*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?
- 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?

The notice under Square Numbers is inaccurate: The number 36, which is a square number because 6 * 6 = 36, is not comprised of one solid color and/or one color broken into parts. It is instead comprised of two different colors each representing a different prime # raised to the power of 2 (2 and 3)

2 to the 2nd power is 4 (that represents the two separated sections of orange) while the 3 to the 2nd power is 9 (that is represented by the two congruent sections of yellow) 4 * 9 =36 . If you take one of each factors 2 & 3 and multiply them, you will have the factor 6 twice and 6 x 6 yields the square number of 36.

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2*2=4

3*3=9

4*9= 36

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I remember when I turned 36 as a teacher, and a student asked me how old I was, to which I said ” I am the smallest number with 9 factors” . About 30 seconds later she said ” you’re 36″ . She was one of my best students.

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