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Maths Kit Key Stage 2 BOOK I: Activities, Extensions, and Teachers’ Quick Reference Notes www.techniquest.org 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 www.techniquest.org continued overleaf... 3 TABLE OF CONTENTS: FOLLOW-UP ACTIVITIES 4 3D Noughts and Crosses....................51 4x4 ............................................55 Birthday Cake....................................61 Cola Crate........................................69 Crazy Paving.....................................77 Dominoes..........................................85 Garden Path......................................91 Handshakes.......................................97 Matchsticks......................................105 NIM ..........................................111 Packing Balls...................................117 Packing Parcels................................123 Pentominoes....................................129 Pyramids.........................................135 Pythagoras Puzzle............................141 Reach the Goal................................147 Spheres..........................................155 Tetra Solid.......................................159 Times Table......................................165 Towers of Brahma.............................171 Towers of Hanoi...............................177 www.techniquest.org 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. www.techniquest.org 5 Pentominoes Balls for Packing Spheres and 3D Noughts & Crosses Towers of Hanoi NIM Handshakes Dominoes Matchsticks Birthday Cake 4x4 3D Noughts & Crosses Cola Crate Stored at bottom are boxes with accessories for the activities. Match these items with photos in Books I & II. Packing Spheres Times Table Packing Parcels Reach the Goal Garden Path Towers of Brahma Tetra Solid Crazy Paving Box with pieces for “Spheres” activity Pythagoras Puzzle Pyramid NIM Stored at bottom are boxes with accessories for the activities. Match these items with photos in Books I & II. Spheres Stacking System for Kit One (top) and Kit Two (bottom) 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. www.techniquest.org 7 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 51. 8 www.techniquest.org 4x4 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? www.techniquest.org 9 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 55. 1 8 5 8 3 7 3 9 7 6 5 3 6 7 8 5 3 4 9 6 2 7 5 2 7 1 3 4 10 2 1 4 9 8 1 2 7 8 4 9 6 5 1 6 2 4 9 www.techniquest.org 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. www.techniquest.org 11 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 60o. 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 20o to 180o. 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 10 15 20 30 40 60 90 120 180 Formula: To calculate the number of images observed, divide 360o by the angle. For example, to calculate the number of images observed at 90o, divide 360o by 90o. 360/90 = 4. There should be 4 images. See follow-up activities on page 61. 12 www.techniquest.org 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? www.techniquest.org 13 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. 14 See follow-up activities on page 69. www.techniquest.org 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 77. www.techniquest.org 15 16 www.techniquest.org 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. www.techniquest.org 17 For example, in this game a 6-3 has been played. 6 + 3 = 9. 9 can be divided by 3 three times, so this move scores 3 points. 3-3 is played. The end numbers are 6 + 3 + 3 = 12. 12 is divisible by 3 four times, giving a score of 4. 3-4 is played. This gives end numbers of 6 + 4 = 10. 10 is divisible by 5 twice, giving a score of 2. See follow-up activities on page 85. 18 www.techniquest.org 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, 2 + 3 = 5. So, the next answer is 3 + 5 = 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? www.techniquest.org 19 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 91. 20 www.techniquest.org 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 97. www.techniquest.org 21 22 www.techniquest.org 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 105. www.techniquest.org 23 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 www.techniquest.org 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 111. www.techniquest.org 25 26 www.techniquest.org 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 Row Number Square Based Pyramid Number of Spheres Row Number Number of Spheres 1 1 1 1 2 1+3 = 4 2 5 3 1+3+6 =10 3 14 4 4 5 5 6 6 Compare the triangular based pyramid with the square based pyramid. What do you notice? www.techniquest.org 27 28 www.techniquest.org 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 123. www.techniquest.org 29 30 www.techniquest.org 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 129. www.techniquest.org 31 32 www.techniquest.org 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 135. www.techniquest.org 33 34 www.techniquest.org 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. www.techniquest.org 35 Extensions: This right-angled triangle uses whole numbers to demonstrate Pythagoras’ theorem. a2 + b2 = c2 (4 x 4) + (3 x 3) = (5 x 5) 16 + 9 = 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 141. 36 www.techniquest.org 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. www.techniquest.org 37 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 147. 38 www.techniquest.org 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. www.techniquest.org 39 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 155. 40 www.techniquest.org 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 159. www.techniquest.org 41 42 www.techniquest.org 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. www.techniquest.org 43 Extensions: Complete the table below. 1 2 3 4 5 6 7 8 2 4 6 8 10 12 14 16 3 6 9 12 15 18 21 4 8 12 16 20 24 5 10 15 20 25 6 12 18 24 7 14 21 8 16 9 10 11 12 9 10 11 12 Are there any patterns? What do you notice along the main diagonals? Please see follow up activities on page 165. 44 www.techniquest.org 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. www.techniquest.org 45 Extensions: Investigate the number of moves needed with different numbers of disks. Complete the table below. Number of Disks Number of Moves 1 1 2 3 4 5 6 7 Can you predict the number of moves needed for 8 disks? Please see follow up activities on page 171. 46 www.techniquest.org 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 177. www.techniquest.org 47 Solution 48 www.techniquest.org 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? ______________________________________________________ ______________________________________________________ www.techniquest.org 49 50 www.techniquest.org 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. www.techniquest.org 51 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 www.techniquest.org 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? www.techniquest.org 53 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 www.techniquest.org 4x4 Follow-up Activity One: Magic Squares This is a magic square. 4 3 8 9 5 1 2 7 6 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? www.techniquest.org 55 4x4 Follow-up Activity One: Magic SquaresAnswers a) 3 2 7 8 4 0 1 6 5 b) All of the rows, columns and main diagonals add up to 12. 56 www.techniquest.org 4x4 Follow-up Activity Two: Magic Squares This is a magic square. 4 3 8 9 5 1 2 7 6 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? www.techniquest.org 57 4x4 Follow-up Activity Two: Magic SquaresAnswers 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 www.techniquest.org 4x4 Follow-up Activity Three: Magic Squares This is a magic square. 4 3 8 9 5 1 2 7 6 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? www.techniquest.org 59 4x4 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 4/9. 60 www.techniquest.org 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. www.techniquest.org 61 Birthday Cake Follow-up Activity One: Rangoli Patterns Answers 62 www.techniquest.org 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. A 10 units 10 units 6 units B 3 units 15 units 10 units C 8 units 10 units 7 units D 10 units 15 units www.techniquest.org 63 Birthday Cake Follow-up Activity Two: Making 2D Shapes Answers Commands may vary depending on the programme used. 64 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 90 www.techniquest.org 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 www.techniquest.org 65 Birthday Cake Follow-up Activity Two: Rotational Symmetry continued Shapes Sheet 66 A B C D E F www.techniquest.org Birthday Cake Follow-up Activity Two: Rotational Symmetry continued Outline Sheet A B C D E F www.techniquest.org 67 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 90o. d) Name of Number of Number of Fraction of Angles Order of Polygon Sides Times One Turned Rotation Polygon Revolution Fits into Outline Rectangle 4 2 1/2 180o 2 Equilateral 3 3 1/3 120o 3 6 6 1/6 60o 6 5 5 1/5 72o 5 8 8 1/8 45o 8 4 4 1/4 90o 4 Triangle Regular Hexagon Regular Pentagon Regular Octagon Square 68 www.techniquest.org 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. 10 9 8 7 6 5 4 3 2 1 0 1 2 3 www.techniquest.org 4 5 6 7 8 9 10 11 12 13 14 69 Cola Crate Follow-up Activity One: Coordinate Shapes Answers a) to e) 10 9 4 8 2 7 6 3 5 4 1 3 5 2 1 0 70 1 2 3 4 f) Shape 1: Rectangle Shape 2: Square Shape 3: Triangle Shape 4: Hexagon Shape 5: Pentagon 5 6 7 8 9 10 11 12 13 14 www.techniquest.org Cola Crate Follow-up Activity Two: Coordinate Puzzles Work out these coordinates by carrying out the additions or subtractions. For example, 4 + 3 = 7 16 - 8 = 8 gives the coordinates (7, 8) a) 43 - 39 = 16 - 16 = ( , ) b) 100 - 97 = 20- 17 = ( , ) c) 20 - 19 = 33 - 29 = ( , ) d) 6 + 9 - 12 = 24 - 19 = ( , ) e) 8 - 5 + 1 = 16 - 9 = ( , ) f) 41 - 36 = 16 -11 = ( , ) g) 16 - 9 = 27 - 23 = ( , ) h) 73 - 68 = 14 - 11 = ( , ) i) 81 - 77 = 36 - 20 - 16 = ( , ) 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. www.techniquest.org 71 Cola Crate Follow-up Activity Two: Coordinate Puzzles continued 7 6 5 4 3 2 1 0 72 1 2 3 4 5 6 7 www.techniquest.org Cola Crate Follow-up Activity Two: Coordinate Puzzles continued 7 6 5 4 3 2 1 0 1 www.techniquest.org 2 3 4 5 6 7 73 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) 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 Pupils responses to k) and l) will depend on the image they draw. 74 www.techniquest.org 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) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 b) Now plot the following on the graph. (-1, 1), (1, 2), (1, -3), (-1, 1) c) What shape have you drawn? www.techniquest.org 75 Cola Crate Follow-up Activity Three: Coordinate PictureAnswers a) and b) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 c) The shape is a fish. 76 www.techniquest.org 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? www.techniquest.org 77 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 www.techniquest.org 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) Make four rectangles Make 1 triangle and 2 quadrilaterals d) f) www.techniquest.org Make 3 right-angled triangles e) Make 2 squares and 2 rectangles Make 3 triangles and 1 quadrilateral c) Make 2 triangles, 1 pentagon and 1 quadrilateral g) Make 2 triangles and 2 quadrilaterals 79 Crazy Paving Follow-up Activity Two: Making Shapes Answers a) b) d) e) f) g) c) There may be other solutions. 80 www.techniquest.org 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) www.techniquest.org ii) 81 Crazy Paving Follow-up Activity Three: Tangrams continued Tangram 82 www.techniquest.org 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 www.techniquest.org 83 Crazy Paving Follow-up Activity Three: Tangrams Answers continued: c) i) ii) 84 www.techniquest.org 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: Both numbers on the domino are even Group 2: Both numbers on the domino are odd Group 3: 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. www.techniquest.org 85 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? 2 2 4 Even 2 4 6 Even 2 6 8 Even 4 4 8 Even 4 6 10 Even 6 6 12 Even Total Odd or even? Group 2: both numbers on the domino are odd Numbers on dominoes 1 1 2 Even 1 3 4 Even 1 5 6 Even 3 3 6 Even 3 5 8 Even 5 5 10 Even Group 3: There is one even and one odd number on the domino Numbers on dominoes 86 Total Odd or even? 1 2 3 Odd 1 4 5 Odd 1 6 7 Odd 2 3 5 Odd 2 5 7 Odd 3 4 7 Odd 3 6 9 Odd 4 5 9 Odd 5 6 11 Odd www.techniquest.org 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? www.techniquest.org 87 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 3/8. 88 www.techniquest.org Dominoes Follow-up Activity Three: Domino Fractions Imagine this domino represents 1/2: a) Solve these problems: i) + = iii) ii) + = + = iv) + = + = v) b) Is it possible to write any of the solutions as dominoes? Remember, the numbers on dominoes only go up to 6. www.techniquest.org 89 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. 90 i) 3/4 is ii) 5/6 is www.techniquest.org 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 1 2 1 3 3 4 5 6 7 bottom row Total number of rectangles www.techniquest.org 91 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 1 2 3 4 5 6 7 1 3 6 10 15 21 28 bottom row Total number of rectangles 92 www.techniquest.org Garden Path Follow-up Activity Two: Buying a Garden Path A rectangular block, two squares long by one square wide, costs £2.00. 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 £2.50. 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? www.techniquest.org 93 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 = £48.00 b) This would need 5 blocks by 9 blocks = 45 blocks. 45 x £2.00 = £90.00 c) This would need 11 blocks by 4 blocks = 44 blocks. 44 x £2.00 = £88.00 d) Garden path (b) is the most expensive. e) This would need 9 blocks by 6 blocks = 54 blocks. 54 x £2.50 = £135.00 f) This would need 5 blocks by 10 blocks = 50 blocks. 50 x £2.50 = £125.00 g) This would need 8 blocks by 6 blocks = 48 blocks. 48 x £2.50 = £120.00 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 = £288.00 Using 3 x 1 blocks: this would need 12 by 8 blocks = 96 blocks. 96 x £2.50 = £240.00 It would be cheaper to build the garden path using 3 x 1 blocks. 94 www.techniquest.org 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 1 1 2 1 3 2 4 5 6 7 8 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? www.techniquest.org 95 Garden Path Follow-up Activity Three: Golden Rectangle Answers a) Continue creating these shapes until you have completed the table. Pattern number 1 2 3 4 5 6 7 8 Side length of 1 1 2 3 5 8 13 21 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; 1 + 1 = 2). c) For pattern 9, the number of squares will be the number for patterns 7 and 8 added together. 13 + 21 = 34. 34 squares will be added for pattern 9. d) Add the last two number of squares together to find the next number. Rule: nth term = (n - 1)th term + (n - 2)th term e) This is known as the Fibonacci sequence. 96 www.techniquest.org 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 8. www.techniquest.org 97 Handshakes Follow-up Activity One: Number Triangles Answers a) 1 6 2 1 5 6 3 4 3 2 4 3 5 2 6 5 4 1 b) 3 5 4 4 7 6 7 2 2 7 3 5 4 6 3 2 6 5 c) Using the numbers 3 to 8, you can make the totals 15, 16, 17 and 18. 98 www.techniquest.org Handshakes Follow-up Activity Two: Number Pyramid This is the top line of a number pyramid. 0 0 1 1 1 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: 0 0 1 0 0 1 1 0 1 1 1 1 1 0 0 Complete the patterns for these top lines: a) 1 0 1 0 1 b) 0 0 1 0 0 c) 1 1 0 0 1 d) 1 0 1 1 0 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? www.techniquest.org 99 Garden Path Follow-up Activity Two: Number Pyramid Answers a) 1 0 0 0 1 1 0 1 1 1 b) 1 0 1 0 0 1 c) 0 1 1 0 0 1 d) 1 0 0 0 1 0 1 0 0 1 1 0 0 1 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 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). 100 www.techniquest.org 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 3 4 5 6 7 8 c) Find a rule for working out the number of lines for any number of points. www.techniquest.org 101 Handshakes Follow-up Activity Three: Mystic Roses continued 102 www.techniquest.org 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 3 3 4 6 5 10 6 15 7 21 8 28 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) www.techniquest.org 2 = n (n - 1) 2 103 104 www.techniquest.org 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) + + = 12 www.techniquest.org 105 Matchsticks Follow-up Activity One: Square Patterns Answers For instance: a) 9 + 6 + 4 = 19 b) 9 + 4 - 6 = 7 c) 6 + 2 - 4 = 4 d) 2 + 9 - 6 = 5 e) 6 + 9 - 4 = 11 f) 9 + 2 + 6 = 17 g) 4 + 9 - 2 = 11 h) 9 - 4 - 2 = 3 i) 9 - 6 - 2 = 1 j) 9 + 2 + 4 = 15 k) 9 + 6 - 2 = 13 l) 6 + 2 + 4 = 12 106 www.techniquest.org 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 7 42 b) 60 5 c) 100 20 9 45 a) d) For these, you need to insert two operations from the list below. A B 40 1 f) 3 28 g) 5 16 h) 30 10 12 30 e) i) ÷ 10 ÷5 + 10 +4 x6 x3 –6 –3 +1 You will have to use one of these operations twice. www.techniquest.org 107 Matchsticks Follow-up Activity Two: Function Machines Answers a) 7 x 6 = 42 b) 60 ÷ 12 = 5 c) 100 x 5 = 20 d) 9 x 5 = 45 e) 40 ÷ 10 - 3 =1 f) 3 + 10 = 28 g) 5 x 3 + 1 = 16 h) 30 ÷ 5 + 4 = 10 i) 12 x 3 - 6 = 30 108 x 6 www.techniquest.org Matchsticks Follow-up Activity Three: Matchstick Patterns Make these patterns using matchsticks. Make the next two patterns. a) Complete this table: Pattern Number 1 2 3 4 5 6 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? www.techniquest.org 109 Matchsticks Follow-up Activity Three: Matchstick Patterns Answers a) Pattern Number 1 2 3 4 5 6 Number of 4 7 10 13 16 19 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 + 1 110 www.techniquest.org 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 www.techniquest.org l) 340 m) 639 111 Nim Follow-up Activity One: Using an Abacus Answers a) 264 Two hundred and sixty-four b) 382 Three hundred and eighty-two c) 551 Five hundred and fifty-one d) 794 Seven hundred and ninety-four e) 603 Six hundred and three f) 920 Nine hundred and twenty g) 27 Twenty-seven h) 163 One hundred and sixty-three i) 400 Four hundred j) k) l) m) 112 www.techniquest.org Nim Follow-up Activity Two: Roman Numerals Our Roman Numerals Numerals (Arabic) When using Roman Numerals: If a smaller numeral is after a larger numeral, the smaller one is added to the larger one. 1 I 2 II 3 III 4 IV 5 V 6 VI 7 VII 8 VIII 9 IX For example: 10 X = 10 + (5 - 1) 50 L = 14 100 C 500 D 1000 M For example: XVI = X + V + I = 10 + 5 + 1 = 16 If a smaller numeral is in front of a larger numeral, the smaller one must be subtracted from the larger. XIV = X + (V - I) Work out these Roman Numerals: a) XIII b) LV c) IC d) XXX e) DLXI f) MCLX g) CLXVIII h)MX www.techniquest.org 113 Nim Follow-up Activity Two: Roman Numerals Answers a) 13 b) 55 c) 99 d) 30 e) 561 f) 1160 g) 168 h) 1010 114 www.techniquest.org Nim Follow-up Activity Three: Roman Numerals Write these numbers as Roman Numerals: a) 325 b) 40 c) 1159 d) 3006 e) 840 Try these additions. Write the solutions as Roman Numerals. f) XII + XIII g) LX + XVII h) CVIII + DLIX i) CCCL + CLXIII j) MDCIV + DLXI Try these subtractions. Write the solutions as Roman Numerals. k) CLX - LXVII l) LXXX - XIX m) DCCIX - CVII n) MX - DCC o) MDCC - LXXXIV www.techniquest.org 115 Nim Follow-up Activity Three: Roman Numerals Answers a) CCCXXV b) XL c) MCLIX d) MMMVI e) DCCCXL f) 12 + 13 = 25 (XXV) g) 60 + 17 = 77 (LXXVII) h) 108 + 559 = 667 (DCLXVII) i) 350 + 163 = 513 (DXIII) j) 1604 + 561 = 2165 (MMCLXV) k) 160 - 67 = 93 (XCIII) l) 80 - 19 = 61 (LXI) m) 709 - 107 = 602 (DCII) n) 1010 - 700 = 310 (CCCX) o) 1700 - 84 = 1616 (MDCXVI) 116 www.techniquest.org Packing Balls Follow-up Activity One: Number Patterns Here is a number pattern. 1 + 0 = 1 2 + 1 = 3 3+2+1=6 Continue the next two patterns and write the number sentences for each one. Look at the numbers you have added together. a) What do you notice about them? b) What are these numbers called? c) What do you think the next three totals will be? Check by drawing the patterns. d) Record your answers in the table. Pattern Number 1 2 3 4 5 6 Total Number of Dots www.techniquest.org 117 Packing Balls Follow-up Activity One: Number Patterns Answers a) The total is found by adding together all of the consecutive numbers from 1 to the pattern number. b) These are called triangular numbers. c) Pattern four has a total of 10 dots, pattern five has a total of 15 dots and pattern 6 has a total of 21 dots. d) Pattern Number 1 2 3 4 5 6 Total Number of Dots 1 3 6 10 15 21 118 www.techniquest.org Packing Balls Follow-up Activity Two: Dot Patterns Look at these dot patterns. a) Work out pattern 5 and pattern 6. b) Complete this table. Look for a pattern in the table. Pattern Number 1 2 3 4 5 6 Total Number of Dots c) Predict how many dots will be in pattern 7. Check your prediction. d) Predict how many dots will be in pattern 20. e) Explain how you would the number of dots for any number pattern. www.techniquest.org 119 Packing Balls Follow-up Activity Two: Dot Patterns Answers a) Pattern 5 would have 9 dots and pattern 6 would have 11 dots. b) Pattern Number 1 2 3 4 5 6 Total Number of Dots 1 3 5 7 9 11 c) Pattern 7 would have 13 dots. d) Pattern 20 would have 39 dots. e) To work out the number of dots, multiply the pattern number by 2 and then subtract 1. Rule: 120 Number of dots = 2n - 1 www.techniquest.org Packing Balls Follow-up Activity Three: Pascal’s Triangle Look at the pattern in Pascal’s triangle below. 1 1 1 1 1 1 2 3 4 1 3 6 1 4 1 a) Complete the next two rows. b) What is the total for each row? Complete this table. Row Number 1 2 3 4 5 6 7 Total c) Can you find a pattern in the row totals? d) Can you predict the total for the next row? Check your prediction. www.techniquest.org 121 Packing Balls Follow-up Activity Three: Pascal’s Triangle Answers a) 1 1 1 1 1 1 6 1 2 3 4 5 1 3 6 10 15 1 1 4 10 20 1 5 15 1 6 1 b) Row Number 1 2 3 4 5 6 7 Total 1 2 4 8 16 32 64 c) The total for each row doubles every time the row number increases; the total increases by a power of two. d) The total for row 8 will be 128. 122 www.techniquest.org Packing Parcels Follow-up Activity One: Finding the Volume a) Roll a die. Write the number on the die in the first column of the table. b) Roll the die again. Write this number in the second column of the table. c) Roll the die for a third time and write this in the third column of the table. d) Build a cuboid using building cubes with these dimensions. e) Work out the volume of the shape. f) Repeat this for three more shapes. Shape Example Width Length Height 1st die roll 2nd die roll 3rd die roll 2 3 4 Volume 24cm3 1 2 3 4 www.techniquest.org 123 Packing Parcels Follow-up Activity One: Finding the Volume Answers a) to f) the answers to this activity will depend on the numbers rolled. 124 www.techniquest.org Packing Parcels Follow-up Activity Two: Volume of Cuboids a) Find the volume of these cuboids. 1 cm 2 cm 4 cm 2 cm cm m 2c 2 cm 4 cm 3 5 6 cm 3 cm cm 5 cm How many different cuboids can you make with a volume of: b) 12cm3 c) 20cm3 d) 24cm3 e) 30cm3 f) 60cm3 www.techniquest.org 125 Packing Parcels Follow-up Activity Two: Volume of Cuboids Answers a) i) Volume = 12cm3 ii) Volume = 8cm3 iii) Volume = 60cm3 iv) Volume = 60cm3 b) Volume of 12cm3 = 12 x 1 x 1 = 6 x 2 x 1 = 3 x 4 x 1 =3x2x2 b) Volume of 20cm3 = 4 x 5 x 1 = 10 x 2 x 1 = 20 x 1 x 1 =2x5x2 b) Volume of 24cm3 = 24 x 1 x 1 = 12 x 2 x 1 = 8 x 3 x 1 = 4 x 6 x 1 =4x3x2 =2x2x6 b) Volume of 30cm3 = 6 x 5 x 1 = 3 x 5 x 2 = 15 x 2 x 1 = 30 x 1 x 1 = 1 x 3 x 10 b) Volume of 60cm3 = 2 x 3 x 10 = 1 x 4 x 15 = 4 x 3 x 5 = 1 x 5 x12 = 5 x 6 x 2 = 1 x 6 x 10 = 1 x 1 x 60 = 2 x 2 x 15 = 1 x 1 x 30 = 1 x 3 x 20 126 www.techniquest.org Packing Parcels Follow-up Activity Three: Making Boxes Make a box using the diagram below. 15 cm 15 cm 1 cm 1 cm Cut out the shaded areas. Fold along the dotted lines. a) What is the volume of this box? b) By cutting away a square from each corner, what is the largest volume box you can make? Complete the table. Size of Square Length of Base Width of Base Height Volume 1 13 13 1 169cm3 2 3 4 5 6 7 www.techniquest.org 127 Packing Parcels Follow-up Activity Three: Making Boxes Answers a) The volume of the box is 13 x 13 x 1 = 169cm3 b) Size of Square Length of Base Width of Base Height Volume 1 13 13 1 169cm3 2 11 11 2 242cm3 3 9 9 3 243cm3 4 7 7 4 196cm3 5 5 5 5 125cm3 6 3 3 6 54cm3 7 1 1 7 7cm3 If you only cut whole numbers, the largest volume box is 243cm3. This is made when a 3cm square is cut away from each corner. Can you find a bigger volume? 128 www.techniquest.org Pentominoes Follow-up Activity One: Tetrominoes Tetrominoes are shapes made up of four squares, joined edge to edge. e.g. a) How many other tetrominoes can you draw? Now investigate the following: b) Work out the perimeters for the tetrominoes. Do any of them have the same perimeter? c) Which tetromino has the smallest perimeter? d) Work out why this is. www.techniquest.org 129 Pentominoes Follow-up Activity One: Tetrominoes Answers a) b) Perimeter of: Red tetromino = 10 Orange tetromino = 10 Yellow tetromino = 8 Green tetromino = 10 Blue tetromino = 10 c) The yellow tetromino has the smallest perimeter. d) This is because there are four pairs of touching edges in the yellow tetromino, as opposed to three pairs in all of the other shapes. 130 www.techniquest.org Pentominoes Follow-up Activity Two: Pentominoes Pentominoes are shapes made up of five squares, joined edge to edge. e.g. a) Draw the other pentominoes. How many are there altogether? Now investigate the following: b) Work out the perimeters for the pentominoes. Do any of them have the same perimeter? c) Which pentomino has the smallest perimeter? d) Work out why this is. www.techniquest.org 131 Pentominoes Follow-up Activity Two: Pentominoes Answers a) There are 12 pentominoes altogether. b) All pentominoes have a perimeter of 12, except the yellow one, which has a perimeter of 10. c) The yellow pentomino has the smallest perimeter. d) This is because there are five pairs of touching edges in the yellow pentomino. 132 www.techniquest.org Pentominoes Follow-up Activity Three: Investigating Pentominoes Here are all of the possible pentominoes. a) Predict which of these pentominoes could be folded to make an open box. b) Cut out the pentominoes and test your predictions. www.techniquest.org 133 Pentominoes Follow-up Activity Three: Investigating pentominoes Answers a) and b) The pentominoes that can be folded to make open boxes are: 134 www.techniquest.org Pyramid Follow-up Activity One: Making Pyramids Using straws, make these four pyramids. a) Complete this table. Shape of Base Number of Base Straws Total Number of Straws c) What is the rule for finding the total number of straws needed for a pyramid? d) How many straws would you need for a heptagonal-based pyramid? e) How many straws would you need for an octagonal-based pyramid? www.techniquest.org 135 Pyramid Follow-up Activity One: Making Pyramids Answers a) Shape of Base Number of Base Straws Total Number of Straws Triangle 3 6 Square 4 8 Pentagon 5 10 Hexagon 6 12 c) The total number of straws is twice the number of straws in the base. Rule: Total number of straws = 2n d) A heptagonal-based pyramid would have 7 straws in the base. Therefore the total number of straws is 2 x 7 = 14. e) An octagonal-based pyramid would have 8 straws in the base. Therefore the total number of straws is 2 x 8 = 16. 136 www.techniquest.org Pyramid Follow-up Activity Two: Making Pyramids Using straws, make these four pyramids. a) Complete this table. Shape of Base Number of Base Total Number of Vertices Vertices (Corners) c) What is the rule for finding the total number of vertices for a pyramid? d) How many vertices would there be for a heptagonal-based pyramid? e) How many vertices would there be for an octagonal-based pyramid? www.techniquest.org 137 Pyramid Follow-up Activity Two: Making Pyramids Answers a) Shape of Base Number of Base Total Number of Vertices Vertices (Corners) Triangle 3 4 Square 4 5 Pentagon 5 6 Hexagon 6 7 c) The total number of vertices is th number of vertices in the base plus 1. Rule: Total number of vertices = n + 1 d) A heptagonal-based pyramid would have 7 vertices in the base. Therefore, the total number of vertices would be 7 + 1 = 8. e) An octagonal-based pyramid would have 8 vertices in the base. Therefore, the total number of vertices would be 8+ 1 = 9. 138 www.techniquest.org Pyramid Follow-up Activity Three: Pyramids Here are four pyramids. A B C D a) Name these pyramids. b) Complete this table. Pyramid Number of Faces Number of Number of Edges Vertices (Corners) c) Can you find the rule for finding the number of edges. www.techniquest.org 139 Pyramid Follow-up Activity Three: Pyramids Answers a) Pyramid A: Triangular-based pyramid Pyramid B: Square-based pyramid Pyramid C: Pentagonal-based pyramid Pyramid D: Hexagonal-based pyramid b) Pyramid Number of Faces Number of Number of Edges Vertices (Corners) Triangle-based 4 4 6 Square-based 5 5 8 Pentagon-based 6 6 10 Hexagon-based 7 7 12 c) To find the number of edges, add the number of faces to the number of vertices and subtract 2. Rule: 140 Number of edges = number of faces + number of vertices - 2 www.techniquest.org Pythagoras Puzzle Follow-up Activity One: Area a) Using peg boards and elastic bands, create four shapes and measure their area. b) Find the area of the following letter shapes. c) Shade these grids to show the area. 6cm2 www.techniquest.org 9cm2 4cm2 3cm2 141 Pythagoras Puzzle Follow-up Activity One: AreaAnswers a) The answers to this question will depend on the shapes created. b) H: 11cm2 T: 6cm2 E: 10cm2 F: 8cm2 c) 6cm2 142 9cm2 4cm2 3cm2 www.techniquest.org Pythagoras Puzzle Follow-up Activity Two: Investigating Area Investigate the following statements. Are they true or false? a) All rectangles with an area of 12cm2 have the same perimeter. b) Multiplying the length and width of a rectangle by 10 makes the area larger. c) Every rectangle with an area of 36cm2 has a perimeter of less than 34cm. www.techniquest.org 143 Pythagoras Puzzle Follow-up Activity Two: Investigating Area Answers a) False: they will have different perimeters. For instance, a 3x4 rectangle would have a perimeter of 14cm. A 6x2 rectangle would have a perimeter of 16cm. They would both, however, have the same area. b) True: the area does get larger. c) False: for example, a 1x36 rectangle will have a perimeter of 74cm. 144 www.techniquest.org Pythagoras Puzzle Follow-up Activity Three: Investigating Cubes All of these shapes are made from four cubes. a) What is the volume of each shape? b) Using estimation, put these shapes into order according to their surface area, starting with the largest. c) Work out the surface area for each shape. www.techniquest.org 145 Pythagoras Puzzle Follow-up Activity Three: Investigating Cubes Answers a) All shapes have a volume of 4 units3. b) This will depend on the pupils’ estimations. c) All shapes have a surface area of 18 units2, except the first one, which has a surface area of 16 units2. 146 www.techniquest.org Reach the Goal Follow-up Activity One: Collecting Carrots Mark out the path of the rabbit following the given compass bearings. i) N - E - N - W - W - N - W N W E S ii) N - E - N - E - S - E www.techniquest.org 147 Reach the Goal Follow-up Activity One: Collecting Carrots continued iii) W - N - W - N - E - N - E - N - W - N a) For each path, work out how many carrots the rabbit collects. b) The farmer gets 5p for each carrot he sells at market. Work out how much money she will make for each of these vegetable patches once the rabbit has eaten its share. c) How much does the farmer make from all three vegetable patches? 148 www.techniquest.org Reach the Goal Follow-up Activity One: Collecting Carrots Answers a) i) The rabbit collects 3 carrots. ii) The rabbit collects 1 carrot. iii) The rabbit collects 2 carrots. b) i) There are 5 carrots in the patch and the rabbit eats 3. This leaves 2 carrots for the farmer. She gets 5p per carrot, so the farmer gets 2 x 5 = 10p. ii) There are 5 carrots in the patch and the rabbit eats 1. This leaves 4 carrots for the farmer. She gets 5p per carrot, so the farmer gets 4 x 5 = 20p. iii) There are 9 carrots in the patch and the rabbit eats 2. This leaves 7 carrots for the farmer. She gets 5p per carrot, so the farmer gets 7 x 5 = 35p. c) Altogether the farmer gets 10p + 20p + 35p = 65p. www.techniquest.org 149 150 www.techniquest.org Reach the Goal Follow-up Activity Two: Frogs All of these frogs want to swap places so that the black frogs are on the right and the white frogs are on the left. Rules: 1) A frog can jump onto an empty leaf which is next to it. 2) A frog can jump over one frog of the other colour to land on an empty leaf. 3) A frog cannot move backwards. a) Using counters for frogs, can you change their positions. Keep a record of their moves. b) What is the minimum number of moves needed? c) Complete the table below using different numbers of frogs. Number of Black Counters Number of White Counters 1 1 2 2 3 3 4 4 5 5 6 6 Number of Moves d) Can you work out the rule for the number of moves for any number of frogs? www.techniquest.org 151 Reach the Goal Follow-up Activity Two: Frogs Answers a) Winning strategy: Make sure that 2 counters of the same colour are never brought together until the end. b) The minimum number of moves needed for 3 black frogs and 3 white frogs is 15. c) Number of Black Number of White Number of Moves Counters Counters 1 1 3 2 2 8 3 3 15 4 4 24 5 5 35 6 6 48 d) When there are an equal number of black frogs and white frogs, the rule for working out the minimum number of moves is: multiply the number of black counters by the number of white counters and then subtract 2. Rule: 152 Minimum moves = ((black counters + 1) (white counters +1) - 1) www.techniquest.org Reach the Goal Follow-up Activity Three: Moving Counters Here is a 3 x 3 grid with a white counter in the top corner. You can only move the counter using knight’s moves. A knight’s move is either 2 squares down or up and then 1 square across. For example, Or a knight’s move is 2 squares across and then 1 square either down or up. For example, a) Is it possible to move to every square in the grid using only knight’s moves? b) Draw a 4 x 4 grid. Is it possible to move to every square in this grid using only knight’s moves? c) Draw a 5 x 5 grid. Is it possible to move to every square in this grid using only knight’s moves? www.techniquest.org 153 Reach the Goal Follow-up Activity Three: Moving Counters Answers a) Using only knight’s moves, the centre square of a 3 x 3 grid will be left. b) Using only knight’s moves, one corner square of a 4 x 4 grid will be left. c) Yes. See the 5x5 grid below. 7 2 4 1 154 6 5 15 6 3 8 5 2 9 12 3 14 11 4 7 1 8 13 10 3 10 21 16 5 20 15 4 11 22 9 2 25 6 17 14 19 8 23 12 1 24 13 18 7 www.techniquest.org Spheres Follow-up Activity One: Drawing Circles Using a pair of compasses, experiment with drawing circles to create patterns. For example: www.techniquest.org 155 Spheres Follow-up Activity One: Drawing Circles Answers The answers for this activity will depend on what the pupils have drawn. 156 www.techniquest.org Spheres Follow-up Activity Two: Investigating Circumferences Look at this circle pattern. a) Measure the radius of the circles. b) By how much does the radius increase with each circle? c) Estimate the circumference of each circle. d) Measure the circumference of each circle, using a piece of string. e) By how much does the circumference increase with each circle? www.techniquest.org 157 Spheres Follow-up Activity Two: Investigating Circumferences Answers a) The radii of the circles are: 1cm, 2cm, 3cm, 4cm, 5cm and 6cm. b) The radius increases by 1cm with each circle. c) To estimate, multiply the radius by 3. d) The calculated circumferences would be (to 2 decimal places): 3.14, 6.28, 9.42, 12.57, 15.71 and 18.85. Any measurement close to these is good. e) Mathematically speaking, the circumference increases by 3.14 or “pi”. 158 www.techniquest.org Tetra Solid Follow-up Activity One: Tetrahedra a) Using four different colours (e.g. red, blue, yellow and green) – and using each colour to fill one of the small triangles within each large triangle – how many different triangles can you make? b) Cut out each of these large triangles. By putting them together, how many different tetrahedra can you make? www.techniquest.org 159 Tetra Solid Follow-up Activity One: Tetrahedra Answers You can draw 24 triangles with different colour patterns. However, many of these will be rotations. If rotations are not allowed, only eight different triangles are possible. If reflections are also not allowed, the number of different triangles comes down to four. 160 www.techniquest.org Tetra Solid Follow-up Activity Two: Tetrahedra Game Cut out each of the coloured triangles to play this game. Rules: • Divide the 16 coloured triangles evenly and randomly between players. • Take turns to place the coloured triangles into the grid on the following page. • Start at the top of the triangle and work down. • You can only place a colour on the triangle next to a matching colour, for example, red to red, green to green. The middle colours of the triangles do not have to match • The losing player is the one who cannot place any of their triangles into the grid. • Is it possible to put all of the coloured triangles into the grid? www.techniquest.org 161 Tetra Solid Follow-up Activity Two: Tetrahedra Game continued 162 www.techniquest.org Tetra Solid Follow-up Activity Three: Polyhedra Puzzle a) Can a spider walk along all of the edges of a cube in one continuous path, without walking over any edges more than once. b) What about these other polyhedra? tetrahedron octahedron dodecahedron icosahedron www.techniquest.org 163 Tetra Solid Follow-up Activity Three: Polyhedra Puzzle a) The spider cannot walk along all of the edges of the cube. It can walk along a maximum of 9 edges. b) Tetrahedron: No, the spider can only walk along five edges. Octahedron: Yes. Dodecahedron: No, the spider cannot walk along all of the edges. Icosahedron: No, the spider cannot walk along all of the edges. For the spider to be able to walk along all of the edges of a solid shape in one continuous path, no more than two vertices can have an odd number of edges radiating from it. 164 www.techniquest.org Times Table Follow-up Activity One: Dice Probability A six-sided die has the numbers 2, 3, 4, 6, 8 and 12. Roll the die. a) What is the probability that I will roll a 2? b) What is the probability that I will roll an even number? c) What is the probability that I will roll a factor of 12? d) What is the probability that I will roll a multiple of 3? e) I roll the die twice. What is the probability that the product of the two numbers rolled will be 12? www.techniquest.org 165 Times Table Follow-up Activity One: Dice Probability Answers a) There is one 2 on the six-sided die, so the probability is 1/6. b) There are five even numbers, so the probability is 5/6. c) 2, 3, 4, 6 and 12 are all factors of 12. Therefore the probability is 5/6. d) 3, 6 and 12 are all multiples of 3. Therefore the probability is 3/6 or 1/2. e) To get a product of 12, you could roll: 2 and then 6, 6 and then 2, 3 and then 4, or 4 and then 3. Therefore there are four different ways of getting a product of 12. There are 36 possibilities (6 for the first roll x six for the second). This make 4/36 or 1/9. 166 www.techniquest.org Times Table Follow-up Activity Two: Number Patterns Fill in the missing numbers and say which multiplication table the sequence is a part of. a) a) b) This is part of the ________ times table. 6 ________ times table. This is the This is part of the ________ times table. 25 c) 30 65 50 This is part of the ________ times table. 40 d) 18 3 36 24 This is part of the ________ times table. 30 24 18 e) Which numbers appear in more than one table? www.techniquest.org 167 Times Table Follow-up Activity Two: Number Patterns Answers a) This is part of the 3 times table. 3 6 9 12 15 18 45 50 24 20 21 24 55 60 16 12 18 16 27 30 65 70 b) This is part of the 5 times table. 25 30 35 40 c) This is part of the 4 times table. 40 36 32 28 8 4 d) This is part of the 2 times table. 30 28 26 24 22 20 e) 12 appears in the 2, 3 and 4 times tables. 18 appears in the 2 and 3 times tables. 20 appears in the 2 and 4 times tables. 24 appears in the 2, 3 and 4 times tables. 30 appears in the 2, 3 and 5 times tables. 40 appears in the 4 and 5 times tables. If you continue these tables, more of these numbers will appear. 168 14 12 www.techniquest.org Times Table Follow-up Activity Three: Money Multiplication I have baked six cherry buns, two doughnuts and four chocolate brownies to sell at a fair. a) How many cakes have I baked altogether? b) The cherry buns are 20p each. If I sell four of them, how much money have I made? c) The doughnuts cost twice as much as the cherry buns. How much are they? d) Someone buys all four chocolate brownies and pays £2. They get 20p change. How much does one brownie cost? e) I have been given ten fairy cakes to sell. I sell them all for 50p. How much does one fairy cake cost? f) How many times more expensive are the brownies than the fairy cakes? g) How much does a doughnut cost? h) How many more times expensive are the cherry buns than the fairy cakes? i) How many fairy cakes can be bought for the cost of two doughnuts? www.techniquest.org 169 Times Table Follow-up Activity Three: Money Multiplication Answers a) 6 + 2 + 4 = 12 cakes altogether. b) 20p x 4 = 80p c) 2 x 20p = 40p d) £2 - 20p = £1.80; £1.80 divided by 4 = 45p e) 50p divided by 10 = 5p f) Fairy cakes cost 5p and brownies cost 45p. 45 divided by 5 = 9. They are 9 times more expensive. g) Doughnuts cost 40p. h) Cherry buns are 20p and fairy cakes are 5p. 20p divided by 5p = 4. They are four times more expensive. i) One doughnut costs 40p. Two doughnuts cost 2 x 40p = 80p. Fairy cakes cost 5p. 80p divided by 5p = 16. You can buy sixteen fairy cakes for the cost of two doughnuts. 170 www.techniquest.org Towers of Brahma Follow-up Activity One: Dice Doubles Roll two dice. a) Add the numbers together from the two dice. b) Double the total. First Second Number Number www.techniquest.org Total Double 171 Towers of Brahma Follow-up Activity One: Dice Doubles Answers a) and b) Answers will depend on the numbers thrown. The totals will range from 2 to 12 and the doubles will range from 4 to 24. 172 www.techniquest.org Towers of Brahma Follow-up Activity Two: Doubles a) Colour each flower a different colour. b) Match the flower number to the pot that is its double. Colour the flower and the matching pot the same. 4 6 8 14 12 7 20 16 20 15 14 34 10 40 www.techniquest.org 8 19 17 38 30 28 173 Towers of Brahma Follow-up Activity Two: Doubles Answers 4 7 6 8 12 8 14 15 174 16 30 20 20 17 14 28 10 19 34 38 40 www.techniquest.org Towers of Brahma Follow-up Activity Three: Probability Doubles Here are the flowers in their pots. Pick one flower. a) What is the probability that the flower is odd? b) what is the probability that the double on the pot is higher than 20? c) What is the probability that the flower is a prime number? d) What is the probability that the double is a multiple of 5? e) What is the probability that the flower and its double are both lower than 15? 4 7 6 8 12 8 14 16 15 30 www.techniquest.org 20 20 17 14 28 10 19 34 38 40 175 Towers of Brahma Follow-up Activity Three: Probability Doubles Answers a) There are four odd flowers - 7, 15, 17 and 19. Therefore the probability is 4/10 or 2/5. b) There are five pots higher than 20 - 28, 30, 34, 38 and 40. Therefore the probability is 5/10 or 1/2. c) There are three prime numbers - 7, 17 and 19. Therefore the probability is 3/10. d) There are three multiples of 5 - 20, 30 and 40. Therefore the probability is 3/10. e) There are three flowers and pots lower than 15 - 4-8, 6-12 and 7-14. Therefore the probability is 3/10. 176 www.techniquest.org Towers of Hanoi Follow-up Activity One: Square Patterns Study this square pattern. 1 2 3 a) How many small squares do you think would be in pattern 4? Check your prediction by drawing the shape. b) Complete this table. Pattern number 1 2 3 Number of Small 1 4 9 4 5 6 Squares c) How did you find the missing numbers? Look at the number pattern in the bottom row of your table. d) Find out what these numbers are called. e) How many squares would be in pattern 7? f) How can you work out the number of squares for any pattern? www.techniquest.org 177 Towers of Hanoi Follow-up Activity One: Square Patterns Answers a) Pattern 4 would have 16 squares. b) Complete this table. Pattern number 1 2 3 4 5 6 Number of Small 1 4 9 16 25 36 Squares c) The missing numbers are found by squaring the pattern number. d) These numbers are called square numbers. e) Pattern 7 would be 72 or 7 x 7 = 49. There would be 49 squares in pattern 7. f) To find the number of squares for any pattern, square the pattern number. Rule: 178 Number of squares = n2 www.techniquest.org Towers of Hanoi Follow-up Activity Two: Square Number Addition Some numbers can be made by adding two square numbers together. For example: 13 = 22 + 32 a) Find all of the square numbers less than 100 that can be made by adding 2 square numbers together. Complete this table to help you. + 12 22 32 42 52 62 72 12 22 32 42 52 62 72 Using your table, write these numbers as the sum of three square numbers. b) 26 = + c) 49 = + + d) 51 = + + e) 101 = + f) 42 = + + www.techniquest.org + + 179 Towers of Hanoi Follow-up Activity Two: Square Number Addition Answers a) Find all of the square numbers less than 100 that can be made by adding 2 square numbers together. Complete this table to help you. + 12 22 32 42 52 62 72 12 2 5 10 17 26 37 50 22 5 8 13 20 29 40 53 32 10 13 18 25 34 45 58 42 17 20 25 32 41 52 65 52 26 29 34 41 50 61 74 62 37 40 45 52 61 72 85 72 50 53 58 65 74 85 98 Using your table, write these numbers as the sum of three square numbers. b) 26 = 32 + 42 + 12 c) 49 = 62 + 32 + 22 d) 51 =52 + 52 + 12 e) 101 = 72 + 42 + 62 f) 42 = 12 + 42 +52 180 www.techniquest.org Towers of Hanoi Follow-up Activity Three: Investigating Square Numbers a) Choose a number less than 100. Square your number. Add twice your original number. Add 1. Write down your answer. b) Using the same starting number, add 1 to the number. Square your number. Write down your answer. c) What do you notice about your answers for a) and b)? d) Repeat a) and b) using a different number. What do you notice about your two answers? e) Can you think of any number where this will not work? www.techniquest.org 181 Towers of Hanoi Follow-up Activity Three: Investigating Square Numbers Answers c) The answers for both a) and b) will be the same. Although the questions are phrased differently, a) and b) are the same. Therefore they will always produce the same result. a) is n2 + 2n + 1 whereas b) is (n + 1)2. This is the same calculation. d) The answers for a) and b) will be the same. e) This should work for any number (n). 182 www.techniquest.org