NNC Year 6 Algebra. 61 minutes. 59 marks. Page 1 of 32

Transcription

1 NNC Year 6 Algebra 6 minutes 59 marks Page of 32

2 Q. Here is a sequence of towers built from cubes. These are the plans of each tower. The numbers show how many cubes are in each vertical column. How many cubes are required to build Tower 5? Page 2 of 32

3 What is the number of the Tower in this sequence which uses 65 cubes? Q2. Ann makes a pattern of L shapes with sticks. Ann says : I find the number of sticks for a shape by first multiplying the shape-number by 4, then adding 3. Work out the number of sticks for the shape that has shape-number 0. mark Page 3 of 32

4 Ann uses 59 sticks to make another L shape in this pattern. What is its shape-number? Here is Ann s rule again: I find the number of sticks for a shape by first multiplying the shape-number by 4, then adding 3. Write a formula to work out the number of sticks for any L shape. Use S for the number of sticks and N for the shape-number. S = Page 4 of 32

5 Q3. Find the value of t in this equation. 4 + t = 9t Q4. Mrs Jones prints books. Page 5 of 32

6 Jon pays 4.35 for his book, including the cover. How many pages are in his book? Here are the print prices again. 3p per page and 75p for the cover. Write a formula for the total cost of printing a book with cover. t stands for the total cost in pence. Use n for the number of pages. t = Q5. Fill in the empty boxes to complete the pattern. n + 6 7n + 6 4n + 3 7n + 3 n 4n Page 6 of 32

7 Q6. In this pattern white hexagons surround shaded hexagons. How many white hexagons are needed to surround a line of 00 shaded hexagons? W represents the number of white hexagons. S represents the number of shaded hexagons. What is the rule connecting W and S? mark Page 7 of 32

8 Q7. Find the value of y in the equation. 8 + y = 7 + 5y Q8. Here is a sequence of patterns made from octagons and squares. The sequence continues. Page 8 of 32

9 How many squares will there be in the pattern that has 40 octagons? q represents the number of squares. n represents the number of octagons. What is the rule connecting q and n? mark Page 9 of 32

10 Q9. A, B and C stand for three different numbers. The mean of A and B is 40 The mean of B and C is 35 A + B + C = 00 Calculate the values of A, B and C. Q0. Here is a sequence of shapes made from squares and circles. The sequence continues in the same way. Page 0 of 32

11 The formula for the number of circles (c) n shape number (n) is c = 3n Use the formula to work out the shape number which has 04 circles. Write the formula for the number of squares (s) in shape number (n). S =... mark Page of 32

12 Q. k, m and n each stand for a whole number. They add together to make 500 k + m + n = 500 m is three times as big as n. k is twice as big as n. Calculate the numbers k, m and n. Page 2 of 32

13 Q2. Liam thinks of a number. He multiplies the number by 5 and then subtracts 60 from the result. His answer equals the number he started with. What was the number Liam started with? Q3. k stands for a whole number. k + 7 is greater than 00 k 7 is less than 90 Find all the numbers that k could be.... Page 3 of 32

14 Q4. Kate has some rectangles. They each measure 6 centimetres by 50 centimetres. Not actual size She makes this design with four of the rectangles. Work out the lengths x and y. x = cm mark y = cm mark Page 4 of 32

15 Q5. k stands for a number. Complete the number sentences below. One has been done for you. 5 more than k is...k less than k is... 3 more than twice k is... 6 more than half of k is... Q6. Here are three equations. a + b + c = 30 a + b = 24 b + c = 4 What are the values of a, b and c? a = b = c = Page 5 of 32

16 Q7. Look at these equations. a = 2b b = 3c Which equation below is also true? Put a ring round the correct one. b = 2a a = 2b + 3c a = 5c a = 6c a + b = 5 mark Q8. The box below shows all the possible values for x. x is a whole number. 40 < x < 45 x could be 4, 42, 43 or 44 Write all the possible values for k. k is a whole number. 29 < 2k < 35 k could be Page 6 of 32

17 Write all the possible values for w. w is a whole number. 8 < 3w + < 24 w could be 3 marks Q9. The cost to hire a boat on a lake is worked out using the information below. (a) Four friends hire a boat for five hours. They share the cost equally. How much does each person pay? Page 7 of 32

18 Chen s family hires a boat and pays a total of 5 How many hours did they have the boat for? hours mark Q20. Here are Alfie and Emma with their parents. You can use the table below to predict how tall children will be when they are adults. There is one formula for boys and a different one for girls: Boy s predicted height Girl s predicted height 0.4(x + y) (x + y) + 29 x is the father s height in cm. y is the mother s height in cm. (a) Calculate the predicted height of Alfie when he is an adult. cm mark (b) When Emma is an adult, she is predicted to be taller than her mother. How much taller? cm mark Page 8 of 32

19 Q2. There are n counters in Alfie s bag. Alfie puts 3 more counters in the bag. Write an expression for the number of counters that are in the bag now. mark (b) Megan has two boxes. There are m counters in each box. She puts all her counters together in a pile, then removes 5 of them. Write an expression for the number of counters that are in the pile now. mark Q22. Here are an equilateral triangle and a regular pentagon. Not actual size Each side of the triangle is 0 cm Each side of the pentagon is d cm Page 9 of 32

20 The perimeter of the pentagon is 4 centimetres more than the perimeter of the triangle. What number does d represent? Q23. Write the missing numbers so that 2a + 5b = 30. One is done for you. 2a + 5b = 30 when a = 0 and b = 6 2a + 5b = 30 when a = 5 and b = 2a + 5b = 30 when a = 5 and b = mark mark Q24. n stands for a whole number. 2n is greater than 30 5n is less than 00 Write all the numbers that n stands for.... Page 20 of 32

21 M. (a) Award TWO marks for the correct answer of 35, even if the working is incorrect. If answer is incorrect, award ONE mark for evidence of formulation of a number sequence AND attempt at analysis, eg: etc. (5 ) + (4 2) +(3 3) + etc., 3, 6, 0, 5, etc. OR attempt at solution by drawing. Up to 2 (b) Award TWO marks for the correct answer of 9, even if the working is incorrect. If answer is incorrect, award ONE mark for evidence of formulation of a number sequence AND attempt at analysis, eg: OR OR attempt at solution by drawing. Up to 2 [4] M2. (a) Award TWO marks for the correct answer of 43, even if there are errors in the working. If the answer is incorrect, award ONE mark for evidence of an appropriate calculation of multiplication by 4 and addition of 3, eg: 3 + (4 0) OR by drawing OR other methods. Up to 2 (b) 4 Page 2 of 32

22 (c) Award TWO marks for expressions such as: S = 4N + 3 S = 3 + 4N S = N + N + N + N + 3 If the answer is incorrect, award ONE mark for evidence of multiplying N by 4 in the expression, eg: 4N 4 N N.4 N + N + N + N OR award ONE mark for evidence of adding 3 in the expression, eg: N + 3 Do not accept S = = N Up to 2 [5] M3. Award TWO marks for the correct answer of /2 OR 0.5 If answer is incorrect, award ONE mark for evidence of appropriate method which results in: 8t = 4, or equivalent. Up to 2 [2] M4. (a) Award TWO marks for correct answer of 20 OR 95 (if book is assumed to have two covers) If the answer is incorrect, award ONE mark for evidence of appropriate strategy, eg: = = Up to 2 Page 22 of 32

23 (b) Award TWO marks for correct algebraic expression equivalent to t = 3n + 75, OR t = 3n + 50, eg: t = 3 n + 75 t = 75 + n3 If expression is incorrect award ONE mark for evidence of 3 n, eg: 3n OR evidence of addition of 75 (or 50) to an expression involving n, eg: n + 75 No mark is awarded for the expression in words. Accept inclusion of p in expression, eg: 3p n + 75p Accept use of N as well as n. Answer to 20b must be consistent with answer to 20a, ie if 2 covers are assumed in 20a, they should be assumed in 20b. Up to 2 [4] M5. Award TWO marks for all three boxes correct as shown below. 4n + 6 n + 3 7n If only two boxes are correct, award ONE mark. No marks are awarded for only one correct box. Accept n for n. Accept 7n + 0 for 7n and similar alternatives. Accept 3 + n for n + 3 and similar alternatives. Up to 2 [2] M6. (a) Award TWO marks for the correct answer of 204. If the answer is incorrect, award ONE mark for evidence of an appropriate method, eg up to 2 Page 23 of 32

24 (b) Appropriate descriptions of the rule connecting W and S, eg W = 2S + 4 ; Black hexagons times = white hexagons ; W take away 4 is twice S ; S 2 (+4) = W. Descriptions may be in words or symbols or a mixture of the two. Do not accept vague or arbitrary descriptions, eg The number of shaded hexagons goes up so do the white ones, W increases by 2 each time, S increases by each time. [3] M7. Award TWO marks for the correct answer of 0.25 OR ¼ If the answer is incorrect, award ONE mark for evidence of an appropriate method, eg algebraic manipulation to reach 4y = OR 4y = OR trial and improvement showing at least two convergent attempts OR trials with two values that differ by less than and which straddle the correct value. Accept equivalent fractions. Up to 2 [2] M8. (a) Award TWO marks for the correct answer of 2 If the answer is incorrect, award ONE mark for evidence of an appropriate method, eg Calculation need not be completed for the award of the mark. Up to 2 (b) q = 3n + Accept equivalent expressions, eg q = n 3 + q = n + n + n + q = + n3 Accept the answer in words, eg q is 3 times n then add. [3] Page 24 of 32

25 M9. Award TWO marks for the correct answer as shown: A = B = C = All three numbers must be correct for the award of the mark. If the answer is incorrect, award ONE mark for evidence of an appropriate method, eg A + B = 80 B + C = 70 A + 2B + C = B = 50 Accept for ONE mark the correct three numbers but written in the incorrect boxes. Up to 2 [2] M0. Award TWO marks for the correct answer of 35 If the answer is incorrect, award ONE mark for evidence of an appropriate method, eg 04 = 3n 05 = 3n n = 05 3 Answer need not be obtained for award of the mark. Up to 2 (b) 2n + Accept equivalent expressions eg n + n + Accept the answer written in words, eg twice the shape number add one. [3] M. Award TWO marks for all three answers correct, as shown: k = m = n = If the answer is incorrect, award ONE mark for evidence of appropriate method, eg 2n + 3n + n Page 25 of 32

26 OR a trial and improvement method, eg = = = 200 Accept for ONE mark any permutation of the correct answers, eg k = 750, m = 250, n = 500 Answer need not be obtained for the award of ONE mark. A trial and improvement method must show evidence of improvement. Up to 2 (U) [2] M2. Award TWO marks for the correct answer of 5 If the answer is incorrect, award ONE mark for evidence of appropriate working, eg 60 4 = wrong answer Calculation must be performed for the award of ONE mark. OR a trial and improvement method. eg = = = 40 OR 5x 60 = x x = wrong answer A trial and improvement method must show evidence of improvement, but a final answer need not be reached for the award of ONE mark. Up to 2 (U) [2] Page 26 of 32

27 M3. Award TWO marks for all three numbers, as shown: 94, 95, 96 Accept numbers written in any order. All three numbers and no incorrect numbers must be given for the award of TWO marks. If the answer is incorrect, award ONE mark for: two numbers correct and none incorrect OR three numbers correct and one incorrect OR 93, 94, 95, 96, 97 Up to 2 (U) [2] M4. (a) 34 (b) 82 [2] M5. Award TWO marks for all three expressions correct, eg k 2 2k k Accept equivalent or unconventional notation, eg k + k + 3 OR k + 6 OR 6 + k 2 If the answer is incorrect, award ONE mark for two expressions correct. Up to 2 [2] Page 27 of 32

28 M6. Gives all three correct values, ie a = 6, b = 8, c = 6 2 Gives at least one correct value or Gives three values that satisfy the second and third equations eg a = 8, b = 6, c = 8 (satisfies a + b = 24 and b + c = 4 : note that a c = 0) [2] M7. Equation circled as shown: b = 2a a = 2b + 3c a = 5c a + b = 5 Accept unambiguous indication [] M8. Gives all three possible values for k, in any order, eg 5, 6, 7 Gives both possible values for w, in either order, eg 6, 7 As evidence of a correct method: Gives a completely correct response to at least one question part OR Makes not more than three errors or omissions throughout the question, eg: For the st part: 5, 6, 7, 8 [one error] For the 2nd part: 7 [one omission] For the st part: 4, 5, 6 [one error, one omission] For the 2nd part: 6, 7, 8 [one error] For the st part: 5 [two omissions] For the 2nd part: 7 [one omission] Page 28 of 32

29 OR Includes non-integers within an otherwise correct response for at least one question part, eg: For the st part: 5, 5.5, 6, 6.5, 7 For the st part: 4.5 < k <7.5 Do not accept Ignores exclusivity of inequality, eg: For the st part: 4.5, 5, 5.5, 6, 6.5, 7, 7.5 [3] M9. (a) 5.50 or 22 seen! Money For 2m and m, do not accept misreads of numbers given as words, eg: four instead of five 2 OR Shows or implies a complete, correct method, eg: c = = = 2 (error) 2 4! For m, accept answers with incorrect or ambiguous units as evidence of a correct method, eg: p 5.5! Correct embedded solutions For m, condone a response which shows 5.50 embedded irrespective of how it is obtained Do not accept incomplete methods, eg: = = 3.63 (b) 3 [3] Page 29 of 32

30 M20. (a) 78 (b) 5 [2] M2. (a) n + 3 or 3 + n! Algebra! Alternative letter used, eg, for part (a), accept m used instead of n, if the expression is otherwise correct: m + 3 (b) 2m 5! Condone unsimplified or unconventional algebra, eg, for part (b): m + m 5 m2 5 [2] M Accept equivalent fractions and decimals, eg: 2 or Shows or implies a complete, correct method, eg: 5d = d = 34 d = = 40 (error) = = 8.4 (error) Page 30 of 32

31 = Do not accept incorrect methods, eg: where the perimeter of the pentagon is treated as being 4cm less than the perimeter of the triangle: 30 4 = = 5.2 [2] M23. (a) 4! Algebra (b) 0 [2] M24. Award TWO marks for four numbers correct as shown: 6 AND 7 AND 8 AND 9 If the answer is incorrect, award ONE mark for: three numbers correct and none incorrect OR all four numbers correct and one incorrect Numbers may be given in any order. Up to 2m U [2] Page 3 of 32

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