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1 Review of Year 8 This chapter reviews the Year 8 component of the mathematics syllabus and includes outcomes from Number and Algebra, Measurement and Geometry, and Statistics and Probability. You should
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1 Review of Year 8 This chapter reviews the Year 8 component of the mathematics syllabus and includes outcomes from Number and Algebra, Measurement and Geometry, and Statistics and Probability. You should be able to: complete data investigations calculate ratios and rates identify congruent figures including triangles, stating the conditions work with numbers and algebraic terms involving indices calculate the perimeter and area of plane and compound shapes calculate time using mied units operate with fractions, decimals and percentages in worded problems calculate area and circumference of circles and surface area and volume of cylinders analyse sample data using mean, mode and median and make inferences use Pythagoras theorem to perform calculations in right-angled triangles use algebraic techniques to simplify, epand and factorise simple algebraic epressions calculate volume and capacity solve linear equations and simple inequations solve probability problems involving simple events and use Venn diagrams graph and interpret linear relationships on the number plane. A Data Eercise 1A 1 Define the statistical term sample. Would a census or a sample be used to investigate the number of people who use a particular brand of toothpaste? Why? 3 Describe the sample you would use if you wanted to gather support for improved skateboard facilities at your local park. 4 For the scores 11, 14, 15, 19, 19, 1, find the: a mean b mode c median d range. 5 For the scores in this stem-and-leaf plot, find the: a mean b mode c median d range. 6 The back-to-back stem-and-leaf plot compares the marks gained by classes A and B in their half-yearly Mathematics eam. a Find the mean, mode, median and range for each class. b Which class performed better? Eplain your answer. 7 From a school of 800 students, a random sample of 50 students was selected. There were 13 left-handed students in the sample. a What fraction of the sample was left-handed? b Estimate how many students at the school were left-handed. Stem Leaf Class A Class B Leaf Stem Leaf B Ratios and rates Eercise 1 B 1 Epress the ratio 5 min : 1 1_ 4 h in simplest form. REVIEW OF YEAR 8 Which of the following is equivalent to 3 : 5? A 1 : 36 B 45 : 75 3 Find if 4 = Simplify the following ratios. a 5 cm : 0.6 m b 360 m : 0.5 km Insight Mathematics 9 stages 5.1/5. Australian Curriculum 5 The ratio of teachers to students is : 11. Calculate the number of students if there are 10 teachers. 6 A scalene triangle has side lengths in the ratio : 5 : 4. a If the shortest side is 1.4 cm, find the lengths of the other two sides. b Calculate the perimeter of the triangle. 7 Ian jogs 3.5 km in 0 minutes. Epress this as a rate of km/min. 8 Which is the better buy, A or B? A 1. L of Fizz Whiz Cola at $1.05 B.5 L of Fizz Whiz Cola at $.0. 9 The scale of a model aeroplane is 1 : 10. If the wingspan of the model is 17 cm, calculate the actual wingspan of the real aeroplane. 10 The actual height of a building is 85 m. If a model of the building is constructed using a scale of 1 : 1500, calculate the height of the model. C Congruence Eercise 1C 1 Which transformation(s) could have been used to produce each pair of congruent figures? a b 1 For each pair of triangles, state the congruency test used to show that the triangles are congruent. a b 1 c d 3 For each pair of triangles, state why the triangles are not congruent. a b c REVIEW OF YEAR 8 Chapter 1 Review of Year 8 3 4 Which triangles in each group are congruent? Give a reason for your answer. a 8 b For each pair of triangles, state why the triangles are congruent. Hence find the values of the pronumerals. a b 35 c 50 y y y z z z D Inde laws Eercise 1D 1 Write the following in inde form. a b c d Write the base and inde of each number. a 3 8 b 5 c 8 4 d Write the following in epanded form. a 6 4 b 7 3 c 6 d Evaluate: a 9 b 3 6 c 5 4 d Simplify, leaving your answers in inde form. a b (7 ) 6 c d e ( 5 ) Determine whether these calculations are true or false. a = 8 11 b = 1 4 c = 1 11 d = Evaluate: a 6 b 6 1 c 6 0 d (7 ) 0 REVIEW OF YEAR 8 8 Simplify: a a a a b 6 r r r r c y y y y 9 Epand: a t b 5a 4 c p 6 d 15e 5 10 If a = 3, evaluate: a a b 4a c (4a) d 4a 0 4 Insight Mathematics 9 stages 5.1/5. Australian Curriculum E Perimeter and area Eercise 1E 1 Estimate the width of your classroom. Convert the following lengths to millimetres. a 0.7 m 3 Convert the following lengths to centimetres. a 0.34 m b km b 0.07 km 4 Calculate the perimeter of a rectangle with width 11.9 cm and length 6.3 cm. 5 Calculate the perimeter of each shape. All measurements are in centimetres. a 38. b A regular octagon has a perimeter of cm. Calculate the length of each side. 7 Find the length of each side marked with a pronumeral, then calculate the perimeter. All measurements are in centimetres. 8 By counting squares, find the area of the shape y w z Find the areas of the following shapes. a b c d 8 cm 15 mm 5 m 11 cm 4 mm 7 cm m 10 Find the areas of the following composite shapes. a b c 4 cm 10 cm 50 m 40 m 10 m 18 m 18 m REVIEW OF YEAR 8 Chapter 1 Review of Year 8 5 11 Find the shaded area in each shape. a b 4 m 1 m m m m 0 cm 50 cm 7 m 1 Complete these conversions. a 5 cm = mm b 800 cm = m c 640 mm = cm d 11.6 m = cm e 43.8 cm = mm f 8400 cm = m g 8 cm = mm h 7. m = cm i 9000 mm = cm F Time Eercise 1F 1 How many hours in days? Complete the following conversions. a 40 s = min 3 Convert 10 min to hours and minutes. 4 Calculate the following. a 3 h 35 min + 5 h 48 min b 300 min = h b 3 h 1 min 1 h 4 min 5 If Sergio caught the bus at 6:35 am, at what time did he arrive at work, given the bus trip took 4 min? 6 High tide is at 5:0 am and low tide is at 9:08 am. Calculate the time difference between high and low tide. 7 Convert 1_ 3 h to hours and minutes. 8 Round the digital clock display 03:16:41 to the nearest minute. Epress the time in hours and minutes. G Fractions, decimals and percentages Eercise 1G REVIEW OF YEAR 8 1 Shade 7 10 of this diagram. In a class of 0 students, 1_ 4 play soccer, 1_ 5 play netball and the remainder play football. What fraction of the class plays football? 6 Insight Mathematics 9 stages 5.1/5. Australian Curriculum 3 a Convert to a mied numeral. b Convert 3 5 _ 8 to an improper fraction. 4 a Complete: 155 = Arrange in descending order: 4_ 5, 8 _ 15, 3 b Simplify a State the reciprocal of _ 3. b Calculate 3 _ 8 of 59 kg. 7 Liam earns $600 per week. He banks 1_ 5, spends _ 3 on rent and food, and keeps the remaining money for personal use. a How much does Liam bank each week? b How much does he spend weekly on rent and food? c What fraction of Liam s weekly wage is for personal use? d How much is kept for personal use? 8 State the value of in Epress as a decimal a Write 4. as a mied numeral. Epress 1 6 as a decimal correct to decimal places. a Round to the nearest hundredth. b Write 3 3 _ 8 as a decimal. b Round to the nearest whole number. Simplify the following. a b c Simplify the following. a (.1 + 3) ( ) b ( ) + (0.4 9) a Ahmed earns $4.60 per hour. How much does he earn if he works for 10 1_ hours? b Sylvanna won $ in a lottery. She decided to share it equally between eight people. How much did each person receive? Shade 75% of this diagram. Write 48 out of 100 as a percentage. a Convert 37% to a fraction. b Convert 57% to a decimal. 19 Epress the following as percentages. a 3.8 b 5 _ Convert to percentages and arrange in ascending order: 4_ 5, 70%, 0.65 Convert: 7 a 100 to a percentage b 15% to a simplified fraction c 45% to a decimal. a Calculate 15% of $360. b Find 5% of 48 m. Epress 13 kg as a percentage of 5 kg. a Increase 100 by 30%. b Decrease 30 by 5%. REVIEW OF YEAR 8 Chapter 1 Review of Year 8 7 H Circles and cylinders Eercise 1H 1 Name the features of each circle shown in orange. a b c d What fraction of a circle is represented by this sector? Write the formula for the circumference of a circle when given the diameter. 4 Calculate the circumference of a circle with a diameter of 11.4 cm, correct to 1 decimal place. 5 Write the formula for the circumference of a circle when given the radius. 6 Calculate the circumference of a circle with a radius of 6.8 cm, correct to decimal places. 7 Write the formula for calculating the area of a circle when given the radius. 8 Calculate the area of a circle, correct to 1 decimal place, given: a radius = 7 cm b diameter = 3.9 cm 9 Calculate the area of this shape correct to 1 decimal place cm Calculate the volume of each cylinder. a A = 0.8 m b c 1.6 m 58 cm 19 cm 15.3 cm 8.5 cm REVIEW OF YEAR 8 I Eercise 1I Mean, mode, median and sampling 1 Write the outlier in each data set. a 0, 71, 7, 7, 75 b 3, 4, 5, 5, 7, 8, 89 8 Insight Mathematics 9 stages 5.1/5. Australian Curriculum Consider the three data sets given. A 8, 10, 11, 11, 13, 96 B 7, 7, 8, 9, 11, 1 C 4, 7, 8, 9, 9, 9, 9 In which data set(s) is the following measure not a central value? a mean b mode c median 3 The sees of 5 students chosen at random from Year 9 are female, male, male, female, male. For this data, find, where possible, the: a mean b mode c median. 4 a The weights, in kilograms, of seven 1-year-old horses of the same breed were 40, 40, 430, 440, 460, 470, 650. For these weights, find the: i mean ii mode iii median. b Which measure would be the most appropriate to represent the weight of 1-year-old horses of this breed? 5 Five samples of 0 students were chosen from all students in a school. The students were asked to state the number of tet messages they had sent the day before. The mean number of tets per day for each sample is shown in the table. Using the information given, what is the best estimate of the mean number of tets sent per day by students of this school? Give the answer to the nearest whole number. J Sample number Mean number of tets Pythagoras theorem Eercise 1J 1 Consider the following triangles. i Which side is the hypotenuse? ii Write an epression for Pythagoras theorem for the triangle. a b P c b Q a State whether each triangle is right-angled. a 8 cm b R 7 cm 13 cm 4 cm 6 cm 3 a Find the value of 9. b Calculate the value of 70 correct to 1 decimal place. 10 cm 4 Find the length of the hypotenuse in each triangle, correct to 1 decimal place. a 5.8 cm b 4.7 cm 7.6 cm 15.3 cm REVIEW OF YEAR 8 Chapter 1 Review of Year 8 9 5 Find the length of the third side of each triangle, correct to 1 decimal place. a b 9.4 cm 1.3 cm 44. cm 7. cm 6 Find the value of the pronumeral in each of the following triangles, correct to 1 decimal place. a b p 8 cm 1 cm 10.8 cm 9.3 cm 7 Calculate the length of the diagonal of a square with side length 36 cm, correct to decimal places. K Algebra Eercise 1K 1 Simplify: a b 7y y c 3a + 4a d 9ac 3ca Simplify: a 5 1n b 5 a c 8m 3 d 5p 7 3 Simplify: a 10a b 1m 3 c 4 Simplify: a 4w + y 5w 5y abc a b 6m + m 8m d 1m 3 5 Simplify: 4a a 7 + a 7 b a a 3 5 c q q d 6p p 5 6 Epand: a a(a n) b mn(n 5) c 4p(3p + ) d p(4y w) 7 Epand and simplify. a 3(5a + 3) 4(8 4a) b 3(y 4) + 4y(5 ) REVIEW OF YEAR 8 8 Factorise: a mn + mn b pq aq c 4p 1d d 5f 15 9 Factorise each by taking out a negative factor. a 3k + 9 b 4p 1d 10 Insight Mathematics 9 stages 5.1/5. Australian Curriculum 10 Define the following terms. a pronumeral b coefficient 11 If Q = 7 and p = 4, evaluate: a 4Q + p b Qp 8 c 6p 5Q d 3(Q p) + 7p 8Q 1 Write an algebraic epression for each. a The product of si and d plus twenty-three b The difference between and seven multiplied by three and the result divided by eight L Volume and capacity Eercise 1L 1 a If ABFE is the top face of the rectangular prism, name the bottom face. b Name the front and back faces. c Name the two side faces. a Draw a net of the cube shown. b Use the net to calculate the total surface area of the cube. 3 Calculate the surface area of each prism. a 15. cm 6.8 cm 3 cm b A D 3.8 cm E H 18.8 cm 1 cm 8 cm 6 cm B C 5 cm F G 4 Construct prisms with the following cross-sections. a b 5 Draw the cross-section of each prism if it is cut along the orange dotted line shown. a b REVIEW OF YEAR 8 Chapter 1 Review of Year 8 11 6 Calculate the volume of each solid. a b A = 133 mm A = 47.5 cm 7.4 cm 5 mm 7 Calculate the volume of this cylinder to the nearest cm cm 8 Calculate the volume of each solid. a 38.7 cm b 15.3 cm 7.8 cm 10.3 cm 9 Complete the following capacity conversions. a 1 cm 3 = mm 3 b 1 kl = ml c 1 kl = cm 3 d 1 m 3 = L e 1 m 3 = kl f 5.3 kl = cm 3 M Equations and inequations Eercise 1M 3.5 cm 1 Show each step required to get from the epression back to. Solve the following equations. a + 11 = 17 b + 9 = 6 c 4 = 36 d 9 = 63 e 3y + 18 = 9 f 5 4p = 47 g 4d + 8 = 3d 1 h c = 3 3c i 3(m + 6) = (m 1) j 8(q 5) = 3(10 + 3q) 3 Solve the following equations. 4p a 5 = 6 b = Is the given value for the pronumeral a solution to the equation? 9.4 cm 5.8 cm REVIEW OF YEAR 8 a 5d + 1 = 8; d = 3 b = 4; = 3 _ 5 5 Write an equation and solve this problem. The sum of a certain number and 3 is 114. What is the number? 6 Solve the following inequations. a b m 7 4 1 Insight Mathematics 9 stages 5.1/5. Australian Curriculum N Probability and Venn diagrams Eercise 1N 1 A hat contains 1 red, 1 blue, 1 green and 1 yellow ticket. One ticket is chosen. a List the sample space. b What is the probability of selecting the red ticket? Ten cards with the numbers 1 to 10 written on them are shuffled and one card is chosen. a List the sample space. b What is the probability that the card selected has 7 written on it? 3 Complete this table. Fraction Decimal Percentage a 0.7 b 5% c 5 _ 8 4 A bag contains 4 green, 9 red and 7 blue marbles. One marble is selected at random. a How many marbles are in the bag? b How many marbles are red? c What is the probability of selecting a red marble? 5 One card is selected at random from a normal deck of 5 cards. What is: a P(diamond)? b P(red card)? c P(king)? 6 a Write a statement describing a probability of 0. b Estimate a percentage probability for the phrase even chance. c Write a phrase to describe a probability of about 85%. 7 A die with the numbers 1 6 is rolled once. Describe an event that would be: a certain b impossible c of even chance. 8 A spinner has 5 equal-sized sectors coloured green, yellow, orange, brown and white. It is spun once. What is the probability of getting: a white? b any colour ecept white? c yellow or orange? d any colour ecept yellow or orange? 9 a In a group of 9 girls, 15 play netball, 11 play oztag and 8 play both. Draw a Venn diagram to show this. b How many girls: i play netball but not oztag? ii play oztag but not netball? iii play netball or oztag or both? iv play netball or oztag but not both? v play neither netball or oztag? REVIEW OF YEAR 8 Chapter 1 Review of Year 8 13 O Coordinate geometry and straight lines Eercise 1O 1 Plot these points on a number plane: A(0, 3), B(, 3), C(3, 4), D( 3, ), E(, 5) a Plot the points A( 3, 6), B(3, 6) and C(3, 0). b If ABCD is a square, find the coordinates of point D. 3 a Use this pattern of matches to complete the table. Shape 1 Shape Shape 3 Shape 4 Shape 5 Shape number Number of matches b Write a rule describing the number of matches required to make each shape. c Using to represent the shape number and y to represent the number of matches, write a set of coordinate points describing this information. d Graph these points on a number plane. e Mark in the net two points and write their coordinates. 4 Bulk minute steak for barbecues is sold for $7.50 per kilogram with a minimum purchase of kg. The following table shows weight versus cost for various quantities of minute steak. Weight (kg) Cost ($) a Using to represent the number of kilograms and y to represent the cost in dollars, write a set of coordinate points describing this information. b Graph these points on a number plane and draw a straight line through them. c Use the graph to find how much 16 kg of minute steak would cost. d Use the graph to find how much minute steak could be purchased for $90. 5 Complete the table and draw the graph of y = 3. REVIEW OF YEAR y The graph on the right shows a straight line. a Use the graph to complete this table of values y b Write the rule describing this straight line. The rule is of the form y = ±. y Insight Mathematics 9 stages 5.1/5. Australian Curriculum Indices This chapter deals with indices and the distributive law. After completing this chapter you should be able to: simplify algebraic products and quotients using the inde laws simplify epressions involving the zero inde evaluate numerical epressions involving negative (integral) indices simplify algebraic epressions involving negative (integral) indices apply the inde laws to epressions with negative indices apply the distributive law to the epansion of algebraic epressions. NSW Syllabus references: 5.1 N&A Indices, 5. N&A Indices, 5. N&A Algebraic techniques (part) Outcomes: MA5.1-1WM, MA5.1-3WM, MA5.1-5NA, MA5.-1WM, MA5.-3WM, MA5.-6NA, MA5.-7NA ACMNA09, ACMNA1, ACMNA13, ACMNA31 A The inde laws The inde laws for numbers were established in Year 7: 3 Inde, power or eponent Base The plural of inde is indices. 1 When multiplying numbers with the same base, add the indices. For eample: = = 3 10 When dividing numbers with the same base, subtract the indices.for eample: = = 3 3 When raising a power of a number to a higher power, multiply the indices. For eample: (3 6 ) 4 = = 3 4 If we use letters to represent numbers, the rules can be generalised: a m a n = a m + n a m a n = a m n (a m ) n = a mn EXAMPLE 1 Show by writing in epanded form that m 4 m 3 = m 7. Solve Think Apply m 4 m 3 = m 7 m 4 m 3 = (m m m m) (m m m) = m m m m m m m = m 7 Eercise A The power of a number is how many of that number are multiplied together. Epand means to spread out. Here it means by writing with multiplication signs. Epand each term then write the answer in inde form. 1 Complete the following. a m 3 m 4 = (m m m) ( ) b m 5 m 3 = m m m m m = m m m = 1 = = c (m ) 3 = (m m) (m m) (m m) = = Show by writing in epanded form that: a m m 4 = m 6 b m 6 m = m 4 c (m ) 4 = m 8 16 Insight Mathematics 9 stages 5.1/5. Australian Curriculum EXAMPLE a Use a calculator to evaluate these epressions when a = 3. i a 4 a 3 ii a 7 b Does the value of a 4 a 3 equal the value of a 7? a i a 4 a 3 = = 81 7 = 187 Solve/Think Apply Substitute the value of the variable into each epression and evaluate using a calculator. ii a 7 = 3 7 = 187 b Yes, a 4 a 3 = a 7. Compare the numerical answers. 3 a Use a calculator to complete the following when a =. i a 5 a 4 = 5 4 ii a 9 = 9 = 3 = = b Does the value of a 5 a 4 equal the value of a 9? 4 a Use a calculator to evaluate the following epressions when m = 5. i m 8 m ii m 6 b Does the value of m 8 m equal the value of m 6? 5 a Use a calculator to evaluate the following epressions when n = 3. i (n 4 ) ii n 8 b Does the value of (n 4 ) equal the value of n 8? EXAMPLE 3 Use the inde laws to simplify the following. a y 7 y 3 b y 18 y 17 c (b 5 ) 3 Inde comes from the Latin word 'indicare': to point, disclose, show; as in using your inde finger. Solve Think Apply a y 7 y 3 = y 10 y 7 y 3 = y When multiplying powers with the = y 10 same base, add the indices. b y 18 y 17 = y 1 = y y 18 y = y = y 1 = y When dividing powers with the same base, subtract the indices. c (b 5 ) 3 = b 15 (b 5 ) 3 = b 5 3 When raising a power of a number to = b 15 a higher power, multiply the indices. 6 Complete the following using the inde laws. a n 3 n 5 = n + b m 7 m 3 = m c (k ) 5 = k = n = m = k Chapter Indices 17 7 Use the inde laws to simplify the following. a m 3 m 6 b q 8 q 7 c t 10 t 9 d b 15 b b 4 e v v 5 v 7 8 Use the inde laws to simplify the following. a a 1 a 10 b 15 5 c w 8 w d b 6 b 5 e z 0 z 19 9 Use the inde laws to simplify the following. A (b 4 ) b (h 5 ) 3 c (k 8 ) d (z 10 ) 6 e (n ) 4 10 Use the inde laws to simplify the following. a m 4 m b 9 6 c (b 4 ) 6 d m 3 m 6 m 4 e (v 7 ) 10 f n 8 n 7 g b 8 b h (y 5 ) 5 i t 10 t 0 t j a 1 a 6 EXAMPLE 4 Eplain why the inde laws
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