# A serendipitous Pythagorean #LessonStarter

#LessonStarter is a Twitter hashtag, used particularly by Matt Skoss, to collect together ideas that might start (or take over!) a lesson. A #LessonStarter is usually a provocative image, but could also be an intriguing mathematical prompt. For me, lesson starters are often spontaneous. Today, a few serendipitous moments meant that I had a lesson starter just before walking into class.

This morning I woke up to a tweet from Bryn Humberstone.

I love using topical moments — like today’s date — as a lesson starter. While I understand the fatigue that days like Pi Day (14 March) can cause, I think it can be a great opportunity to briefly introduce some maths that might not be in the lesson plan. (Sara Van Der Werf has a recent and positive blog post on math(s) holidays that you should read.) Plus, I like my students to witness the joy I have in mathsy moments — even if they think I’m weird!

After the tweet, I opened my email to this image from brilliant.org and the 100 Day Summer Challenge.

To top it off, my plan for today’s #math1070 class was to review last week’s task: choose one of the four images below and tell me what you Notice and Wonder about it. You might be able to see how my lesson starter was going to unfold!

I started with today’s date: 15/8/17. I said it was an interesting mathematical date. Could anyone work out why? I then showed this image. We had a quick refresher of the Pythagorean theorem. This image already throws up some interesting questions: is it possible to always colour the squares on the hypotenuse with a complete square of one colour and a double thickness ‘half-border’ of the other?

Next, I showed the image from brilliant.org. I was careful to draw the squares on the sides of the 3-4-5 triangle, and connect Pythagoras’ theorem to the area of the squares that can be drawn on each side of the triangle. Then I asked them what they noticed and wondered. What do you notice and wonder? We had a little discussion based on this prompt, including generalising the side lengths.

Finally, I showed the image from their weekly task. Those who chose this image as their prompt generated a lot of noticings and wonderings; they are at the bottom of this post. For now, I directed their attention to the number of isosceles trapeziums making up the larger trapeziums on each side. The counts are 9, 16 and 25. What do you wonder now?

We explored whether this was a 3-4-5 triangle (it is) using the long base of the smaller trapezium as one unit. The overwhelming question in the room was why Pythagoras’ Theorem works for non-square shapes. I acknowledged that it works for similar shapes, and left it there. (By now, the lesson starter was threatening to take over the lesson!)

I finished with a quote from Chapter 7 of Tracy Zager’s book, from Peter Hilton: “Computation involves going from a question to an answer.  Mathematics involves going from an answer to a question.” I hope that these images and initial discussion prompted many more questions that they could go on to explore in their own time.

Incidentally, it was another serendipitous find this morning, shared on Twitter by John Golden, that sharply reminded me of this quote. John shared this blog post by Pat Ciula. It uses the same trapezium image to launch a complete different exploration. I love this post; do check it out for yourself.

This wasn’t really what I had planned for today’s class, but when these moments come along, you need to grab them!

#### ‘Noticing and Wondering’ from my students

These are from last week’s task, submitted in advance of today’s class. I’ve made some minor edits.

I notice (that) …

• Different colours are used: red and pink.
• A triangle has been made in the middle.
• The triangle formed is a right-angled triangle.
• All three shapes are the same.
• All three shapes are made up of smaller versions of the overall shape.
• The shapes are created by wooden blocks all put together.
• Each individual piece is a trapezium.
• There are three larger trapeziums made out of individual tessellating trapeziums.
• That the four-sided polygons go in an anti-clockwise movement ranging from smallest to largest.
• The blocks are arranged in different ways in each trapezium.
• The three trapezoids are similar trapezoids.
• There are nine sides forming the outer perimeter of the shape.
• In the centre of the three larger trapezium shapes, the points of the bases of these shapes make a right-angled triangle.
• If you were to move the largest trapezium shape so it was opposite where it is now, the shape in the middle made by the sides of the trapeziums would still be a triangle.
• Each tile is the same shape – trapezoids. In each tile, three sides (top and the two edges) are equal, and the base (the longest side) is twice the length as the other three sides.
• If the length of the base of the tile is one unit, then the length of the middle triangle’s three sides are 3 units, 4 units and 5 units. The layers of the tiles within the trapezoids are also 3, 4, and 5.
• Small trapezium is made out of 9 smaller trapeziums. Bottom trapezium is made out of 16 smaller trapeziums. Larger trapezium is made out of 25 smaller trapeziums.
• The number of tiles on each side of the triangle follows Pythagoras Theorem: 32 + 42 = 52, which is 9 + 16 = 25.
• The image includes an aspect of scale.
• The small pink polygons shapes make a negative internal outline of a right-angled scalene triangle.
• That the fewer pink polygons on each section related directly to the length of the sides of the internal outline of the triangle, that is, fewer pink polygons, shorter sides.
• I noticed that the depth and width of the outer shapes was linked, that is, left side shape is 3 pieces deep and 6 ‘lengths’ wide and consists of 9 pieces, bottom shape is 4 pieces deep and 8 ‘lengths’ wide and consists of 16 pieces and finally upper right shape is 5 pieces deep and 10 ‘lengths’ wide and consists of 25 pieces.

I wonder …

• Is the angle exactly 90 degrees?
• How many trapeziums there are?
• How long did it take to make?
• Why are there two different colours?
• How many blocks would it take to fill the right-angled triangle?
• Are sides ‘a’ and ‘b’ equal to ‘c’ in length?
• If the bottom section would fill the triangle perfectly?
• Are the small trapeziums arranged in a particular way for a reason? Or is it random to fit the desired shape?
• Do the larger shapes go down in size proportionally and is there a specific number to make up each shape?
• Would the area of the trapeziums be bigger or smaller than the area of the triangle in the middle?
• What the two remaining internal angles would be (excluding the right-angle).
• Why using trapezoids to represent Pythagoras Theorem also works? Is it because the ratio of the area of the trapezoid to the square is fixed?
• I wonder if the sizes of the trapezium differed, what the effect would be on the right-angled triangle on the middle? Is it possible to make an isosceles or equilateral triangle in the centre?
• How many triangles can each Isosceles Trapezoid be split into?
• How many other shapes are present in the Isosceles Trapezoids? For example – I have already noticed that triangles can be found — what other shapes are there?
• What is the reasoning for this image? Why was it created?
• How I could start a conversation with the students regarding this shape and what they may come back with?

# Skyscrapers

This is a quick post mainly for the benefit of my ‘Developing Mathematical Thinking’ (#math1070) students.

#### Introducing the puzzle

Skyscrapers are one of my favourite logic puzzles. They are a Japanese creation, introduced at the first World Puzzle Championship1 in 1992.

Skyscrapers are a type of Latin Square puzzle. A Latin Square in an n × n grid filled with n different symbols, each occurring exactly once in each row and exactly once in each column. (Sudoku is another type of Latin Square puzzle).

In a Skyscraper puzzle the objective is to place a skyscraper in each square, with heights between 1 and n, so that no two skyscrapers in a row or column have the same height. The numbers (clues) on the outside of the grid tell us how many skyscrapers are visible from that position. (I like to imagine that I’m standing on that number and along the street that is the row/column.) Shorter skyscrapers aren’t visible behind taller ones.

We use logical deductions to solve the puzzle. For example, in the puzzle below, the clue ‘4’ tells us that the skyscrapers must appear in ascending height order in that row/column. Similarly, the clue ‘1’ tells us that the tallest skyscraper must be adjacent to the clue. That leads us to the partially-filled grid below. If you want to solve it yourself, the solution is at the bottom of this post. You can also play them online at Brain Bashers.

#### Hands-on skyscrapers

It is fairly easy to turn a skyscraper puzzle into a hands-on activity — just choose objects of different heights. Teachers often use linking cubes. You can also be more creative; at David Butler‘s One Hundred Factorial gathering at the University of Adelaide in May, we experimented with video cassettes (remember them?) and cups of different sizes.

A while back, I wanted to make several hands-on sets for 5 × 5 grids to use with groups of school students. They needed to be cheap, lightweight, compact and portable. So, I made paper cylinders that nestle inside each other. You can download and print the skyscraper cylinders. The tabs are meant to show where to overlap and tape. You can use them with these puzzles (print A3 size): Puzzle 1, Puzzle 2, Puzzle 3, Puzzle 4, Puzzle 5.

#### Skyscrapers in the classroom

My plan for MATH 1070 was curtailed by our short week (Week 3). I had planned the activity with these goals:

• Form visibly random groups with four students so that students could meet a few more classmates.
• Work collaboratively towards a common goal (and contrast this with the competitive nature of Prime Climb last week.)
• Practice claims and warrants as part of the focus on Maths Disputes: ‘I think <claim: this number goes here> because <warrant: my reason>.’

There are a variety of reasons to use skyscrapers in the classroom; you might like to read these posts by teachers: Mary BourassaMark Chubb, Sarah Carter. Any activity introduced into the classroom should be intentional. You might like to think about these dot points. Mark has a fuller list in his blog post.

• If giving these puzzles to individual students is different than to groups of students.
• If a physical model is different than a pen-and-paper version.
• If you’ll use it as part of a lesson or as a ‘time filler’.
• What you’ll do if students give up easily.

If you give them a go, let me know what you think!

# Redux: #NoticeWonder and #PrimeClimb

Last year I wrote a post about using the two simple questions ‘What do you notice?’ and ‘What do you wonder?’ with my maths pre-service teachers to dig into the mathematically-rich image that accompanies Dan Finkel‘s game, Prime Climb.

This year, I wanted to turn this into a student-driven rather than teacher-led activity. I also wanted to create opportunities for even deeper mathematical exploration. In this post I’ll briefly outline how this unfolded.

#### Using Tiny Polka Dots for visible random groupings

A goal for this year is to do better at helping students be good mathematical colleagues. I’ve been making heavy use of Chapter 12 of Tracy Zager’s book to guide this endeavour. I also wanted to begin using visibly random groups to build our mathematical community. However, I’m mindful it can be confronting to work with complete strangers, so I had students pair up first with someone they knew. (Pairing students up was also to serve another purpose that will become apparent later.)

I randomised the pairs by handing out cards from another of Dan’s games, the delightful Tiny Polka Dots. I deliberately used cards with different representations of 1, 2, 7 and 9 (with 8 as a back-up). Pairs with a different representation of the same number formed a group, one at each table. I gave them the rest of the cards for their number, and we did a quick ‘Notice and Wonder’ on the different representations.

#### Different-sized grids for Prime Climb

Rather than show students the Prime Climb hundreds chart arranged in columns of 10 (the image earlier in this post), I wanted to foster noticing and wondering by having them construct the charts themselves with mini-cards of the numbers: physically handling, examining, ordering and organising. The (roughly) six students at each table arranged the cards into charts with the number of columns corresponding to their Tiny Polka Dots card (with 1 and 2 corresponding to 11 and 12, respectively). Click on the images below to make them bigger. What do you notice? What do you wonder?

To add impetus to the discussion, I relayed that part of the weekly task (contributing towards their course grade) was to individually write a forum post with at least five things that they noticed and wondered about their charts. It was heartening to see students collaboratively generating lists of their many observations.

#### Playing Prime Climb to ‘make thinking visible’

The final part of this activity was to have students play Prime Climb in a way that compels them to articulate their thinking. One observation from playing Prime Climb at One Hundred Factorial with David Butler was that playing in pairs had an initially unexpected benefit of making thinking visible. (Side note: Bodyscale Prime Climb—where the numbers are A4 sized and the player is the pawn—is the most wonderful way to experience this game. Walking the board gives a different perspective to the relationship between numbers. You also need at times to shout your thinking across the board to your partner, which really makes thinking visible! Read more in David’s blog post.) We used a modification of the rules devised by David. This is the printable version that I wrote based on the rules that come with the game.

Collaboration and competition reached ear-splitting levels, so much so that we were asked to quieten down from those in the classrooms around us. (I sheepishly and profusely apologised later.) This was definitely a good start to building community.

#### Further exploration

While the in-class activities concluded with playing Prime Climb, there is so much more exploration provided by the cards. David, and later I, explored the different patterns that emerge when the cards are arranged in different ways. Others chimed in on Twitter with ideas; click the links above or look at the images below for more. The Prime Climb colouring is such a rich medium!