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KS3 Mathematics

KS3 Mathematics. A3 Formulae. A3 Formulae. Contents. A. A3.2 Using formulae. A. A3.1 Introducing formulae. A3.3 Changing the subject of a formula. A. A3.4 Deriving formulae. A. Formulae. What is a formula?. In maths a formula is a rule for working something out. .

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KS3 Mathematics

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  1. KS3 Mathematics A3 Formulae

  2. A3 Formulae Contents • A A3.2 Using formulae • A A3.1 Introducing formulae A3.3 Changing the subject of a formula • A A3.4 Deriving formulae • A

  3. Formulae What is a formula? In maths a formula is a rule for working something out. The plural of formula is formulae. For example, How can we work out the number of days in a given number of weeks? number of days = 7 × number of weeks

  4. Writing formulae in words Number of table legs = number of tables____ What formula would we write to work out the number of table legs in the classroom? × 4 We can now work out the number of table legs for any given number of tables. For example, if there are 16 tables in the classroom: Number of table legs = 16 × 4 = 64

  5. Writing formulae in words Mark is 5 years older than his sister, Kate. What formula would we write to work out Mark’s age if we are given Kate’s age? Mark’s age = Kate’s age____ + 5 For example, if Kate is 8 years old: Mark’s age = 8 + 5 = 13 If Kate is 49 years old: Mark’s age = 49 + 5 = 54

  6. Writing formulae in symbols p= 4s Formulae are normally written using letters instead of words. Each letter in a formula represents a numerical amount. For example, The perimeter of a square = 4 × the length of one of the sides. s We can write this as, s s s Where p is the perimeter of the square and s is the length of a side.

  7. Writing formulae in symbols W = 7h + 200 Sally earns a basic salary of £200 a week working in an office. She also earns £7 an hour for each hour of overtime she does. What formula could we use to work out Sally’s weekly income? Weekly income, in pounds = 7 × number of extra hours + 200 We can write this in symbols as: How much does Sally earn if she works 9 hours overtime? W = 7 ×9 + 200 = £263

  8. Examples of formulae Remember, a formula is a special type of equation that links physical variables. Here are some examples of well-known formulae from maths. The area, A, of rectangle of length, l, and width, w, is given by the formula: A = lw The perimeter, P, of a rectangle is given by the formula: P = 2(l + w)

  9. Writing formulae m c = p Write a formula to work out: The cost, c, of b boxes of crisps at £3 each. c = 3b The distance left, d, of a 500 km journey after travelling k km. d = 500 –k The cost per person, c, if a meal costing m pounds is shared between p people.

  10. Writing formulae a + b + c w = 3 The number of seats in a theatre, n, with 25 seats in each row,r. n = 25r The age of a boy Andy, a, if he is 5 years older than his sister Betty, b. a = b + 5 The average weight, w, of Alex who weighs a kg, Bob who weighs b kg and Claire who weighs c kg. 1 of 20

  11. Castle entrance prices Stony Castle Entrance fee Adult £3 £2 Child T = What formula could we use to work out the total cost in pounds, T, for a number of adults, A and a number of children, C, to visit the castle? 3A + 2C

  12. Castle entrance prices T = 3A + 2C Using this formula, how much would it cost for 4 adults and a class of 32 children to visit the castle? We substitute the values into the formula: T = 3 ×4 + 2 × 32 = 12 + 64 = 76 It will cost £76.

  13. Newspaper advert C = To place an advert in a local newspaper costs £15. There is then an additional charge of £2 for each word used. Write a formula to work out the total cost in pounds, C, to place an advert containing n words. 15 + 2n How much would it cost to place an advert containing 27 words? C = 15 + 2 × 27 = 15 + 54 = 69 It will cost £69.

  14. A3 Formulae Contents • A A3.1 Introducing formulae • A A3.2 Using formulae A3.3 Changing the subject of a formula • A A3.4 Deriving formulae • A

  15. Substituting into formulae h w l The surface area S of a cuboid is given by the formula S = 2lw + 2lh + 2hw where l is the length, w is the width and h is the height. What is the surface area of a cuboid with a length of 1.5 m, a width of 32 cm and a height of 250 mm?

  16. Substituting into formulae What is the surface area of a cuboid with a length of 1.5 m, a width of 32 cm and a height of 250 mm? Before we can use the formula we must write all of the amounts using the same units. l=150 cm, w = 32 cm and h = 25 cm Next, substitute the values into the formula without the units. S = 2lw + 2lh + 2hw = (2 × 150 × 32) + (2 × 150 × 25) + (2 × 25 × 32 ) = 9600 + 7500 + 1600 = 18 700 cm2 Don’t forget to write the units in at the end.

  17. Substituting into formulae The distance D, in metres, that an object falls after being dropped is given by the formula: D = 4.9t2 where t is the time in seconds. Suppose a boy drops a rock from a 100 metre high cliff. How far will the rock have fallen after: a) 2 seconds b) 3 seconds c) 5 seconds? When t = 2, When t = 3, When t = 5, D = 4.9 × 22 D = 4.9 × 32 D = 4.9 × 52 = 4.9 × 4 = 4.9 × 9 = 4.9 × 25 = 19.6 metres = 44.1 metres = 122.5 metres

  18. Substituting into formulae and solving equations w l The formula used to find the perimeter P of a rectangle with length l and width w is P = 2l + 2w What is the length of a rectangle with a perimeter of 20 cm and a width of 4 cm?

  19. Substituting into formulae and solving equations Substitute P = 20 and w = 4 into the formula: P = 2l + 2w 20 = 2l + (2 × 4) Solve this equation Simplifying: 20 = 2l + 8 Subtracting 8: 12 = 2l Dividing by 2: 6 =l l = 6 So the length of the rectangle is 6 cm.

  20. Substituting into formulae and solving equations A = bh 1 2 h b The area A of a triangle with base b and perpendicular height h is given by the formula What is the length of the base of a triangle with an area of 48 cm2 and a perpendicular height of 12 cm?

  21. Substituting into formulae and solving equations A = bh 1 1 1 2 2 2 4 = b 48 = b × 12 Substitute A = 48 and h = 12 into the formula: Solve this equation Dividing by 12: Multiplying by 2: 8 = b b = 8 So the base of the triangle measures 8 cm.

  22. A3 Formulae Contents • A A3.1 Introducing formulae A3.2 Using formulae • A A3.3 Changing the subject of a formula • A A3.4 Deriving formulae • A

  23. Using inverse operations A = B + 5 B = A– 5 Andy is 5 years older than his brother, Brian. Their ages are linked by the formula: where A is Andy’s age in years and B is Brian’s age in years. Using this formula it is easy to find Andy’s age given Brian’s age. Suppose we want to find Brian’s age given Andy’s age. Using inverse operations, we can write this formula as:

  24. The subject of a formula Look at the formula, V = IR where V is voltage, I is current and R is resistance. V is called the subject of the formula. The subject of a formula always appears in front of the equals sign without any other numbers or operations. Sometimes it is useful to rearrange a formula so that one of the other variables is the subject of the formula. Suppose, for example, that we want to make I the subject of the formula V = IR.

  25. Changing the subject of the formula V = IR ×R ÷R V I = R V is the subject of this formula The formula: can be written as: I V The inverse of this is: I V I is now the subject of this formula or

  26. Matchstick pattern Look at this pattern made from matchsticks: Pattern Number, n 1 2 3 4 Number of Matches, m 3 5 7 9 The formula for the number of matches, m, in pattern number n is given by the formula: m = 2n + 1 Which pattern number will contain 47 matches?

  27. Changing the subject of the formula × 2 + 1 ÷ 2 – 1 m – 1 n = 2 m is the subject of this formula The formula: m = 2n + 1 can be written as: n m The inverse of this is: n m n is the subject of this formula or

  28. Changing the subject of the formula 47– 1 n = 2 46 n = 2 m– 1 n = 2 To find out which pattern will contain 47 matches, substitute 47 into the rearranged formula. n = 23 So, the 23rd pattern will contain 47 matches.

  29. Changing the subject of the formula F = + 32 9C 9C 5(F – 32) 5(F – 32) 5 5 F – 32 = 9 9 = C C = To make C the subject of the formula: Subtract 32: 5(F – 32) = 9C Multiply by 5: Divide by 9:

  30. Equivalent formulae

  31. A3 Formulae Contents • A A3.1 Introducing formulae A3.2 Using formulae • A A3.4 Deriving formulae A3.3 Changing the subject of a formula • A • A

  32. Connecting Dots

  33. Tiling patterns

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