# A lesson plan (of sorts) for quadrilaterals

This tweet sums up today’s class.

We’ve had a two week break, culminating with a public holiday Monday. In our first class back, I wanted to add some supporting activities for the difficulties we were having in our last class. Let’s just say that I’m not sure that what we did today helped! Luckily, I have another opportunity with the other half of the class tomorrow. I thought I’d write a blog post to help organise my thoughts (and provide some kind of detailed post-class lesson plan), and then press send so that others can have a peek into my thought process (for what it’s worth!), and perhaps offer suggestions. If you want to focus on what didn’t work so well, skip right to the last section.

### Previously …

We’ve been developing mathematical reasoning using the key elements of noticing patterns, conjecturing and generalising, crafting claims, and exploring why (proving). (I’ve leant quite heavily in places on Chapter 10 of Tracy Zager’s Becoming the Math Teacher You Wish You’d Had for what follows.)

In our last class (before the break), we were working on claims. For each claim, I asked students to think about whether the statement is true, and, if so, when the statement is true. To help with this, I asked students to find counterexamples (examples where the claim doesn’t hold true). Then, they needed to revise the conditions of the claim to make it always true.

For example: ‘If you subtract a number from 5, your answer will be less than or equal to 5.’ This is definitely not always true. A counterexample is -7;  if we subtract -7 from 5, the answer is 12, which is not less than or equal to 5. With some more investigation we can see that this statement is always true if we revise the conditions on the claim to be ‘for all numbers greater than or equal to zero’.

Here are some of the others that students worked on:

1. $a \times \frac{1}{a} = 1$
2. A rectangle is a square.
3. The sum of four even numbers is divisible by 4.
4. One-third is one of three pieces.
5. $2-x = x-2$
6. An even number divided by an even number is an even number.
7. $p+12 = s+12$
8. Division always makes a number smaller.
9. The square of a number is greater than or equal to the number.
10. When you add two numbers you get the same result as when you multiply them.

These kinds of statements are used in the instructional routine Always, Sometimes, Never. For more on this, I thoroughly recommend Chapter 10 of Tracy’s book. If you don’t have access to the book, you can still check out the links at the companion website.

Of the wide variety of statements, students struggled (perhaps most) with the nature of statements 2 and 4. The discussion around ‘A rectangle is a square’ led me to think that our experience with classifying objects (in particular, shapes) and identifying properties they had in common needed to be richer. Which led me to the plans for today’s class …

### Today’s class

If I’m honest, I kind of skirt around geometry, in particular the properties of shapes — too many definitions that differ depending on which part of the world you are from. But shapes offer lots of opportunities for classifying and characterising, and for nailing down some precise terminology. So, I started by sharing in a matter-of-fact way that I was a little uneasy with what we were about to do but that, as teachers, we can’t simply avoid a topic.

The progression that follows includes ideas from Pauline Carter and Kath Ireland, as well as from Van de Walle et al. (2018) Teaching Student-Centred Mathematics Vol III (Grades 6-8). Underlying all of this are the Van Hiele levels of geometric thought. I won’t elaborate, but a Google search should help — and Christopher Danielson’s excellent Teacher Guide for his Which One Doesn’t Belong? book gave me a useful way to describe the different levels to my pre-service teachers at the end of the session.

#### 1. Names of shapes

I gave this handout and asked students to name as many of the shapes as they could. (I chose not to pre-cut shapes for the students so that orientation was initially fixed.) The purpose of talking about the shapes was to draw out some terminology and some misconceptions. For example, we talked about ‘diamonds’ and the orientation of shapes. We talked about concave and convex. But, I refused to arbitrate on the precise names of shapes at this stage — that was for them to determine!

We kept this page up throughout the workshop, and added notation for parallel sides, congruent sides, right angles,and congruent angles.

#### 2. Sorting and mystery definitions

Next we worked on dividing our quadrilaterals into two groups. The items in one group need to have something in common. That property must not present in the items in the other group. An example is below. Can you work it out?

(The items on the left do not have interior right angles. The items on the right do.) I asked students to make up puzzles for each other. This seemed to go well. In retrospect, I should have had a few more examples up my sleeve. I could have also asked them to write theirs down.

#### 3. Guess Who?

We moved into a few rounds of ‘Guess Who?’ with shapes. We reviewed the board game (in the photo above). Key points: each player selects and keeps hidden a mystery person. The other player asks ‘Yes/No’ questions to determine who the mystery person is. We talked about what makes good questions (for example, properties that divide the remaining people as evenly as possible into two groups).

When playing with quadrilaterals, we used sheet protectors (see the photo) to eliminate candidates. This ‘open play’ was also useful in checking each other’s deductions, and in making the game a little more collaborative. This felt successful, though I’d like to listen in a little more closely to some of the conversations.

#### 4. Card/shape sort

To hone in on the nested properties of quadrilaterals, I gave them these cards and asked which properties each of their quadrilaterals had.

Students were able to sort them with relative ease, and so we moved onto the next task.

#### 5. Nested diagrams and family trees

I handed out the following Euler diagram (A3 sized). The task was to put the property cards and the shapes in the relevant regions. We talked about the seven regions, and noted the overlap of the red and the purple. We discussed how an object in the very middle of the diagram has all of the properties and an object at the edge of the diagram (outside the blue ring) has none of the properties.

Some students unintentionally ignored the property cards and made up their own. Few realised that the property cards were numbered in a ‘useful’ way, but I’m not worried about that.

Each region now contains a class of shapes. What are the names of each class of shapes?

This seemed to come far more quickly. We also connected our nested diagram to a ‘quadrilateral family tree’ (credit to David Butler for his post on this).

Each class of shape inherits the properties of those above it. What I missed here (which is important for what comes next) was nailing down the description of the shapes. What differentiates (say) a kite from a rhombus? As David Butler helped me think about later, what distinguishing features do each class of shapes have?

#### 6. Always, Sometimes, Never True

And finally we come to the troubles … back to claims that are ‘always, sometimes, never’ true.

1. A rectangle is a rhombus.
2. A rhombus is a rectangle.
3. A square is a rectangle.
4. A rectangle is a square.
5. A parallelogram is a quadrilateral.
6. A trapezium is a parallelogram.
7. An equilateral quadrilateral is a square.
8. A parallelogram is a rectangle.
9. An equilateral parallelogram is equiangular.
10. A rhombus is a square.
11. A square is a rhombus.
12. A kite is a square.
13. A square is a kite.

I found that it helped to say each statement three times, as follows:

• A rectangle is always a rhombus.
• A rectangle is sometimes a rhombus.
• A rectangle is never a rhombus.

Only one of these can be true. Which is it? If it is sometimes true, then let’s think of examples (and later, properties) where it is true and examples where it is not.

Our problems were with the ‘sometimes’ statements. Consider the statement: ‘A trapezium is a parallelogram.’  As David Butler put it, the answer to a question like ‘When is a trapezium a parallelogram?’ is ‘When it’s a parallelogram’. Not very satisfying!

And ‘not very satisfied’ is a good description of how some students felt. ‘Magic’ is probably what others thought; I think they viewed it as redefining what it meant to be ‘a trapezium’ or ‘a parallelogram’.

Now I wonder if it is helpful instead to start considering ‘sometimes’ statements by first leaning heavily on their definitions and then looking for distinguishing features of isn’t and is.

For example, consider the statement ‘A rectangle is a rhombus’, which we agree is sometimes true. Let’s review our definitions:

• A rectangle is has two pairs of parallel sides and angles that are 90 degrees.
• A rhombus has two pairs of parallel sides and all sides are the same length.

So, when isn’t a rectangle a rhombus? When the pairs of opposite sides of the rectangle are of different lengths. When is a rectangle a rhombus? When all sides are of the same length. If we add this condition to our definition of a rectangle then, to be a rhombus, our rectangle must have two pairs of parallel sides, right angles, and all sides of equal length. (Note that I’ve just described a square, so a rectangle is a rhombus when the rectangle is a square.)

### And now what?

I’ve written about 1700 words to discover that most of what happened today seems okay, but could be tweaked. So in tomorrow’s repeat of this workshop, I’ll include definitions of these quadrilaterals and I’ll reframe the ‘sometimes’ statements as I outlined above. Also, when naming shapes, I’ll use the prompt ‘What is the most precise name that we can give this shape?’ Finally, I’ll try to articulate more clearly the role of these activities in ‘developing (our) mathematical thinking’ — the whole point of the course.

I’ll report back with an addendum to this post to see whether any of these ideas help it go smoother.

# 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!

# Tangling and untangling

This is the seventh in a series of posts about my course ‘Developing Mathematical Thinking’, a maths content elective for pre-service teachers training in primary and middle maths. All posts in the series are hereWARNING: It’s a long post.

Edited to fix the confusion between × (multiply) and x (the letter).

I have been itching to try Conway’s Rational Tangles with a group of students. I first read about this problem a couple of years ago in Fawn Nguyen’s excellent post. It looked super interesting, but I was still somewhat confused with how it works (not to mention why it works). So I was excited to be in Fawn’s ‘Conway Tangles’ Math Micro-Session at the NCTM Annual Meeting in San Francisco this year, where it started to make some more sense.

This week I tried the activity with my #math1070 students. I waited until the last week of the course because: (1) now that we know each other better, I thought they’d tolerate me muddling through it, (2) their resilience and problem-solving skills for more challenging and ill-stated problems have increased. (Note that the ‘ill-stated’ part is my fault, not that of the problem!) I was upfront with them about how I was both excited and nervous about the session. It was a bit sketchy with the first group of students, but I was able to make some adjustments with the second group.

Below is a mash up of how I did it this week and how I would improve it in the future. This outline is based on Fawn’s write up, but I also pulled in ideas from Tom Davis’ thorough notes for a Maths Teacher Circle, along with the three-part outline from nrich maths: Twisting and Turning, More Twisting and TurningAll Tangled Up. We spent ~1.5 hours on the activity. Perhaps half of that was outside, with students doing it themselves.

#### Getting started: the rules

Have four volunteers come to the front. Each person holds the end of a rope so that the two ropes are horizontally parallel. This is the starting position. This state has a value of 0.

There are only two moves that can be made: TWIST (T) and ROTATE (R).

TWIST is when the person at the bottom left moves under the orange rope to the top left, as shown below. This new state has a value of 1. We notate this as $0 \xrightarrow{T} 1$. I tell the students that every time a TWIST operation is performed, the value of the ropes increases by 1. So, TWIST is +1.

A ROTATE is when every person moves clockwise to the next position, as shown below. (Note that this is from the starting position.) I say that I am not going to tell them the value of this new state. $0 \xrightarrow{R} ??$ I’m also not going to tell them what ROTATE does; that’s for them to figure out.

#### The ‘aim’

Our aim is to (in my words) ‘tangle the crap out of the ropes’ by performing any number of TWISTS and ROTATES and then work out how to untangle the ropes back to horizontally parallel (with value 0). But, remember that there are only two available moves: TWIST and ROTATE. ‘Untwist’ and ‘Unrotate’ are not possible moves.  (I wrote ‘aim’ because this isn’t the only goal, but it’s the one that students will initially want to work towards.)

At present, we have two questions:
(1) What does ROTATE do?
(2) How to get out of any tangle?

#### A first go at experimenting with ROTATE

Before I let them loose with some ropes, we try a few more systematic experiments.

We reset the ropes to 0, and try ROTATE followed by ROTATE (RR). We discover that the ropes end up horizontally parallel again. That is, $0 \xrightarrow{R} ?? \xrightarrow{R} 0$. We decide not to ever do RR unless we want to waste energy. We test this further by resetting the ropes to 0, and trying TWIST followed by RR. As expected, we end at a state with value of 1. We summarise: $0 \xrightarrow{T} 1 \xrightarrow{R} ?? \xrightarrow{R} 1.$

The other possibility after an initial rotate is to try a twist. So we reset the ropes to 0, and try ROTATE followed by TWIST (RT). This action is kind of strange; the ropes stay vertically parallel no matter how many twists we do: $0 \xrightarrow{R} a \xrightarrow{T} a \xrightarrow{T} \cdots \xrightarrow{T} a.$

There are already some interesting conclusions that we could come to as a group, but I decide (based on experience) that it might make more sense if everyone is participating instead of watching.

#### Trying it for themselves

Students get into groups of at least five: four on the ropes and at least one person recording the steps. I distribute the ropes (\$1 each at Kmart; bargain!). We go outside. Maths classes in university are never held outside, so this is novel for all of us.

I suggest that to help answer Question 1 (What does ROTATE do?) we might want to break Question 2 down into further sub-questions.

(1) What does ROTATE do?
(2a) Work out how to get out of one TWIST; two TWISTS; three TWISTS; four TWISTS; any number of TWISTS.
(2b) Work out how to get out of a mixed up sequence, like TTRTTTRT, shown below.

Everyone starts with (2a), and works it out fairly quickly (15 minutes?). Their strategy to untangle is to always start with a ROTATE (otherwise we would be further tangling the rope), then to look at the ropes and ‘see’ what to do next. Eventually they write down how to get out of these positive integer states (T, TT, TTT, TTTT, …) and see a pattern. Try it for yourself! (Or look on page 5 of Tom Davis’ notes.)

In general, my students find it hard to conjecture what ROTATE does. I talk to each individual group in turn. To get them started, I write down something they’ve just done: $0 \xrightarrow{T} 1 \xrightarrow{R} a \xrightarrow{T} 0$. We can work backwards from T to realise that a=-1.  We also realise that when we start with just twists, the value of the state keeps increasing from 0. OBSERVATION: To return back to 0, ROTATE must involve a negative somehow.

I suggest perhaps ROTATE is ×(-1). We look at $0 \xrightarrow{T} 1 \xrightarrow{R} -1 \xrightarrow{R} =1$. This works!
We test it on one TWIST: $0 \xrightarrow{T} 1 \xrightarrow{R} -1 \xrightarrow{T} 0$. This also works.

We predict what should happen with two TWISTS: $0 \xrightarrow{T} 1 \xrightarrow{T} 2 \xrightarrow{R} -2$. To get this untangled, we should be able to do TT. We try it with the ropes. Groan as it doesn’t work. (Note that some students have already forgotten that they know how to get out of two twists, from (2a).)

There is more conjecturing about ROTATE. For example, some students try ROTATE is -2. Later in the class discussion we realise that ROTATE can’t involve just adding or subtracting as RR would take us further away from 0 (positive or negative), and we know that $0 \xrightarrow{R} ?? \xrightarrow{R} 0$. OBSERVATION: ROTATE must involve multiplication or division (or perhaps some other operation).

Most are still stuck. I ask them if they’ve done (2b) and untangled TTRTTTRT. If so, I tell them that the tangled state has value 3/5. OBSERVATION: ROTATE must involve a negative and fractions somehow. Some more cautious conjecturing eventuates.

If they are still stuck, I tell them that TTRTTR has value -2/3.

After working on it for ~45 minutes, some of them give up and demand the answer. I know there is more problem-solving work to come so I tell those that haven’t worked it out that ROTATE is ×(-1/a), where a is the previous state value.  To summarise:

• $x \xrightarrow{T} x+1$
• $x \xrightarrow{R} -\left(\frac{1}{x}\right)$

#### Efficiently getting out of any tangle

I ask them to come up with a scheme to efficiently get out of any tangle. (Later we decide that we aren’t sure that it is the minimum number of moves, but it seems efficient.) It works a bit like this: Get as close to zero with a numerator of 1 and a positive denominator (like 1/m) then ROTATE. This leaves you with a negative integer, –m, and you can TWIST your way m times back to 0.

#### Wrapping up

Back in the classroom as one group, we summarise what we discovered, and make a few more observations.

• We go back and think about starting with a single ROTATE. Now that we know what ROTATE does, we see that the state becomes -1/0. This is like infinity. Another ROTATE brings it back to 0. When we start with a single ROTATE, TWIST leaves it exactly the same: $\infty + 1 = \infty$. So we can have a tangle value of infinity. This is all kind of cool.
• We wonder if every rational number can be reached through tangles, and then be untangled.
• We wonder about how to prove the minimum number of moves to get out of each tangle.
• We talk briefly about function notation: $T(x) = x+1$ and $R(x) = -\left(\frac{1}{x}\right)$. We confirm that $R(R(x)) = x$, so two RR leave the state unchanged. We talk about composition of functions, and how RTT is represented by $T(T(R(x))).$

We talk about how this activity is suitable for a range of students and different areas of focus:

• problem solving and team work just by trying to untangle a tangle (no investigation into TWIST and ROTATE)
• practicing fluency with fractions
• older students can work with function notation and tackle some of the more challenging questions.

I reflect later how there is so much more depth in this activity than I had realised. I also realise that because it has so many different dimensions—physical manipulation, symbolic notation, numerical calculations, pattern recognition, conjecturing, teamwork, leadership—it gave students opportunities to shine in different ways.

# How many triangles?

It’s been quiet on the blog, but a lot has been happening. University classes in Adelaide have just resumed after a two week mid-semester break. To warm up, I gave my MATH 1070 students the following problem. I found it via Tanya Khovanova who states that it was an entrance problem for the 2016 MIT PRIMES STEP Program. (Read more on Tanya’s blog.)

I drew several triangles on a piece of paper. First I showed the paper to Lev and asked him how many triangles there were. Lev said 5 and he was right. Then I showed the paper to Sasha and asked him how many triangles there were. Sasha said 3 and he was right. How many triangles are there on the paper? Explain.

Here are some solutions from my students, all considered to be correct. The ones in blue originally appeared in Tanya’s blog post (although several students came up with these too). Additional ideas are shown in red below. The black rectangle shows the piece of paper. Two of the rectangles contain instructions instead of diagrams.

Update (8 October 2017): A few new ideas from this week’s classes are shown in purple.

I loved this as an opener to encourage creative problem solving. Thanks Tanya!

# Notice and wonder: the Prime Climb hundreds chart

This is the sixth in a series of posts about my course ‘Developing Mathematical Thinking’, a maths content elective for pre-service teachers training in primary and middle maths. All posts in the series are here.

This is the final post detailing how I introduced ‘Notice and Wonder’ to my pre-service teachers. We’ve used it for sense making. We’ve then looked at photos from the world around us and brainstormed what we noticed and wondered. The students later took their own photos and identified the mathematical ideas that they saw. (The photos and reflections were so much fun to look through!) Next, we transferred our ‘Notice and Wonder’ skills to more mathematical settings, including one of Dan Meyer’s Three-Act Maths Tasks, Toothpicks. I’ll now tell you about the consolidating task in which I had students tell me what they notice and wonder about an image bursting with mathematical ideas.

Update 6 August 2017: This post describes another way to incorporate ‘Notice and Wonder’ with the Prime Climb hundreds chart.

Note: although this looks like a long post, the first 200 words are an introduction; the last 1500 words are a summary of student ideas.

#### The Prime Climb hundreds chart

Prime Climb is a beautiful board game in which players deepen their understanding of arithmetic through gameplay. To be quite honest, I’ve never played!  But that hasn’t stopped me appreciating the gorgeous hundreds chart that accompanies the game. A version is below; you can buy this image on a stunning poster here.

This hundreds chart compels us to notice and wonder. Take a moment and brainstorm for yourself. (Dan Finkel, creator of the game Prime Climb, talks about this image in his wonderful TED talk, ‘Five Principles of Extraordinary Math Teaching‘. It’s worth taking the ~15 minutes to watch.)

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

I asked my group of nearly fifty pre-service teachers to each tell me five things that they noticed, and one thing that they wondered. As a group, that’s potentially 250 different things that they notice, and 50 things that they wonder! Here is a collated list of about 100 of their ideas (with slight amendments to incorrect terminology), loosely grouped under my own section headings. I asked them to do this as individually. I’m sure that in a group discussion they would have built on and extended each other’s ideas. Next time!

Enjoy the read; I certainly did.

#### I notice that …

Colour and structure

• Circles are numbered 1-100.
• The chart is organised into a 10×10 system.
• The numbers are ascending.
• The numbers in each column increase by ten as you go down the list.
• Colour has something to do with number, and vice versa.
• There are different colours: blue, orange, yellow, red, green.
• Some circles have only one colour.
• With the exception of the whole red circles, each other colour appears as a whole circle only once.
• Each circle is made up of one or more colours.
• Colour is used to demonstrate relationships between numbers.
• Every second number has orange in it (and similar statements about other colours).
• All even numbers are yellow/orange.
• Friendly numbers (5s and 10s) have blue in them.
• Circles with blue end in 5 or 0.
• There are a lot of red-only circles/numbers.
• There are 21 solid red-only circles/numbers.
• Red is the most prominent colour.
• Purple is the least-used colour.
• Completely green numbers are multiples of 3 (and similar statements about other colours).
• The rings are broken into fractions that vary between a whole and 1/6.
• Some of the red sections have little white numbers in them.
• All the small white numbers that appear ‘randomly’ on the bottom of the circles are all odd numbers.
• The red full circles only occur on odd numbers.
• Numbers with orange in them (multiples of 2) are in a vertical pattern, as are numbers with blue in them (multiples of 5). But numbers with green in them (multiples of 3) are in a a diagonal pattern (right to left) when viewed from top to bottom.
• If you place your finger on a number with purple, then move your finger up one row and then move it three columns to the right, you will end up on another number with purple (works with most purple numbers unless it is too close to the edge).
• The greatest number of coloured sectors around a number is six.
• The greatest number of different colours included in the sectors surrounding any number is three.
• No number/circle has all the colours present.
• There doesn’t seem to be a pattern in the colours.

The number 1

• The number 1 has no colour, because it is neither a prime or a composite number.
• The number 1 has its own colour and is not part of any particular pattern in the chart. Every whole number has a divisor of 1.
• 1 is not a prime number, which is why it is not coloured.

Prime numbers

• The circles with full colours are prime numbers.
• All prime numbers have a single unbroken circle.
• 97 is the largest prime number less than 100.
• Prime numbers have their own specific colour up to the value of 7.
• Red circle numbers are also prime numbers from 11 upwards.
• Other than 2, all prime numbers between 1 and 100 are odd numbers.
• There are 25 prime numbers between 1 and 100.
• If there is a little number written at the bottom of a circle for a greater number then it means that greater number is divisible by a prime number. For example the number 92 has a small 23 written at the bottom of the circle, this indicates that 92 is divisible by the prime number 23.
• There is only one prime number between 91 and 100. All other blocks of ten have at least two prime numbers.
• The ‘3’s column has the most prime numbers between 1 and 100.

Composite numbers

• Numbers that aren’t prime are a mix of colours. For example, 15 is 5×3 where 5 is blue and 3 is green, so 15 is half blue and half green.
• All multiples of 6 have to have orange (2) and green (3) in them.
• Any number ending in 4,6,8 or 0 isn’t a prime number.
• Some non-prime numbers are made up of factors which are just (only) prime numbers.

Square numbers

• All square numbers are comprised of one colour in several parts.
• The sum of all the square numbers is 385.

Multiplication-oriented

• We can use the colours around each number and multiply their ‘representing numbers’ together to make the number in the middle.
• The circle fragments symbolise how many times multiplication has occurred. For example, the number 8 has three yellow circle fragments, indicating 2×2×2.
• The colours of each circle represent the numbers in which the greater number can be divided by. For example number 95 is coloured blue and red. These colours represent 5 and the prime number 19. When multiplied their sum is 95.

Divisor and factor-oriented

• There are only 2 numbers on this chart that are represented by a circle split into sixths. They are 64 and 96.
• No more than six factors are required to make numbers up to 100.
• Odd numbers more commonly have factors that are prime numbers.
• The circles are divided into sections depending on how many divisors they have.
• The factors of each number are obvious through the colouring.
• Different coloured sections in the circle mean that the number is divisible by more than one number.
• Odd numbers generally have fewer factors, even if they aren’t prime.

Prime factors

• The colours that surround the number represent the prime factors of the number. For example, number 96 has five orange segments and one green segment, which suggests that the prime factors for the number 96 are 2×2×2×2×2×3.

Other

• All numbers divisible by 11 have the number 11 in a subscript, and are in a diagonal line.
• Consider numbers with the same digits (11, 22, …). The sum of the digits are all even numbers.
• There are no explicit instructions or ‘key’ to explain what the chart is actually displaying.
• The sum of the first nine prime numbers is 100.
• If you squint your eyes, you start to see colour patterns rather than noticing numbers, which is how I noticed some of my previous points.

#### I wonder …

Colour and structure

• Why 1 is the only number that is grey?
• Why some circles have extra numbers in white?
• What do the sections of the circles mean?
• Why are different numbers cut into different ‘fractions’? Is there an underlying reason for this?
• Why do some numbers have parts in their colour, even if those parts are the same colour? For example, number 64 has six parts of orange, and orange is associated only with 2.
• How did they work out to segment the outside circle of 24 into four segments? And why are three of them orange and one green?
• What colour is used the most?
• Would the chart be easier to read if all prime number had their own colour rather than the first 10?
• Why do 96 and 64 have the most divisions?
• Are there multiple ‘solutions’ to this problem?

Patterns

• If there is a pattern? And if I could figure it out?
• Is there are pattern between the numbers and the number of parts in its coloured circle that can be used to work it out for any number?
• Why didn’t they write the number of times that a particular number goes into the large number inside the appropriate colour section?
• Why are the numbers coloured in randomly (no specific pattern)?
• Can you use this number chart and extend it to find every single prime number without manual and tedious calculations?
• Is there a systematic way of determining the greatest number of sectors or different colours that can surround any number in a set (1 to 1,000,000 for example) without having to sit down and multiply prime numbers?

Extending the chart

• If this went to 1000, what number would have the most number of different colours?
• If this went to 1000, would we start to see more and more red compared to other colours?
• I wonder what the next 100 numbers would look like prime factorised in this way. I would imagine that the amount of red visible would decrease.
• What would this look like if extended to 200?
• If it went to 200, would the numbers have more than four or five colours?
• How many prime numbers would there be in the next set of 100 numbers, as in from 101 to 200?
• When is the first row of 10 with no prime numbers?

Other

• What maths learning this could be used for?
• What are hundreds charts used for?
• Could a chart like this be used to help introduce maths to young children before they use rote memorisation?
• If knowing primes and composite numbers can help in everyday life?
• What would this look like if we created an image like this based on addition?
• If this chart would be as easily translated if squares or triangles or some other shape was used in place of circles?
• What does this diagram represent? Who was it made for?
• Why did someone choose this representation?