe @karencampe.bsky.social · 1d

Howie, this is absolute gold! Love these explainers of what the different means *mean*
...
So, the arithmetic mean is the average of addends, the geometric mean is the "average" of scale factors, then the harmonic mean must be ... ?
#iTeachMath

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Karen Campe @karencampe.bsky.social · 2d

Follow the link from here to get to the printable models.
cc: @mathhappens.bsky.social

#MathsToday #iTeachMath

www.mathhappens.org/conic-sectio...

Conic Section Models - MathHappens

www.mathhappens.org

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Karen Campe @karencampe.bsky.social · 2d

Number of diagonals? Or diagonals that are bisected/ congruent/ etc.

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Karen Campe @karencampe.bsky.social · 3d

Dislike....

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Karen Campe @karencampe.bsky.social · 4d

Awesome!
Here's the area model post if you're interested...
karendcampe.wordpress.com/2023/08/16/a...

Area Arrangements

Have you ever wondered if there’s a better way to teach tricky arithmetic and algebra procedures to your students? Think about multiplying fractions and polynomials, long division, simplifying radi…

karendcampe.wordpress.com

Karen Campe @karencampe.bsky.social · 4d

No worries!
Do you have students discover/notice that products of diagonals are the same, or do you just present it as part of the method?

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Karen Campe @karencampe.bsky.social · 4d

Very nice.
The key is that in the area model, the product of diagonals is equal.
[I was momentarily confused by using abcd for those 4 values ... because they aren't same as ABC from quadratic]
The area model can do so much!! Wrote a whole post about it a couple years ago.

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Karen Campe @karencampe.bsky.social · 4d

Huge... and it might have been earlier. I was asked about her work when interviewing for teaching jobs in late 1980s/early 1990s.

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Karen Campe @karencampe.bsky.social · 4d

Yes true! That's why I'm so excited about your "addend first" strategy.

Students who "get" factoring will find the factors either way, but Ss who struggle get another way into it with addends first.

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Karen Campe @karencampe.bsky.social · 4d

^^ yes absolutely.
And your "addend first" idea is accessible to students who can't do numerical factoring readily.

But also yes to @dsrussosusan.bsky.social 's point that sets of addends are infinitely many, but sets of factors are finite.

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Karen Campe @karencampe.bsky.social · 4d

Yes exactly. I was wondering if David's "adding first" method would involve AC.

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Karen Campe @karencampe.bsky.social · 4d

So, I'd try
-10•36, sum = 26
-12•30, sum = 18
-18•20, sum = 2
I must have missed 9...
-15•24, sum = 9 ⭐️

Then
18x^2 – 15x + 24x – 20
3x(6x – 5) + 4(6x – 5)
(6x – 5)(3x + 4)

Done.

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Karen Campe @karencampe.bsky.social · 4d

Thanks for this explanation!
The "AC Method" is to multiply A•C. Then find two numbers whose product is AC and sum is B.
"Split" the B term into the sum you found, then factor by grouping.

18x^2 + 9x – 20
AC = -360
B = 9

We write:
___ • ___ = -360
___ + ___ = 9
then try to fill in blanks

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