Notation is Human-made

As a precursor to:

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Equality is a weird thing…

  • Should everyone in this class get an equal grade?
  • Should everyone in this class be graded equally?
  • Should the teacher treat everyone in this class equally?
  • Should you treat everyone in this class equally?
  • Should everyone in this class do an equal amount of work in class? At home?

 

Notation is not given to us by God or the universe. We, at some stage, invented it. Robert Recorde, from Tenby (Wales), invented the equals sign after getting sick of having to write out “is equal to” every time… Took 200 years to be fully accepted – slow-moving meme.

What other equals signs should exist? Think of a gap in the market that needs filling…

Possible ideas:

  • Is a cat equal to a dog?
  • Are two cats equal?
  • Are two triangles with same side lengths equal?
  • Are two triangles with the same angles equal?
  • Is Mr Judge equal to your Math teacher?
  • Is your age the same as your neighbors?
  • Is 4 + 1 equal to 3 + 2?
  • Are women equal to men?
  • Are two meter-sticks equal?
  • Is a + a equal to 2a?
  • Is 3x equal to 9?

 

Student notation:

12

We have to use the Rafael and the Olivia signs as much as possible. Maybe in 200 years’ time they will be standard…

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Geometry –> Algebra. A progression

Step 1: Handshake problem

If 10 people all shake hands with each other, how many handshakes in total?

Purpose: to relate every object in an algebraic expression (in this case 1/2n(n-1) ) to a verbal/geometric reason.

Step 2: Moser Circle problem

If I have 10 dots on the circumference of a circle, and join them all up, how many regions are there?

Purpose: to realise that not all patterns have easily identifiable algebraic rules. to TRUST NOTHING.

Step 3: Maximum Right-Angles?

If I have a 100-sided polygon, what is the maximum number of right-angles there can be?

Purpose: to combine the motivation for algebra from Step 1 and the skepticism from Step 2, to go on a deep and joyous journey.

Possible prompts:

  • Can you prove that a 100-sided shape definitely cannot have 99 right-angles? Can you generalize?
  • You think you know the pattern… Can you find an 11-sided shape with 9 right-angles? And look – here is a 12 sided shape with 9 right-angles!

Students gave the following answers:

  • There is on answer since there are infinite different polygons
  • Between 50 and 100
  • 52
  • 66
  • 67

Possible Extensions:

  • What if you replaced right-angles with acute angles? Obtuse angles?
  • What if you allowed the polygon to have lines that overlap?
  • What if you allowed both interior and exterior right-angles?
  • What if you only allowed exterior right-angles?
  • What if the polygon has to be convex?
  • How many different 9 sided shapes have 7 right angles?
  • What if you…

 

Student work

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Great whiteboard

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Not the best use of a table…
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When your theory breaks, you either give up or just get more excited!
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Great to see slogans being repeated back to you by students…
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Teacher approximation
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… which a student independently came up with. 

Miller’s Quadsector Theorem

We spent a happy half-hour exploring the difference between proof and illustration on Geogebra.

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  • Angle bisectors of a triangle meet at a single point. A single point?! Madness.
  • Angle trisectors meet to form an equilateral triangle. Morley proved this.
  • I asked students to construct this on Geogebra. Professor Miller found trisectors too tricky, so went for quadsectors instead – bisect and bisect again. (fascinating and unknowing nod to the futile attempts to trisect an angle using straight-edge and compass). Excitement fizzed around the room at the new conjecture. Professor Miller is very good at Geogebra, after only one lesson.
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Seems legit?
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Hmm. What’s going on?
  • Another group managed to quint-cept the angles. No idea how. They also found that the central triangle seems to be equilateral.
  • A final group suggested that the pattern could continue for any n-cepted angles?

Excellent playing with initial circumstances, and students pumped up at the possibility of creating new knowledge. Good times.

Update: Professors Stitch and Madhava explored Morley’s Thoerem, but trisecting sides rather than angles…

Exploring Chicago 1: Pilsen

C expertly led us through Pilsen, a Mexican neighborhood of Chicago. We travelled on the pink line to get there – a surprising proportion of students had never been on it before. Is this just because the city is sprawling, or signifying a segregated city?

We started our tour by a brand new mural. Fence in the background – the ever-present symbol of the border.

Local artist Hector Duarte’s house, covered entirely by a huge and beautiful mural:

Next, the Mexican Art Museum. The students in the group with Mexican heritage led the tour, showing greater knowledge of their folklore than I know of British stories. We learnt of the grape boycott, the merging of the Virgin Mary with preexisting beliefs, the way Mexicans honor their dead.

In allyship with local residents affected by gentrification, C was wearing a badge saying something like “Forced out of Pilsen”. One of the museum staff asked her about it, and seemed offended that she was wearing a badge without actually being forced out herself. Gentrification is a very sensitive topic here.

Another incredible mural, on 18th Street:

Lunch at 5 Rabanitos – cilantro and avocado make everything amazing. Live music from passing buskers. Continued to walk along 18th street, picking up Conchas from one of the multitude of bakeries.

Thank you C for an excellent visit to Pilsen! Can’t wait for the next adventure.