Working systematically - Primary Mathematics



Theme: Problem Solving – Finding all possibilities

|Year Target |Group Target |Key Resources / Models and Images |Outcomes |

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|Yr 1 |Know that there is sometimes more than one |Smart board resources |• Which dominoes have a total of 5 spots? 7 spots? 10 |In how many different ways can you make 10p using only 2p|

| |answer to a problem |unit plans: |spots? |and 1p coins? |

| | |Autumn unit 3 | | |

| | |Summer unit 8 |What numbers could you add to give a total of 4? Are |Which three coins make 11p? How else could you make 11p? |

| | |Summer unit 9 |there other ways to get a total of 4? | |

| | | | |James paid 13p for chews. What coins could he use? What |

| | |Nrich multiple pack |• How many different ways can you score 4 by rolling two |if he paid 17p? |

| | | |dice? What about 6? | |

| | |ICT files | |In how many different ways can you make 30p using only |

| | | |• Ann is 2 years older than Tom. |silver coins? |

| | |Four pin bowling |How old could each of them be? | |

| | |Down path | |Tell me some addition questions that have 80p as an |

| | |Gobstopper |• Investigate different ways of putting 7 buttons in 3 |answer. |

| | |Ice cream |boxes. | |

| | |Pick a pair |[pic] |Here are 20 counters. Arrange them in equal rows. Is |

| | |Snakes and ladders |Now try 10 buttons. Or 11? |there a different way to arrange them in equal rows? |

| | |Beanbag buckets | | |

| | |Ride at the fair |• Write as many different ways as you can of making the |Show me two numbers that have a difference of 3. Can you |

| | |Sum up |number 12. |think of another pair of numbers with a difference of 3? |

| | |Bird eggs | | |

| | |Line of symmetry |• How many different ways can you colour two squares |Build me two towers that have a difference of four cubes |

| | |Card sharp |using a red pen and a blue pen? |in their heights. |

| | |Jack and the beanstalk | | |

| | |Monster |Break this rod of eight cubes into three pieces. How many|How many different ways can you score 4 by rolling two |

| | |Fireworks |cubes are in each piece? Can you do it in a different |dice? What about 6? |

| | |Goldfish |way? | |

| | |Ones and twos | |An orange costs 17p. Which three coins would pay for it? |

| | |Christmas tree |Here are five rectangles of the same size. How many | |

| | |At the toyshop |different bigger rectangles can you make using two or |Which three coins make 32p? How else could you make 32p? |

| | |Ben’s numbers |more of the rectangles? | |

| | |Spot the shapes |[pic] |Put 1, 2 or 3 in each circle so that each side adds up to|

| | |5rectangles | |5. You can use each number as often as you like. Find |

| | |Arithmagon1 |Put numbers in the shapes that add to 12. |different ways of doing it. |

| | |Duck pond |[pic][pic][pic][pic]12 | |

| | |Quack quack | | |

| | | |Use the numbers 15 to 20. Choose a pair of numbers to |Using 2p, 5p and 10p coins, how many different ways can |

| | |Christmas candles |make this sentence true: |you make 20p? |

| | |Easter chicks | | |

| | |Invisible ink |[pic] is one more than [pic] |Pick 3 cards from a set numbered 1-5. How many different |

| | | | |sums can you make? Have you got them all? |

| | | |How many different pairs can you find that make the |1 |

| | | |sentence true? | |

| | | | | |

| | | |Use the number cards 1 to 9. Which pairs of numbers total| |

| | | |10? | |

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|Mathematical |Work with objects, numbers and pictures in a | | | |

|challenges for able |systematic way to solve finding all | | | |

|pupils in Key stages 1|possibilities problems. | | | |

|and 2 | | | | |

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|Finding all | | | | |

|possibilities problem | | | | |

|solving pack | | | | |

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|Guidance booklet | | | | |

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|Further examples of | | | | |

|pitch and | | | | |

|expectations: | | | | |

|Foundation to year 1 | | | | |

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|Year 1 | | | | |

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|Information | | | | |

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|- Divide and rule1 | | | | |

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|- Divide and rule2 | | | | |

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|- teaching mental | | | | |

|calculation strategies| | | | |

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|- teaching written | | | | |

|strategies | | | | |

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|- exemplification of | | | | |

|standards | | | | |

| |Use simple lists and practical resources to | | | |

| |organize answers. | | | |

| |Spot and describe simple patterns. | | | |

| |Explain how different answers or solutions | | | |

| |are different’ | | | |

|Year Target |Group Target |Key Resources / Models and Images |Outcomes |

|Yr 2 | |Organise the recording of possibilities in an|Smart board resources |Solve puzzles and problems such as: |use pinboards to make different shapes with five straight|

| | |ordered list or table. |unit plans: | |sides (pentagons). Which have a right angle? |

| | | |Y2 autumn unit 4 |• How many dominoes have an odd total of spots? | |

| | | |Y2 spring unit 8 | |Fatima paid 57p for a yogurt. What coins could she use? |

| | | |Y2 summer unit 8 |• Using three dice, find different ways of scoring 12. | |

| | | | | |Fnd as many ways as possible to complete a missing-digit |

| | | |Nrich multiple pack |• Explore different ways of adding three odd numbers to |calculation such as [pic]1 [pic][pic][pic][pic]0 |

| | | | |make 11. | |

| | | |ICT files | |How many different pairs of numbers can you remember that|

| | | | |• Write as many different ways as you can of making the |have a total of 20? How can you be sure you have |

| | | |Problem solving materials: |number 30. |remembered them all? |

| | | |Ben’s numbers | | |

| | | |Four pin bowling |• Find ways of rearranging the digits so that the sum of |Look at these multiples of 10. Which pair of numbers has |

| | | |Down path |each row, column and diagonal is the same. |a total of 100? Are there any other possibilities? 10, |

| | | |Gobstopper |[pic] |20,30, 40, 50, 60, 70, 80, 90 |

| | | |Ice cream |Now try with different digits: for example, 4, 5, 6. | |

| | | |Pick a pair | |[pic][pic][pic][pic]100. |

| | | |Snakes and ladders |How many different numbers can be made with the place | |

| | | |Beanbag buckets |value cards 20, 40, 3 and 5 |What two numbers could go in the boxes? Are there any |

| | | |Ride at the fair | |other possibilities? |

| | | |Sum up |Look at this number sentence: [pic][pic][pic][pic]7. What|[pic][pic][pic][pic]20 [pic][pic][pic][pic]5 |

| | | |Bird eggs |could the two missing numbers be? What else? | |

| | | |Line of symmetry | |• Use 1, 4 and 5, and the signs +, – and =. |

| | | |Card sharp |Tell me all the pairs of numbers that make 7. How do you |What different answers can you make? |

| | | |Jack and the beanstalk |know you have told me them all? | |

| | | |Monster | |Choose three of these numbers: 14, 15, 16, 17. Add them |

| | | |Fireworks |Here are five identical triangles. |up. What different totals can you make? |

| | | |Goldfish |Use some or all of the triangles to make a bigger | |

| | | |Ones and twos |triangle. |How many different ways can you find to pay 50p using |

| | | |Christmas tree |[pic] |only silver coins? |

| | | |At the toyshop |Is there another way to do it? | |

| | | |Spot the shapes | |Place two red squares, two green squares and two blue |

| | | |Christmas candles |Give the children three digit cards, including 0, for |squares in a line so that the squares make a symmetrical |

| | | |Easter chicks |example: 3 , 6 , 0 |pattern. How many different ways are there of doing it? |

| | | |Invisible ink |What numbers can you make using two or three of these | |

| | | |Lollipops |digits? |Write as many different ways as you can of making 12. |

| | | |Maisie maze | | |

| | | |Horses |How many dominoes have a total number of spots that is |Using digit cards 1-9, choose three for the square boxes |

| | | |Creating shapes |odd? |and use [pic]or - in the circles to make this number |

| | | |Triangles and pentagons | |sentence correct. |

| | | |6triangles |This shape is made from four identical squares touching | |

| | | |Farm problem |edge to edge.Make different shapes from four identical |[pic][pic][pic][pic][pic][pic]11 |

| | | | |squares touching edge to edge. Record each different | |

| | | | |shape that you make. |Three birds laid some eggs. Each bird laid an odd number |

| | | | | |of eggs. Altogether they laid 19 eggs. How many eggs did|

| | | | |Jo has three 20p and two 15p stamps. What values can he |each bird lay? Find different ways to do it. |

| | | | |make using one or more of the stamps? | |

| | | | | |On a 100-square, how many two digit numbers have a digit |

| | | | | |sum of 9? What is the biggest? |

| | | | | | |

| | | | | |I have only one sort of coin in my purse. I have 20p. |

| | | | | |Find different ways that this is possible. |

|Mathematical |Know when all the possibilities have been | | | |

|challenges for able |found. | | | |

|pupils in Key stages 1| | | | |

|and 2 | | | | |

| | | | | |

|Finding all | | | | |

|possibilities problem | | | | |

|solving pack | | | | |

| | | | | |

|Guidance booklet | | | | |

| | | | | |

|Further examples of | | | | |

|pitch and | | | | |

|expectations: | | | | |

| | | | | |

|year 2 | | | | |

| | | | | |

|Information | | | | |

| | | | | |

|- Divide and rule1 | | | | |

| | | | | |

|- Divide and rule2 | | | | |

| | | | | |

| | | | | |

|- teaching mental | | | | |

|calculation strategies| | | | |

| | | | | |

| | | | | |

|- teaching written | | | | |

|strategies | | | | |

| | | | | |

|- exemplification of | | | | |

|standards | | | | |

| |Check for repeats of solutions. | | | |

| |Use lists or tables to find answers to other | | | |

| |questions | | | |

| |Spot simples patterns and make predictions. | | | |

|Year Target |Group Target |Key Resources / Models and|Outcomes |

| | |Images | |

|Yr 3 |Begin to make decisions about how best to organise|Springboard materials: |• How many different ways can you choose two dominoes |In my purse I have [pic]1 coins, 10p coins and 1p coins. |

| |solutions. |Unit 8; unit 9 |with a total of 8 spots? |Find all the possible amounts I can make by choosing |

| | | | |three of these coins. |

| | |Unit plans |• Using three 1 to 6 dice, what even totals can you get? | |

| | |Spring unit 8 |What if you used other dice? |Complete this division fact in as many ways as you can: |

| | |Summer unit 8 | |20 [pic][pic][pic][pic] |

| | | |• Explore different ways of adding four odd numbers to |What multiplication facts did you use to help you do |

| | |Nrich multiple pack |make 20. |this? |

| | | | | |

| | |ICT files |• Use 2, 4 and 5, and the signs +, x and =. |Look at this statement: [pic][pic][pic][pic]35. What |

| | | |How many different answers can you make between 40 and |could the missing numbers be? |

| | |Problem solving materials:|200? | |

| | |Spaceships | |A farmer has cows and chickens on the farm. Altogether |

| | |Suzie snake |• Fit shapes together to make a symmetrical shape. |the animals have 24 legs. How many cows and chickens |

| | |Treasure hunt |For example, make symmetrical patterns from a set of |could there be on the farm? |

| | |Stamps |three shapes such as: | |

| | |Maisie mouse | |Use three of the digits 2, 3, 4, 5 and 6, to create |

| | |Kieron’s Cats |[pic] |multiplication calculations (e.g. 34[pic]6). |

| | |Card tricks |Discuss the lines of symmetry in the patterns. |What products can you make? What is the largest/smallest |

| | |Next door numbers | |product? |

| | |Fireworks |Look at this number sentence: [pic][pic][pic][pic]19. | |

| | |Sheepdog trials |What could the two missing numbers be? What else? Can you|• Count all the rectangles in this diagram (26). |

| | |Three digits |tell me all the pairs of numbers that make 19? How do you| |

| | |Three monkeys |know you have got them all? |Take two identical right-angled triangles. |

| | |Number puzzle | | |

| | |Right angled triangles |Look at this calculation: [pic] 5[pic]8 |Investigate the different shapes you can make by fitting |

| | |Create new shapes |[pic] [pic] [pic]. Write a digit in each box so that the |two sides together edge-to-edge. Write the name of each |

| | |Farm problem |calculation is correct. How else can you do it? What |shape. |

| | | |patterns do you notice? | |

| | | | |Katie has these digit cards. She makes different 2-digit |

| | |Excel files |Repeat with [pic] 2[pic]7 [pic] [pic] [pic]. |numbers with them. |

| | |Zids and zods | |[pic] |

| | |Pentabods and bipods |How many 3p and 4p stamps might we use for a 19p letter? |Write all the 2-digit numbers Katie can make with them. |

| | | |And a 29p letter? |Write a number in each box to make this correct. |

| | | | |300 ÷ 2 = χ × χ |

| | | |How many ways can you colour half of a 2 by 2 square? | |

| | | | |What do the digits in the number fifteen add up to? |

| | | |Write numbers in the boxes to make this correct. |How many other numbers have digits with the same total |

| | | |350 + χ + χ = 420 |but no zeros? Order them from smallest to largest |

| | | | | |

| | | |The difference between the heights of two children is 37 | |

| | | |cm. What could their heights be? | |

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| | | |Find the different totals you can make by adding pairs of| |

| | | |these numbers: | |

| | | |47    50 8    29 | |

|Mathematical challenges for |Describe how I know I have found all the possible | | | |

|able pupils in Key stages 1 and|solutions. | | | |

|2 | | | | |

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|Finding all possibilities | | | | |

|problem solving pack | | | | |

| | | | | |

|Guidance booklet | | | | |

| | | | | |

|Further examples of pitch and | | | | |

|expectations: | | | | |

| | | | | |

|year 3 | | | | |

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|Teaching maths in year 3 | | | | |

|booklet | | | | |

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|Information | | | | |

| | | | | |

|- Divide and rule1 | | | | |

| | | | | |

|- Divide and rule2 | | | | |

| | | | | |

|- teaching mental calculation | | | | |

|strategies | | | | |

| | | | | |

|- teaching written strategies | | | | |

| | | | | |

|- exemplification of standards | | | | |

| | | | | |

| |Check that my solutions meet all the criteria of | | | |

| |the problem. | | | |

| |Begin to suggest extensions by asking ‘what if…?’ | | | |

|Year Target |Group Target |Key Resources |Outcomes |

|Yr 4 |Develop my own systems for solving finding all |Springboard materials: |• Find three consecutive numbers which add up to 39. What|Investigate the different routes from A to B using only |

| |possibilities problems, e.g. start with the |Unit 6; |other numbers up to 50 can you make by adding three |the directions north-west and north-east and record their|

| |smallest number. | |consecutive numbers? |results systematically in a table. |

| | |Autumn: unit 4 | | |

| | |Spring unit 8 |• Draw three rings. |What numbers could go in the boxes to make these correct?|

| | |Summer unit 8 |Use each of the numbers from 1 to 9. |[pic][pic][pic][pic]20 30 [pic] [pic] - [pic] |

| | | |Write them in the rings so that each ring has a total of |Write what the two missing digits could be. |

| | |Nrich multiple pack |15. Find different ways to do it. |χ62 + χ95 = 757 |

| | | | |¼ of χ = ½ of χ |

| | |ICT files |• You can make 6 by using each of the digits 1, 2, 3 and | |

| | | |4 once, and any operation: for example,6 = (21 + 3) ÷ 4 |Using a set of digits cards 1-9, complete the number |

| | |Problem solving materials:|or 6 = (3 x4) ÷ (1 x2) |sentence to make a total that is a multiple of 5. How |

| | |Spaceships |Use each of the digits 1, 2, 3 and 4 and any operation to|many ways van you do it? |

| | |Suzie snake |make each number from 1 to 40. Can you go further? | |

| | |Treasure hunt | |[pic][pic] + [pic][pic] = [pic][pic] |

| | |Stamps |• Each ۞ represents a missing digit. | |

| | |Maisie mouse |a. Choose three digits from this set: 1, 3, 4, 8. |Using digits 1-9, complete the number sentence: |

| | |Kieron’s Cats |Replace each ۞ to make this statement true: |[pic] = 0.5. How many ways can you do it? What if |

| | |Card tricks | |[pic] 16 is one of the numbers you make? What if the |

| | |Next door numbers |۞۞ – ۞ = 38 |fraction was equivalent to 0.25? |

| | |Fireworks | | |

| | |Sheepdog trials |b. Find the missing digits. |My calculator display says 1.2. What was the question? |

| | |Three digits |4[pic] + [pic]8 = 74 3[pic] –[pic]9 = 9 |What other possibilities are there? |

| | |Three monkeys | | |

| | |Jack and the beanstalk |3[pic] +[pic]7 = 120 7[pic] [pic][pic]8 |How many different nets of a closed cube can you find? |

| | |Three rings |[pic]1[pic][pic] | |

| | |3 by 3 grid | |The perimeter of a rectangle is 24 cm. What could its |

| | | |c. Find different ways of completing: |area be? How many different areas can you find? |

| | |fraction mysteries | | |

| | |multiplication mystery |[pic][pic] x [pic] = 252 |Here are 6 cards with some numbers or some signs on. |

| | |subtraction mystery | |24 4 6 x ÷ = |

| | | |• Use a computer program to solve number puzzles: for |Choose 5 different cards to make a correct number |

| | | |example, to fill 4 carriages on a train with 24 people. |sentence. How many different correct number sentences can|

| | | | |you make? |

| | | |Look at this number sentence:[pic] [pic][pic][pic]1249. | |

| | | |What could the missing numbers be? |47 54 28 95 19 |

| | | | |Choose pairs of numbers from the list above. How many |

| | | |Tell me all the pairs of whole numbers that make 15. How |different totals can you find? How many different |

| | | |do you know you have got them all? |differences can you find? |

| | | | | |

| | | |The difference between a pair of two-digit numbers is 13.|Use this 3 by 4 rectangle to find two fractions that add |

| | | |What could the pair of numbers be? |up to 1. How many ways can you do it? |

| | | | | |

| | | |What numbers are missing? | |

| | | |[pic][pic][pic] [pic] 36 [pic][pic][pic][pic]18 | |

| | | |[pic] ÷ 4 [pic] [pic] [pic] ÷ [pic] [pic] 6 | |

| | | | | |

| | | |Put 24 interlocking cubes together to make a 2 by 3 by 4 | |

| | | |cuboid, and then work out what other cuboids they can | |

| | | |make using 24 cubes. | |

|Mathematical challenges for |List answers in a systematic way and explain why I| | | |

|able pupils in Key stages 1 and|chose to solve the problem the way I did. | | | |

|2 | | | | |

| | | | | |

|Finding all possibilities | | | | |

|problem solving pack | | | | |

| | | | | |

|Guidance booklet | | | | |

| | | | | |

|Further examples of pitch and | | | | |

|expectations: | | | | |

| | | | | |

|year 4 | | | | |

| | | | | |

|Information | | | | |

| | | | | |

|- Divide and rule1 | | | | |

| | | | | |

|- Divide and rule2 | | | | |

| | | | | |

|- teaching mental calculation | | | | |

|strategies | | | | |

| | | | | |

|- teaching written strategies | | | | |

| | | | | |

|- exemplification of standards | | | | |

| | | | | |

|Calculator activities | | | | |

| | | | | |

|Reasoning about numbers | | | | |

| | | | | |

|Shape and space activities | | | | |

| |Check solutions against given criteria | | | |

| |Recognise patterns and relationships between | | | |

| |solutions and begin to generalise. | | | |

|Year Target |Group Target |Key Resources |Outcomes |

|Yr | |Solve finding all possibilities problems by |Autumn: unit 8; unit 12 |• Choose any four numbers from the grid. |(8, 10) and (10, 8) are two vertices of a right-angled |

| | |organising solutions into lists or tables. |Spring: unit 2 |Add them up. |triangle. What are the coordinates of the third vertex? |

| |5 | |Summer: unit 6b unit 12 | |Are there any other possibilities? |

| | | | |Find as many ways as possible of making 1000. | |

| | | |ICT files | |(6, 5) and (8, 5) are two vertices of a square, they find|

| | | | |• With 12 squares you can make 3 different rectangles. |all three possibilities for the pair of missing vertices.|

| | | |Problem solving |Find how many squares can be rearranged to make exactly 5| |

| | | |materials: |different rectangles. |Use the digits 2, 3, 5 and 7 and the [pic]symbol once |

| | | |Make 1000 | |each to create a multiplication calculation, for example |

| | | |Four by four |• Find ways to complete: |572 [pic]3 or 35 [pic]72. How many different products can|

| | | |Arithmagons 2 |∆∆+ s ۞۞ + u ( = 1 |you make? What is the largest product? What is the |

| | | |Three digits |• Find:two consecutive numbers with a product of 182; |smallest product? |

| | | |Age old problems |three consecutive numbers with a total of 333. | |

| | | |Zids and Zods | |Two numbers have a difference of 1.58. One of the numbers|

| | | |A bit fishy |• Each ۞ represents a missing digit. |is 4.72. What is the other? Is this the only answer? |

| | | |Eggs (excel eggs) |Use your calculator to solve: | |

| | | |Spendthrift | |Write what the four missing digits could be: |

| | | |King Arnold |۞۞ x 6 ۞ = 6272 |[pic][pic][pic][pic]10 [pic]3[pic] |

| | | |Handshakes | | |

| | | |How many triangles? |• A pentomino is a shape made from five identical squares|Place eight squares together (edge to edge) to make a |

| | | |Polyhedra chain |touching edge to edge. |shape with two lines of symmetry. How many different |

| | | |Spot the shapes 2 | |shapes can you make? |

| | | |Planet Zargon |Divide this shape into two pentominoes. | |

| | | |Sleigh ride |Do it in four different ways. |How many different shapes they can be made from five |

| | | |3 by 3 grid | |squares touching edge to edge and which form the net of |

| | | |Five squares |The product is 400. At least one of the numbers is a |an open cube? |

| | | |Symmetry |multiple of 10. What two numbers could have been | |

| | | |2-3-5-7 |multiplied together? Are there any other possibilities? |[pic] = 0.2. How many ways can you do it? What if |

| | | | | |[pic] 15 is one of the numbers you make? |

| | | |Oranges and lemons |Find all the factors of 30. Explain how you know you have| |

| | | |Library area |found them all. |Using a calculator, explore Which two numbers have a |

| | | |Roses for sale | |product of 912? Are there any other possibilities? |

| | | |Bunches of grapes |The product of two numbers is 24. What could the numbers |Find different ways to complete the number sentence if |

| | | |Ages to ages |be? Find as many pairs of numbers as you can with a |the answer is 12. |

| | | |Prehistoric monster |product of 160. | |

| | | |Ages and ages | |Place a digit in the box so that the number 3.[pic]4, |

| | | |Fruit bowl |My age is a multiple of 8. Next year my age will be a |when rounded to the nearest whole number, is 3. How many |

| | | |Arithmagons 3 |multiple of 7. How old am I? |possibilities are there? |

| | | | | | |

| | | |fraction mysteries |Find as many rectangles as you can with whole-number |Find different ways of completing this calculation: |

| | | |multiplication mystery |sides and an area of 36cm2. Which has the smallest |240[pic][pic][pic][pic] |

| | | |subtraction mystery |perimeter? | |

| | | | | |Find all the different totals you can using two numbers |

| | | | | |from the set of numbers: 3.75, 13.75, 1.82, 0.76, 3.93. |

| | | | | | |

| | | | | |0.[pic] [pic][pic][pic]2. |

| | | | | | |

| | | | | |Find all possible ways to complete the calculation by |

| | | | | |placing one digit in each box. |

|Mathematical challenges for able |Decide on the best way to solve a problem | | | |

|pupils in Key stages 1 and 2 |involving finding all possibilities. | | | |

| | | | | |

|Finding all possibilities problem| | | | |

|solving pack | | | | |

| | | | | |

|Guidance booklet | | | | |

| | | | | |

|Further examples of pitch and | | | | |

|expectations: | | | | |

| | | | | |

|year 5 | | | | |

| | | | | |

|Information | | | | |

| | | | | |

|- Divide and rule1 | | | | |

| | | | | |

|- Divide and rule2 | | | | |

| | | | | |

|- teaching mental calculation | | | | |

|strategies | | | | |

| | | | | |

|- teaching written strategies | | | | |

| | | | | |

|exemplification of standards | | | | |

| | | | | |

|Calculator activities | | | | |

| | | | | |

|Reasoning about numbers | | | | |

| | | | | |

|Shape and space activities | | | | |

| |Explain my thinking when I solve a finding all | | | |

| |possibilities problems and begin to decide whether| | | |

| |it was effective. | | | |

| | | | | |

|Year Target |Group Target |Key Resources / |Outcomes |

|Yr 6 |Must |When solving problems or puzzles,|Springboard: |• Use digit cards 1 to 9. |30 children are going on a trip. It costs [pic]5 including|

| | |I record my work clearly and |Unit 25; unit 26; unit |Replace each ۞ to make a correct product. |lunch. Some children take their own packed lunch. They pay|

| | |logically using jottings or |28; unit 29; | |only [pic]3. The 30 children pay a total of [pic]110. How |

| | |diagrams (if needed) and I can | |۞۞ x ۞ = ۞۞۞ |many children take their own packed lunch? |

| | |explain why I chose to solve the |Unit plans | | |

| | |the problem the way I did. |Autumn: unit 8 |Find different ways to do it if: |Imagine you have 25 beads. You have to make a three-digit |

| | | |Spring: unit 4 unit 7 |you can only use digits 1-6 |number on an abacus. You must use all 25 beads for each |

| | | |unit 11 |the single digit is a 7 |number you make. How many different three-digit numbers |

| | | |Summer: unit 4; unit 5; | |can you make? How can you be sure that you have counted |

| | | |unit 7; unit 9; unit 10 |• Find two consecutive numbers with a product of 1332; two|them all? |

| | | | |numbers with a product of 899. | |

| | | |ICT files | |The result of dividing one number by another is 4 [pic] |

| | | |Problem solving |• Use a calculator to solve these. |What were the two numbers? Are there any other |

| | | |materials: |a. Each ۞ represents a missing digit. Solve: |possibilities? |

| | | |Make 1000 |۞2۞ x ۞۞ = 11 316 | |

| | | |Joins |b. One whole number divided by another gives |Here are some digit cards. |

| | | |Moneybags |1.1818181. What are the two numbers? |[pic] |

| | | |Spot shapes | |Write all the three-digit numbers, greater than 500, that |

| | | |Five numbers |How many different shapes can be made by placing two |can be made using these cards. |

| | | |Maze |identical equilateral triangles edge to edge? What about | |

| | | |Jacks book |3, 4, 5, ... identical equilateral triangles? |Find two square numbers that total 100 |

| | | |Four by four | | |

| | | |Three digits |The area of a rectangle is 32 cm2. What are the lengths of|The product of two prime numbers is 437. What could the |

| | | |Age old problems |the sides? Are there other possible answers? |prime numbers be? Two square numbers have a product of |

| | | |Zids and Zods | |576. What are they? |

| | | |A bit fishy |Write the three prime numbers which multiply to make 231. | |

| | | |Eggs (excel eggs) |χ× χ× χ = 231 |How many different ways can you complete this three-digit |

| | | |Spendthrift | |number so that it is a multiple of 9? |

| | | |King Arnold | |2 χ χ |

| | | |Handshakes |Jason threw some darts at this board. Every dart landed on| |

| | | |How many triangles? |the board. Jason scored exactly 100. |Write in what the missing numbers could be. |

| | | |Polyhedra chain |How many darts did he throw? Which numbers did they land |170 + χ = 220 – χ |

| | | |Planet Zargon |on? | |

| | | |Sleigh ride | |This is cuboid is made from cm cubes. |

| | | |3 by 3 grid | |It is 4 cm by 3 cm by 2 |

| | | |Cube nets |17 multiplied by itself gives a 3-digit answer. |cm. How many cm cubes are needed to make it? |

| | | |Oranges and lemons |What is the smallest 2-digit number that can be multiplied|Another cuboid is made from centimere cubes. It has a |

| | | |Library area |by itself to give a 4-digit answer? |volume of 30 cubic centimetres. What could the length, |

| | | |Roses for sale |[pic] |height and width be? |

| | | |Bunches of grapes | | |

| | | |Ages to ages |How many squares are there altogether on a chess board? |How many different symmetrical shapes can you make using |

| | | |Prehistoric monster |(The answer is not 64!) |the three shapes? |

| | | |Ages and ages | |How many of the shapes have only one line of symmetry? How|

| | | |Fruit bowl | |many have two lines of symmetry? |

| | | |Arithmagons 3 | | |

| | | | | | |

| | | |fraction mysteries | | |

| | | |multiplication mystery | | |

| | | |subtraction mystery | | |

|Mathematical challenges for able |Should |I organise information | | | |

|pupils in Key stages 1 and 2 | |systematically and look for | | | |

| | |patterns and relationships | | | |

|Finding all possibilities problem| |between numbers or shapes to | | | |

|solving pack | |solve problems more efficiently. | | | |

| | | | | | |

|Guidance booklet | | | | | |

| | | | | | |

|Further examples of pitch and | | | | | |

|expectations: | | | | | |

| | | | | | |

|year 6 | | | | | |

| | | | | | |

|Information | | | | | |

| | | | | | |

|- Divide and rule1 | | | | | |

| | | | | | |

|- Divide and rule2 | | | | | |

| | | | | | |

|- teaching mental calculation | | | | | |

|strategies | | | | | |

| | | | | | |

|- teaching written strategies | | | | | |

| | | | | | |

|exemplification of standards | | | | | |

| | | | | | |

|Calculator activities | | | | | |

| | | | | | |

|Reasoning about numbers | | | | | |

| | | | | | |

|Shape and space activities | | | | | |

| | | | | | |

| |Could |I work systematically, using my | | | |

| | |knowledge of number facts and the| | | |

| | |relationship between operations | | | |

| | |to solve problems efficiently and| | | |

| | |I can explain and justify my | | | |

| | |conclusions in words and symbols | | | |

| | | | | | |

|Year Target |Group Target |Key Resources / |Outcomes |

|Yr 6 |Could |I work systematically, using my |Springboard: |Using a 3 by 3 array on a pinboard, identify the eight |• Use each of the digits 1, 2, 3, 4, 5 and 8 once to make |

| | |knowledge of number facts and the|Unit 25; unit 26; unit |distinct triangles that can be constructed (eliminating |this sum correct: |

| | |relationship between operations |28; unit 29; |reflections, rotations or translations). Classify the |۞۞ + ۞۞= ۞۞ |

| | |to solve problems efficiently and| |triangles according to their side, angle and symmetry | |

| | |I can explain and justify my |Unit plans |properties. |• Use only the digits 2, 3, 7 and 8, but as often as you |

| | |conclusions in words and symbols |Autumn: unit 8 | |like. Make each sum correct. |

| | | |Spring: unit 4 unit 7 |Find all the DIFFERENT quadrilaterals can be made by | |

| | | |unit 11 |joining the dots on the circle? |۞۞+ ۞۞= 54 ۞۞+ ۞۞= 155 |

| | | |Summer: unit 4; unit 5; | |۞۞+ ۞۞= 69 ۞۞+ ۞۞= 105 |

| | | |unit 7; unit 9; unit 10 | |۞۞+ ۞۞= 99 ۞۞+ ۞۞= 110 |

| | | |Problem solving | | |

| | | |materials: | |Two prime numbers are added. The answer is 100. |

| | | |Make 1000 |Find all possible solids that can be made from four cubes.|Find all the possible solutions? |

| | | |Joins |Record the solids using isometric paper. | |

| | | |Moneybags | |Using the digit cards: 2, 4, 7 and 8, how many different |

| | | |Spot shapes | |multiples of 6 can you make? How do you know you have them|

| | | |Five numbers |Investigate the number of different ways that a 2 by 2 by |all? |

| | | |Maze |2 cube can be split into two pieces: | |

| | | |Jacks book |a. of the same shape and size; |• Find two consecutive numbers with a product of 702. |

| | | |Four by four |b. of different shapes and sizes. | |

| | | |Three digits | |Find the smallest number with exactly 3 factors. |

| | | |Age old problems |k, m and n each stand for a whole number. They add |Now find the smallest number with exactly 4 factors. What |

| | | |Zids and Zods |together to make 1500. |about other numbers of factors? For example, can you find |

| | | |A bit fishy |k + m + n = 1500 |a number with exactly 13 factors? |

| | | |Eggs (excel eggs) |m is three times as big as n. k is twice as big as n. | |

| | | |Spendthrift |Calculate the numbers k, m and n. |Follow these instructions: |

| | | |King Arnold | |Draw a rectangle with an area of 18 cm². |

| | | |Handshakes |6 green apples cost 75p. |Each vertex of the rectangle is at a grid point of a |

| | | |How many triangles? |10 red apples cost 90p. |coordinate grid (-20 to 20) (all the vertices have whole |

| | | |Polyhedra chain | |number coordinates). |

| | | |Planet Zargon |Jason bought some bags of green apples and some bags of |One of the vertices is at the point (0, 0). |

| | | |Sleigh ride |red apples. He spent £4.20. How many bags of each type of |How many different rectangles satisfy these conditions? |

| | | |3 by 3 grid |apples did he buy? |Can you explain how you know that you have found them all?|

| | | |Cube nets | | |

| | | |Oranges and lemons |p and q each stand for whole numbers. p + q = 1000 |The same number is missing from each box. Write the same |

| | | |Library area |p is 150 greater than q. Calculate the numbers p and q. |missing number in each box. |

| | | |Roses for sale | |χ × χ × χ = 1331 |

| | | |Bunches of grapes |An isosceles triangle has a perimeter of 12cm. One of its | |

| | | |Ages to ages |sides is 5cm. What could the length of each of the other |Susan says: ‘When you cut a piece off a shape, you reduce |

| | | |Prehistoric monster |two sides be? |its area and perimeter.’ |

| | | |Ages and ages |Two different answers are possible. Give both answers. |Is Susan’s conjecture sometimes true, always true or never|

| | | |Fruit bowl | |true? Explain how you know. |

| | | |Arithmagons 3 | | |

| | | |Chalk problem; | |Liam thinks of a number. He multiplies the number |

| | | |8pointquadrilateral | |by 5 and then subtracts 60 from the result. |

| | | |fraction mysteries | | |

| | | |multiplication mystery | |His answer equals the number he started with. What was the|

| | | |subtraction mystery | |number Liam started with? |

| | | | | | |

|Mathematical challenges for able | | | | | |

|pupils in Key stages 1 and 2 | | | | | |

| | | | | | |

|Finding all possibilities problem| | | | | |

|solving pack | | | | | |

| | | | | | |

|Guidance booklet | | | | | |

| | | | | | |

|Further examples of pitch and | | | | | |

|expectations: | | | | | |

| | | | | | |

|year 6 into year 7 | | | | | |

| | | | | | |

|Information | | | | | |

| | | | | | |

|- Divide and rule1 | | | | | |

| | | | | | |

|- Divide and rule2 | | | | | |

| | | | | | |

|- teaching mental calculation | | | | | |

|strategies | | | | | |

| | | | | | |

|- teaching written strategies | | | | | |

| | | | | | |

|exemplification of standards | | | | | |

| | | | | | |

|Calculator activities | | | | | |

| | | | | | |

|Reasoning about numbers | | | | | |

| | | | | | |

|Shape and space activities | | | | | |

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