A1 SS Number patterns on a spreadsheet This sheet will help you to create your own number patterns on a spreadsheet. Do the steps one at a time. You will soon feel more confident with a spreadsheet program. 1 Open the Excel program. It will show you a grid of rectangles. These are called cells. Each cell has an address. Its address is a letter (for the column) and a number (for the row). 2 Click in cell A1. In cell A1 type 1. 3 Click in cell A2. In cell A2, type 0. 4 In cell A3, type =A2+A$1. This gives A3 the value of 1. (This formula says take the number in the cell above (e.g. A2 is above A3) and add the number from cell A1.) 5 Put the cursor into A3. Hold down the mouse button and move the cursor down to A12. This will select the cells from A3 to A12. They will change colour. Release the mouse button. 6 In the menu bar at the top of the screen, find Edit. In that menu find Fill. Select Fill down. This puts the formula from A3 into all the selected cells below. The result is the set of numbers 1 to 10. (The formula adds 1 (from A1) to the previous number, so the pattern goes up in 1s.) 7 Change the number in A2 to 10. The pattern will now start at 10 and go to 20. 8 Make the pattern start at 100 and go up in 1s. 9 Change the number in A2 to 0. 10 Click on A1. Change the 1 to 2. The pattern should now start at 0 and go up in 2s. These are even numbers. 11 Change the number in A2 to make a pattern of odd numbers. 12 Make the pattern start at 0 and go up in 3s. You now know how to make the pattern start at any number and go up by any number. 13 Type 20 in cell A2. Change number in cell A1 to 1. This will make the pattern go down by 1. (It subtracts 1 from the previous number each time.) 14 Make the pattern start at 30 and go down by 3 each time. 15 Change cell A2 back to 1 and cell A1 back to 2. You can change the formula in A3 in other ways. Change it to =A2*A$1. Then select cells A3 to A12, and use Fill down (steps 5 and 6 above.) The * means multiply, so this will double the number from the cell above. 16 See what happens if A1 is 1. 17 See what happens if A1 is 1. 18 You can combine two operations in a pattern. In cell B1 type 1. Now change the formula in A3 to =A2*A$1+B$1. Then select cells A3 to A12, and use Fill down (steps 5 and 6 above.) 19 Explore the effect of multiplying by one number (in cell A1) and adding another (in cell B1). 20 Explore the effect of adding one number and subtracting another. 21 When multiplying and subtracting there are some combinations that make the pattern repeat the same number. (An example is multiply by 3 and subtract 2.) Find some others.
A2 SS Formula guessing game This sheet will help you to use the spreadsheet called Formula guessing. When opened it looks a bit like this. Enter two numbers for the formula. term number 1 2 3 4 5 6 7 pattern number 6 10 14 18 22 26 30 4 2 Multiply by 4 and add 0 Keep trying After a GREAT! result, delete the numbers in C6 and F6. This shows the term numbers (1, 2, 3, 4, etc.) This also shows pattern numbers (6, 10, 14, 18, etc.). You have to find the formula that changes each term number into the pattern number. Put your formula numbers into cells C6 and F6. (They are the only cells you can use.) This pattern goes up in 4s, so the multiply by number is 4. This part is correct. But the number you add is not 0. So you have to Keep trying. When you get both numbers correct, the answer is GREAT! To get a new problem, delete one of the numbers in C6 or F6. The spreadsheet has three parts. To choose another part click on its label at the bottom of the screen. The one shown above is called mx + c. This is because you multiply by one number and add another number. There is also one called mx c. For this one you multiply by one number and subtract another number. There is also one called c mx. For this one you multiply by one number and subtract the answer from another number.
A3 SS Squares on a spreadsheet This sheet will explain how to create the square numbers on an Excel spreadsheet. There are two ways. Squares by adding odd numbers 1 In cell A2, type 1. In cell A3 type =A2+2. 2 Put the cursor into A3. Hold down the mouse button and move the cursor down to A12. This will select the cells from A3 to A12. They will change colour. Release the mouse button. 3 In the menu bar at the top of the screen, find Edit. In that menu find Fill. Select Fill down. This puts the formula from A3 into all the selected cells below. The result is the odd numbers from 1 to 21. (The formula adds 2 to the previous number, so the pattern goes up in 2s.) 4 Now you will see how to add these to get the square numbers. In cell B2, type =B1+A2. This will put 1 into cell B2. 5 Use steps 2 and 3 above to select cells B2 to B12, and then fill down the formula in B2. Each cell now takes the square number above it, and adds the next odd number, to get the next square number. Squares by direct formula 6 In cell C2 type 0. In cell C3 type =C2+1. Then select and fill down from C3 to C12. This will give the numbers 0 to 10. 7 In cell D2, type =C2*C2. This formula says multiply the number in C2 by itself. Since C2 is 0, it will give 0. 8 Then select and fill down from D2 to D12. This will give the numbers in column C, each multiplied by itself. These are the square numbers. Making a two-way table of squares 9 In A2, put 0. In A3, type =A2+10. Then select and fill down from A3 to A11. This will give the numbers 0 to 90, by 10s. 10 In A1, put 0. In B1, type A1+1. Then select and fill across from B1 to K1. This will give the numbers 0 to 10. 11 To get the numbers from 1 to 100 in the table, type =B$1+$A2 in cell B2. Then select all the cells from B2 to K11. Fill down and Fill across. Each cell now has the sum of the left column and the top row, which makes all the numbers from 0 to 100. 12 To change these into square numbers, change the formula in B2 to =(B$1+$A2)^2. This says to take that sum (the number from 1 to 100) and square it. 13 Then select all the cells from B2 to K11. Fill down and Fill across. A challenge There are quite a few square numbers that are the difference between two others. How many can you find?
A4 SS Diophantine equations 1 The first person known to have made up problems about numbers was called Diophantus. He lived in the first century CE. Here is one of his problems. Solve it using any method you can. Find two numbers such that their sum is 20 and the sum of their squares is 208. 2 Most of Diophantus problems were about pairs of whole numbers. Here are some other problems of this type. (For many there will be more than one solution.) a There are some 4-legged chairs and 3-legged stools. Between them they have 29 legs. How many of each must there be? b I have some $5 notes and $2 coins. They are worth $27. How many of each do I have? c Using only $5 notes and $2 coins how can you make each amount from $7 to $30? d Tickets cost $10 (adult) and $3 (child). We took $140. How many of each did we sell? e Tickets cost $10 (adult) and $3 (child). What numbers of dollars can we make? Spreadsheet The rest of this sheet show you how to use the spreadsheet Diophantine equations. When opened it looks like this. x= 4 y= 3 total = 2 9 ax+by 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 1 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 2 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 3 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 4 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 5 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 6 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 7 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 8 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 9 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 10 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100 11 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98 101 104 12 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 13 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100 103 106 109 112 14 59 62 65 68 71 74 77 80 83 86 89 92 95 98 101 104 107 110 113 116 15 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117 120 16 67 70 73 76 79 82 85 88 91 94 97 100 103 106 109 112 115 118 121 124 17 71 74 77 80 83 86 89 92 95 98 101 104 107 110 113 116 119 122 125 128 18 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117 120 123 126 129 132 19 79 82 85 88 91 94 97 100 103 106 109 112 115 118 121 124 127 130 133 136 20 83 86 89 92 95 98 101 104 107 110 113 116 119 122 125 128 131 134 137 140 The left column is one set of values. In this example it is the number of 4-legged chairs. This is shown by x = 4. The top row is another set of values. In this example it is the number of 3-legged stools. This is shown by y = 3. The total number of legs is 29. This is shown by Total = 29. The two solutions are shown in colour. We can find them by reading the table. We need 5 chairs and 3 stools, or 2 chairs and 7 stools. Use the spreadsheet to solve other Diophantine problems.
A5 SS Remainders on a spreadsheet This sheet will help you to use the spreadsheet Remainders. When opened it will look a bit like this. 9 multiples of 5 5 10 15 20 25 30 35 40 45 divisor 3 2 1 0 2 1 0 2 1 0 squares 1 4 9 16 25 36 49 64 81 divisor 3 1 1 0 1 1 0 1 1 0 triangles 1 3 6 10 15 21 28 36 45 divisor 3 1 0 0 1 0 0 1 0 0 Fibonacci 1 1 2 3 5 8 13 21 34 divisor 3 1 1 2 0 2 2 1 0 1 cubes 1 8 27 64 125 216 343 512 729 The top line shows the numbers 1 to 20. (Not all are shown above.) The second line shows the multiples of a number that you can change. (Multiples of 5 are shown.) At the left of the third line is the divisor. (The divisor is 10 in the sample above.) This is the number by which you divide to find the remainder. (Dividing by 10 leaves a remainder that is the units digit.) The rest of the third line shows the remainders. Next are the first twenty squares, and the remainders after dividing them by the divisor. There are unexpected patterns also in remainders from triangle numbers, Fibonacci numbers and cubes. These appear in the rows under the labels. 1 Try looking for patterns with a divisor of 2. This will show odd and even numbers using 1 and 0. Use different multiples. Try to work out why the patterns happen. 2 Look for patterns with a divisor of 3. Use different multiples. Try to explain the patterns. 3 Look for patterns with a divisor of 4. Use different multiples. Try to explain the patterns. 4 Look for patterns with a divisor of 5. Use different multiples. Try to explain the patterns. 5 Look for patterns with a divisor of 6. Use different multiples. Try to explain the patterns. 6 Look for patterns with a divisor of 7. Use different multiples. Try to explain the patterns. 7 Look for patterns with a divisor of 8. Use different multiples. Try to explain the patterns. 8 Look for patterns with a divisor of 9. Use different multiples. Try to explain the patterns. 9 Look for patterns with a divisor of 11. Use different multiples. Try to explain the patterns. 10 There are many other patterns to explore. The last two lines are for you to add your own special sets of numbers. (You will need to unprotect the sheet first. There is no password.)
A6a SS An any-year calendar This sheet is a set of instructions for making a calendar that will work for the last 20 and the next 30 years. An any-year calendar The materials are on the next sheet. 1 Cut around the dotted lines. There are four dotted lines on the calendar; cut these carefully. There will be one calendar, and two sliding strips. One strip has the names of months, and one has the dates. 2 Carefully slide the month strip into the narrow slits, so the month names are showing at the front. 3 Carefully slide the date strip into the wider slits, so the dates are showing at the front. Here is what you should have made. On the year numbers, some years are listed twice. These are leap years. Use the first listing up to Feb. 29, and the second listing for the rest of the year. Year numbers YEAR Months Month names S M T W T F S Dates Numbers 4 To see this month s calendar: Find the correct year. Don t forget the leap years. Slide the month strip so that the JAN is under the date for this year. Now slide the date strip so the 1 is under the name of this month. You should now see the correct calendar. (You may need to forget one or more numbers at the end of the month!) 5 To see the calendar for any month, repeat the same steps. Don t forget the leap years. 6 There is a spreadsheet (Calendar) that does all these steps for you and will show you the monthly calendar for any month in any millennium, past or present. Year February 2000 M T W T F S S Month 1 2 3 4 5 6 number 7 8 9 1 0 1 1 1 2 1 3 2 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
An any-year calendar 79 80 80 81 82 83 83 84 84 85 86 87 88 88 89 90 91 92 92 93 94 95 96 96 97 98 99 00 00 01 02 03 04 04 05 06 07 08 08 09 10 11 12 12 13 14 15 16 16 17 18 19 20 20 21 22 23 24 25 26 27 28 28 29 30 31 32 32 33 34 S M YEAR MONTHS T W T F S DATES 1 2 3 4 5 6 7 2 3 4 5 6 7 8 9 10 1112 13 14 9 10 11 12 13 14 15 16 17 18 19 20 21 16 17 18 19 20 21 22 23 24 25 26 27 28 23 24 25 26 27 28 29 30 31 30 31 A6b SS NOV DEC NOV DEC MAY AUG MAR JUN SEP JUL OCT MAY AUG MAR JUN SEP JUL FEB APR JAN FEB APR