#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.

Today’s date is a Pythagorean triple (written as 15/8/17 or 8/15/17). To ponder: How long until the next one? #iteachmath #MTBoS

— Bryn Humberstone (@brynhumberstone) August 14, 2017

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!

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‘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?

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.

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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?
• Why was this created?
• How long did it take to create?
• Who came up with this representation? It’s really cool!

If you read this far, well done! But to quote the last student, it is really cool, isn’t it?

This is the fifth 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.

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In my last two posts I’ve been explaining how I’ve introduced the ‘Notice and Wonder’ routine to my pre-service teachers. We started with the value of ‘Notice and Wonder’ for sense making. We then looked at photos from the world around us and brainstormed what we noticed and wondered. My intention was that students would gain experience with everyday situations before transferring their ‘Notice and Wonder’ skills to mathematical settings. In this post I’ll write about the next stage of this journey. But to do this, I want to first tell you about a great task and how I’ve never managed to do it full justice.

An activity that fizzled — because of me

It’s all in the presentation

I’ve mentioned before that I was fortunate to attend a micro-conference in June led by Anthony Harradine. This was a master-class in having people think and work mathematically. Anthony emphasised three key ingredients for a successful problem-solving experience for students.

1. Pick a problem where students are likely to already have the required ideas and skills. My interpretation is that the problem-solving process is already cognitively demanding and so students shouldn’t also be grappling with calculations that they find difficult.
2. How the problem is presented matters a great deal. Let students have ideas about the problem. (And, if needed, find a way to make them have the ideas that you need them to have!) Acknowledge and value all their ideas. If their ideas don’t suit your purpose, put them on an imaginary shelf to be pulled down and tackled later. (This is similar to Dan Meyer acknowledging all questions that students have in the initial stages of a Three Act task, and returning to them at the end to see if they can now be answered.)
3. The way that students work on the problem is important. How much structure will you provide? Will students work individually or collaboratively? How will students share their resolutions? Will you provide a full resolution? What will you leave them to think about?

Back to Paper Strips. While planning Week 3 — and looking for a more mathematical setting for students to develop their ‘notice and wondering’ skills — it occurred to me that in the past, I’d had two out of these three elements for Paper Strips. But I was missing a vital ingredient: the right presentation that let students have more ideas for themselves.

The revival: notice and wonder to the rescue

This year I told students that we were going to do a visualisation activity, and that I would walk them through a set of instructions.

• Imagine a long thin strip of paper stretched out on the table in front of you.
• Hold each end.
• Now fold the paper by moving your left hand over to your right.
• Make a crease along the folded edge with your left hand.
• Now hold the creased end with your left hand.
• Fold it again by moving your left hand towards your right.
• Make another crease.
• Now slowly imagine the paper unfolding.
• What does it look like?

This presentation is exactly as outlined in the Shell Centre’s Problems with Patterns and Numbers, of which John Mason is a co-author. And compare it to the earlier description. Similar, right? But not the same.

Rather than continue on and ask students to investigate the number of creases and folds, I lingered on ‘What does it look like?’ I asked students to draw what they thought it looked like. Sketches ranged from the simple to the complicated. A typical sketch looked like this, although a few others were 3D.

I then handed out strips of paper and repeated the instructions. They compared their record of their mental image with the physical model. I asked them if there was anything else that they noticed about their physical model that was missing in their drawing.

Students then brainstormed other features they noticed about their strip of paper. As a group we noticed creases, folds, sections, sections of equal size, up creases and down creases, the pattern of the creases (two folds gives down, up, up), the dimensions of the strip of paper. We agreed on definitions for many of these terms. We thought about whether our diagrams could be more accurate. For example, were our sections of equal size? Have we distinguished in our diagram between up and down creases? Is accuracy even important here?

Next I suggested that students make more folds, and brainstorm anything new that they noticed. I also asked them to record what they wondered. How did the paper-folding process affect the features of the paper strip that we identified earlier?

The very first idea volunteered is shown below. Look at the profile that forms when the sections between two ‘up creases’ (∨ shape) are placed flat. How does this pattern change with more folds?

Other students added on to this idea. They noticed what would happen to the profile when we unfolded the paper. What do you think will happen?

Another group had wondered how the original dimensions of the paper strip affected the size of sections after each fold. They noticed that sometimes it resulted in a square, and sometimes in rectangles of particular sizes.

These were the first three observations in one group and I loved them. I hadn’t even noticed them as I’d only been considering the strip as a 2D object. Other ‘notice and wonderings’:

• Is there a formula for the number of creases for a particular number of folds?
• The number of sections seems to double with the number of folds.
• The number of sections seems to be one more than the number of creases.
• It looks like there is always one more down crease than up crease.
• The more folds you do, the shorter the ‘bottom layer’ and so that affects things. (The physical process differs from the theoretical process.)
• There seems to be a pattern between the number of consecutive down folds (∧) and the number of folds.
• Is it possible that the maximum number of down folds in a row is three?
• The number of creases appears to be one less the number of folds.
• Could we predict the crease pattern after another fold?
• …

Many of the ideas I wanted them to notice came out of the brainstorming exercise — and so many more interesting questions that I hadn’t even considered. Fifty brains are definitely better than one!

This was the endpoint for this activity. Notice how we didn’t resolve any of these questions as a group. Some students worked out formulae or explanations, but I asked them to keep those private for now. There will be time later when we dig into justifications to revisit this problem.

And now for the whole truth

When I started writing this post, I had thought that ‘Notice and Wonder’ was the key to making this activity shine. It’s the truth, but it’s not the whole truth. Looking back through last year’s photos, I had written a big ‘Notice and Wonder’ next to my instructions for students on the whiteboard. And yet it still kind of flopped.

As The Classroom Chefs say, how you plate a meal is important. And as with meals, how a maths problem is presented is everything! Jennifer Wilson’s latest blog post also reminds me that, throughout their book, John and Matt constantly encourage us to savour our meals, that is, to slow down.

Go back through the Entrée stories you just read, and look specifically at the questions each teacher asked the students. Notice how no teacher was in a hurry; they let students discuss a topic or an idea until they were satisfied that the students fully understood it.

Slow maths. Let students notice and wonder for themselves. Don’t rush them towards what you want them to focus on. As Anthony Harradine said: ‘Let students have ideas about the problem. And, if needed, find a way to make them have the ideas that you need them to have.’. For Paper Strips it was the right presentation, combined with slowing down, that meant I didn’t need to find a way for students to notice what I wanted them to see. They saw that — and so much more.

This is the fourth in a planned 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.

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In my previous post, I talked about how I used sense making as a powerful motivator for the ‘Notice and Wonder’ routine. My next step was to have my pre-service teachers experience ‘Notice and Wonder’ for themselves.

Entry task

I deliberately chose to start with an everyday, seemingly non-mathematical image. Study the image below. What do you notice? What do you wonder?

I displayed the image and asked the two magic questions. There was silence. Inwardly I was thinking ‘Oh, crap — this is going to be disastrous.’. I think it was at this stage that I reminded them that non-mathematical ‘notice and wonderings’ are as important as mathematical ones. After another quiet moment the buzz started. I quickly walked to the back of the room and made myself invisible. Once the intensity of the discussion subsided, each group shared some of their ideas. I wish I could remember them all, as there were as many non-mathematical ideas as mathematical ones, but their wonderings were questions like:

• Why does a shot cost the same as a small?
• Do you need to buy a coffee to get a free babycino for kids?
• Where was the photo taken? (If you are Australian, you might recognise some clues. But there are also red herrings such as the units, which are in ounces.)
• How can you get a shot of tea?
• Which size is best value for money?
• Do they really mean that skim milk is free? Or just doesn’t attract an extra charge?
• How many people buy the large size when they really want a medium because it’s ‘just an extra 50 cents’?
• Why do we measure coffee in ounces in Australia?
• How did they decide the pricing structure?
• Are the diagrams to scale?
• Why does anyone buy coffee at a petrol station? (Yes — this is where I took this photo, while filling up my car.)

We then reflected on what had happened. I supplemented their ideas with those from Max Ray-Riek’s fabulous book, Powerful Problem Solving. Chapter 4 is dedicated to ‘Noticing and Wondering’ and can downloaded as a sample chapter from Heinemann here. Here is an extract:

These activities are designed to support students to: