Maths Kit. Key Stage 2. BOOK I: Activities, Extensions, and Teachers Quick Reference Notes.

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1 Maths Kit Key Stage 2 BOOK I: Activities, Extensions, and Teachers Quick Reference Notes

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3 Maths Kits Book I: Activities, Extensions, Teachers Quick Reference Notes TABLE OF CONTENTS: ACTIVITIES How to use the Key Stage 2 Maths Kits...5 Stacking Diagram...6 3D Noughts and Crosses...7 4x4...9 Birthday Cake...11 Cola Crate...13 Crazy Paving...15 Dominoes...17 Garden Path...19 Handshakes...21 Matchsticks...23 NIM...25 Packing Balls...27 Packing Parcels...29 Pentominoes...31 Pyramids...33 Pythagoras Puzzle...35 Reach the Goal...37 Spheres...39 Tetra Solid...41 Times Table...43 Towers of Brahma...45 Towers of Hanoi...47 What I learned sheet...49 continued overleaf

4 TABLE OF CONTENTS: FOLLOW-UP ACTIVITIES 3D Noughts and Crosses x Birthday Cake...61 Cola Crate...69 Crazy Paving...77 Dominoes...85 Garden Path...91 Handshakes...97 Matchsticks NIM Packing Balls Packing Parcels Pentominoes Pyramids Pythagoras Puzzle Reach the Goal Spheres Tetra Solid Times Table Towers of Brahma Towers of Hanoi

5 How to use the Key Stage 2 Maths Kit Setting up the Kit Place the Maths Kit trays on work tables and it is ready to use. What to do Using the Kit: The Key Stage 2 Maths Kit comprises self-contained activities, with built-in easy-tofollow instructions. The kit is designed to be used in a session of about an hour. Children work in pairs, spending five minutes on each activity. Optional Follow Up Activities: There are three follow-up activities of increasing difficulty for each kit activity. These follow-up activities can be completed independent of the kit itself, but extend the maths concepts they demonstrate. These follow-up activities are fully photocopiable and help children further develop their maths skills. Support Material BOOK I: Lesson Plans and Teachers Quick Reference Notes Session with the Maths Kit: about an hour. For each activity, the book gives: Maths Concepts: Key Stage 2 National Curriculum points. The Activity: a description and photograph. Background: a brief explanation of the relevant maths. Extensions: suggestions for additional activities for use with the kit. What I Learned Sheet: a fill-in sheet to help children to assess their session with the Maths Kit. Optional Follow-Up Activities: three extension activities for each kit activity. BOOK II: National Curriculum (DCELLS), History National Curriculum (DCELLS): assessment criteria are defined for each activity and a check list provided. Applications: information is provided on real-life applications of the maths concepts, where appropriate. History: The origins of the activity are provided, where appropriate. Risk Assessments are provided for each element of the Maths Kit. 5

6 Pentominoes Towers of Hanoi Handshakes Matchsticks Balls for Packing Spheres and 3D Noughts & Crosses NIM Dominoes Birthday Cake 4x4 Cola Crate Packing Spheres 3D Noughts & Crosses Stored at bottom are boxes with accessories for the activities. Match these items with photos in Books I & II. Times Table Reach the Goal Packing Parcels Garden Path Towers of Brahma Crazy Paving Pythagoras Puzzle NIM Spheres Tetra Solid Box with pieces for Spheres activity Pyramid Stored at bottom are boxes with accessories for the activities. Match these items with photos in Books I & II. Stacking System for Kit One (top) and Kit Two (bottom)

7 3D Noughts and Crosses Maths Concepts Develops spatial awareness, a strategic approach, symmetry and pattern, and supports the development of mathematical thinking. Activity Children select a colour and take it in turns to place balls into the grid, with the aim of being the first player to get three balls of the same colour in a row along any vertical, horizontal or diagonal row. Background This activity requires the children to constantly assess their surroundings and develop strategies for getting their three coloured balls in a row. Extensions: 3D Noughts and Crosses: When playing this game, is it possible to find a winning strategy? Does it make a difference who starts? 2D Noughts and Crosses: This game is played on a 3 x 3 grid. Traditionally, the first player plays an X and the second player plays a 0. Turns are taken to try and complete the horizontal, vertical or diagonal row. The first player to complete a row wins. With practice, observation and analysis of a few games, it is possible to become unbeatable. Other Variations of 2D Noughts and Crosses: Experiment with different sized grids. Does this affect the likelihood of a drawn game? Note: Both 2D and 3D versions of noughts and crosses are competitive games. Some children do not necessarily respond well in a competitive situation. Also, some children in striving to win may miss some of the mathematical subtleties. An interesting approach is to ask children to work together, to analyse their games in order to discover and share winning strategies. This approach is more inclusive and less divisive than having an all-out competition. 7

8 Other Similar Problems Magic Square Tic-Tac-Toe: Instead of X s and o 0 s, the numbers 1 to 9 are used. Each number can be used only once. Players take it in turns to write down one of the numbers between 1 and 9 in the grid. The winner is the first player to get the numbers in any row, column or diagonal to add up to 15. Teeko: This game is played on a 5 x 5 grid by two players, using 4 counters each. Players take it in turns to place their four counters on the grid. They then take it in turns to move their counters one space at a time in any direction until the first player has a line of four counters. Four in a Row: This can be played on a 7 x 7 grid. The goal of this game is to arrange four counters in a row, horizontally, vertically or diagonally. See follow-up activities on page

9 4 x 4 Maths Concepts Supports number recognition and manipulation, symmetry and pattern, and aids the development of mathematical thinking. This activity also demonstrates the need for constant checking of results. Activity Children turn the cubes so that the numbers in each column, row and along the diagonals all add up to 10. Background One possible solution is show below. How many other solutions are there? Do you notice any patterns or symmetries within this solution? This solution uses one of each digit in each line. Are there any solutions that use the same digit more than once in each line? How many other solutions are there? 9

10 Extensions: Symbol 4 x 4: Instead of numbers, this version uses four distinct symbols. The aim of this activity is to fit one of each symbol into every row, column and diagonal. There are no numbers here, but the approach to this challenge requires the same kind of strategic thinking. Sudoku: This is a current, popular puzzle found in many newspapers. The sudoku grid is made up nine 3 x 3 sub-grids. The aim is to enter a number from 1 to 9 in each cell of the sub-grid so that each row and column of the whole and each sub-grid contains only one instance of each number. See follow-up activities on page

11 Birthday Cake Maths Concepts Supports reflection, symmetry and pattern, left and right-handedness, and the development of mathematical thinking. Activity Children move the mirror and notice how the number of reflections of the candle varies as the angle of the mirror changes. Background As the mirrors are moved, more and fewer reflections of the candle are produced. Children can try making a cake for a four-year old and a six-year old and measure the angle in each case. Extensions: Join two plastic mirrors together with adhesive tape to form a hinge. Stick black tape along the bottom edge of the mirrors. Place a protractor on a piece of paper. Draw a line parallel to the base, as in the diagram below. 11

12 Place the hinge of the mirror at the centre of the protractor. Open and close the mirror; look into the mirrors and notice how the number of reflections change. Notice how the reflection of the line produces different shaped polygons. Place an object between the mirrors. Form an angle of 60 o. Notice the vertex forms a three-legged star, Notice the reflections of the line form a hexagon. Draw up a table with three columns labelled as below and rows labelled from 20 o to 180 o. Use the formula below to predict how many objects you will observe and then carry out the study to test your predictions. Angle in degrees Predicted number of images Actual number of images observed Formula: To calculate the number of images observed, divide 360 o by the angle. For example, to calculate the number of images observed at 90 o, divide 360 o by 90 o. 360/90 = 4. There should be 4 images. See follow-up activities on page

13 Cola Crate Maths Concepts Supports counting, checking, symmetry and pattern, and aids the development of mathematical thinking. Activity Children must arrange the 18 bottles in the crate so that each row and each column contains an even number (2, 4, 6) of bottles. Background Below is one possible solution to this activity. Are there any other solutions? Are all of the solutions similar? Are any of the solutions symmetrical? 13

14 Extensions: 5 x 5: Arrange 25 counters in rows of 5 x 5. Remove 5 counters so that 4 counters are left in each row, column and diagonal. Three Queens: Place eight queens on a 64 square chess board so that no two queens are in the same column, row or diagonal. Here is one possible solution. See follow-up activities on page

15 Crazy Paving Maths Concepts Develops visualisation skills and the development of mathematical thinking. It may encourage understanding of the link between pattern of shape and pattern of number, with specific reference to angles. Activity Children fit the six different shapes into the space in the frame so that no two shapes of the same colour are next to each other. Background Here is one possible solution to the puzzle. Are there any others? Extensions: Using combinations of pieces, make the following shapes: Square Rectangle Isosceles triangle Equilateral triangle Rhombus Parallelogram Trapezium Six pointed star See follow-up activities on page

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17 Dominoes Maths Concepts Supports number recognition and manipulation, clustering and counting and the development of mathematical thinking. Practical experience in exploring the pattern and behaviour of number is essential to the development of understanding. Activity Children fit all of the dominoes into the shaded area so that the dots in the columns add up to 2 and the dominoes on the bottom row add up to 8. Background One possible solution is shown below. Are there other solutions? Extensions: 3s and 5s: Using a set of dominoes, share them randomly between players. Players take it in turns to place their dominoes on the table, as in a typical game. Dominoes can only be placed next to matching numbers. A score is achieved by adding the totals at both ends of the dominoes. If the total is divisible by either 3 or 5, then one point is scored for each multiple. 17

18 For example, in this game a 6-3 has been played = 9. 9 can be divided by 3 three times, so this move scores 3 points. 3-3 is played. The end numbers are = is divisible by 3 four times, giving a score of is played. This gives end numbers of = is divisible by 5 twice, giving a score of 2. See follow-up activities on page

19 Garden Path Maths Concepts Develops visualisation skills. This activity encourages understanding of the links between patterns of shape and patterns of number and it also supports the development of mathematical thinking. Activity Children identify the number of different ways of arranging paving in each of the driveways. They then try and predict how many different ways there will be to arrange slabs on the next driveway. Background The number of ways to arrange the paving slabs follow the Fibonacci sequence, i.e., one slab, one arrangement; two slabs, two arrangements; three slabs, three arrangements; four slabs, five arrangements, as shown. The next number in the sequence is calculating by adding the previous two answers together. For example, = 5. So, the next answer is = 8 (arrangements). or or or or or or or Extensions: The Fibonacci Sequence: Using the rule for calculating Fibonacci numbers, how many can you calculate in 15 minutes? Fibonacci s Rabbits: Suppose a newly-born pair of rabbits, one male and one female, are put in a field. Rabbits are able to mate at the age of one month, so at the end of the second month a female can produce a pair of rabbits. Suppose our rabbits remain healthy and that the female always produces one new pair (one male, one female) every month from the second month. How many pairs will there be after one year? 19

20 Fibonacci s Rabbits: Solution At the end of the first month, they mate, but there is still only one pair. At the end of the second month, the female produces a new pair, so now there are two pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making three pairs of rabbits in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making five pairs in total. The breeding pattern follows the Fibonacci sequence, with 1 pair, 1 pair, 2 pairs, 3 pairs and ending with 5 pairs at the end of four months. How many pairs of rabbits will there be after 12 months? See follow-up activities on page

21 Handshakes Maths Concepts Supports counting, combinations, sequence, symmetry and pattern, and the development of mathematical reasoning. Activity Children calculate the number of handshakes for various group sizes, if everyone in the group shakes hands once with everyone else. Background The solution is outlined below. Two people: Person 1 shakes hands with 1 person Person 2 shakes hands with 0 people Number of handshakes: 1 Three people: Person 1 shakes hands with 2 people Person 2 shakes hands with 1 person Person 3 shakes hands with 0 people Number of handshakes: 3 Four people: Person 1 shakes hands with 3 people Person 2 shakes hands with 2 people Person 3 shakes hands with 1 person Person 4 shakes hands with 0 people Number of handshakes: 6 Five people: Person 1 shakes hands with 4 people Person 2 shakes hands with 3 people Person 3 shakes hands with 2 people Person 4 shakes hands with 1 person Person 5 shakes hands with 0 people Number of handshakes: 10 Extensions: Study the number of handshakes for two people, three people, four people and five people. Try and predict the number of handshakes for six people. Can you calculate the number of handshakes for seven people? See follow-up activities on page

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23 Matchsticks Maths Concepts Supports problem recognition, spatial awareness, shape and pattern, and the development of mathematical thinking. Activity Children move the matches in each pattern to solve the problems. Background In addition to mathematical reasoning, these problems require lateral thinking to achieve the solutions. This may benefit pupils who are less confident with applying their maths skills. Extensions: How Many Squares? Arrange matchsticks to form a grid of 3 x 3 squares. How many squares are there altogether? Hint: There are more than 9! Try a 5 x 5 grid. How many squares are there now? Make a table of results. Can you predict how many squares there will be in a 6 x 6 grid? See follow up activities on page

24 Extensions: continued How Many Triangles? 3 matches make one 1 x 1 x 1 triangle. 9 matches make one 2 x 2 x 2 triangle and four 1 x 1 x 1 triangles. How many triangles are there in a 3 x 3 x 3 triangle? How many of each size? How many matches have been used? Can you predict how many matches will be needed to make a 4 x 4 x 4 triangle? How many of each sized triangle will there be? Warning: this is quite a complex problem. 24

25 Nim Maths Concepts Develops spatial awareness and strategic approach, decision making and supports number recognition and the development of mathematical thinking. Activity Children take it in turns to remove sticks from the three rows, with the aim of making their opponent remove the last stick. Background This activity challenges pupils to plan ahead and try to limit the different possible moves their opponent can make. Extensions: Try these two alternative games using counters: Does it make a difference how the lines are arranged? Can you find a strategy for winning? Please see follow up activities on page

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27 Packing Balls Maths Concepts Supports counting and combinations. This activity also develops spatial awareness and supports the development of mathematical thinking. Activity Children arrange the spheres to form a square based pyramid and a triangular based pyramid. Background This activity requires pupils to think about sequences and relationships between numbers. It then challenges them to predict the size of the following rows of spheres. Extensions: Complete the table to demonstrate the progression as layers of spheres are added to each arrangement. Triangular Based Pyramid Square Based Pyramid Row Number Number of Spheres Row Number Number of Spheres = = Compare the triangular based pyramid with the square based pyramid. What do you notice? 27

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29 Packing Parcels Maths Concepts Develops visualisation skills and increases familiarity with vocabulary of shape. It also supports the development of mathematical thinking. Activity The aim of this activity is to pack all nine parcels into the crate. Background This activity requires logical thinking and problem solving skills, as well as pupils mathematical skills. To solve this problem, the single cubes need to be lined up corner to corner along a diagonal, as shown below. Extensions: Try arranging these shapes differently. Is it possible to use all of the shapes to make a cuboid instead of a cube? Please see follow up activities on page

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31 Pentominoes Maths Concepts Supports counting, combinations, and spatial awareness. This activity also develops mathematical thinking. Activity Children arrange a selection of pentominoes to fit into the blue square. Background This activity requires careful selection of pieces and the solutions require trial and error. This can require perseverance and patience! Please see follow up activities on page

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33 Pyramid Maths Concepts Develops visualisation skills and increases familiarity with vocabulary of shape. It also supports the development of mathematical thinking. Activity Children arrange the polyhedra into the bases to make two pyramids. Background There are several key observations for this activity: Two square based pyramids can be combined to form an octahedron. The different solids are arranged to form a larger version of the original square based pyramid. The angles on both the tetrahedron and the octahedron are the same. The vertices on both the tetrahedron and the octahedron are the same. The tetrahedron and the octahedron are both regular shapes. Please see follow up activities on page

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35 Pythagoras Puzzle Maths Concepts Develops visualisation skills and increases familiarity with vocabulary of shape. It also supports the development of mathematical thinking. Activity Children arrange the five pieces into the biggest square. Then use the same pieces to make the two smaller squares. Background This activity requires pupils to use trial and error to find the solution. It also requires the use of logic. Below is the solution to the problem; this shows the two small squares fitting into the larger square. 35

36 Extensions: This right-angled triangle uses whole numbers to demonstrate Pythagoras theorem. a 2 + b 2 = c 2 (4 x 4) + (3 x 3) = (5 x 5) = 25 Using squared paper, explore other similar triangles. How many squares long are the other two sides of a triangle whose longest side is 13 squares long? Please see follow up activities on page

37 Reach the Goal Maths Concepts Develops spatial awareness. This activity also develops strategic approach, decision making and mathematical thinking. Activity Children move the rugby ball to the goal by sliding the tiles around the grid. Background Sliding one tile into one space counts as one move. The 3 x 3 grid should be completed in 13 moves; the 5 x 5 grid should be completed in 29 moves. 37

38 Extensions: Use the grid below to move counters in the same way. Remember to use one counter of a different colour and leave one space on the grid. How many moves are needed with a 6 x 6 grid? What about with a 7 x 7 grid? Please see follow up activities on page

39 Spheres Maths Concepts Develops visualisation skills and encourages the understanding of links patterns of shape and patterns of number, with specific reference to angles. It also supports the development of mathematical thinking. Activity Children arrange the spheres to form three pyramids of different sizes. Background Some of these diagrams might help. 39

40 Extensions: Cut out the nets below and make the shapes. Can you fit them together to make a pyramid? Please see follow up activities on page

41 Tetra Solid Maths Concepts Develops visualisation skills. This activity will also increase familiarity with vocabulary of shape and develop mathematical thinking. Activity Children arrange the tetrahedra and octahedra to build two pyramids. Background This activity is likely to be solved using trial and error and it will often require perserverence to find the solution. Extensions: Count the number of shapes in the small pyramid. How many tetrahedra are there? How many octahedra are there? Now count the number of shapes in the large pyramid. How many tetrahedra are there? How many octahedra are there? Imagine you are building the next size pyramid. Predict how many shapes you would need. How many tetrahedra would you need? How many octahedra would you need? Please see follow up activities on page

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43 Times Table Maths Concepts Develops visualisation skills and challenges recall of the times tables in an informal setting. It also supports the development of mathematical thinking. Activity Children arrange the dice to represent the products of the times tables. Background One method for solving this problem is to select a times table, for example, 4 times, and look for a die with a 4, then one with 8 and so on. This approach is likely to be adopted by pupils who are not so confident with a particular table. Another method is to take a die and look for a number that is the product of 4 and another number. This number should be placed uppermost. This will then be repeated for all products, and then the dice sorted into ascending order. 43

44 Extensions: Complete the table below Are there any patterns? What do you notice along the main diagonals? Please see follow up activities on page

45 Towers of Brahma Maths Concepts Suports counting and combinations. This activity also supports the development of mathematical thinking. Activity Children must move the pyramid of disks from one pole to another, moving one disk at a time and never placing a larger disk on top of a smaller one. Background This activity requires pupils to think about sequences and relationships. It will also require trial and error to complete. A solution is outlined below using three disks. 45

46 Extensions: Investigate the number of moves needed with different numbers of disks. Complete the table below. Number of Disks Number of Moves Can you predict the number of moves needed for 8 disks? Please see follow up activities on page

47 Towers of Hanoi Maths Concepts Supports counting and combinations and develops mathematical thinking. Activity Children move the disks one at a time from one pole to another, to eventually make the red disks and blue disks exchange places. Background This activity requires pupils to think about sequences and relationships. It will also require trial and error to complete. The solution using two discs of each colour is shown overleaf. It takes 11 moves to complete this puzzle. Extensions: Try the classic version of this puzzle. Set up all of the disks on one pole, with the largest disk at the bottom decreasing to the smallest disk on the top. Transfer all of the disks from one pole to another, moving only one disk at a time and never placing a larger disk on top of a smaller disk. Please see follow up activities on page

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49 What I Learned... Which activity was most interesting? Which activity surprised you most? Write one thing you learned. Which activity would you most like to do again? 49

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51 3D Noughts and Crosses Follow-up Activity One: Barrier Game For this activity, you will need to work with a partner. Set up a barrier between your workspaces, e.g. a book. Take five cubes and build a shape without your partner watching. Describe your shape to your partner, one step at a time, so that he/she can build the same shape, e.g. The base has three cubes in a horizontal line. Your partner should then build their shape without you seeing it. When you have finished, remove the barrier and compare the shapes. a) Are the two shapes the same? b) If not, why not? Swap over and try again. 51

52 3D Noughts and Crosses Follow-up Activity One: Barrier Game Answers a) and b) Pupils who give clear instructions to a partner that asks a lot of questions are more likely to build shapes that are the same. 52

53 3D Noughts and Crosses Follow-up Activity Two: Viewing Shapes Build a shape with five cubes and place it in the centre of the table. For example: a) Draw the view from the four sides of your shape. b) How are your drawings different? c) What will you see if you look down on the shape? Draw this view. d) Try a different arrangement with the five cubes. Does this shape look different from different angles? e) Are there any shapes you can build with five cubes that look the same from every side? 53

54 3D Noughts and Crosses Follow-up Activity Two: Viewing Shapes Answers The pupils answers for this activity will depend on the shape they have built. 54

55 4 x 4 Follow-up Activity One: Magic Squares This is a magic square All of the columns, rows and main diagonals add up to 15. a) Using the numbers 0 to 8, make a magic square where all the columns, rows and main diagonals add up to the same number. b) What number does each row, column and main diagonal add up to? 55

56 4 x 4 Follow-up Activity One: Magic Squares Answers a) b) All of the rows, columns and main diagonals add up to

57 4 x 4 Follow-up Activity Two: Magic Squares This is a magic square All of the columns, rows and main diagonals add up to 15. a) Will the square still be magic if you add 10 to each of the number within the square? Check your prediction. b) What about multiplying each number by 3? Predict and check. c) What about doubling each number? Predict and check. d) Think of a rule of your own and test it. Does the square stay magic? 57

58 4 x 4 Follow-up Activity Two: Magic Squares Answers a) The square will still be magic, if all of the numbers have 10 added to them. b) If all of the numbers are multiplied by 3, the square is no longer magic. c) If all of the numbers are doubled, the square is no longer magic. Rule: For the square to stay magic, the same number must be added to all numbers in the same way. For example, + 2. d) So long as the pupils follow the rule above, their square will stay magic. 58

59 4 x 4 Follow-up Activity Three: Magic Squares This is a magic square All of the columns, rows and main diagonals add up to 15. a) I randomly place a counter on one of the squares. What is the probability the square will be an odd number? b) I pick up my counter and randomly place it on another square. What is the probability that the number in the square can be divided by 2? c) ) I pick up my counter and randomly place it on another square. What is the probability that the number in the square is a multiple of 3? c) ) I pick up my counter and randomly place it on another square. What is the probability that the number in the square is higher than 5? 59

60 4 x 4 Follow-up Activity Three: Magic Squares Answers a) There are nine numbers altogether and five of these are odd (1, 3, 5, 7, 9), so the probability is 5/9. b) There are nine numbers altogether and four of these can be divided by 2 (2, 4, 6 8), so the probability is 4/9. c) There are nine numbers altogether and three of these are multiples of 3 (3, 6, 9). Therefore the probability is 3/9 or 1/3. d) There are nine numbers altogether and four of these are higher than 5 (6, 7, 8, 9), so the probability is 4/

61 Birthday Cake Follow-up Activity One: Rangoli Patterns Rangoli Patterns are a type of Indian art. The completed patterns are often symmetrical. Complete these Rangoli Patterns to make them symmetrical. Write down their order of symmetry each time. 61

62 Birthday Cake Follow-up Activity One: Rangoli Patterns Answers 62

63 Birthday Cake Follow-up Activity Two: Making 2D Shapes a) Using a Roamer or LOGO, make the following shapes, always starting from the bottom left hand corner and proceeding clockwise. b) For each shape, write down the instructions. 10 units A 10 units B 6 units 3 units C 10 units 15 units 8 units 10 units 7 units D 10 units 15 units 63

64 Birthday Cake Follow-up Activity Two: Making 2D Shapes Answers Commands may vary depending on the programme used. Shape A: FD 10 RT 90 FD 10 RT 90 FD 10 RT 90 RD 10 RT 90 Shape B: FD 3 RT 90 FD 15 RT 90 FD 3 RT 90 FD 15 RT 90 Shape C: FD 6 RT 135 FD 10 RT 135 FD 8 RT 90 Shape D: FD 7 RT 90 FD 10 LT 90 FD 3 RT 90 RD 5 RT 90 FD 10 RT 90 FD 15 RT

65 Birthday Cake Follow-up Activity Three: Rotational Symmetry Cut out the shapes on the shapes sheet. Fit the shapes into their outline on the outline sheet. Rotate the shape. a) How many times will it fit in its outline before returning to its original position? b) What fraction of one revolution was turned each time it fitted? c) What angle is this? d) Complete the table below for all of the shapes. Name of Number of Number of Fraction of Angles Order of Polygon Sides Times One Turned Rotation Polygon Revolution Fits into Outline 65

66 Birthday Cake Follow-up Activity Two: Rotational Symmetry continued Shapes Sheet A B C D E F 66

67 Birthday Cake Follow-up Activity Two: Rotational Symmetry continued Outline Sheet A B C D E F 67

68 Birthday Cake Follow-up Activity Three: Rotational Symmetry Answers a) The square will fit four times before returning to its original position. b) It was turned 1/4 of a revolution each time it fitted. c) This angle is 90 o. d) Name of Polygon Number of Sides Number of Times Polygon Fits into Outline Fraction of One Revolution Angles Turned Order of Rotation Rectangle 4 2 1/2 180 o 2 Equilateral Triangle Regular Hexagon Regular Pentagon Regular Octagon 3 3 1/3 120 o /6 60 o /5 72 o /8 45 o 8 Square 4 4 1/4 90 o

69 Cola Crate Follow-up Activity One: Coordinate Shapes Draw the shapes below by plotting the coordinates onto the grid. For each shape, join each point to the one before. a) Shape 1: (2, 1), (5, 1), (5, 5), (2, 5), (2, 1) b) Shape 2: (1, 9), (4, 9), (4, 6), (1, 6), (1, 9) c) Shape 3: (11, 7), (13, 5), (9, 5), (11, 7) d) Shape 4: (6, 8), (7, 9), (8, 9), (9, 8), (8, 7), (7, 7), (6, 8) e) Shape 5: (6, 1), (6, 3), (8, 4), (10, 3), (10, 1), (6, 1) f) Name each shape you have drawn

70 Cola Crate Follow-up Activity One: Coordinate Shapes Answers a) to e) f) Shape 1: Rectangle Shape 2: Shape 3: Shape 4: Shape 5: Square Triangle Hexagon Pentagon 70

71 Cola Crate Follow-up Activity Two: Coordinate Puzzles Work out these coordinates by carrying out the additions or subtractions. For example, = = 8 gives the coordinates (7, 8) a) = = (, ) b) = = (, ) c) = = (, ) d) = = (, ) e) = 16-9 = (, ) f) = = (, ) g) 16-9 = = (, ) h) = = (, ) i) = = (, ) j) Plot these points on the grid on the next page, joining each plot to the previous one. k) Draw your own picture on a grid. l) Write your own coordinate puzzle for drawing your picture. 71

72 Cola Crate Follow-up Activity Two: Coordinate Puzzles continued

73 Cola Crate Follow-up Activity Two: Coordinate Puzzles continued

74 Cola Crate Follow-up Activity Two: Coordinate Puzzles Answers a) (4, 0) b) (3, 3) c) (1, 4) d) (3, 5) e) (4, 7) f) (5, 5) g) (7, 4) h) (5, 3) i) (4, 0) j) Pupils responses to k) and l) will depend on the image they draw. 74

75 Cola Crate Follow-up Activity Three: Coordinate Picture a) Plot the following points on the grid below, joining each point to the previous one. (-5, -1), (-4, 0), (-5, 1), (-4, 3), (-2, 4), (0, 4), ((2, 3), (3, 1), (4, 3), (5, 4), (4, 0), (5, -4), (4, -3), (3, -1), (2, -3), (0, -4), (-2, -4), (-4, -3), (-5, -1) b) Now plot the following on the graph. (-1, 1), (1, 2), (1, -3), (-1, 1) c) What shape have you drawn? 75

76 Cola Crate Follow-up Activity Three: Coordinate Picture Answers a) and b) c) The shape is a fish. 76

77 Crazy Paving Follow-up Activity One: A Tight Fit! a) Cut out the triangle template. Can you draw around it to make a tessellating pattern? b) Try to make tessellating shapes with each of the other shapes. Are there any shapes that do not make tessellating patterns? c) Can you combine two or more shapes to make tessellating patterns? 77

78 Crazy Paving Follow-up Activity One: A Tight Fit! Answers a) Equilateral triangles will create tessellating patterns. b) Equilateral triangles, squares, rectangles and regular hexagons all create tessellating patterns. Regular pentagons do not create tessellating patterns. c) Tessellating patterns can be made from: - Equilateral triangles and squares - Regular hexagons, squares and equilateral triangles There may be other combinations. 78

79 Crazy Paving Follow-up Activity Two: Making Shapes Draw two straight lines across each of the shapes (from one side to another) to make the shapes described below. The lines do not have to pass through the middle of the shape. For example: Make four squares a) b) c) Make four rectangles Make 1 triangle and Make 3 right-angled 2 quadrilaterals triangles d) e) Make 2 squares and 2 rectangles Make 2 triangles, 1 pentagon and 1 quadrilateral f) g) Make 3 triangles and 1 quadrilateral Make 2 triangles and 2 quadrilaterals 79

80 Crazy Paving Follow-up Activity Two: Making Shapes Answers a) b) c) d) e) f) g) There may be other solutions. 80

81 Crazy Paving Follow-up Activity Three: Tangrams A tangram is a Chinese puzzle. a) Cut out the tangram on the following page. Name each shape. b) Make the following shapes: i) A square using two pieces. ii) A rectangle using three pieces. iii) A parallelogram using two triangles. iv) A parallelogram using three pieces. v) A parallelogram using four pieces. vi) A triangle using three pieces. c) Use all seven pieces to make these pictures. The pieces must not overlap. i) ii) 81

82 Crazy Paving Follow-up Activity Three: Tangrams continued Tangram 82

83 Crazy Paving Follow-up Activity Three: Tangrams Answers a) The tangram is made up of five triangles, a square and a parallelogram. b) i) A square using two pieces ii) A rectangle using three pieces iii) A parallelogram using two triangles iv) A parallelogram using three pieces v) A parallelogram using four pieces vi) A triangle using three pieces 83

84 Crazy Paving Follow-up Activity Three: Tangrams Answers continued: c) i) ii) 84

85 Dominoes Follow-up Activity One: Odd and Even Totals You will need a set of dominoes for this activity. Remove any dominoes that have a blank tile. Sort your dominoes into three groups: Group 1: Group 2: Group 3: Both numbers on the domino are even Both numbers on the domino are odd There is one even and one odd number on the domino a) For each group, add the two numbers on the dominoes together. b) Say whether the total is odd or even. 85

86 Dominoes Follow-up Activity One: Odd and Even Totals Answers a) and b) Group 1: both numbers on the domino are even Numbers on dominoes Total Odd or even? Even Even Even Even Even Even Group 2: both numbers on the domino are odd Numbers on dominoes Total Odd or even? Even Even Even Even Even Even Group 3: There is one even and one odd number on the domino Numbers on dominoes Total Odd or even? Odd Odd Odd Odd Odd Odd Odd Odd Odd 86

87 Dominoes Follow-up Activity Two: Domino Probability Here are some dominoes. a) I select a domino at random. What is the probability that it will have at least one even number on it? b) I replace the domino and select another at random. What is the probability that the sum of the two numbers on the domino will be even? c) I replace the domino and select another at random. What is the probability that the product of the two numbers on the domino will be odd? d) I replace the domino and select another at random. What is the probability that the product of the two numbers on the domino will be a multiple of 3? e) I replace the domino and select another at random. What is the probability that the product of the two numbers on the domino will be more than 6? 87

88 Dominoes Follow-up Activity Two: Domino Probability Answers a) Six of the eight dominoes have at least one even number, so the probability is 6/8 or 3/4. b) Three of the dominoes have an even total (1-3, 1-5 and 4-6). Therefore the probability is 3/8. c) Two of the dominoes have an odd product (1-3 and 1-5). Therefore the probability is 2/8 or 1/4. d) Four of the dominoes have a product that is a multiple of 3 (1-3, 1-6, 3-4 and 4-6). Therefore the probability is 4/8 or 1/2. e) Three of the dominoes have products that are higher than 6 (3-4, 4-6 and 4-5). Therefore the probability is 3/

89 Dominoes Follow-up Activity Three: Domino Fractions Imagine this domino represents 1/2: a) Solve these problems: i) ii) + = + = iii) iv) + = + = v) + = b) Is it possible to write any of the solutions as dominoes? Remember, the numbers on dominoes only go up to

90 Dominoes Follow-up Activity Three: Domino Fractions Answers a) i) 1/2 + 1/4 = 3/4 ii) 1/2 + 1/3 = 5/6 iii) 1/3 + 1/4 = 7/12 iv) 1/5 + 1/6 = 11/30 v) 1/6 + 1/4 = 10/24 b) The answers to i) and ii) can both be written using dominoes. i) 3/4 is ii) 5/6 is 90

91 Garden Path Follow-up Activity One: Stacking Blocks Look at these blocks. There are two rules for the way they are stacked. 1. There can only be one rectangle on the top row. 2. Each row must have one more block than the one above. a) How many rectangles are there in the bottom row? b) How many rectangles are there altogether? c) How many rectangles would there be in the bottom row if there were 15 rectangles altogether? d) Complete this table: Number of rectangles in bottom row Total number of rectangles

92 Garden Path Follow-up Activity One: Stacking Blocks Answers a) There are three rectangles in the bottom row. b) There are six rectangles altogether. c) There would be five rectangles in the bottom row if there were 15 rectangles altogether. d) Number of rectangles in bottom row Total number of rectangles

93 Garden Path Follow-up Activity Two: Buying a Garden Path A rectangular block, two squares long by one square wide, costs How much would it cost to pave each of these garden paths? a) 12 squares long by 4 squares wide. b) 10 squares long by 9 squares wide. c) 11 squares long by 8 squares wide. d) Which of these paths is the most expensive? A rectangular block, three squares long by one square wide, costs How much would it cost to pave each of these garden paths? e) 27 squares long by 6 squares wide. f) 15 squares long by 10 squares wide. g) 24 squares long by 6 squares wide. h) Which of these paths is the most expensive? i) There is a path measuring 12 squares wide by 24 squares long. Is it cheaper to lay the garden path using 2 x 1 rectangular blocks or 3 x 1 blocks? 93

94 Garden Path Follow-up Activity Two: Buying a Garden Path Answers a) This would need 6 blocks by 4 blocks = 24 blocks. 24 x 2.00 = b) This would need 5 blocks by 9 blocks = 45 blocks. 45 x 2.00 = c) This would need 11 blocks by 4 blocks = 44 blocks. 44 x 2.00 = d) Garden path (b) is the most expensive. e) This would need 9 blocks by 6 blocks = 54 blocks. 54 x 2.50 = f) This would need 5 blocks by 10 blocks = 50 blocks. 50 x 2.50 = g) This would need 8 blocks by 6 blocks = 48 blocks. 48 x 2.50 = h) Garden path (e) is the most expensive. i) Using 2 x 1 blocks: this would need 12 by 12 blocks = 144 blocks. 144 x 2.00 = Using 3 x 1 blocks: this would need 12 by 8 blocks = 96 blocks. 96 x 2.50 = It would be cheaper to build the garden path using 3 x 1 blocks. 94

95 Garden Path Follow-up Activity Three: Golden Rectangle The Golden Rectangle was used by the ancient Greeks and Egyptians to construct their buildings. It was thought to produce the most pleasing shape and to have magical properties. A Golden Rectangle can be any width, but its length has to be just over 3/5 longer than its width. Here is how to draw a Golden Rectangle: Draw a single square. Add another square. You now have a 2x1 rectangle 2x2 Add a third square to fit the longer side of the rectangle. square Add another square to fit the longer side of the new rectangle. 3x3 square This rectangle is now 5x3, so the next square to add would be 5x5. The further you go with this investigation, the closer you will be to drawing a Golden Rectangle. a) Continue creating these shapes until you have completed the table. Pattern number Side length of new square b) What do you notice about the numbers in the table? c) Can you predict how many squares will need to be added for pattern number 9? d) Can you find a rule? e) What are these numbers called? 95

96 Garden Path Follow-up Activity Three: Golden Rectangle Answers a) Continue creating these shapes until you have completed the table. Pattern number Side length of new square b) You can find out the next number of squares, by adding the previous two numbers of squares together (e.g. for pattern 3, add the number of squares for patterns 1 and 2; = 2). c) For pattern 9, the number of squares will be the number for patterns 7 and 8 added together = squares will be added for pattern 9. d) Add the last two number of squares together to find the next number. Rule: n th term = (n - 1) th term + (n - 2) th term e) This is known as the Fibonacci sequence. 96

97 Handshakes Follow-up Activity One: Number Triangles For each of these triangles, the three lines of numbers must add up to the same total. a) Use the numbers 1 to 6 to complete these triangles. Total 9 Total 10 Total 11 b) Use the numbers 2 to 7 to make these triangles add up to the same on each side. Total 12 Total 13 Total 14 c) Find out what totals you could make if you use the numbers 3 to

98 98 Handshakes Follow-up Activity One: Number Triangles Answers a) b) c) Using the numbers 3 to 8, you can make the totals 15, 16, 17 and

99 Handshakes Follow-up Activity Two: Number Pyramid This is the top line of a number pyramid To find the numbers for the rows below you need to use the following rules: Place a 0 under: 0 1 and 1 0 Place a 1 under: 0 0 and 1 1 Like this: Complete the patterns for these top lines: a) b) c) d) e) Look at the pyramids you have created. Is there a rule to decide if the triangle will end in a 0 or a 1? 99

100 Garden Path Follow-up Activity Two: Number Pyramid Answers a) b) c) d) e) If the top row is symmetrical, the triangle will end with a 1, as in a) and b). If the top row is not symmetrical, the triangle could end with either a 1, as in c), or a 0, as in d)

101 Handshakes Follow-up Activity Three: Mystic Roses This is a mystic rose. a) Can you work out how it is drawn? Every point on the edge links to every other point on the edge. Investigate drawing mystic roses using the templates on the next page. b) Record your results in the table below. Number of Points Number of Lines c) Find a rule for working out the number of lines for any number of points

102 Handshakes Follow-up Activity Three: Mystic Roses continued 102

103 Handshakes Follow-up Activity Three: Mystic Roses Answers a) Mystic roses are drawn by connecting every point around the circle to every other point around the circle. b) Number of Points Number of Lines c) To find the number of lines, multiply the number of points by the number of points minus one. Then divide this product by 2. Rule: Number of lines = Number of points x (number of points - 1) 2 = n (n - 1)

104 104

105 Matchsticks Follow-up Activity One: Square Patterns Write either 2, 4, 6 or 9 in the boxes to make the sums correct. Each answer must be made up of different numbers. You cannot use the same number more than once in a calculation. a) + + = 19 b) + - = 7 c) + - = 4 d) + - = 5 e) + - = 11 f) + + = 17 g) + - = 11 h) - - = 3 i) - - = 1 j) + + = 15 k) + - = 13 l) + + =

106 Matchsticks Follow-up Activity One: Square Patterns Answers For instance: a) = 19 b) = 7 c) = 4 d) = 5 e) = 11 f) = 17 g) = 11 h) = 3 i) = 1 j) = 15 k) = 13 l) =

107 Matchsticks Follow-up Activity Two: Function Machines Using any of the four basic operations, can you get from the number in Box A to the number in Box B? Write the operation and the number in the middle box, e.g. x 3 A B a) b) c) d) For these, you need to insert two operations from the list below. A B e) 40 1 f) 3 28 g) h) i) x 6 x3 6 3 You will have to use one of these operations twice

108 Matchsticks Follow-up Activity Two: Function Machines Answers a) 7 x 6 = 42 b) = 5 c) 100 x 5 = 20 d) 9 x 5 = 45 e) = 1 f) 3 x = 28 g) 5 x = 16 h) = 10 i) 12 x 3-6 =

109 Matchsticks Follow-up Activity Three: Matchstick Patterns Make these patterns using matchsticks. Make the next two patterns. a) Complete this table: Pattern Number Number of Matchsticks b) How many matchsticks will be needed for the 7th pattern? c) How many for the 10th pattern? d) Can you find the rule to find the number of matchsticks for any pattern number? 109

110 Matchsticks Follow-up Activity Three: Matchstick Patterns Answers a) Pattern Number Number of Matchsticks b) The 7th pattern will need 22 matchsticks. c) The 10th pattern will need 31 matchsticks. d) To find the number of matchsticks, multiply the pattern by 3 and then add 1. Rule: Number of matchsticks = (Pattern number x 3) +1 = 3n

111 Nim Follow-up Activity One: Using an Abacus We can use an abacus to represent values. For example: 213 is represented by: H T U Write these values in words and in figures. a) b) c) d) e) f) g) h) i) Draw abacuses for these numbers and write the numbers in words: j) 425 k) 697 l) 340 m)