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Reverse and Add 1 Write down a two digit number, e.g. 62 2 Reverse the digits to form another two digit number, e.g. 26 3 Add the two numbers, e.g. 62 + 26 88 4 Repeat for other two digit numbers. 5 What do you notice? 6 Can you explain why this happens? Convince a friend Reverse and Subtract 7 Write down a two digit number, e.g. 62 8 Reverse the digits to form another two digit number, e.g. 26 9 Subtract the smaller number from the larger, e.g. 62 - 26 36 10 Repeat for other two digit numbers. 11 What do you notice? 12 Can you explain why this happens? Convince a friend Dot-to-dot Triangles A B C D E Draw a set of five labelled dots – [Don’t copy the ones I have drawn] Make sure that you don’t choose three dots in a row Make several copies of your set of dots On each set draw a triangle(s) by joining any three dots. How many different triangles can you draw? How can you be sure you have found all the possible triangles? Convince a friend Can You prove it? Sue Waring The Mathematical Association Staircase Numbers A staircase is formed by adding consecutive integers. Examples:- 12 because 3 + 4 + 5 = 12 (3–step) 5 53 because 26 + 27 = 53 (2-step) 80 because 14 + 15 + 16 + 17 + 18 = 80 (5-step) 4 Investigate staircase numbers and try to answer the following questions. 3 1. What do you notice about (i) 3-step numbers, (ii) 5-step, (iii) 7-step numbers? 2. What about “odd-step” staircase numbers? 3. Can you explain this result? 4. Do “even-step” staircase numbers behave in the same way? 5. Explain. 6. Describe how to find a staircase number. 7. Is it possible to find more than one staircase? 8. Are there any numbers which are not staircase numbers? If so, which? Write a report about your investigation to Convince a penfriend Triangle Numbers 1 3 6 10 The first four triangle numbers are shown above. Use the following questions to guide your investigation and write a report. 1. By drawing similar diagrams find the next three triangle numbers and continue this table. Position Triangle Number (n) (T) 1 1 2 3 3 .. 4 Can You prove it? Sue Waring The Mathematical Association 2. Write down the next three triangle numbers after the last one you have drawn. 3. Explain how you could obtain the 20th triangle number. 4. Find the most efficient way of doing this. 5. Test it on some triangle numbers you have found. 6. Try to write your method as a formula for finding the nth triangle number T. 7. Test your formula for n = 2, 3 and 4. 8. Explain how and why your formula works for the fifth triangle number. 9. Will it always work? How can you be sure? Convince a penfriend Calendar Squares In the diagram showing the calendar for the month of May, a group of four numbers in a square has been highlighted. Adding diagonally opposite numbers in this square gives:- 15 + 23 = 38 and 16 + 22 = 38 MAY Repeat for other squares. Sun Mon Tue Wed Thu Fri Sat Repeat for other squares. 1 2 3 4 5 6 Describe what you notice about the totals. Explain why this happens. 7 8 9 10 11 12 13 Convince a penfriend 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Partitioning Three objects can be partitioned in three ways:- 2 sets A set of 2 objects and a set of 1 object and A set of 1 object and a set of 2 objects and 3 sets each of 1 object and and In how many ways can a) four objects b) five objects be partitioned? Convince a penfriend that you have examined all possibilities Can You prove it? Sue Waring The Mathematical Association