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Teacher Notes This “Stairs and Steps “ investigation is concerned with: 1.Deriving a formula for the sum of the first n odd numbers. 2.Introducing triangular numbers and realising how they are generated from a pattern of dots/cubes. 3.Deriving a formula for the n th triangular number and realising that this is equivalent to adding successive natural numbers. You may need cubes or the printable spotty paper at slide 10 depending on the group. It is envisaged that slides 8 and 9 will be used as extension material with a top group. There is some much simpler work on triangular numbers in the presentation “Lines and Dots”

A 3 stair goes up and down 3 and is made of 9 cubes. A 4 stair goes up and down 4 and is made of 16 cubes. Use cubes or spotty paper to make some different size stairs and record your results in a table like so. Look for patterns and find a formula linking the number of cubes to the height of the stairs. Height up1234567 N o of cubes

Height up1234567 N o of cubes14916253649 How many cubes will there be in an up and down 10 stair? 10 2 = 100 How many cubes will there be in an up and down n stair? n2n2 5 3 1 5 3 1 7 Comment on the number of cubes needed to form each layer of the stairs. They are consecutive odd numbers. The first three odd numbers add to 9 What do the first four odd numbers add up to? 16 What do the first 10 odd numbers add up to? 10 2 = 100 What do the first 20 odd numbers add up to? 20 2 = 400 What do the first n odd numbers add up to? n2n2

A 2 step stair goes up 2 and is made of 3 cubes. A 3 step stair goes up 3 and is made of 6 cubes. Use cubes or spotty paper to make some different size stair steps and record your results in a table like so. Look for patterns and try to find a rule linking the number of cubes to the height of the steps. Height up1234567 N o of cubes

Height up1234567 N o of cubes13610152128 How many cubes will there be in a 10 step stair ? 28 + 8 + 9 + 10 = 55 Comment on the number of cubes needed to form each layer of the stairs. 1 2 1 2 3 1 2 3 4 They are made up of the counting or natural numbers. The first 4 numbers add to 10 What do the first 7 numbers add up to? 28 What do the first 10 numbers add up to? 55

Number1234567 Triangular Number13610152128 These numbers are called triangular numbers and are often seen as a pattern of dots rather than cubes, not that it makes any difference. Comment on the number of cubes needed to form each layer of the stairs. 1 2 1 2 3 1 2 3 4 They are made up of the counting or natural numbers. The capital letter “T” along with a subscript is often used to denote them. T 3 = 1 + 2 + 3 = 6 T 6 = 1 + 2 + 3 + 4 + 5 + 6 = 21 We can use shorthand notation for larger numbers: For example T 50 = 1 + 2 + 3 + … + 48 + 49 + 50 where … means the missing numbers.

We could write the n th triangular number as T n = 1 + 2 + 3 + … + n - 2 + n - 1 + n T n = 1 + 2 + 3 + + + n – 2 + n – 1 + n T n = n + n - 1 + n - 2 + + + 3 + 2 + 1 2T n = n + 1 + n + 1 + n + 1 + + + n + 1 + n + 1 + n + 1 There is a very nice way to derive the formula for the n th triangular number, by addition. Here is clue. Can you work it out? 2T n = n(n + 1) T n = n(n + 1)/2 So using the formula to check the 7 th triangular number: T 7 = (7 x 8)/2 = 28 Evaluate (a) T 10 (b) T 18 (c) T 40 T 10 = (10 x 11)/2 = 55 T 18 = (18 x 19)/2 = 171 T 40 = (40 x 41)/2 = 820 Use the formula to mentally add up the first 100 numbers. T 100 = (100x 101)/2 = 5050

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