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Which of these are true? Every even number greater than 3 is the sum of two prime numbers? Every number is the sum of no more than 4 square numbers? The sum of 7 and 3 is always 10? If there are over 30 people in a room it is more likely than not that two of them have the same birthday?

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Christian Goldbach (1690-1764) He was born in Konigsberg, Prussia on March 18, 1690. His father was a Protestant minister in Konigsberg. He attended the Royal Albertus University. He became a mathematics professor and historian at St. Petersburg Academy of Sciences in 1725. He tutored Czar Peter II in 1728. Goldbach's conjecture remains unproven to this day. Goldbach's conjecture remains unproven to this day.

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If you analyse HMI and OFSTED reports over the last 90 years the concerns noted most often are (in order): The concern over investigational maths is directed at the opportunities provided for the use of higher order thinking skills. The difference between: knowing about maths and then using knowledge to work like a mathematician. Investigational mathematics Technical aspects of writing Speaking/Listening Reading for meaning Writing for meaning Standards of behaviour

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Why is special? 26

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Three key tasks face us if we are to challenge children to think in the way mathematicians do: To enable them to ask questions as well as answer them To enable them to look at data and hypothesise /suggest reasonable explanations To enable them to make connections between things they already know

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Combining Bloom’s Taxonomy and the Key Thinking Skills REMEMBERING Recognise, list, describe, identify, retrieve, name …. Information-processing skills: locate, collect and recall relevant information UNDERSTANDING Interpret, give examples, summarise, infer, paraphrase ….. Information-processing skills: interpret information to show they understand relevant concepts and ideas analyse information eg sort, classify, sequence, compare and contrast understand relationships APPLYING Implement, carry out, use … Enquiry skills: ask relevant questions pose and define problems plan what to do and how to research predict outcomes, test conclusions and improve ideas ANALYSING Compare, give attributes, organise, break down into different parts… Reasoning skills : give reasons for opinions draw inferences and make deductions use precise language to explain what they think, EVALUATING Check, make a critical judgement, hypothesise... Evaluation skills : evaluate information they are given judge the value of what they read, hear and do develop criteria for judging the value of their own and others' work or ideas CREATING Design, construct, plan, produce, use the new knowledge in a different situation... Creative thinking skills : generate and extend ideas suggest possible hypotheses be imaginative in their thinking, look for alternative innovative outcomes *Bloom’s Taxonomy in brown *Key Thinking Skills in blue

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To what extent are there opportunities for children to: Generate and extend ideas Suggest possible hypotheses/ explanations Look for alternative and/or innovative outcomes Make connections between different elements

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Pierre de Fermat 1601 - 1665 Pierre de Fermat was a French lawyer and government official most remembered for his work in number theory; in particular for Fermat's Last Theorem. It was his theorem, that deemed him fitting of the title, "The greatest French mathematician of the seventeenth century’’, and yet he was only a part-time mathematician. Pierre de Fermat was a French lawyer and government official most remembered for his work in number theory; in particular for Fermat's Last Theorem. It was his theorem, that deemed him fitting of the title, "The greatest French mathematician of the seventeenth century’’, and yet he was only a part-time mathematician. Commemorative stamp to mark the re-proof of the theorem by Andrew Wiles

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Fermat’s Last Theorem x ² + y ² = z ² Thus 3 ² + 4 ² = 5 ² (9 + 16 = 25) B U T.................. x ³ + z ³ never = z ³

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Sir Michael Atiyah Sir Michael Atiyah is an honorary professor at University of Edinburgh and has been awarded maths’ equivalent of the Nobel Prize “The creative side of maths is all to do with dealing with ideas and trying to understand how things are related in various ways. At school, often you’re simply given the questions: the creative part of maths is searching for exactly what question to ask, what unknown you’re going to explain.’’

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The Park Opening Times April 1 st to September 30 th Monday to Friday: Gates open at 8.30am and close at 9 pm Weekends: Gates open at 8.00 am and close at 9 pm October 1 st to March 31st Monday to Friday: Gates open at 8.30 am and close at 5 pm Weekends: Gates open at 8.00 am and close at 6 pm The Park Opening Times April 1 st to September 30 th Monday to Friday: Gates open at 8.30am and close at 9 pm Weekends: Gates open at 8.00 am and close at 9 pm October 1 st to March 31st Monday to Friday: Gates open at 8.30 am and close at 5 pm Weekends: Gates open at 8.00 am and close at 6 pm Weather for Wednesday March 2 nd March 2nd Number of people in The Park 36 32 28 24 20 16 12 8 4 Time of day 8 - 8.30 am 8.30 - 9 am 9 - 9.30 am 9.30 - 10 am 10 - 10.30 am 10.30 - 11 am

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Create a mathematical representation of a supermarket visit of twenty minutes (from time entered store and leaving after having checked out). The information to be shown is time passing, distance travelled and number of items bought in each minute. BEWARE – you will be asked to describe/explain the information contained. BEWARE – you will be asked to describe/explain the information contained.

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The answer is 35 What is the (mathematical) question?

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2004 KS2 SATS Question Use these clues to guess the number. It’s more than 20 It’s less than 40 It’s a multiple of 5 It’s a multiple of 3

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5 5

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5 5 15

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30

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8 8

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8 8 19

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11

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7 7

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7 7 84

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48

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12

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Start with the sequence of non-zero digits 1 2 3 4 5 6 7 8 9 The problem is to place plus or minus signs between them so that the result of thus described arithmetic operation will be 100 One answer: 12 + 3 - 4 + 5 + 67 + 8 + 9 = 100 Can we find another?

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Dr Alison Steadley Alison Steadley was Professor of Philosophy and Creative Thinking at Philadelphia University ‘’If we want to be creative, we must be able to explore connections between different areas. This means we should have a good knowledge base as the ingredients for creativity depend on the store of ideas and knowledge that are available to us.’’

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This number 9 goes, according to a rule in my head, with one of these. Which one, and why? 6 27 25 15 12 99 11

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The challenge is to get 4 people across this bridge at night with a torch in no more than 15 minutes. WARNING If anyone is on the bridge the torch must be on the bridge also. WARNING The bridge will collapse if more than two people stand on it. BE CAREFUL The torch must be on the bridge. If they cross in pairs they can only travel at the speed of the slowest. TO MAKE IT DIFFICULT If anyone steps on the bridge they must then go all the way across. Person 1: can only travel slowly and needs 8 minutes to cross Person 2: can travel a little quicker, but still takes 5 minutes to cros s Person 3: is quick and can cross in only 2 minutes Person 4: is really quick and can get across in only one minute

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Key Thinking Skills Questions Template An ‘easy-to-view’ selection of maths questions that might be useful starting points to initiate specific types of thinking. Information-processing Why did ……happen? What does…… mean? List the (numbers, shapes etc that….) How many…… are/were there? What is a ……? Who has the widest/ shortest/ biggest/ smallest……? When did ……… happen? What happened after/before……? Which of these …….. happened first/latest etc.)? Which way would I go to……? How much would……cost? Reasoning Looking at the chart/ timeline/graph etc. can you tell me……? Can you sort these ……into groups? What is the pattern? What caused …… to happen? Can you draw diagram that shows………? Can you use the diagram to find out……? Can you explain how ……? In what order did …………. happen? Can you give an example of……? What kind of a ……is this? Which one doesn't belong in this group? What is the purpose of…? What is the relationship between …and…? Evaluation Explain why you…….… Explain how……… What would you have done/do if…………? Why? Which is the best solution to the problem of …… ? Why do you think so? How could ……be improved? What has changed from the beginning of …. and the end? What does it mean that …… happened? Is there another way to find a solution to……? What generalization can you make from this information? Enquiry How is …like…? How are……and ……different? How could we find out if……? What questions could we ask in order to find out…? What do we know that..…? What do we need to know in order to be able to……? How could we prove that……..? Is it true that….? How do you know? If we changed/altered ….. what would happen to ……? Creative Can you create a pattern using……? What will/would happen if ……? If we changed/altered … what would happen to …? Can you create a way to ……? Can you find a solution to the problem of…? How are these things the same/different…? Is there another way to show that…. ? How could...be used in another way? How might we link/use these things together..? How many ways can you find to explain why …….might have happened? How many ways can you find to……..?

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1/4 1/2 1/16 3/16

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What might this pie chart be telling us?

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Why is special? 26

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‘’ Can you tell me this: Why does a ball bounce? ’’

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Pascal's Triangle This pattern is named after the French mathematician Blaise Pascal who brought the triangle to the attention of Western mathematicians (it was known as early as 1300 in China, where it was known as the "Chinese Triangle"). The triangle itself is made by arranging numbers. Each number in the triangle is the sum of the pair of numbers directly above it (to the above left and above right). The first four rows are as follows: This pattern is named after the French mathematician Blaise Pascal who brought the triangle to the attention of Western mathematicians (it was known as early as 1300 in China, where it was known as the "Chinese Triangle"). The triangle itself is made by arranging numbers. Each number in the triangle is the sum of the pair of numbers directly above it (to the above left and above right). The first four rows are as follows:

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Blaise Pascal (1623-62) Pascal invented the first digital calculator to help his father with his work collecting taxes. He worked on it for three years between 1642 and 1645. The device, called the Pascaline, resembled a mechanical calculator of the 1940s. Pascal invented the first digital calculator to help his father with his work collecting taxes. He worked on it for three years between 1642 and 1645. The device, called the Pascaline, resembled a mechanical calculator of the 1940s. If all men knew what each said of the other, there would not be four friends in the world. People are usually more convinced by reasons they discovered themselves than by those found by others. I have discovered that all human evil comes from this; man's being unable to sit still in a room. We know truth, not only by reason, but also by the heart.

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Alan can do the job in 6 hours. David can do the same job in 5 hours. What part of the job can they do by working together for 2 hours? ‘’Do they like talking to each other?’’ ‘’Do they need tools that they couldn’t both use at once?’’ ‘’Just tell me what the job is and then I’ll try and tell you the answer.’’

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Debbie buys one pizza for £2.15. How much will 5 pizzas cost her? ’’Is there an offer on?’’ ‘’ Sometimes you get a discount if you buy three or more’’ What are the chances of throwing a six on a dice? ‘’Quite good if you blow into your hand six times before you roll it.’’

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If it takes one woman nine months to have a baby how long does it take three women to have three babies?

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The Three Little Pigs (From an American website) Show your work. Pig 3 got a good deal on his on his phone bill. It cost him $2 the first month, $4 the second month, and $6 the third month. At this rate, what will his bill be in the 5th month? After an exciting game of leap hog, Pig 3 had an idea. To help pay for their homes, they could open a lemonade stand. They could sell lemonade for 10 cents a glass. If they sold 10 glasses, how much would they make? Pig 1 felt something was wrong. "We're being followed!" he screamed. "Let's run for home!" The pigs ran and ran. They ran 4 miles in 2 minutes. How many miles did they run each minute? Both pigs went squealing down the road to their brother, who like all big brothers said, "I told you so!" And they sat down to watch TV. Their favorite show, Pigmalion, comes on at 8:00 p.m. It was 7:30 p.m. How long did they have to wait for their program? Anyway, this wolf wasn't stupid. He knew he couldn't blow down the brick house without popping a lung so he thought...."I'll just get in my 1963 Volkswagen and run this house down!" If it's 1999, how old was the car?

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The Three Little Pigs How they used their time each day before and after the tragic death of the Big Bad Wolf. Before Number of Hours 14 12 10 8 6 4 2 After Number of Hours 14 12 10 8 6 4 2

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Mrs. Belton’s Class (Yr3 children) Collective height on the morning of September 15 th : 36 metres and 87 centimetr es This is the distance from the right hand side of the classroom door to the wall with the heater on it, down that wall to the bottom, along the wall with the smartboard on it, along the wall with the door into the playground on it and then back along the wall with the classroom door on it until you reach the marker we have placed there. We have put a second marker further along that wall to show where we think we will get to after measuring our heights again at the end of March.

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‘Fermi’ questions Enrico Fermi was famous for thinking about physics out loud – devising problems and then going through a process of reasoning and guessing to arrive at very plausible estimates. The most well-known example is as follows (the thinking process is outlined as well). HOW MANY PIANO TUNERS ARE IN NEW YORK CITY? The number of piano tuners in some way depends on the number of pianos. The number of pianos must connect in some way to the number of people. 1) Approximately how many people are in New York City? 10,000,000 2) Does every individual own a piano? No 3) Would it be reasonable to assert that "individuals don't tend to own pianos; families do’’? Yes. 4) About how many families are there in a city of 10 million people? Perhaps there are 2,000,000 families in NYC. 5) Does every family own a piano? No. Perhaps one out of every five does. That would mean there are about 400,000 pianos in NYC. 6) How many piano tuners are needed for 400,000 pianos? If we assume that "on the average" every piano gets tuned once a year, then there are 400,000 "piano tunings" every year. 7) How many piano tunings can one piano tuner do? Let's assume that the average piano tuner can tune four pianos a day. Also assume that there are 200 working days per year. That means that every tuner can tune about 800 pianos per year. How many piano tuners are needed in NYC? The number of tuners is approximately 400,000/800 or 500 piano tuners.

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Why is special? 26

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