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The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence Name: HW1 – Grade F/G Word formulas and number sequences 1. Teacher Assessment If x = 5 and y = 2, find the value of 3x – 4y. ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ Answer ............................................... (Total 2 marks) 2. (a) Here is a number machine. Input Output –3 x2 +7 .................. The input is –3. What is the output? ...................................................................................................................................... Answer …………………………………......... (1) (b) Here are two different number machines. They both have an input of 7 and an output of 11. (i) Complete the boxes to make this machine work. Input Output x .......... 7 – .......... 11 (1) (ii) Complete the boxes to make this different machine work. Input 7 Output x .......... – .......... 11 (1) (Total 2 marks) The Robert Smyth School 1 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence 3. A bus company works out its fares using the formula. Fare = Rate per Mile × No. of miles travelled. The bus company sets its rate at 20p per mile. (a) Laura travels 9 miles on the bus. What fare does she pay? ..................................................................................................................................... ..................................................................................................................................... Answer £....................................................... (2) (b) Moreen pays a fare of £3. How far does he travel? ..................................................................................................................................... ..................................................................................................................................... Answer .......................................................miles (2) (Total 4 marks) 4. The charges on a light railway are worked out by this formula. 30p per mile plus 25p (a) Abdul travels 7 miles on the railway. How much is he charged? ...................................................................................................................................... ...................................................................................................................................... Answer £ ............................................................. (2) (b) Belle is charged £3.85 How far does she travel? ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer ....................................................... miles (2) (Total 4 marks) The Robert Smyth School 2 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence 5. Harry hires a mini-bus for a day. The mileage cost is worked out using the formula Mileage cost = number of miles travelled x rate per mile The rate per mile is £2. The number of miles travelled is 80. (a) Work out the mileage cost. ..................................................................................................................................... ..................................................................................................................................... Answer £ .................................................... (2) (b) The total cost of hiring the mini-bus is given by the formula Total cost = mileage cost + fixed charge The fixed charge is £30. Work out the total cost. ..................................................................................................................................... ..................................................................................................................................... Answer £ .................................................... (2) (Total 4 marks) 6. Dario and Ewan buy clothes in a sale. BARGAIN SWEATERS ALL ONE PRICE CUT-PRICE JEANS Dario buys a sweater and a pair of jeans. These cost him £9.20. Ewan buys 3 sweaters and a pair of jeans. These cost him £15.10. What is the cost of a sweater? ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... Answer £ ..................................................................... (Total 3 marks) The Robert Smyth School 3 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence 7. Here is a sequence of equations. x + 5 = 20 x + 4 = 19 x + 3 = 18 (a) Write down the next two lines of the pattern. Answer .................................................... .................................................... (2) (b) Write down the value of x. Answer .................................................... (1) (Total 3 marks) 8. Part of a sequence of numbers is shown below. ..... (a) 37 31 25 19 13 ..... Write down the number that comes before 37 in this sequence. ..................................................................................................................................... Answer......................................................................................................................... (1) (b) Write down the number that comes after 13. ..................................................................................................................................... Answer......................................................................................................................... (1) (c) Write down the rule for continuing this sequence. ..................................................................................................................................... Answer......................................................................................................................... (1) (Total 3 marks) 9. (a) A sequence of numbers begins 40, 37, 34, 31,........ (i) What is the next number in the sequence? ........................................................................................................................... Answer .................................................... (1) (ii) Describe in words the rule for continuing the sequence. ........................................................................................................................... ........................................................................................................................... (1) The Robert Smyth School 4 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence (b) The rule for another sequence is Next number = Multiply the previous number by 3 then subtract 3 (i) A sequence begins 2, 3, 6, 15, ...... What is the next number in the sequence? ........................................................................................................................... ........................................................................................................................... Answer .................................................... (1) (ii) Another sequence, using the same rule, starts with 4. What is the next number in this sequence? ........................................................................................................................... ........................................................................................................................... Answer .................................................... (1) (iii) Another sequence, using the same rule, starts with –6. What is the next number in this sequence? ........................................................................................................................... ........................................................................................................................... Answer .................................................... (1) (Total 5 marks) 10. The rule for a sequence is Next number = Multiply the previous number by 3 then subtract 3 A sequence starts with –6. What are the next two numbers in this sequence? ............................................................................................................................................... ............................................................................................................................................... Answer .......................... and .......................... (Total 2 marks) The Robert Smyth School 5 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence 11. (a) A sequence begins 1, 2, 5, ... The rule for continuing this sequence is Multiply the last term by 3 and subtract 1 What is the next number in the sequence? ..................................................................................................................................... Answer ............................................................. (1) (b) A different sequence begins 4, 1, –2, –5, ... Write down the rule for this sequence. ..................................................................................................................................... ..................................................................................................................................... (1) (Total 2 marks) 12. Here is a sequence of numbers. 128 (a) 64 32 x 8 4 y 1 Write down the values of x and y. Answer x =................................ , y = ................................ (2) (b) Write down the rule for continuing the sequence. ..................................................................................................................................... ..................................................................................................................................... (1) (Total 3 marks) Success: The Robert Smyth School Target: 6 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence HW2 – Grade D/E Expressions, patterns and formulae 1. (a) Teacher Assessment Matt buys 12 boxes of orange juice at 43 pence each. (i) Calculate the total cost. ............................................................................................................................ ............................................................................................................................ Answer £ ........................................................... (2) (ii) He pays with a £10 note. How much change will he receive? ............................................................................................................................ ............................................................................................................................ Answer £ ........................................................... (1) (b) Aisha buys c apples at 16 pence each. (i) Write down an expression for the total cost in terms of c. Answer ........................................................... pence (1) (ii) She now buys d pears at 20 pence each. Write down an expression for the total cost of the apples and pears. ............................................................................................................................ ............................................................................................................................ Answer .......................................................... pence (2) (Total 6 marks) 2. (a) At a cafe a cup of coffee costs 75p. Write down an expression for the cost, in pence, of x cups of coffee. ..................................................................................................................................... Answer ..............................................................pence (1) (b) (i) The cafe sells twice as many cups of tea as it does cups of coffee. Write down an expression for the number of cups of tea sold when x cups of coffee are sold. ........................................................................................................................... Answer ....................................................................... (1) The Robert Smyth School 7 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence (ii) Each cup of tea costs 50p. Write down an expression for the cost, in pence, of the cups of tea sold. ........................................................................................................................... Answer ..............................................................pence (1) (c) The cafe also sells cakes and buns. (i) Write down an expression for the cost, in pence, of y cakes at 40p each and 3 buns at 60p each. ........................................................................................................................... Answer ..............................................................pence (2) (ii) The total cost of the y cakes and 3 buns is £4.60 Find the number of cakes sold. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... Answer ....................................................................... (3) (Total 8 marks) 3. Grace is x years old. Harry is 3 years older than Grace. (a) Write down an expression for Harry’s age. Answer ……………………………………… (1) (b) Jack is twice as old as Harry. Write down an expression for Jack’s age. Answer ……………………………………… (1) (Total 2 marks) The Robert Smyth School 8 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence 4. Sticks are used to make a sequence of patterns. The first three patterns are shown. Pattern 1 (a) Pattern 2 Pattern 3 Pattern 4 Draw pattern 4 in the space provided. (1) (b) Complete the table showing the number of matches in each pattern. Pattern number 1 2 3 Number of sticks 8 11 14 4 5 (1) (c) What pattern do you notice in the number of sticks? ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (1) (Total 3 marks) 5. Sam makes a sequence of patterns with sticks. Here are his first three patterns. Pattern 1 (a) Pattern 2 Pattern 3 Draw Pattern 4. (1) The Robert Smyth School 9 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence (b) Complete the table. Pattern number 1 2 Number of sticks 3 5 3 4 (2) (c) How many sticks will Sam use for Pattern 5? ..................................................................................................................................... Answer ............................................... (1) (d) There are 33 sticks in Pattern 16. How many sticks are there in Pattern 17? ..................................................................................................................................... Answer ............................................... (1) (Total 5 marks) 6. Patterns are made of sticks. Pattern 1 (a) Pattern 2 Pattern 3 Pattern 4 Complete the table for Pattern 4. Pattern number 1 2 3 Number of sticks 5 7 9 4 (1) (b) Sketch Pattern 5. (1) The Robert Smyth School 10 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence (c) Which pattern will have 25 sticks? ..................................................................................................................................... ..................................................................................................................................... Answer .................................................................. (2) (d) Here is the rule for working out the number of sticks. Multiply pattern number by 2 and add 3 How many sticks will be in Pattern 100? ..................................................................................................................................... Answer .................................................................. (1) (Total 5 marks) 7. Patterns are made from shaded and unshaded squares. Pattern 1 Pattern 2 Pattern 3 Pattern 4 (a) Draw Pattern 6. (b) How many shaded squares will there be in Pattern 20? Pattern 5 (1) ..................................................................................................................................... Answer ......................................................................... (1) (c) How many unshaded squares will there be in Pattern 20? ..................................................................................................................................... ..................................................................................................................................... Answer ......................................................................... (1) (d) Which pattern will have 30 squares altogether? ..................................................................................................................................... ..................................................................................................................................... Answer ......................................................................... (1) (Total 4 marks) The Robert Smyth School 11 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence 8. (a) Find the value of 3x + 4y (i) when x = 2 and y = 5 ........................................................................................................................... ........................................................................................................................... Answer ........................................................................ (2) (ii) when x = 6 and y = –3 ........................................................................................................................... ........................................................................................................................... Answer ........................................................................ (2) (b) Find the value of a3 + b2 when a = 2 and b = 5 ..................................................................................................................................... ..................................................................................................................................... Answer ........................................................................ (2) (Total 6 marks) 9. (a) Use the formula a = 5b + 2c to work out a when b = 3 and c = –4 ..................................................................................................................................... ..................................................................................................................................... Answer ......................................................................... (2) (b) Use the formula a = 5b + 2c to work out c when a = 16 and b = 2 ..................................................................................................................................... ..................................................................................................................................... Answer ......................................................................... (3) (Total 5 marks) 10. Use the formula v = u + 10t to work out u when v = –4 and t = 7 ............................................................................................................................................... ............................................................................................................................................... Answer u =………………………………………………… (Total 3 marks) Success: The Robert Smyth School Target: 12 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence HW3 – Grade C Finding the nth term 1. (a) The first term of a sequence is –2. The rule for continuing the sequence is Teacher Assessment Add 7 then Multiply by 4 What is the second term of the sequence? ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer …………………………………………… (1) (b) This rule is used to continue a different sequence. Multiply by 2 then Add 5 The second term of this sequence is 3. What is the first term? ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer …………………………………………… (3) (Total 4 marks) The Robert Smyth School 13 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence 2. A pattern using pentagons is made of sticks. (a) Diagram 1 Diagram 2 Diagram 3 5 sticks 9 sticks 13 sticks How many sticks are needed for Diagram 5? ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. Answer ............................................ (2) (b) Write down an expression for the number of sticks in Diagram n. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. Answer ............................................ (2) 3. Patterns are made from shaded and unshaded squares. 1st pattern (a) 2nd pattern 3rd pattern 4th pattern How many shaded squares are there in the nth pattern? …………..........…............……...................…………….....…………...................... Answer ………………………………….. (1) (b) How many unshaded squares are there in the nth pattern? …………..........…............……...................…………….....…………...................... …………..........…............……...................…………….....…………...................... Answer ………………………………….. (1) (Total 2 marks) The Robert Smyth School 14 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence 4. The diagrams show a sequence of rectangles. Area = 2 Diagram 1 (a) Area = 6 Diagram 2 Area = 12 Diagram 3 Area = 20 Diagram 4 Write down an expression for the area of Diagram n. ..................................................................................................................................... ..................................................................................................................................... Answer ...................................................................... (2) (b) A rectangle in the sequence has an area of 110 cm2. What is its perimeter? ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer ................................................................ cm (2) (c) Which diagram has a perimeter of 26cm? ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. Answer ............................................ (3) (Total 7 marks) 5. (a) The nth term of a sequence is 4n + 1 (i) Write down the first three terms of the sequence. .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... Answer ......................., ......................, ....................... (2) The Robert Smyth School 15 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence (ii) Is 122 a term in this sequence? Explain your answer. .......................................................................................................................... .......................................................................................................................... (1) (b) Tom builds fencing from pieces of wood as shown below. Diagram 1 4 pieces of wood Diagram 2 7 pieces of wood Diagram 3 10 pieces of wood How many of pieces of wood will be in Diagram n? ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer ....................................................... (2) (Total 5 marks) 6. (a) Write down the first three terms of the sequence whose nth term is given by 5n 4n 7 ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (2) (b) Which term of the sequence has a value of 1? ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (2) (Total 4 marks) Success: The Robert Smyth School Target: 16 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence HW4 – Grade C Finding the nth term & rearranging formulae 1. The nth term of a sequence is given by the expression n2– 3 Write down the first three terms of the sequence. Teacher Assessment ............................................................................................................................................... ............................................................................................................................................... Answer .................... , ..................... , ...................... (Total 2 marks) 2. (a) Write down the first three terms of the sequence whose nth term is given by 5n 3. ..................................................................................................................................... ..................................................................................................................................... Answer........................................................................................................................ (2) (b) The letters p and q represent integers. p is an odd number and q is an even number. (i) Which of these statements describes the number 2p + q? always even always odd could be odd or even Explain your answer. ........................................................................................................................... ........................................................................................................................... (2) (ii) 1 q is always even. 2 Give an example to show that Anna is wrong. Anna says that p + ........................................................................................................................... ........................................................................................................................... (1) (Total 5 marks) The Robert Smyth School 17 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence 3. (a) p is an odd number. Is 2p + 1 an odd number, an even number or could it be either? Tick the correct box. odd even either (1) (b) The nth term of a sequence is 4n + 1 (i) Write down the first three terms of the sequence. ........................................................................................................................... ........................................................................................................................... Answer .......................................................... (2) (ii) Is 122 a term in this sequence? You must explain your answer. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... (1) (Total 4 marks) 4. A sequence of numbers is shown. 2 (a) 5 8 11 14 Find an expression for the nth term of the sequence. ...................................................................................................................................... ...................................................................................................................................... Answer ......................................................................... (2) (b) Explain why 99 will not be a term in this sequence. ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... (2) (Total 4 marks) The Robert Smyth School 18 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence 5. Here is a number sequence. 1 (a) 3 6 10 15 Write down the next two numbers in the sequence. ...................................................................................................................................... Answer …………………………………......... (2) (b) Describe a rule for continuing the sequence. ...................................................................................................................................... ...................................................................................................................................... (1) 6. Part of a number grid is shown below. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 The shaded shape is called L3 because it has the number 3 at the top. The sum of the numbers in L3 is 26. (a) Calculate the sum of the numbers in L22 ..................................................................................................................................... Answer …………………………………......... (1) (b) Fill in the empty squares of Ln n (2) (c) Write down an expression, in terms of n, for the sum of the numbers in Ln Simplify your expression. ..................................................................................................................................... ..................................................................................................................................... Answer …………………………………......... (2) The Robert Smyth School 19 The Robert Smyth School Mathematics Faculty Topic 5 - Expressions, formulae and sequences Innovation & excellence (d) If the sum of the numbers in Ln is 143, find the value of n. ..................................................................................................................................... ..................................................................................................................................... Answer …………………………………......... (2) (Total 7 marks) 7. Rearrange y = mx + c to make x the subject of the formula. ..................................................................................................................................... ..................................................................................................................................... Answer x = .................................................... (Total 4 marks) 8. Make p the subject of the formula t = 5p + 40 ....................................................................................................................................... ....................................................................................................................................... Answer p = ................................................................. (Total 2 marks) 9. Make r the subject of the formula p = 3 + 2r ……………...........………………………….....……………………………………. ……………...........………………………….....……………………………………. ……………...........………………………….....……………………………………. Answer r = …………………………………. (Total 2 marks) 10. Make c the subject of the formula d= c +e 5 ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... Answer ....................................... (Total 2 marks) Success: The Robert Smyth School Target: 20
