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MEP: Demonstration Project Y9B, Unit 11 UNIT 11 Algebraic Manipulation Activities 11.1 Sums in Squares 11.2 Areas of Rectangles 11.3 Horseshoes 11.4 Flower Beds Notes and Solutions (2 pages) © The Gatsby Charitable Foundation Activities MEP: Demonstration Project Y9B, Unit 11 ACTIVITY 11.1 1. Draw a 2 × 2 square on the number grid like the one shown. Add up the 4 numbers in the square and record your results in a table as shown below. Repeat for several other squares. Smallest Number in Square (x) 34 Sums in Squares 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 Total of the Four Numbers (T) Largest Number in Square (y) 158 45 9 10 19 20 29 30 39 40 49 50 59 60 69 70 79 80 89 90 99 100 2. Show that the formula T = 4 x + 22 is true for each of your starting numbers, x. 3. Explain why the formula will always work. 4. Determine a formula for T in terms of y. Extension Repeat for a 3 × 3 square and for a 4 × 4 square. © The Gatsby Charitable Foundation MEP: Demonstration Project Y9B, Unit 11 ACTIVITY 11.2 Areas of Rectangles Calculate the area of each of the rectangles by calculating the area of each part: A. B. 2 cm 7 cm 5 cm x cm 1 cm 3 cm 4 cm C. 4 cm D. x 3x x 2 y 3 cm E. F. 6 2x 6x 2 3x x y G. 2x 8 3 x © The Gatsby Charitable Foundation 5 MEP: Demonstration Project Y9B, Unit 11 ACTIVITY 11.3 Horseshoes Farriers can buy ready-made horseshoes, but many make their own so that the fit is better. The horseshoe is made up from a straight strip of iron which is forged into the familiar horseshoe shape. The problem that the farrier has to solve is to determine the length of iron (l) needed to make the appropriate sized shoe. Two formulae are commonly used: A l = 2w + 2 B l = w + d + 1.5 where w is the width of the shoe, in inches, and d is the diagonal measured in inches from the toe to the heel, as shown. Data for Horseshoes 3. Horse Width (w) Diagonal (d) Crystal 5.15 5.50 Honey 5.25 5.75 Frosty 5.50 5.80 William 5.75 6.00 Smudger 6.00 6.40 1. Determine the length of iron required to shoe each horse using (a) formula A, (b) formula B. Comparing the lengths required by each formula, what do you notice? 2. What is the condition on d and w which ensures that the two formula are the same? The cost of the shoe is directly related to the length of iron, l, used. If Crystal has a new set of horseshoes 8 times a year, what saving is made in iron using formula B rather than formula A? What is the percentage saving? Ready-made shoes often use 8 nail holes symmetrically placed so that they can be used on either foot, but a handmade shoe is made specifically for the left or right foot, with 4 nails on the outside half and 3 on the inside half. 4. The nails for Honey's front left shoe are shown opposite, with distances between them as illustrated. Determine the appropriate values to take for x and y. © The Gatsby Charitable Foundation MEP: Demonstration Project Y9B, Unit 11 ACTIVITY 11.4 Flower Beds KEY Flower bed Paving slab The council wish to create a long display made up of 100 flower beds each surrounded with hexagonal paving slabs, according to the pattern shown above. (In this pattern 18 slabs surround 4 flower beds.) 1. How many slabs will be needed? 2. Determine a formula that the council can use to decide the number of slabs needed for any number of flower beds. Extensions There are many other ways of surrounding flower beds with hexagonal paving slabs. 1. Find formulae for the number of slabs needed for configurations of each of the layouts below, both of which have n hexagonal flower beds. (a) 2. (b) Invent your own examples and find general formulae for the number of slabs needed for any number of them. © The Gatsby Charitable Foundation