MEI Conference 2012 Short Open-Ended Investigations for KS3 Kevin Lord Kevin.lord@mei.org.uk
10 Ideas for Short Investigations These are some of the investigations that I have used many times with a variety of groups. Besides providing opportunities for students to apply mathematics and practise their problem-solving skills, these activities help to reinforce number and calculation skills, geometry, probability and statistics ideas. If sufficient time is allowed most of these investigations can be extended and will lead to generalisations and algebraic expressions. 1. Number Chains 2. HOTPOs 3. Magic Total 34 4. 20-line Game 5. Patterns on a times-table square 6. Angles in Stars 7. Euler s Theorem for Networks 8. Snakes and Ladders Investigation 9. Isle of Wight Problem 10. Cereal Packet Free Gift
NUMBER CHAINS Numbers can form surprising chains when very simple rules are applied to their digits. Look at the numbers in this chain.... START 16 31 8 40 4 20? 2 Can you see a rule which might link the numbers together? What numbers do you think come next? Once you have discovered the rule, think about these questions and make some predictions. How many numbers do you think will be in the chain? What do you think will be the highest number in the chain? Do you think the chain will go on forever, or stop at some point? Do you think the numbers will begin to repeat themselves? Carry on the chain and investigate what happens to numbers a) Describe the rule for linking the numbers in this chain in your own words so that someone else would understand. (Use an example or diagram to help show how the rule works.) b) Write down how many numbers are in the chain. c) Write down any numbers which are missing and find out what happens if you start with one of these. d) Try a slightly different rule and investigate what happens this time.
HOTPO A little number investigation, which practices dividing by 2 and multiplying by 3, known as the Half or Triple Plus One - HOTPO Starting with any number, U n If U n is EVEN, then divide by 2 U n+1 = U n /2 If U n is ODD, then triple and add 1 U n+1 = 3U n + 1 And continue For example, U n = 13 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, (4, 2,1 begins to repeat after 1 is reached) Where do the sequences end? (Is this true for all numbers?) How long is the sequence for different starting numbers? What is the Highest Value reached in a sequence? Could you make a map of the routes of different number?
MAGIC TOTAL 34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Above is a 4 by 4 square numbered 1 to 16. By choosing just one number in each row or column as shown by the shaded squares you can check the total is 34. Is this always true? What other patterns of 4 squares can you find that also add up to 34? Try to think about the symmetry of the pattern of squares you choose. Extension What is the magic total for other sizes of square, such as a 5 by 5? Do rectangular grids also have a magic total? How would you explain the rules for rectangles?
Name MAGIC TOTAL 34 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8 5 6 7 8 9 10 11 12 9 10 11 12 9 10 11 12 13 14 15 16 13 14 15 16 13 14 15 16 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8 5 6 7 8 9 10 11 12 9 10 11 12 9 10 11 12 13 14 15 16 13 14 15 16 13 14 15 16 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8 5 6 7 8 9 10 11 12 9 10 11 12 9 10 11 12 13 14 15 16 13 14 15 16 13 14 15 16 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8 5 6 7 8 9 10 11 12 9 10 11 12 9 10 11 12 13 14 15 16 13 14 15 16 13 14 15 16
Number Line Game Start with a number line, numbered 0 to 20. You can add or subtract 2 numbers on the line so that the answer is also on the line. For example 6 + 4 = 10 10 take away 9 makes 1 1. Cross out the numbers you use. Circle the numbers you end on. Then cross these out as you start with them for the next calculation. You decide any other rules for the game. How will you know who has won? You could play Collaboratively Try to fill in the whole line. You could play Competitively - Stop your partner from going. How could this be adapted to be less / more challenging? What observations can you make about this game? What are good strategies for winning the game?
Patterns on a Times-table Square Starting with a 12x12 times-table square investigate the patterns made by colouring in all the numbers with a particular property. Example 1 : Multiples of 4 x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84 8 8 16 24 32 40 48 56 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144 Example 2: Factors of 24 x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84 8 8 16 24 32 40 48 56 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144
x 1 2 3 4 5 6 7 8 9 10 11 12 x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 14 16 18 20 22 24 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84 7 7 14 21 28 35 42 49 56 63 70 77 84 8 8 16 24 32 40 48 56 64 72 80 88 96 8 8 16 24 32 40 48 56 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144 12 12 24 36 48 60 72 84 96 108 120 132 144 x 1 2 3 4 5 6 7 8 9 10 11 12 x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 14 16 18 20 22 24 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84 7 7 14 21 28 35 42 49 56 63 70 77 84 8 8 16 24 32 40 48 56 64 72 80 88 96 8 8 16 24 32 40 48 56 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144 12 12 24 36 48 60 72 84 96 108 120 132 144 x 1 2 3 4 5 6 7 8 9 10 11 12 x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 14 16 18 20 22 24 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84 7 7 14 21 28 35 42 49 56 63 70 77 84 8 8 16 24 32 40 48 56 64 72 80 88 96 8 8 16 24 32 40 48 56 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144 12 12 24 36 48 60 72 84 96 108 120 132 144
STARS If a star has 5 points then you can draw it using 5 straight lines, without removing your pencil from the paper. For stars that have an odd number of points, you can draw them in this way, with the number of points equal to the number of lines. What is the sum of the interior angles of the 5 points of a pentagram (five pointed star)? What is the sum of the exterior angles of the pentagram? What is the sum of the interior angles of the points on a 7 or 9 pointed star? Could you write a procedure using Logo to draw any star with an odd number of points?
Euler s Theorem for Networks Investigate the relationship between the number of regions of a network and the number of lines and nodes. For example, here is a simple network diagram. There are 4 nodes and 5 lines. It divides the paper into 3 regions, 2 inside and one outside. 1 2 3 a) For the 8 networks below sketch them and shade in the regions. b) Record your results in a table with the following headings. Network No of Regions No. of Nodes No. of Lines 1 2 c) Look for a relationship between the number of Regions, R, Lines, L and Nodes, N.
SNAKES and LADDERS The game of Snakes and Ladders has a long history and originated in Indian where it was a game for teaching children how to live a moral and good life. Good deeds allowed you to climb the ladders and bad deeds meant that you slipped back down the snakes to the bottom. Boards are not all the same, some have more squares, some have more snakes or more ladders. How would you compare two different board to decide which one is harder or easier to reach the top? Can you calculate the length of each snake and each ladder? Why is the average length of snake or ladder useful?
The Isle of Wight The Problem When I was at school I remember being told that all the people in the world could fit on the Isle of Wight. Is this still true? I expect that the area of the Isle of Wight has not changed very much in 30 years but what about the population of the world? Modelling What information do you need to solve this problem? How will you get this information? What assumptions about the Isle of Wight, and the people in the world will you make in order to tackle this problem? What is the calculation you will need to do to solve the problem?
Hints and Possible approaches 1. Use the print out of a map to estimate area of Isle of Wight using simple shapes. How could you do this more accurately? 2. Use some metre sticks to make a square metre and carry out an experiment to estimate the number of people who could fit in a square metre. What is the average number of people per square metre? How accurate do you think your value is? 3. Use the internet to research the population of the world and verify the area of the Isle of Wight. World Population in 2011 = 6.93 billion people World Population in 1980 = 4.44 billion people Area of Isle of Wight = 384 km 2 4. Calculation required : Number of people = Area x Population density Or Area required = Population/ Pop. Density 5. At the end remember to discuss the validity of the result, by discussing again the assumptions you made at the start. For example Not all people are the same size, most would be bigger than children It is not possible to stand on all parts of the island 6. Can you find other islands that could be home to the whole world? Mathematics skills that may be used Estimation and approximation Area of rectangles and triangles Compound measures like density Standard form Simple averages and using averages Converting units m 2 to km 2 Interpreting graphs
Cereal Packet Free Gift Cereal manufacturers often use the marketing ploy of including a set of small toys in their cereal boxes to get children to pressure their parents to buy the cereals to get the toy. Suppose there are 6 toys in the set to collect. How many boxes do you think you will have to buy before you have the full set? What does your answer depend on, what assumptions are you making? Modelling Let s assume that your chance of getting each toy are the same = 1/6. We can roll a die to simulate the randomness of which toy is inside the cereal packet. Simulation Roll a fair die and record in the tally chart the number shown? Stop when all 6 numbers have been scored? Count up you total number of rolls. This is the number of packets you would have had to buy. Repeat the simulation 3 or 4 times to see if your results are similar. First Trial Second Trial Third Trial Fourth Trial Score Tally Freq. Tally Freq. Tally Freq. Tally Freq. 1 2 3 4 5 6 Total Total Total Total How could you set up the simulation if, there were 10 toys to collect? the probability of getting one of the toys was much lower than the rest? Investigate the relationship between the number of toys to collect and the number of packets that have to be bought.