6 Monomi e polinomi ESERCIZI PRACTICE WITH CLIL Mathemagics! 1. Write number 9 on a piece of paper (without showing your friends); then, fold the piece of paper and put it in an envelope. 2. On a table, open a tombola calling board. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 Then, call a friend and give him/her the following instructions. a. Lay a toothpick on the calling board, in a chosen position, so that it brings together three numbers in a row (for example: 56, 57 and 58). b. Sum the digits of the three numbers (in our example: 5 + 6 + 5 + 7 + 5 + 8 = 36). c. Now, place the toothpick in another part of the calling board, so to touch three new numbers (for example: 83, 74 and 65). d. Again, sum the digits of the three numbers chosen (in this example: 8 + 3 + 7 + 4 + 6 + 5 = 33). e. Multiply the two numbers resulting from the additions (so 36 33 = 1188). f. Sum the digits of the number thus obtained and repeat the operation until number is reduced to a single digit (1 + 1 + 8 + 8 = 18 1 + 8 = 9). 3. Open the envelope and show your friend that your predicted number matches exactly number 9. Let us reveal the trick The success in this game depends on a characteristic of arithmetic progressions that is a sequence of numbers in which the consecutive terms are formed by adding a constant quantity with the preceding term. The constant quantity is called common difference (R). For example: 1, 2, 3, 4, ... in fact 1, 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, common difference R = 1 21, 32, 43, 54, ... in fact 21, 21 + 11 = 32, 32 + 11 = 43, 43 + 11 = 54, common difference R = 11 Indeed, the property to look at is the following: «The sum of three consecutive terms of a generic arithmetic progression is always equal to 3 times the middle number . We can easily check the property in the following given examples: 32 + 43 + 54 = 129 and 3 43 = 129 Thus, as the first three terms are: n, n + R, n + 2R If we apply the property of operation and of monomials, their sum will be: n + (n + R) + (n + 2R) = 3n + 3R = 3 (n + R) By observing the tombola calling board, you will quite easily realize that, any set of three close terms, in straight line and in any direction, is made of the consecutive terms of an arithmetic progression. Consequently, the sum of the sets of three terms chosen with a toothpick is a multiple of 3. We have drawn 2 sets; therefore, by multiplying the two, we will have a multiple of 9. In fact, if the second set has the first term m and the common difference Q, you will get: 3 (n + R) 3 (m + Q) = 9 (n + R) (m + Q) This number will be a multiple of 9 and, thus, by applying the well-known divisibility rule, the repeated sum of its digits will certainly give 9 as result. 363