PS1 Use and apply mathematics to solve problems, communicate and reason Year 1 PS1.1 Number stories 1 PS1.2 Difference arithmagons PS1.3 Changing orders PS1.4 Making shapes PS1.5 Odd or even? PS1.6 Odd shape out PS1.7 Making a total 1 PS1.8 Doubling problem 1 PS1.9 Combining lengths PS1.10 Spending an exact amount 1 Framework objectives Choose and use appropriate number operations and mental strategies to solve problems. Solve simple mathematical problems or puzzles; recognise and predict from simple patterns and relationships. Suggest extensions by asking What if? or What could I try next?. Explain methods and reasoning orally. Investigate a general statement about familiar numbers or shapes by finding examples that satisfy it. Explain methods and reasoning orally. Use mental strategies to solve simple problems set in real life, money or measurement contexts, using counting, addition, subtraction, doubling and halving, explaining methods and reasoning orally. Recognise coins of different values. Find totals and change from up to 20p. Work out how to pay an exact sum using smaller coins. Vocabulary answer, how did you work it out?, what is the same different?, what if?, what could we try next?, number story, number sentence, add, take away, difference between, order, shape, square, rectangle, triangle, circle, star, cube, cuboid, pyramid, cylinder, odd, even, score, largest, smallest, double, total, pattern, how tall?, how much taller?, height, buy, spend, price PS1.1 Number stories 1 Drag images from the piles to model the problem. To take away images, you can drag them into the bin. What number sentence can we write for this number story? Use the pen tool to write the matching number sentence. Click repeatedly on the Undo button to reset the screen. Repeat the activity for a variety of addition and subtraction facts to 10. There are piles of three images on the screen: an apple, a girl and a tree. Pose an appropriate addition or subtraction word problem based on one or more of the images, e.g. Dina has 6 apples. She uses 4 of them to make an apple pie. How many apples does she have left?; There were 3 apples on the tree. Then 2 more apples grew. How many apples are on the tree now? Ask: Do you need to add or take away to find the answer? Why? What is the answer? How did you work it out? Simplification: Use Sheet 2. Write addition and subtraction questions to 5 on the board, e.g. 3 + 1. Each time help children to use the tree and apple images to create a number story based on the question and to confirm the answer. Extension: Include some problems involving addition and subtraction facts beyond 10 and some 2-step problems, e.g. The tree had 12 apples on it. 3 fell to the ground. Then another 5 fell to the ground. How many apples are left on the tree? 12 www.cambridge.org
PS1.2 Difference arithmagons missing corner number. The difference between 7 and the missing number is 4. What could the missing number be? Why? Establish the two possibilities: 3 and 11. The difference between 2 and the missing number is 1. What could the missing number be? Why? Establish the two possibilities: 3 and 1. So what must the missing number be? Drag down the remaining curtain to reveal two more arithmagons. Discuss and solve them in a similar way. When the final arithmagon is complete, ask: What happens when one middle number is zero? Children could work independently to create their own arithmagons to investigate this question further. Explain that each middle number on the triangle s sides is the difference between the next-door corner numbers. Point to a star and ask, e.g. What is the difference between 8 and 3? How did you work it out? To confirm the answer click the star, then the Cut button. Repeat for the other two stars. Drag back the right-hand curtain to reveal another arithmagon. Identify and reveal the missing middle number, then discuss the Simplification: Use Sheet 2 to carry out the activity for smaller differences. The stars are in the same positions so the same discussion points can be used. Extension: Use Sheet 3 to carry out the activity for larger numbers. The same discussion points can be used. PS1.3 Changing orders Sheet 3 displays a set of cards. To provide a more kinaesthetic approach to the whiteboard activities, copies of this sheet can be printed and cut up to create a set of cards for each child to experiment with. Open Sheet 1 and explain that on the screen there is a line of 3 animals arranged in the following order: tiger, bear, monkey. You want to find out how many different orders the 3 animals can be arranged in. Drag one of each animal from the piles of images and invite a child to arrange them in a different order. Repeat until you have 4 suggested orders. Are all the orders different? How can we find out what other orders there are? Drag the ordered lines into 3 sets according to which animal is first, i.e. tiger first; bear first; monkey first. To drag a whole line together, first select the whole line by clicking on the blank sheet close to the line and dragging a box around the animals. Establish that for each set there are 2 possible orders for the last 2 animals in the line, e.g. for bear first: tiger then monkey, or monkey then tiger. Help children to use this to deduce the two extra orders. Simplification: Children try to find all the possible orders by trial and discounting repeated orders only. Extension: Use Sheet 2 to carry out the activity for a tiger, a bear and 2 monkeys. Children will need the cards from 2 copies of Sheet 3 to be able to make all the possibilities. 13 www.cambridge.org
PS1.4 Making shapes Sheet 2 displays a set of right-angled triangles. To provide a more kinaesthetic approach to the whiteboard activity, copies of this sheet can be printed and cut up to create a set of triangles for each child to experiment with. Open Sheet 1 and point to the two triangles. What is the same different about these two shapes? Drag a few triangles from the piles to show children that there are lots of them. Also show how to rotate a triangle by clicking on it, then on the Rotate button on its toolbar and dragging the triangle with a circular motion. Explain that you want to arrange triangles from the piles to make different shapes. (They can use as many triangles as they want each time.) Who can see how to make a square? Invite a child to drag the triangles together to show their idea. How do you know this is a square? Can you make a different square? Again invite a child to show their idea. How are the squares the same different? Click repeatedly on the Undo button to reset the screen. Repeat the activity to find and discuss two different examples of a rectangle and then a triangle. Then encourage children to make their own shapes using the starting triangles. How would you describe this shape? Simplification: Children find one example of a square, a rectangle and a triangle. Extension: Children find three different examples of a square, a rectangle and a triangle. PS1.5 Odd or even? Drag the ladybird back to zero. Explain that you will now think about which numbers both the frog and the ladybird land on. I think all the numbers are even. Do you think I m right? Why? Model the frog s jumps as before, but this time colour the squares he lands on. To colour a square click on the grid border to reveal the grid toolbar, on a colour Fill button on the toolbar, then on the square. Repeat for the ladybird, colouring the squares she lands on in a different colour. Where she lands on the same square as the frog, use the Full/half button on the grid toolbar to colour half of the square. Are all the numbers they both land on even? Would this be the same for numbers bigger than 29? Why? Explain that the frog will jump in 3s from zero on the grid. Will the numbers he lands on be odd, even or both? How did you decide? Check by dragging the frog: What number will he land on first next? Is it odd or even? Establish that the numbers alternate between odd and even. Drag the frog back to zero. Repeat the activity for the ladybird moving in 2s. Establish that the numbers the ladybird lands on are all even. Simplification: Carry out the activity for jumps of 5 and 2. Extension: Use Sheet 2 to carry out the activity on a 0 99 grid. Also discuss the patterns made by the multiples of 3 and 2 on the grid. Is 100 a multiple of 2? 3? How does the pattern help? 14 www.cambridge.org
PS1.6 Odd shape out click on the Rotate button on its toolbar, then drag the shape in a circular motion. To be able to drag a shape to a new position after rotating you will need to click on the Select button first. Drag up the curtain and choose a different set of 3 shapes. As before, state a possible odd one out and discuss the shapes. Repeat several times, aiming to discuss a range of features, e.g. flat and solid shapes, curved and straight faces, edges and sides, same and different heights. You can change a shape s size or colour. To change the colour, select the shape, choose a colour from the menu next to the Fill colour button on its toolbar, then click the shape again. To resize a shape, select it and drag one of its square black resizing handles. Refer to the 3 shapes. I think the light blue shape is the odd one out. Do you agree? Why / why not? Discuss why the blue square could be the odd one out, e.g. the other 2 shapes are standing on a corner have a side length in common. Discuss suggestions for a different odd one out, e.g. the triangle because the other 2 shapes are squares; the orange shape because the other 2 shapes are blue. To aid comparisons the shapes can be dragged or rotated. To rotate a shape click on it, Simplification: Discuss the differences and similarities between pairs of shapes, encouraging correct vocabulary. Extension: Consider 4 shapes at a time. You could also use the activity to discuss right angles. PS1.7 Making a total 1 Explain that on the screen is a fairground game. You throw darts at the target and if you get a total score of exactly 10, you can choose a prize from the shelf. What is the smallest number of darts you would need to score 10? Why? Establish that you cannot score 10 with 1, 2 or 3 darts, but you can with 4 darts. Click on the target, then on the Dart button on its toolbar. Click 4 times on the blank screen to create 4 darts. How could you score 10 with 4 darts? Ask children to investigate this question independently, then discuss results. Display suggestions by clicking on the Select button and dragging the 4 darts onto the appropriate parts of the target. Is the total 10? How did you decide? How could you use doubling to check the total? Establish that for, e.g. 3 + 3 + 2 + 2 you could add double 3 and double 2. Use the pen tool to record each possibility as a number sentence, e.g. 3 + 3 + 2 + 2 = 10. Decide as a class whether to treat, e.g. 3 + 3 + 2 + 2 as the same as 3 + 2 + 3 + 2. What if you had as many darts as you wanted how could you score exactly 10? Ask children to investigate this independently, then use the whiteboard to discuss the results as a class. Simplification: Carry out the activity for a total score of 5. Extension: Use Sheet 2 to carry out the activity for a total score of 15. 15 www.cambridge.org
PS1.8 Doubling problem 1 Drag ducks and cows from the piles to model and check suggested solutions. Use the table on the board to consider the total numbers of animals in a systematic way and to confirm the solution. If there were no cows 1 cow 2 cows 3 cows and there were double the number of ducks, how many ducks would there be? How many animals would be in the field altogether? Use the pen tool to record the results. What patterns do you notice in the table? What do you think the total will be when there are 4 cows? How does the table help you? (The totals go up in 3s.) Set the context: The only animals in a field are cows and ducks. There are 6 animals in the field in total. The number of ducks is double the number of cows. Use the text on the board to reinforce the key information. How many ducks are in the field? Ask children to investigate the problem independently. You could give them counters in two colours to represent the ducks and cows. Discuss the problem as a class. How did you find your answer? Simplification: Use Sheet 2 first to consider the problem for a total of 3 animals. Then extend to a total of 6 animals. Focus only on trial and error, using images to check suggestions. Extension: Also use Sheet 3 to consider the problem for a total of 15 animals. Encourage children to use the systematic approach to try to solve the problem independently. PS1.9 Combining lengths Explain that you want children to build a robot that is 10 squares tall using a hat, a head, a body, 2 arms, 2 legs and 2 feet. Click repeatedly on the Undo button to reset the screen. Create and discuss children s suggestions. Ask questions such as: How tall is the robot so far? How much taller does he need to be? How do you know? Do the arms change the robot s height? Why not? You could print out each suggestion to enable you to compare possibilities. Sheet 2 displays a set of robot parts. To provide a more kinaesthetic approach to the activity, copies of this sheet can be printed and cut up to create a set of parts for each child to experiment with. Each child will also need a copy of the grid on Sheet 3 in portrait orientation to make their robots on. Open Sheet 1. Drag together parts from the piles on the screen to create a robot standing on a grid line. You can combine leg parts to create longer legs. How tall is my robot? Count squares together to confirm the height. Simplification: Show on the screen the head and hat of a robot that needs to be 10 squares tall. Ask children to find at least one way of completing the robot. Extension: Ask, e.g. How tall would the robot be if he was 2 squares shorter 1 square taller? How can you change the robot s parts to make him that height? 16 www.cambridge.org
PS1.10 Spending an exact amount 1 Then ask children to work independently to find and record addition sentences for as many other possibilities as they can. You could provide children with 1p, 2p, 5p and 10p coins to experiment with. Discuss children s suggestions and mental addition methods. Record possibilities on the board. What is the smallest largest number of sweets you can buy? Why? What if the lolly went up in price to 3p what could you buy for 10p then? Explain that you have a 10p coin and you want to spend it all. You can buy one or more of any of the sweets on the board. What sweets could I buy? How did you decide? Show one or two suggestions on the board by dragging sweets from the piles into groups and using the pen tool to label the groups with appropriate addition sentences, e.g. 5p + 5p = 10p, 5p + 2p + 2p + 1p = 10p. How do you know that the total is 10p? Simplification: Carry out the activity for a total spend of 5p. Extension: Carry out the activity for a total spend of 20p. Before children begin their independent investigation ask: Can you buy several of one type of sweet and spend exactly 20p? How do you know? Are there other answers? How many of each sweet would you need to buy? 17 www.cambridge.org
PS2 Use and apply mathematics to solve problems, communicate and reason Year 2 PS2.1 Number stories 2 PS2.2 Finding missing numbers PS2.3 Sudoku PS2.4 Multiplication arithmagons PS2.5 Halves and quarters PS2.6 How many wheels? PS2.7 Spotting shapes PS2.8 Investigating doubles PS2.9 Investigating dice totals PS2.10 Addition pyramids PS2.11 Making a total 2 PS2.12 Doubling problem 2 PS2.13 Combining weights PS2.14 Totals and change PS2.15 Spending an exact amount 2 Framework objectives Choose and use appropriate operations and efficient calculation strategies to solve problems. Solve simple mathematical problems or puzzles; recognise simple patterns and relationships, generalise and predict. Suggest extensions by asking What if? or What could I try next? Explain how a problem was solved orally and, where appropriate, in writing. Investigate a general statement about familiar numbers or shapes by finding examples that satisfy it. Explain how a problem was solved orally and, where appropriate, in writing. Use mental addition and subtraction, simple multiplication and division strategies to solve simple word problems involving numbers in real life money or measures, using one or two steps. Explain how the problem was solved. Recognise all coins and begin to use.p notation for money. Find totals, give change, and work out which coins to pay. Vocabulary answer, find all, find different, investigate, explain your method, how did you work it out?, what could we try next?, what if?, what is the same different?, discuss, check, number sentence, number story, add, total, subtract, multiply, times, multiplied by, multiple, equal groups of, divide, number operation, double, half, row, column, one half, one quarter, odd, even, total, first, second, third, next, scales, weight, grams, money, cost, how much?, spend, price, pay, buy, bought, change, pound, pence, coin, shape, triangle, score, biggest, largest, smallest, most, fewest PS2.1 Number stories 2 What number sentence can we write for this number story? Use the pen tool to write a matching number sentence. Where more than one number sentence applies, e.g. 3 10 = 30 and 10 + 10 + 10 = 30, record both and discuss how they are related. Click repeatedly on the Undo button to reset the screen. Repeat the activity for a variety of problems involving addition and subtraction facts to at least 10, and multiplication and division facts for 2, 5 and 10. Use the images on screen (a boy and 1p, 2p, 5p and 10p coins) to pose an appropriate addition, subtraction, multiplication or division word problem, e.g. Gus has three 10p coins in his pocket. How much money does he have altogether? Ask: Will you add, subtract, multiply or divide? Why? What is the answer? How did you work it out? Create copies of the images to model the problem click on an image, then on the Copy and Paste buttons. To take away images, you can drag them into the bin. Simplification: Use Sheet 2. Write appropriate addition, subtraction, multiplication and division questions on the board, e.g. 3 10. Each time help children to use the boy and the 1p, 2p and 10p coin images to create a number story based on the question and to confirm the answer. Extension: Use Sheet 3 so you can include 20p and 50p coins in problems. Also include some 2-step problems, e.g. Gus has three 20p coins. He spends 15p. How much does he have left? 18 www.cambridge.org
PS2.2 Finding missing numbers Point to the first input and output. What number operation can change 12 into 24? Establish that because 24 is greater than 12, you must add or multiply to get 24. What must we add to 12 multiply 12 by to make 24? Use the pen tool to record what the machine could be doing: + 12 or 2 (i.e. finding double). Explain that the machine does the same thing to the other inputs too. What will the answer be if the machine adds 12 to 7? multiplies 7 by 2? Click the second arrow on the function machine twice to reveal the output. So what does the machine do? Drag the red panel into the bin to reveal 2. Identify and reveal the third output. Then discuss the two missing inputs. How can we find this number? Why? Establish that dividing by 2 (i.e. halving) undoes multiplying by 2. What are the missing numbers? To reveal an input click on the function machine, on the Show/hide button on its toolbar, then on the appropriate question mark. Move to Sheet 2 and repeat the whole activity for the new ( 9) function machine. Establish that to add/subtract 9 you can add/subtract 10 and adjust. Simplification: Use Sheets 3 and 4 to carry out the activity by finding doubles and halves of numbers to 6, and adding and subtracting 10. Extension: Use Sheets 5 and 6 to carry out the activity by finding doubles and halves of 2-digit numbers to 20, and adding and subtracting 19. PS2.3 Sudoku Explain how Sudoku puzzles work: each red-outlined square must contain each of the numbers 1, 2, 3 and 4, as must each whole row and column. Which column can we complete? Why? Establish that the third column has all but one of the numbers 1, 2, 3 and 4 entered, so you can easily find which number is missing. What is the missing number? Reveal the missing number by clicking on the grid border, on the Show/hide number button on its toolbar, then on the appropriate grid square. Which red-outlined squares can we complete now? Why? Establish that both of the right-hand squares now have all but one of the numbers 1, 2, 3 and 4 entered, so you can easily find which numbers are missing. With children s help, complete both of the right-hand squares. What shall we find now? Why? What number belongs there? Gradually complete the grid in this way. Sheet 4 displays an empty Sudoku grid that you can use to create your own puzzle. To enter each number, click on the Number pad button and drag a number from the pad into the appropriate grid square. To hide a number, click on the grid border, on the Show/hide number button on its toolbar and then on the number. Simplification: Use Sheet 2 to carry out the activity with more clues. Extension: Sheet 3 has fewer clues. Children will need to extend the logic to numbers already used in more than one row or column. 19 www.cambridge.org
PS2.4 Multiplication arithmagons Explain that each middle number on the triangle s sides is the answer when you multiply the next-door corner numbers. Point to a star and ask, e.g. What is 5 times 10? To confirm the answer click the star, then the Cut button. Repeat for the other two stars, taking the opportunity to reinforce that, e.g. 5 3 = 3 5. Drag back the right-hand curtain to reveal another arithmagon. Point to the star in the middle of the side. What is this number? Establish that it could be any answer in the 2 times-table so it is not a good number to find first. Which number should we find first? Establish that the left corner number is the only one with one possible answer. How can you find that number? Establish that you need to find what times 2 is 8. So what is the missing number? Which number should we find next? Discuss and identify the right corner number, then the remaining missing number. Drag down the remaining curtain to reveal two more arithmagons. Discuss and solve them in a similar way. When the final arithmagon is complete, ask: What happens when one middle number is zero? Simplification: Use Sheet 2 to carry out the activity for 2 and 10 times-table facts. The stars are in the same positions so the same discussion points can be used. Extension: Use Sheet 3 to carry out the activity for 2, 3, 4, 5 and 10 times-table facts. The same discussion points can be used. PS2.5 Halves and quarters Click the Background Grids icon in the Shape and Space Toolbox to reveal the grid toolbar. Click on the Fill a whole square button, and invite a child to colour half of the first outlined square by clicking on grid squares. How do you know that is one half? Check that 2 of the suggested parts make a whole by selecting a new colour from the grid toolbar menu and colouring an equal number of squares on the same shape. How many small squares make one half of the larger square? (2) How many different ways do you think there are of colouring one half of the larger square? Invite children to show all the possibilities. Move to Sheet 2 and establish, in a similar way to before, that 2 small squares make one quarter of the rectangle. Do you think we will be able to colour one quarter of each of the 16 rectangles in a different way? Invite children to try. Toggle between Sheets 1 and 2. How can 2 small squares be one half of the square and also one quarter of the rectangle? Establish that the number of squares in one half or one quarter depends upon how many squares make the whole shape. Simplification: Find different ways of colouring one half and one quarter using Sheet 1. To reset the screen after finding halves, click repeatedly on the Undo button. Extension: Invite children to repeat the activities using the Fill the left/right half of a square buttons on the grid toolbar. 20 www.cambridge.org
PS2.6 How many wheels? Explain that you want children to imagine they work in a factory. The factory makes bicycles with 2 wheels each and tricycles with 3 wheels each. Towards the end of the day they are left with 12 wheels. How many bicycles could you make with all 12 wheels? How did you decide? Invite a child to drag the 12 wheels into groups of 2 to demonstrate that you can make 6 bicycles. Use the pen tool to record the result on the table, as 6 in the bicycle column and zero in the tricycle column. Ask children to imagine that they can make all bicycles, all tricycles or a mixture of both using all 12 of the wheels. They work independently to try to find and record all the possibilities. You could provide children with 12 counters to use to represent the wheels. Discuss the possibilities as a class. Invite children to model suggestions by dragging the wheels into groups. Record results on the table. What happens if you make 1 tricycle 3 tricycles? Establish that after making 1 or 3 tricycles an odd number of wheels remains and you cannot use them all by making bicycles. Simplification: Use Sheet 2 to carry out the activity for 8 wheels. Extension: Use Sheet 3 to carry out the activity for 16 wheels. PS2.7 Spotting shapes Repeat for other children s suggestions. Is this triangle different from the others? To compare similar triangles, drag one onto the other. Reflections can be counted as different, but you might like to count rotations as the same. To rotate a triangle to compare it with another example, click on the triangle, on the Rotate button on its toolbar, then drag the triangle in a circular motion. To drag a triangle after rotating, first click on the Select button. Duplicate triangles can be removed from the screen by dragging them into the bin. What other shapes can you see? Create children s suggestions as before. Describe the shape. Do you know its name? What is a triangle? Explain that you want to find as many different triangles as possible in the design on the screen. Invite a child to indicate a triangle in the design. Create their triangle using the polygon tool: click on the Polygon icon in the Shape and Space Toolbox, click on each corner of the triangle, then again on the starting corner to complete it. Drag the triangle onto a blank part of the screen. Simplification: Use Sheet 2 to carry out the activity for a simpler design. Extension: Use Sheet 3 to carry out the activity for a more complex design. 21 www.cambridge.org