# #NoticeWonder with everyday concepts

I often joke that my blog should be called ‘Notice and Wonder in Mathematics’ because I blog about the ‘Notice and Wonder’ prompt often enough!

In case you are not familiar with it, the ‘Notice and Wonder’ prompt involves asking two questions: ‘What do you notice?’ and ‘What do you wonder?’. These are powerful questions to engage students. ‘Notice and Wonder’ helps lower the barrier to entry for all students and encourages sense making.

Notice and Wonder is definitely one of the top five actions that have transformed my teaching. Like Kate, I love how expansive and inclusive these two simple prompts are.

In previous posts, I’ve focused on using Notice and Wonder in problem-solving contexts like Conway’s Rational Tangles or a paper-folding investigation, intriguing prompts like the Prime Climb Hundreds Chart or in the world around us, and as a sense-making activity.

In this post I want to quickly share two recent experiences I’ve had that connect ‘Notice and Wonder’ with the types of concepts or questions that we encounter everyday in our classes, rather than more occasional problem-solving puzzlers.

### Example 1: Revising series notation

In today’s lecture we were revising in preparation for the final exam. My focus was on helping students work out how to get started on questions when they don’t know what to do. On the spur of the moment, I started this question by asking what they noticed.

We noticed:

• The terms go negative, positive, negative, … .
• The denominators are multiplied by 3 each time.
• The numerators go up by 1 each time.
• There are five terms.
• The question has the sigma symbol in it.
• It says it’s a series, which we did in the context of sequences and series.

I asked if we could use these ‘noticings’ to write down one general term that could be used to describe any term in the series. And we were off! The alternating signs caused some consternation, and a wondering about how we could make that happen when the sum would ‘add everything up’. That was fun to tackle. There was another wondering about whether we were required to give a final numerical answer, which focused our attention on the word ‘express’.

What was particularly powerful was having a checklist of features from our ‘noticing’ work that we needed to be sure we incorporated in our final ‘sigma notation’ expression of the series.

### Example 2: Launching a lecture on graphs of quadratic functions

A few weeks ago I was preparing for a lecture that introduced graphs of quadratic functions for the first time. At the last minute, I decided to show this graph and prompt for Notice and Wonder.

We noticed:

1. The graph has two x-intercepts, one positive and one negative.
2. The graph has one y-intercept which is positive.
3. There is a maximum value at (2,9).
4. The curve is in all four quadrants.
5. The shape is ‘downwards’.
6. (They probably noticed more features.)

We then went back through this list and expanded it into things we wondered:

1. Could we have two positive or two negative x-intercepts? What would that look like? (We sketched or talked about some possibilities.)
Instead of two x-intercepts, could we have one or even none? (We sketched some possibilities.)
3. I introduced the term ‘vertex’ for the maximum. We wondered what other possibilities there are, and talked about the vertex being a minimum.
4. We wondered whether the curve could be in exactly one quadrant? Or two quadrants? Or three quadrants? (I was not expecting this!)
5. What other possibilities are there for the shape? I introduced the terms ‘concave up’ and ‘concave down’. We drew a concave up quadratic. We connected these to whether the vertex was a maximum or a minimum. We wondered about putting the shape ‘sideways’, but then discovered that it wouldn’t be a function.

This was a pretty strong start to the lecture as it previewed everything I planned to introduce in the next 90 minutes. I then took it one step further and showed three different forms of the equation of the quadratic. In the spirit of full disclosure, in my haste before class I made a mistake with two of the signs. When we discovered this, there was a nice ‘sense-making’ diversion as we expanded/factored the RHS of each equation. Establishing that these were alternative, equivalent forms of the same function turned out to be useful later too. Below I am showing the corrected versions.

Now we worked on connecting the different representations with the features that we had just noticed.

• From the standard form, we could ‘see’ the y-intercept.
• From the factored form, we could see that the factors related somehow to the x-intercepts.
• From the third form, we could see that the terms related somehow to the vertex. We then named it ‘vertex form’.
• We also discussed the negative sign in front of the x2 and how that related to the shape. I don’t think we discussed it in class, but I can see now how we could develop that idea from the vertex form and the observation that, for any value of x, the y value will be less than or equal to 9.

This short introductory discussion motivated the rest of the lecture in such a way that some of the later material didn’t need to be discussed in the depth that is usually required. I’ve created more time within this topic!

I constantly marvel at how these two simple questions — asked together or independently — have such a positive impact on the learning that happens with my students and for me.

# #NoticeWonder and Rational Tangles

Yesterday we held the first of this year’s Maths Experience days. We invite students in Years 10 and 11 from different schools onto campus for an intensive one-day program. Students find out about mathematical research, talk to professionals who use mathematics in their careers in some way, and participate in hands-on mathematics workshops. Importantly, they also meet and connect with other students who enjoy mathematics.

One of the activities I chose for this year was Conway’s Rational Tangles. I’ve previously written detailed notes about running the activity with pre- and in-service teachers.  For the Maths Experience, apart from the inherent fun of ‘playing’ with ropes, I wanted students to have a collaborative and authentic problem-solving experience. I introduced the activity as one that mirrors mathematical research — full of questions, puzzling moments, uncertainty, frustration and hopefully also joy. I emphasised that we might not solve the problem, but that the experience of working mathematically was our goal, which includes making wild conjectures and having out-of-the-box ideas!

In this post I want to highlight one addition I made to the activity described in my earlier blog post — the inclusion of the ‘Notice and Wonder’ prompt1.

I started the session by showing students the short video below, edited from one I found on Youtube by Tom Hildebrand. Specifically, I turned off the sound, cut out the whiteboard, and sped it up significantly. Then I asked the two magic questions: ‘What do you notice? What do you wonder?’ Take 70 seconds to watch the video, and see what you think.

Here is what they noticed.

Group A

• They are trying to untangle the ropes.
• One person hangs on to one end of the rope for the whole time.
• They rotate 90 degrees clockwise.
• There is a plastic bag.
• Twist involves exactly two people and occurs in exactly the same position.
• They untangle using exactly the same types of moves they used to tangle.

Group B

• Four people holding two ropes.
• Same person holding the same end for the whole activity.
• When rotating, one person moves clockwise. (Later refined to each person moves one position clockwise.)
• The twist movement always involves the two people on the right. The same position goes under each time.
• There was some pattern they kept repeating.
• They did some moves to get a knot. Then they did some more moves and there was no knot.
• There was a bag.
• There were four rotations before the bag appeared and eight rotations after.
• Sometimes there is a different number of twists after a rotate.
• A twist after a rotate goes ‘perpendicular’. (Not sure what that means!)

And here is what they wondered.

Group A

• What’s the deal with the plastic bag?
• What’s the deal with the teacher?
• How did they decide when to stop tangling and start untangling?
• How tangled was the rope?
• What did the teacher and the student pass to each other? (Scissors.)
• How did they work out how to untangle? (I explicitly prompted this question — although I’m sure they were all thinking it.)

Group B

• How did they know how to untangle the ropes? Was it from memory?
• What is the point of rotate? It doesn’t seem to change the rope.
• Does the bag have something to do with the tangling?
• Is it a proper knot? Or just a tangle?
• What is the teacher doing?

There was more conversation that I didn’t manage to capture. (Next time I’ll record it!) Group A spent around 10 minutes on Noticing and Wondering. Group B spent 15-20 minutes. We then largely ran the session as I’ve detailed in the earlier blog post.

What effect do I think Notice and Wonder had? I noticed that students were keen to try the problem for themselves. They made sense of the situation, became intrigued and engaged, and then made the problem ‘theirs’. As a group, students saw that others had interesting ideas. They added on to each other’s thinking. I suspect that it also smoothed the way for working together more intensely once we broke into smaller groups where students didn’t necessarily know one another. It also became more natural for them to Notice and Wonder as the session progressed. All in all, it’s a great modification to a thoroughly engaging activity.

[1] I trialled this with teachers at the MASA conference in April.

# Thank you for the #lessonfail

This blog has been accumulating a layer of dust. I could name a lot of reasons (I only teach four hours a week, I’m busy with other projects, and so on) but the truth of it is that I usually blog when I feel motivated by my teaching. And lately, well, it’s felt a bit ‘meh’, and who wants to read ‘meh’?

Well, it turns out that many of us want to read about lessons that are either uninspiring or failures. We want to know that there are others who have the same experiences that we do. That not every class is sunshine and lollipops.

Yesterday, Annie Perkins tweeted the following.

It led to the hashtag #lessonfail (my suggestion of #lessonmeh was far less popular 😏) and a huge influx of tweets of people sharing their own less-than-perfect teaching moments. Because we are all imperfect humans who make mistakes. Often. And we are constantly learning how to be better.

If you are like me, you view your students’ mistakes as learning opportunities, not learning failures. We need to permit ourselves the same freedom to fail, to learn, and to grow. And we need to talk about it with each other, from beginning teachers to those who have been in the game for a long time. I shared two of mine here. Thanks to those who has shared theirs in the last day or two: Annie FAnnie P, BryanChristine, DavidIlonaMadison, Tracy, and those that I missed. And if you aren’t ready to share, I understand that too. It’s hard to expose our vulnerabilities to others, for whatever the reason. Thank you to everyone for warmly responding to those who share. We are stronger together.

As I’ve mentioned, I love books. Real books, with paper and ink. None of those fancy ebooks. I spend enough time each day staring into screens. Plus, I like to read in the bath and the idea of accidentally dropping a $1000 device doesn’t appeal. (I’ve only ever once dropped a book in the bath. It was a library book. Go figure.) The busier I get, the less I seem to read for pleasure. To redress this, my plan is to read 50 books in 2016. Fiction, mathematics, Australian politics, biographies, non-fiction, anything. Some books are short novellas which you might think of as ‘cheating’. Whatever. Despite the fact that I am counting, the number doesn’t count. It’s just a target to get me to read more. I am tweeting 140 character reviews with #read2016, but I’ll also post the books here in three parts, one every four months. The maths ones (*) might be the subject of separate posts. There were 19 books in Part 1 (January – April), 16 books in Part 2 (May – August) and 18 books in Part 3 (September – December). That makes 53 books in 2016. 1. Black Rock White City, A.S. Patric. It took me ~50 pages to warm to the story, but then it was unputdownable. A decidedly worthy winner of 2016 Miles Franklin Award. #read2016: A couple almost unknowingly clinging to each other through deep wordless grief, and yet a hopeful book. 2. Commonwealth, Ann Patchett. Wonderful writing. Skilled in capturing moments and innermost thoughts in few words. As a promising relationship between a famous author and a waitress begins: ‘He patted the top of her hand, which she had left close by on the bar in case he needed it.’ A glimpse of a left-behind, overworked mother of four young children: ‘The speed at which their mother ran from work to school to the grocery store to home had doubled. She was always arriving, always leaving, never there.’ #read2016: Vignettes spanning 50 years woven together to tell the story of complex blended family relationships. 3. Postcards from Surfers, Helen Garner. Eleven short stories in true Garner style. #read2016: Stories that never use more words than they need. Expertly constructed. 4. Dying: A Memoir, Cory Taylor. I first learned of Cory Taylor on the fabulous ABC ‘Terminally Ill’ program of the ‘You Can’t Ask That‘ series. Cory was frank — the same as in her memoir. #read2016: Clear-eyed. Unsentimental. A deeply reflective view of dying and of life. Moving. 5. Notes on An Exodus, Richard Flanagan. Richard Flanagan won the Man Booker Prize for his remarkable ‘The Narrow Road to the Deep North’. His ‘notes’, along with sketches by Ben Quilty (Archibald Prize winner), paint powerful portraits of Syrian refugees in Lebanon, Greece and Serbia. #read2016 (1): Devastatingly moving portraits of Syrian refugees from two of Australia’s most acclaimed in their crafts. #read2016 (2): A slim volume but not at all light. Honours their dignity + courage. ‘Refugees are not like you and me. They are you and me.’ 6. The Good, the Bad and the Unlikely: Australia’s Prime Ministers, Mungo MacCallum. More than 29 biographies, this also brings together the story of Australian politics. #read2016 (1): A lively and humanising view of each of Australia’s 29 PMs. Witty + concise writing that had me laughing (or snorting!) out loud. #read2016 (2): I learned many things, but am still struck by the news that we had a PM with the middle name of ‘Christmas’. 7. The Curious Story of Malcolm Turnbull, the Incredible Shrinking Man in the Top Hat, Andrew Street. I preferred the first book, ‘The Short and Excruciatingly Embarrassing Reign of Captain Abbott’, but only because Abbott was so laughably hopeless. The sequel certainly reveals some of the ineptitude of Turnbull. #read2016: Grab the popcorn and dig into the spectacle. (If you don’t laugh, you’ll cry.) Street dishes out snark in spades. 8. Salt Creek, Lucy Treloar. An interesting blend of fact and fiction. #read2016 (1): Set on the South Australian Coorong in the 1850s as white settlers first encroach on the lands of the Ngarrindjeri people. #read2016 (2): A beautifully-told, heartbreaking shameful story that matches historical truths. Starts painfully slow; gripping past pg 90. 9. The Long Green Shore, John Hepworth. #read2016: Published 50yrs after writing of Australians in PNG during WWII. Banality alongside barbarity. Matter-of-fact yet almost poetic. 10. The Hate Race, Maxine Beneba Clarke. An Australian of Afro-Caribbean descent, Maxine Beneba Clarke tells what it is like to grow up as a person of colour in Australia. #read2016: Packs a powerful punch, right to the stomach. A difficult, but important read. Particularly now. 11. * Which One Doesn’t Belong? Teacher’s Guide, Christopher Danielson. The premise of ‘Which One Doesn’t Belong?’ is to consider four shapes, and ask the question. In the children’s picture book and its companion teacher’s guide, Danielson focuses on geometry and uses WODB to draw out rich mathematical ideas. The teacher’s guide provides convincing rationale and practical advice. There are plenty more WODB out there; try www.wodb.ca and the hashtag #wodb. #read2016: A delightful way to discuss + explore maths. The writing is crisp, purposeful, insightful + welcoming. A must-have for tchrs. 12. Girt: The Unauthorised History of Australia, David Hunt. Australian history like it should have been taught at school. The ABC Radio podcast Rum, Rebels & Ratbags with Dom Knight is also worth a listen. #read2016: Peppered with witticisms and dripping in places with sarcasm, this is a lively telling of Australian history like no other. 13. * Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms, Tracy Johnston Zager. I have been madly awaiting this book for at least a year. It was so good I read it cover-to-cover almost as quickly as I could. It’s a beautiful, important book. Truly something special for all maths teachers. My full review is in this blog post. 14. * Avoid Hard Work!, Maria Droujkova, James Tanton, Yelena McManaman. There is a lot to like about this book, but ultimately I found it too light. I also have problems with the title. #read2016: A gentle approach to encourage mathematical problem solving with very young children. 15. All That I Am, Anna Funder. #read2016: A fictionalised biography of political activism against the Nazis in WWII. Crushing. Beautifully written. A page turner. 16. Our Souls at Night, Kent Haruf. The first book by Haruf that I’ve read. I’ll definitely be looking for more. From the back blurb: ‘Addie Moore and Louis Waters have been neighbours for years. Now they both live alone, their houses empty of family, their quiet nights solitary. Then one evening Addie pays Louis a visit.’ #read2016: A tender, quiet and impossibly beautiful tale of growing old together with grace. 17. Victoria: The QueenJulia Baird. Loved the narrative-style approach, particularly once I realised it was built around impeccable research. #read2016: A hefty portrait of a formidable + intriguing queen. Flowing, engaging, well researched. Fascinating details of V as a woman. 18. Monkey Grip, Helen Garner. Her acclaimed first novel. Reads like diary entries, with Garner’s perceptive view. #read2016: Explores addiction, to hard drugs + to love. Written in the 70s; curious to see where Garner started. Still wondering if I liked. # Tracy Zager’s new book Tracy Zager’s new book ‘Becoming the Math Teacher You Wish You’d Had‘ is out, and it’s a treat. The central tenet of this important book is to ‘close the gap’ by making maths class more like mathematics, orienting our students towards the habits of mind of professional mathematicians. ‘Good teaching starts with us’ and Tracy companionably guides us through ten practices of mathematicians: taking risks, making mistakes, being precise, rising to a challenge, asking questions, connecting ideas, using intuition, reasoning, proving, working together and alone. Tracy skillfully blends academic research, illuminating classroom dialogues, the thoughts of mathematicians and maths educators, and her own perceptive observations. This seamless mix is a real strength of the book; we not only see what habits are important and why, but how they can be enacted through specific teaching strategies, and the powerful effects they have on our students’ development as confident and capable mathematicians. The reader can’t help but be inspired by the teachers that Tracy holds up as exemplars of good practice. These teachers have so much respect for each of their students as serious mathematical thinkers. I was struck by the extent to which they would go to adapt instruction in response to student ideas and to support them in pursuing their own line of enquiry. Tracy warns early on that the book is long—and it may be—but it is also captivating! The organisation is immensely practical; each chapter can be used as a self-contained guide for a particular mathematical habit. I can see myself repeatedly delving back into specific habits as the teaching year progresses. I read it cover-to-cover over a couple of days while curled up in a secluded cabin, pausing occasionally to stare out into the Australian bush and ponder what I can change in my own teaching. Some of my highlighted passages: • From Chapter 3, Mathematicians Take Risks: ‘When we assign problems that have a single, closed path from start to finish, we’ve eliminated the possibility that students will take mathematical risks. There’s nothing to try if everything is prescribed.(pg 49)In my skills-based courses, I too infrequently give students opportunities to try and be successful with their own approaches. That’s something to work on. • From Chapter 4, Mathematicians Make Mistakes: ‘If we want students to learn from mistakes, we need to teach them how.’ (pg 57). Tracy outlines a three-part goal: to teach students to take mistakes in their stride, to keep going when they’ve made a mistake, and the one I need to focus on: to teach students to make the most of the knowledge and experience they gained by figuring out their mistake‘. How can I help students gain the skills to diagnose and learn from their mistakes, by themselves? • From Chapter 5, Mathematicians Are Precise: ‘Math without inquiry is lifeless, but math without rigor is aimless. There is no tension between teaching students how to solve problems accurately and efficiently and teaching students how to formulate conjectures, critique reasoning, develop mathematical arguments, use multiple representations, think flexibly, and focus on conceptual understanding.’ (pg 80). In my problem-solving course, I deliberately swung the pendulum from the typical procedure-based courses my students had mostly experienced towards creative, collaborative problem-solving. But I also need to find the middleground, where I place as much emphasis on rigour as I do on inquiry. • From Chapter 12, Mathematicians Work Together and Alone: ‘If a major part of doing mathematics involves interacting with other mathematicians, then a major part of teaching students mathematics must be to teach students how, why, and whether to interact with one another mathematically. Students need to learn how to ask for what they need from each other and to be what they need for each other … we need to teach students how to be good colleagues it’s important we honor individual thinking and working time. It’s not reasonable to expect students to collaborate at every moment, and that’s not how mathematicians work.’ (pg 312). This past semester, a few students in my problem-solving course commented that they needed more opportunities to work alone first, and more strategies to work effectively with group members. I’ll definitely be digging further into this chapter next year. And, these phrases are going straight into my repertoire: • ‘Do you have more questions after doing this? What are you wondering about now? (pg 149). • ‘What does ______ have to do with _____?’ (Debbie Nicols, pg 191). • ‘Remember that it’s hard to find mistakes when you assume that you’re right. So go back into it assuming something went wrong.’ (Jennifer Clerkin Muhammad, pg 284). • ‘Would you recommend that strategy to someone you like?’ (pg 118). 😂 There is so much to love about this book. The writing is both encouraging and empowering. It’s labelled K-8 but Tracy offers important insights to help teachers across all year levels; I have been nodding furiously and making notes throughout. This particular passage had me shouting ‘yes!’: ‘We need to give ourselves permission to say, publicly, and with delight, “I never thought about it that way before!” whether it refers to addition, fractions, or place value. It is long past time for us to respect the beauty, power, and importance of elementary mathematics, instead of having contempt for “the basics.”’ (pg 208) Listening carefully to student thinking, especially about ideas I thought I understood, always gives me new insight. It’s why I’ll never tire of teaching. I can confidently say that, alongside ‘Thinking Mathematically‘ (Mason, Burton and Stacey, 1982; 2010), Tracy’s book will become a cornerstone for my teaching. It is a gift to all maths teachers. But don’t just take my word for it; you can preview the book in its entirety here. The companion website promises more, and I can’t wait to look around! Update (22 December 2016): The companion website is now live, and it is packed full of goodies. Be sure to check out the free study guide under ‘Getting Started’, which works for either an individual or group book study. # 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. Additional ideas are shown in red below. The black rectangle shows the piece of paper. Two of the rectangles contain instructions instead of diagrams.

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