iiinova — where your digital self keeps evolving

Reposted by Carl Brandt Long

Emma R. Hasson @dodecalemma.bsky.social · 06/09/2026

Bring some math into your holiday weekend and learn about some ridiculous, fascinating, and absolutely terrible ways to hang a picture. www.scientificamerican.com/article/math...

scientificamerican.com

Mathematicians discover the worst way to hang a painting

New solutions found to picture-hanging problems

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Reposted by Carl Brandt Long

roterote @roterote.bsky.social · 03/10/2026

For a very clear comparison, emulating Magyar would be removing all the MAGA justices and judges from Federal Courts in the first 100 days. That is the scale of democratic chemotherapy we are discussing.

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Carl Brandt Long @carlblong.bsky.social · 02/10/2026

Are you familiar with @fawnnguyen.substack.com's visual patterns collection? share.google/C9Q1eK412LQ4...

share.google

Visual Patterns

Patterns to grow algebraic thinking and reasoning.

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Carl Brandt Long @carlblong.bsky.social · 01/10/2026

Big ideas in class. HW tonight is about reps. Next class, Law of Sines. With ASA, AAS, they need a new tool. We'll also see why SSA can give 2 solns through trig lens. #ITeachMath #MTBoS #MathEd

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Carl Brandt Long @carlblong.bsky.social · 01/10/2026

We do Law of Cosines before Law of Sines. SSA triangles create quadratic eqns, which Ss have studied extensively. They KNOW roots are symmetrical across the line x=vertex. How do these elements connect to the Geom of making triangles? When 0 ,1, 2? Why no more? www.geogebra.org/geometry/q4b...

geogebra.org

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Carl Brandt Long @carlblong.bsky.social · 01/10/2026

Ex: When given impossible triangle (e.g. 3,5,9), what happens when you solve for angle using LoC? Why no soln, geometrically & trigonometrically? Otherwise, how do you know 1 soln? (domain/range of cos^-1)

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Carl Brandt Long @carlblong.bsky.social · 01/10/2026

In #MathsToday, Precalc Ss (11th grade) revisited Geometry through Law of Cosines. Reframed Tri Congruence Thms so that it's not about comparing corr. sides/angles, but that, i.e., given 3 lengths, there is exactly one triangle possible. Today was abt. connecting the algebra, geom, & trig:

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Carl Brandt Long @carlblong.bsky.social · 01/10/2026

This green function is a good fit. I don't understand it, though! www.desmos.com/calculator/c...

desmos.com

Earthscience Water experiment 9-30-26

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Carl Brandt Long @carlblong.bsky.social · 01/10/2026

Letting Desmos create its own vertical shift suggests the room temp is 26.25 degrees. Seems like you'd have noticed that abnormality when you walked in the room, though. Maybe it's an instrument calibration error? www.desmos.com/calculator/e...

desmos.com

Earthscience Water experiment 9-30-26

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Carl Brandt Long @carlblong.bsky.social · 30/09/2026

Nice! I'd argue that you don’t even need multiple colors.

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Carl Brandt Long @carlblong.bsky.social · 30/09/2026

How would you approach it instead?

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Reposted by Carl Brandt Long

Adrienne Hestenes @ahestenes.bsky.social · 30/09/2026

In #MathsToday my seniors read Francis Su’s speech to Harvey Mudd frosh and then wrote a page reaction. They generally enjoyed the reading and honored me with their personal stories and thoughts about math and caring for others. Here is the article they read.

share.google

Science, in Service of Love

My exhortation to the first-year class at Harvey Mudd College

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Carl Brandt Long @carlblong.bsky.social · 29/09/2026

Then, we started watching this video from ancient* times www.youtube.com/watch?v=nBwn... (7:59 to 15:50, but ran out of time around the 11-min mark). *pre-internet

youtube.com

Episode 5: Sines And Cosines Part II - Project MATHEMATICS!

YouTube video by caltech

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Carl Brandt Long @carlblong.bsky.social · 29/09/2026

In #MathsToday, 11th grade pre-calc Ss invented the Law of Cosines (and realized its usefulness). Started w/ right triangle, changed it up. Derivation took a while, but w/ a _little_ guidance, every group found the formula! Tonight, practice. Next class, ambiguous case! #MTBoS #ITeachMath

Three triangles with sides 6 and 8. Included angles are 90, 60, and 120.

Two triangles with sides 6 and 8. Included angles are 40 and 140.
Then one generic triangle with sides/vertices labeled a, b, c, A, B, C. c=?

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Carl Brandt Long @carlblong.bsky.social · 28/09/2026

In #MathsToday, introduced composite functions to #Alg2. Started with Collatz Conjecture, then these 3 prompts. Ss worked in groups for each, leading to full-class sharing/discussions. Keeping it grounded at first, with units in mind. e.g. "Are you adding to the mileage, or to the money?" #MTBoS

The surface area of a sphere, as function of the length of the radius, r, can be represented by S(r)=4πr^2. Match each of the following expressions with its description:
1.05∙S(r) The radius of the sphere is increased by 5%
S(r+5) The surface area of the sphere is increased by 5%
S(1.05r) The surface area of the sphere is increased by 5 square units
S(r)+5 The radius of the sphere is increased by 5 units.

F(m) is the fare, in dollars, for a taxi ride of m miles. Write an expression for each of the following:
The amount paid if the taxi driver charges you for 5 more miles than the actual distance of the trip
The amount each rider pays if the fare is divided equally among 3 passengers
The amount paid if the driver charges you for twice the number of miles you actually travelled
The total amount paid if you tip the driver $2
The total amount paid if you tip the driver 20%

Ulrich’s bakery normally charges C(N) Euros (€) for N pretzels. The expressions below represent various coupons for buying N pretzels. For each expression, provide text for the coupon.
C(N-2)
1/2 C(N)
C(N)-2
C(1/2 N)
0.90∙C(N)

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Carl Brandt Long @carlblong.bsky.social · 28/09/2026

I start out as tan=slope, but then (therefore: rise/1), tan(theta)=the length of the tangent line from (+-1, 0) to where it intercepts the angle's ray. That provides the + or -. i.e. www.desmos.com/calculator/e...

desmos.com

Tan X

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Reposted by Carl Brandt Long

Andrew Stacey @mathforge.org · 24/09/2026

www.bbc.co.uk/news/article... I can't even ... so I won't! Instead, let's make this an alternative headline competition. I'll start: Farmer rounds down sheep, instead of rounding them up. #MathsToday #UKMathsChat #mathsky

bbc.co.uk

Jeremy Clarkson's sheep 'theft' turns out to be miscount on farm

Jeremy Clarkson reported the theft of sheep from his Diddly Squat Farm to police earlier this month.

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Carl Brandt Long @carlblong.bsky.social · 22/09/2026

Thank you for doing thoughtful work! (And accessible-to-high-schoolers work!)

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Carl Brandt Long @carlblong.bsky.social · 22/09/2026

I have a couple lessons that I've redesigned so that the pun at the end works. Nicely done!

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Carl Brandt Long @carlblong.bsky.social · 22/09/2026

Then @monachalabi.bsky.social's video on how definitions affect lives: www.ted.com/talks/mona_c... Big themes today: - You need clear definitions when gather/analyzing data - Balancing those defs can be HARD - Context matters - Data is about humans at every stage Next week we start using numbers!

ted.com

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Carl Brandt Long @carlblong.bsky.social · 22/09/2026

Continuing the theme, we watched part of @chadtopaz.bsky.social's @breakingmath.bsky.social interview about balancing granularity & ability to make statistical conclusions & how you necessarily need to put disparate people into the same box sometimes. youtu.be/nyrBAnQFYWg?... #ITeachMath

youtu.be

How Data Science Exposes Injustice: Chad Topaz on Unlocking Justice

YouTube video by Breaking Math

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Carl Brandt Long @carlblong.bsky.social · 22/09/2026

For #MathsToday, ProbStats Ss had to read Strawberry Jam ch. from The Numbers Game, about how counting in real life is HARD! We went to the library to count red books. Discussed afterwards: -what is "red"? -what makes a book "red"? -how do you sample? -how do you use sample to extrapolate to pop?

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Carl Brandt Long @carlblong.bsky.social · 17/09/2026

I like Open Middle problems for this. They build critical thinking while getting in repetitions. Also force Ss to understand the mechanics rather than just repeat facts. Also, easy enough to create/adapt to any topic. www.openmiddle.com

openmiddle.com

Open Middle - Challenging math problems worth solving

Open Middle offers interactive math problems designed to foster critical thinking. Explore unique challenges that boost problem-solving skills and creativity.

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Carl Brandt Long @carlblong.bsky.social · 16/09/2026

In #MathsToday #Alg2 Ss learned domain, range, inc/dec features of graphs. HW before class was classroom.amplify.com/activity/6aa... In class, formalized the ideas and notation. Then Ss did this to learn domain/range: classroom.amplify.com/activity/5f5... Finished w/ stations to practice/deepen.

classroom.amplify.com

Looking at Graphs v1.26

by Carl Long

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Carl Brandt Long @carlblong.bsky.social · 13/09/2026

This is my first time saving the data. We'll see how it goes!

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Carl Brandt Long @carlblong.bsky.social · 12/09/2026

#ITeachMath #MathEd I can attest that this works to build stronger community and deeper understanding. Its also more fun to teach this way.

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Carl Brandt Long @carlblong.bsky.social · 11/09/2026

I saved the data. In Nov, we'll use it to look at expected value. In Feb, we'll use it to look at sampling distributions, & run some binomial- and chi^2 tests to see if we guessed better than randomly. I can't wait. #ITeachMath #ITeachStats #MTBoS #MathEd

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Carl Brandt Long @carlblong.bsky.social · 11/09/2026

On day 1, it's a discussion abt how humans are bad at randomness - creating & recognizing. (Though most beat my score) Ss noticed that most of the real die rolls had doubles in a row, and did not include all 6 digits.

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Carl Brandt Long @carlblong.bsky.social · 11/09/2026

#MathsToday (2 days late) 1st day of Prob.Stats class. Kids made up a string of 10 fake die rolls. Then rolled die 10 times, wrote down that string. Each S wrote both strings on the board. Class guessed the real string A game: 1 pt for each correct guess, 1 pt for each person fooled by your list.

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Carl Brandt Long @carlblong.bsky.social · 10/09/2026

In Jan of Precalc, 16 months later, we'll study sequences (inc. quadratic). We return to this problem. In Mar, combinations. We return to this problem. In May, series. We return to this problem. We write it in sigma notation. We see how summing the terms and sum formula methods converge.

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Carl Brandt Long @carlblong.bsky.social · 10/09/2026

In terms of curriculum design, our dept treats Algebra 2 and Precalculus as sort of a 2-yr course. (In practicality, it's not. Sections shuffle, some Ss move in and out, teachers do not teach both years.) But the ideas thread through the 2 courses. And the handshake problem is one Ss remember.

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Carl Brandt Long @carlblong.bsky.social · 10/09/2026

I got some to draw diagrams. Connected diags of Polygons. Ss were asked to find a formula. Usually the last method got there. One class got really loud, as Ss argued strategy or got excited when their conjectures worked. That was a big takeaway: Come up with conjecture. Test it. Revise, repeat.

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Carl Brandt Long @carlblong.bsky.social · 10/09/2026

Some groups recognized the pattern for increasing population size: 1,3,6,10,... and continued to class size. Some realized (informally) the 1st person shook n-1 hands, the 2nd n-2, etc. down to 1. Others recognized each person shook n-1 hands, and that we'd double-counted each handshake. #MTBoS

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Carl Brandt Long @carlblong.bsky.social · 10/09/2026

Answers came quickly. All wrong. But wrong in wonderful ways. Me: "Can you test it out?" "Like, shake hands with everyone in the room and count?" Me "Can you test the strategy? What would it be with 3 people?" Usually the answer was 6. I pointed out that there were 3 in the group. They tried. 3.

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Carl Brandt Long @carlblong.bsky.social · 10/09/2026

Yesterday was our first day of school. In #Alg2 #Algebra2, I put the kids into groups of 3 and asked, if we all shook hands with everyone in the room, how many handshakes would that be? I LOVE this problem, esp. on Day 1. 🧵 #ITeachMath #MathEd

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Carl Brandt Long @carlblong.bsky.social · 09/09/2026

This Alpha School? bsky.app/profile/futu...

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Carl Brandt Long @carlblong.bsky.social · 08/09/2026

This is wonderful. Thank you for putting it together!

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Carl Brandt Long @carlblong.bsky.social · 04/09/2026

Cool exercise! This seems to be the sort of thing that Desmos notebooks work really well for: www.desmos.com/notebook/qqx...

desmos.com

Desmos | Notebook

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Carl Brandt Long @carlblong.bsky.social · 28/08/2026

Finished with Draw a 6x4 grid of squares. a) What is the perimeter? What is the area? b) Take away one of the corner squares. What is the perimeter now? c) Remove as many squares as you can while keeping the perimeter =20. What is the area? Good week overall! #ITeachMath #MathsToday #MathEd

Draw a 6x4 grid of squares.
a) What is the perimeter? What is the area?
b) Take away one of the corner squares. What is the perimeter now?
c) Remove as many squares as you can while keeping the perimeter =20. What is the area?

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Carl Brandt Long @carlblong.bsky.social · 28/08/2026

In #MathsToday, "Which Deal is Better" sparked discussion about how percentages stack, and whether you can just choose a starting value arbitrarily (Yes!) docs.google.com/presentation... (Sorry - don't remember the source) 3/4

docs.google.com

Which is a better deal?

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Carl Brandt Long @carlblong.bsky.social · 28/08/2026

Then in #MathsToday, @ddmeyer.bsky.social's Cookies 3act task. Got Ss thinking about 3-d space, inc. conversions from in^3 to ft^3. Also got to discuss limitations of math models (& how we can't expect precision) S: "Do we know how full the trunk is? Fully full?" blog.mrmeyer.com/2016/3acts-n... 2/4

blog.mrmeyer.com

[3ACTS] Nissan Girl Scout Cookies

Treatment #1 A small rectangular prism measures 7 inches x 2.3 inches x 4.6 inches. How many times could it fit in a larger rectangular prism with a volume of 39.3 cubic feet? Treatment #2 Nissan i…

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Carl Brandt Long @carlblong.bsky.social · 28/08/2026

Last Day of 7-8 graders! Lots of fun problems in #MathsToday! Cannonballs is one of my favorites. I use it to generate Qs w/ younger students, and to introduce Sigma notation with older. Good @howiehunt.bsky.social-style "how did you count" discussions, too! docs.google.com/presentation... 1/4

docs.google.com

Cannonballs

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Carl Brandt Long @carlblong.bsky.social · 28/08/2026

I'd love to know how it goes for your kids!

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