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3. Teacher Notes The discussion of the 24 number walls shown on slide 4 is left to the teacher (although the reason for the high and low values is highlighted). The six solutions 37, 43, 45, 47, 49 and 55 occur four times each so a discussion on symmetry would be useful. Also someone should notice that all of the “Top Brick” numbers are odd and this needs to be discussed. The presentation goes on to look at arrangements of higher number walls and factorial notation is introduced. We conclude with a puzzle. There are printable 4 walls and 5 walls at slides 9 and 10. abcde a + bb + cc + dd + e a + 2b + cb + 2c + dc + 2d + e a + 3b + 3c + db + 3c + 3d + e a + 4b + 6c + 4d +e This work could be extended with an appropriate group to look at the general situation and its relationship with Pascal’s Triangle.

4. NUMBERWALLS 4937 131210 2522 47 9473 131110 2421 45 The two number walls shown give different totals for the top brick using the same numbers at the base. Investigate the different totals that the top brick can have using these base numbers. 1.How many different totals are there? 2.Which arrangement gives the highest/lowest total.?

5. 3479 71116 1827 45 3497 71316 2029 49 3749 101113 2124 45 3794 101613 2629 55 3947 121311 2524 49 3974 121611 2827 55 4379 71016 1726 43 4397 71216 1928 47 4739 111012 2122 43 4793 111612 2728 55 4937 131210 2522 47 4973 131610 2926 55 7349 10713 1720 37 7394 101213 2225 47 7439 11712 1819 37 7493 111312 2425 49 7934 16127 2819 47 7943 16137 2920 49 9347 12711 1918 37 9374 121011 2221 43 9437 13710 2017 37 9473 131110 2421 45 9734 16107 2617 43 9743 16117 2718 45 There are 24 possible base arrangements for the number wall as shown below in this systematic construction. Discuss some of the properties of these walls.

6. 48191234 A B Calculate the maximum and minimum “Top Brick” values for the number walls A and B shown. 4891 121710 2927 56 8149 9513 1418 3224 1342 476 1113 3124 436 79 16 NUMBERWALLS

7. We saw that a “4 High” wall gave a total of 24 possible different arrangements of the base numbers. Find the number of arrangements for a 1, 2 and 3 high wall. 4 high 24 5 high ? 6 high ? 3 high 2 high 1 high 1 12 3 123 35 8 21 3 132 45 9 213 34 7 231 54 9 312 43 7 321 53 8 1 2 6 Looking at the sequence formed, can you make a conjecture as to how many arrangements there will be in the 5 and 6 high wall? 1 1 x 2 1 x 2 x 3 1 x 2 x 3 x 4 1 x 2 x 3 x 4 x 5 1 x 2 x 3 x 4 x 5 x 6 120 720 1! 2! 3! 4! 5! 6! NUMBERWALLS

8. 4 high 24 5 high ? 6 high ? 3 high 2 high 1 high 1 12 3 123 35 8 21 3 132 45 9 213 34 7 231 54 9 312 43 7 321 53 8 1 2 6 Looking at the sequence formed, can you make a conjecture as to how many arrangements there will be in the 5 and 6 high wall? 120 720 1! 2! 3! 4! 5! 6! You might want to use the factorial key on your calculator for larger walls. What is the highest wall that you calculator can work out? 69 NUMBERWALLS 1 1 x 2 1 x 2 x 3 1 x 2 x 3 x 4 1 x 2 x 3 x 4 x 5 1 x 2 x 3 x 4 x 5 x 6

9. NUMBERWALLS 3579 81216 2028 48 12345 A puzzle to finish The value of the top brick of the 5 high wall shown is 48. There is one number between 50 and 59 that cannot form the top brick. Can you find it?

10. 3479 71116 1827 45 3497 71316 2029 49 3749 101113 2124 45 3794 101613 2629 55 3947 121311 2524 49 3974 121611 2827 55 4379 71016 1726 43 4397 71216 1928 47 4739 111012 2122 43 4793 111612 2728 55 4937 131210 2522 47 4973 131610 2926 55 7349 10713 1720 37 7394 101213 2225 47 7439 11712 1819 37 7493 111312 2425 49 7934 16127 2819 47 7943 16137 2920 49 9347 12711 1918 37 9374 121011 2221 43 9437 13710 2017 37 9473 131110 2421 45 9734 16107 2617 43 9743 16117 2718 45