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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.

NUMBERWALLS 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.?

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.

A B Calculate the maximum and minimum “Top Brick” values for the number walls A and B shown NUMBERWALLS

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 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 ! 2! 3! 4! 5! 6! NUMBERWALLS

4 high 24 5 high ? 6 high ? 3 high 2 high 1 high 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? ! 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

NUMBERWALLS 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?