Oxford Mathematics 6

diame te r A straight line from one equilateral triangle A triangle with side of a circle to the other, passing three sides and angles the same size. through the centre point. equivalent frac tions Different fractions that digital time Time shown represent the same size in relation to a whole on a clock or watch face with or group. numbers only to indicate the hours and minutes. 1 2 3 4 2 4 6 8 division/dividing The process of sharing e s timate A thinking guess. a number or group into equal par ts, with or even number A number that can be divided without remainders. equally into 2. dot plot A way of representing pieces of data Example: 4 and 8 are even numbers using dots along a line labelled with variables. face The at sur face of a 3D shape. Favourite pet s face cat dog rabbit fac tor A whole number that will divide evenly double/doubles Adding two identical into another number. numbers or multiplying a number by 2. Example: The factors of 10 are 1 and 10 Example: 2+2=4 4×2=8 2 and 5 duration How long something lasts. nancial plan A plan that helps you to Example: Most movies have a duration organise or manage your money. of about 2 hours. ip To turn a shape over horizontally or edge The side of a shape or the line where ver tically. Also known as reection two faces of an object meet. horizontal flip edge vertical flip edge equal Having the same number or value. frac tion An equal par t of a whole or group. Example: One out of two par ts or 1 is shaded. 2 grams or g A unit for measuring the Example: Equal size Equal numbers mass of smaller items. equation A written mathematical problem where both sides are equal. Example: 4+5 = 6+3 = 148 10 0 0 g is 1 kg OX FOR D U N I V E RSI T Y PR E S S

graph A visual way to represent data or isosceles triangle A triangle with two information. sides and two angles of the same size. Pets in our class Pets in our class Cats elpoep fo rebmuN 8 7 6 jump strategy A way to solve number 5 Dogs 4 problems that uses place value to “jump” along 3 2 0 Cats Dogs Rabbits a number line by hundreds, tens and ones. Rabbits Type of pet Example: 16 + 22 = 38 GST or Goods and Ser vices Tax A tax, +10 +10 +1 +1 such as 10%, that applies to most goods and 0 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 ser vices bought in many countries. Example: Cos t + GST (10%) = Amount you pay $10 + $ 0.10 = $10.10 kilograms or kg A unit for measuring the mass of hexagon A 2D shape larger items. with six sides. horizontal Parallel with the horizon or going straight across. kilometres or km A unit for measuring long horizontal distances or lengths. line Orange Grove Tr a n ’s house Glenbrook Way improper frac tion A fraction where the S w a n P a r a d e 5 km numerator is greater than the denominator, L aw son L ane 3 ev irD alaoK such as yaW yballaW daoR allesoR 2 yaW tabmoW integer A whole number. Integers can be daoR um E positive or negative. –5 –4 –3 –2 –1 0 1 2 3 4 5 kite A four- sided shape where two pairs of adjacent sides are inverse operations Operations that are the the same length. opposite or reverse of each other. Addition and subtraction are inverse operations. legend A key that tells you what the symbols on a map mean. Example: 6 + 7 = 13 can be reversed with 13 – 7 = 6 P ar k Ser vice s t ation C ampground R ail w a y Road length The longest dimension of a shape or object. invoic e A written list of goods and ser vices provided, including their cost and any GST. P r iy a’s Pet Store Ta x I nvoice line graph A t ype of Today’s temperature Item Quantity Unit price Cost C° ni erutarepmeT 35 graph that joins plotted 30 Sia mese cat 1 $500 $ 5 0 0.0 0 25 data with a line. 20 Cat food 20 $1.5 0 $ 30.0 0 15 Tota l pr ice of good s $ 530.0 0 10 5 G ST (10%) $ 53.0 0 0 10:00 12:00 02:00 04:00 06:00 Tota l $ 5 8 3.0 0 am pm pm pm pm Time OX FOR D U N I V E RSI T Y PR E S S 149

litres or L A unit for measuring mixed number A number that contains both the capacit y of larger containers. a whole number and a fraction. Example: The capacit y of this 3 bucket is 8 litres. Example: 2 4 mode The number that occurs most often in mas s How heav y an object is. a set of numbers. Example: 2, 3, 2, 5, 2 – the mode is 2 multiple The result of multiplying a par ticular whole number by another whole number. Example: 10, 15, 20 and 10 0 are all multiples of 5. Example: 4.5 kilograms 4.5 grams near doubles A way to add two nearly mean The total of a set of numbers divided identical numbers by using known doubles by however many numbers there are in the set. facts. Example: 5, 3, 6, 2, 4 – the mean is 20 ÷ 5 = 4 me dian The number in the middle of an ordered set of numbers. Example: 4 + 5 = 4 + 4 + 1 = 9 Example: 3, 4, 5, 6, 7 – the median is 5 net A at shape that when folded up makes a 3D shape. metre or m A unit for measuring the length of larger objects. milligram or mg A unit for measuring the mass of lighter items or to use when accuracy of measurements is impor tant. number line A line on which numbers can be placed to show their order in our number system or to help with calculations. 70 0 mg 2L 0 10 20 30 40 50 60 70 80 90 100 1L millilitre or mL A unit for measuring the capacit y of smaller number sentence A way to record containers. calculations using numbers and mathematical 10 0 0 mL is 1 litre symbols. Example: 23 + 7 = 30 millimetre or mm A unit for measuring thelength of ver y small items or to use when numeral A gure or symbol used to accuracy of measurements is impor tant. represent a number. Examples: 1 – one 2 – two 3 – three numerator The top number in a cm 1 2 3 3 4 fraction, which shows how many pieces you are dealing with. There are 10 mm in 1 cm. 150 OX FOR D U N I V E RSI T Y PR E S S

obtuse angle An angle that is larger than Example: Shape pattern a right angle or 9 0 degrees, but smaller than 18 0 degrees. Number pattern 2, 4, 6, 8, 10, 12 pentagon A 2D shape with ve sides. per cent or % A fraction out of 10 0. right angle 62 Example: or 10 0 62 out of 10 0 oc tagon A 2D shape with eight sides. odd number A number is also 62%. that cannot be divided equally into 2. Example: 5 and 9 are odd p e rime te r The distance 7m numbers. around the outside of a 6m shape or area. 5m Example: Perimeter = operation A mathematical process. The 3m four basic operations are addition, subtraction, 10 m 7 m + 5 m + 10 m + 3 multiplication and division. m + 6 m = 31 m y 6 origin The point on a 5 pic tograph A way of representing data using 4 Car tesian plane where the 3 pictures so that it is easy to understand. 2 x - axis and y - axis intersect. x Example: Favourite juices in our class 1 2 3 4 origin –1 –2 –3 –4 –5 –6 outcome The result of a chance experiment. Example: The possible outcomes if you roll a dice are 1, 2, 3, 4, 5 or 6. parallel lines Straight lines that are the same place value The value of a digit depending distance apar t and so will never cross. on its place in a number. parallel parallel not parallel M H Th T Th Th H T O 2 2 7 4 8 2 2 7 4 8 6 7 7 4 8 6 3 4 8 6 3 1 parallelogram A four- sided shape where polygon A closed 2D shape with three or each pair of opposite sides is parallel. more straight sides. pat tern A repeating design or sequence of numbers. polygons not polygons OX FOR D U N I V E RSI T Y PR E S S 151

polyhedron (plural polyhedra) A 3D shape quadrant A quar ter of a circle or one of the with at faces. four quar ters on a Car tesian plane. y 6 5 4 3 2 x –5 –4 –3 –2 –1 1 2 3 4 –1 –2 –3 polyhedra not polyhedra –4 –5 power of The number of times a par ticular quadrant number is multiplied by itself. quadrant 3 Example: 4 is 4 to the power of 3 quadrilateral Any 2D shape with four sides. or 4 × 4 × 4. prime number A number that has just two factors – 1 and itself. The rst four prime numbers are 2, 3, 5 and 7. radius The distance from the centre of a circle to its circumference or edge. prism A 3D shape with parallel bases of the same shape and rectangular side faces. range The difference between the highest and lowest in a set of numbers. Example: 5, 3, 6, 2, 4 – the range is 6 – 2 = 4 reec t To turn a shape over horizontally triangular rectangular hexagonal or ver tically. Also known as ipping prism prism prism ver tical horizontal reection reection probabilit y The chance or likelihood of a par ticular event or outcome occurring. Example: There is a 1 in 8 chance reex angle An angle that is between this spinner will land on red. 18 0 and 36 0 degrees in size. protrac tor An 80 100 100 0 90 1 7 1 0 80 0 7 1 0 1 instrument used to 0 0 1 5 2 3 1 0 0 3 1 remainder An amount left over after dividing measure the size of arms 3 0 0 2 one number by another. 6 0 1 0 2 angles in degrees. 071 01 071 01 081 Example: 11 ÷ 5 = 2 r1 0 vertex rhombus A 2D shape with four sides, all of pyramid A 3D shape with a 2D shape as a the same length and opposite sides parallel. base and triangular faces meeting at a point. right angle An angle of exactly 9 0 degrees. square pyramid hexagonal pyramid 90º 90º 152 OX FOR D U N I V E RSI T Y PR E S S

right-angled triangle A triangle where skip counting Counting for wards or one angle is exactly 9 0 degrees. backwards by the same number each time. 90º rotate Turn around a point. Examples: Skip counting by ves: 5, 10, 15, 20, 25, 30 Skip counting by twos: 1, 3, 5, 7, 9, 11, 13 slide To move a shape to a new position rotational symmetr y A shape has rotational without ipping or turning it. Also known as symmetr y if it ts into its own outline at least translate once while being turned around a xed centre point. 1s t p o s i t i o n Back to the s tar t × • • • × sphere A 3D shape that is • × × × • per fectly round. 2nd p osition split strategy A way to solve number problems round/rounding To change a number to that involves splitting numbers up using place another number that is close to it to make it value to make them easier to work with. easier to work with. 229 can be Example: 21 + 14 = 20 + 10 + 1 + 4 = 35 rounded up to OR rounded down to the nearest 10 the nearest 10 0 230 20 0 + = + + + = scale A way to represent large areas on maps by using ratios of smaller to larger measurements. Example: 1 cm = 5 m scalene triangle A triangle where 2 1 cm 1 cm no sides are the same length and square centimetre or cm no angles are equal. A unit for measuring the area of smaller objects. It is exactly 1 cm long and 1 cm wide. sec tor A section of a circle bounded by two 2 A unit square metre or m radius lines and an arc. for measuring the area of larger 1m 1m spaces. It is exactly 1 m long and arc 1 m wide. radius lines sector square number The result of a number being multiplied by itself. The product can be represented as a square array. semi- circle Half a circle, bounded by an arc 2 Example: 3 × 3 or 3 =9 and a diameter line. straight angle An angle that is exactly semi-circle 18 0 degrees in size. arc 18 0 º diameter line OX FOR D U N I V E RSI T Y PR E S S 153

s trategy A way to solve a problem. In te s s e llation A pattern mathematics, you can often use more than formed by shapes that t one strategy to get the right answer. together without any gaps. Example: 32 + 27 = 59 Jump strategy the rmome te r An instrument for Split strategy measuring temperature. 32 42 52 53 54 55 56 57 58 59 30 + 2 + 20 + 7 = 30 + 20 + 2 + 7 = 59 three - dimensional or 3D subtrac tion The taking away of one A shape that has three number from another number. Also known as dimensions – length, width and depth. width subtracting, take away, difference between and depth minus. See also vertical subtraction 3D shapes are not at. length Example: 5 take away 2 is 3 time line A visual representation of a period of sur vey A way of collecting data or time with signicant events marked in. information by asking questions. 2 9 Januar y 2 5 March 19 May 2 8 June 3 – 6 August 17 December S cho ol Mid - y e ar School st ar t s E as t er C amp S cho ol produc tion holiday s holiday s  nis h e s Strongly agree translate To move a shape to a new position Agree Disagree without ipping or turning it. Also known as Strongly disagree slide symmetr y A shape or pattern has symmetr y when one side is a mirror image of the other. trapezium A 2D shape with four sides and only one set of parallel lines. table A way to organise information that uses columns and rows. Flavour Number of people triangular number A number that can be Chocolate 12 Vanilla 7 organised into a triangular shape. The rst Strawberry 8 four are: tally marks A way of keeping countthatusessingle lines with ever y fth line crossed to make a group. t wo - dimensional or 2D A at shape that has two dimensions – width length and width. term A number in a series or pattern. Example: The sixth term in this pattern is 18. length 3 6 9 12 15 18 21 24 154 OX FOR D U N I V E RSI T Y PR E S S

turn Rotate around a point. volume How much space an object takes up. Example: This object has a volume of 4 cubes. unequal Not having the same size or value. whole All of an item or group. Example: Unequal size Unequal numbers Example: A whole shape A whole group width The shor test dimension of a shape or value How much something is wor th. object. Also known as breadth Example: This coin is wor th 5c. This coin is wor th $1. ver tex (plural ver tices) The point where two edges of a shape or object meet. Also known as a corner x-axis The horizontal reference line showing corner coordinates or values on a graph or map. Favourite sports ver tical At a right angle to the horizon or elpoep fo rebmuN straight up and down. 16 14 vertical 12 line 10 8 6 4 2 0 horizon Rugby Foot- Basket- ball ball ball Sport x-axis y-axis The ver tical reference line showing coordinates or values on a graph or map. Favourite sports ver tical addition A way of T O 16 3 6 y-axis recording addition so that the place - elpoep fo rebmuN 14 + 2 1 value columns are lined up ver tically 12 10 5 7 to make calculation easier. 8 6 4 ver tical subtrac tion A way of T O 2 5 7 recording subtraction so that the 0 Cricket Soccer Net- Rugby Foot- Basket- – 2 1 place -value columns are lined up ball ball ball 3 6 ver tically to make calculation easier. Sport OX FOR D U N I V E RSI T Y PR E S S 155

ANSWERS c Teacher to check, e.g. To get to the UNIT 1: Topic 1 second pentagonal number you add 4 to the rst one (1 + 4 = 5). For the next one Guided practice you add 3 more than that (5 + 7 = 12). Each time, you add 3 more than last time. 1 M Hth Tth Th H T O Number 5 0 0 0 0 0 0 5 000 000 3 0 0 0 0 0 300 000 d 6 0 0 0 0 60 000 7 0 0 0 70 0 0 9 0 0 900 1 0 10 8 8 UNIT 1: Topic 3 2 a 51 604 Guided practice 1 b 200 026 UNIT 1: Topic 2 c 12 010 rebmuN Factors How Prime or composite? Independent practice Guided practice (numbers many it can be factors? Prime Composite 2 2 2 2 1 a 60 000 b 300 000 1 3×3=3 ,3 = 9, 4 × 4 = 4 , 4 = 16, divided by) 2 2 2 2 c 6000 d 1 000 000 5×5=5 ,5 = 25, 6 × 6 = 6 , 6 = 36 1 1 1 neither e 80 000 000 f 50 000 000 2 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10 2 1 and 2 2 ✔ 2 a sixty thousand Independent practice 3 1&3 2 ✔ b three hundred thousand 2 2 1 (Shading may vary.) 7 × 7 = 7 , 8 × 8 = 8 4 1, 2 & 4 3 ✔ c six thousand d one million 2 2 2 2 9 × 9 = 9 , 10 × 10 = 10 , 7 = 49, 8 = 64, 5 1&5 2 ✔ e eighty million f fty million 2 2 9 = 81, 10 = 100 6 1, 6, 2, 3 4 ✔ 3 a 80 487 000 b 10 362 059 2 a 121 7 1&7 2 ✔ c 114 760 209 d 1 400 593 001 b Teacher to check, e.g. The digits 8 1, 8, 2, 4 4 ✔ 4 a As Student Book alternate between odd and even. b 200 000 + 10 000 + 4000 + 800 + 60 + 7 9 1, 9, 3 3 ✔ c 10 000 c 2 000 000 + 500 000 + 60 000 + 10 1, 10, 2, 5 4 ✔ 3 Teacher to check artwork. 7000+ 300 + 20 + 1 15: 1 + 2 + 3 + 4 + 5 = 15, 11 1 & 11 2 ✔ d 5 000 000 + 600 000 + 70 000 + 3000 21: 1 + 2 + 3 + 4 + 5 + 6 = 21, 12 1, 12, 2, 6, 6 ✔ 3, 4 + 200 + 7 28: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28, e 50 000 000 + 7 000 000 + 300 000 + 36: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, 13 1 & 13 2 ✔ 10 000 + 9000 + 200 + 40 45: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45, 14 1, 14, 2, 7 4 ✔ f 400 000 000 + 7 000 000 + 500 000 + 55: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 8000 + 4 15 1, 15, 3, 5 4 ✔ 4 a 66 b 36 16 1, 16, 2, 8, 4 5 ✔ 5 a 9 754 321 b 5 123 479 c Teacher to check, e.g. The numbers 17 1 & 17 2 ✔ c 9 543 217 d 2 314 579 alternate between two odd numbers, then two even numbers. 18 1, 18, 2, 9, 6 ✔ 3, 6 6 a 6 142 793: six million, one hundred and forty-two thousand, seven hundred and Extended practice 19 1 & 19 2 ✔ 1 ninety-three 20 1, 20, 2, 10, 6 ✔ 4, 5 b 280 526 306: two hundred and eighty million, ve hundred and twenty-six Square Multiplication Addition fact fact thousand, three hundred and six number 1×1=1 1 2×2=4 1+3=4 2 3×3=9 1+3+5=9 2 a 2, 3, 5, 7, 11, 13, 17, 19 1 =1 4 × 4 = 16 1 + 3 + 5 + 7 = 16 Extended practice 5 × 5 = 25 1 + 3 + 5 + 7 + 9 = 25 b Teacher to check, e.g. 2 is the only even 2 6 × 6 = 36 1 + 3 + 5 + 7 + 9 + 11 2 =4 = 36 prime number. 1 + 3 + 5 + 7 + 9 + 11 + 1 a + 100 b + 40 000 2 13 = 49 3 =9 1 + 3 + 5 + 7 + 9 + 11 + c – 20 000 d +1 13 + 15 = 6 4 Independent practice 2 1 + 3 + 5 + 7 + 9 + 11 + 2 a $340 000 b $705 000 4 = 16 13 + 15 + 17 = 81 1 1 + 3 + 5 + 7 + 9 + 11 + 2 13 + 15 + 17 + 19 = 100 4 5 6 7 8 9 10 5 = 25 c $825 000 d $1 250 000 2 6 = 36 11 12 13 14 15 16 17 18 19 20 3 Answers may vary. Look for students who justify their answers appropriately. Possible 2 21 22 23 24 25 26 27 28 29 30 7 = 49 7 × 7 = 49 answers: 31 32 33 34 35 36 37 38 39 40 a The digit 5. It means 500 000. $500 000 2 8 × 8 = 64 8 = 64 is a lot of money! 41 42 43 44 45 46 47 48 49 50 b The digit 2. It means 2 whole ones. 2 9 × 9 = 81 9 = 81 I wouldn’t want to write out my times 51 52 53 54 55 56 57 58 59 60 tables any more than that! 2 10 × 10 = 100 61 62 63 64 65 66 67 68 69 70 10 = 100 c The digit 1. It means 10. I really like them but too many might make me ill! 71 72 73 74 75 76 77 78 79 80 2 a Teacher to check, e.g. To get to the next 81 82 83 84 85 86 87 88 89 90 square number, you add the next odd number. 91 92 93 94 95 96 97 98 99 10 0 b 2 11 = 121, 11 × 11 = 121, 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 = 121 2 a 97 b False c 23 c False (49 even composite numbers, 25 odd composite numbers). 3 a 15 b 1, 5, 12, 22, 35 156 OX FOR D U N I V E RSI T Y PR E S S

3 Independent practice 10 9 15 1 a 379 b 599 c 1298 d 228 4 e 3909 f 10 990 2 × 5 3 × 3 3 × 5 2 a 446 b 765 c 874 a The prime factors of 10 b The prime factors of 9 c The prime factors of 15 d 545 e 768 f 1206 are2 and 5. So 2 × 5 = 10 are3 and 3. So 3 × 3 = 9 are3 and 5. So 3 × 5 = 15 3 Examples of strategies that might be used: 21 35 39 a 898 (I said 650 + 250 = 900 then took away 2.) 3 × 7 5 × 7 3 × 13 b 1054 (I took 200 from 1253 = 1053. Then I added 1 back.) d The prime factors of 21 are e The prime factors of 35 are f The prime factors of 39 are 3 and 13. So 3 × 13 = 39 c 3500 (I doubled 1500 then doubled 250.) 3 and 7. So 3 × 7 = 21 5 and 7. So 5 × 7 = 35 d 14 168 (I took 400 away and then took 26 33 34 another 10.) 4 Examples of the way the numbers might be 2 × 13 3 × 11 2 × 17 rounded: g The prime factors of 26 are h The prime factors of 33 are i The prime factors of 34 are 2 and 17. So 2 × 17 = 34 Number fact Rounded I rounded this number number to the 2 and 13. So 2 × 13 = 26 3 and 11. So 3 × 11 = 33 nearest … 4 14 55 49 a Australia has 812 813 000 km thousand 972 kilometres of 2 7 5 × 11 7 7 roads. × × b The Electricit y 1 500 000 hundred Company of China thousand a The prime factors of 14 are b The prime factors of 55 are c The prime factors of 49 are employs 1 502 000 people. 2 and 7. So 2 × 7 = 14 5 and 11. So 5 × 11 = 55 7 and 7. So 7 × 7 = 49 Extended practice c The Mexican soccer $3 000 000 million 1 player, Blanco, 20 18 28 36 earned $2 9 4 3 702 in 2009. 2 10 9 2 4 7 4 9 × × × d The fastest speed 300 km/h hundred recorded at the × Indianapolis 500car race was 2 × 5 3 × 3 2 × 2 299.3 km/ h. × × × a 20 = 2 × 2 × 5 b 18 = 3 × 3 × 2 c 28 = 2 × 2 × 7 d 36 = 2 × 2 × 3 × 3 2 2 2 2 2 e The fastest 10.5 tenth 20 = 2 ×5 18 = 3 ×2 28 = 2 ×7 36 = 2 ×3 100 -metre sprint time for a woman is 2 Diagrams may vary, e.g. the second row of c) could show 3 × 8. 24 10.49 seconds. Teacher to check the resulting prime factors. f The US depar tment 2 000 000 million 30 store Walmar t 27 2 12 × employs 2 100 000 people. 10 × 3 3 × 9 4 × 3 g Each Australian 18 L litre eats an average of 3 × 3 2 × 5 2 × 2 17 L 600 mL of ice-cream a year. a 27 = 3 × 3 × 3 b 30 = 2 × 5 × 3 c 24 = 2 × 2 × 2 × 3 3 h The longest 57 km kilometre 27 = 3 rail tunnel is in 3 Swit zerland. It is 57.1 km long. 24 = 2 ×3 UNIT 1: Topic 4 i The amount of $ 3 billion billion money the movie Guided practice Avatar made was $2 78 3 919 000. j Foreign tourists $29 billion billion or 1 Problem Using rounding it becomes Now I need to Answer 516 174 spend or $ 30 tenbillion 829 a 317 + 199 317 + 200 = 517 take away 1 279 $29 127 000 000 billion 498 732 a year in Australia. 24 62 b 275 – 101 275 – 100 = 175 take away another 1 c 527 + 302 527 + 300 = 827 add another 2 5 Basic: $22 490 d 377 – 98 377 – 100 = 277 add back 2 Deluxe: $31 490 e 249 + 249 250 + 250 = 500 take away 2 Premium: $38 959 f 938 – 206 938 – 200 = 738 take away another 6 g 14 6 4 + 998 14 6 4 + 1000 = 24 6 4 take away 2 2 Problem Expand the numbers Join the par tners Answer 300 + 70 + 500 + 20 300 + 500 + 70 + 20 890 a 370 + 520 2000 + 200 + 3000 + 600 2000 + 3000 + 200 + 600 5800 300 + 40 + 2 + 200 + 30 + 6 300 + 200 + 40 + 30 + 2 + 6 578 b 2200 + 3600 4 00 + 70 + 1 + 200 + 20 + 8 4 00 + 200 + 70 + 20 + 1 + 8 699 700 + 4 0 + 3 + 4 00 + 20 + 6 700 + 4 00 + 4 0 + 20 + 3 + 6 1169 c 342 + 236 800 + 60 + 5 + 700 + 30 + 4 800 + 700 + 60 + 30 + 5 + 4 15 9 9 4 000 + 200 + 70 + 3000 + 200 + 20 4 000 + 3000 + 200 + 200 + 70 + 20 74 9 0 d 471 + 228 e 74 3 + 426 f 865 + 734 g 4270 + 3220 OX FOR D U N I V E RSI T Y PR E S S 157

2 Extended practice Write the multiplication fact par tner ÷ 10 37 × 10 = 370 470 × 10 = 4700 1 Examples of how 200 × 10 = 2000 $2.25 × 10 = $22.50 the numbers might Problem Round the Estimate Which is the likely a 370 37 5.4 × 10 = 5 4 e.g. numbers the answer answer? be rounded: b 470 0 470 c 2000 200 a 5189 – 2995 5000 – 3000 2000 219 4 or 319 4 d $ 2 2.5 0 $ 2.25 b 2958 + 6058 3000 + 6000 9000 9016 or 8016 e 54 5.4 c 8215 – 3108 8000 – 3000 5000 5907 or 5107 3 d 15 963 + 14 387 16 000 + 14 000 30 000 29 350 or 30 350 ÷ 10 0 Write the multiplication fact par tner e 8954 – 3928 9000 – 4000 5000 5026 or 4 026 7 × 100 = 700 $ 4.95 × 100 = $ 495 a 70 0 7 50 × 100 = 5000 120 × 100 = 12 000 f 4568 + 4489 4500 + 4500 9000 8057 or 9057 b $ 495 $ 4.95 87.5 × 100 = 8750 g 13 149 – 7908 13 000 – 8000 5000 6241 or 5241 c 5000 50 d 12 000 120 h 124 963 + 98 358 125 000 + 100 000 225 000 223 321 or 213 321 e 8 75 0 87.5 2 Examples of ways Problem Round the Estimate Calculator answer numbers the answer of rounding: Independent practice 1 e.g. 6190 + 1880 6000 + 2000 8000 8 070 a 4155 + 2896 4000 + 3000 70 0 0 7051 × 10 20 40 80 [double] [double [double again] again] b 9124 – 8123 9000 – 8000 10 0 0 10 01 c 24 065 + 5103 24 000 + 5000 29 000 29 168 a 12 120 24 0 480 960 15 0 300 600 12 0 0 d 19 75 3 – 10 338 20 000 – 10 000 10 000 9 415 b 15 220 440 880 176 0 25 0 500 10 0 0 2000 e 101 582 + 49 268 101 000 + 49 000 150 000 150 850 c 22 500 10 0 0 2000 4000 f 298 0 47 – 198 214 300 000 – 200 000 100 000 99 833 d 25 1 089 274 + 1 000 000 + 2 000 000 2 188 857 e 50 1 099 583 1 000 000 g h 1 499 836 + 1 500 000 + 3 000 000 2 989 803 2 20 40 80 1 489 967 1 500 000 ÷ [halve it] [halve [halve 10 again] again] a 400 40 20 10 5 Unit 1: Topic 5 UNIT 1: Topic 6 b 2000 200 10 0 50 25 c 480 48 24 12 6 d 10 000 10 0 0 500 25 0 125 Guided practice Guided practice e 8800 880 440 220 110 1 a 1234 b 2345 c 3456 1 a 229 b 326 c 2208 d 2119 d 4567 e 5789 f 5678 3 ×5 2 a 589 b 199 c 2149 First multiply Independent practice by 10 Then Multiplication halve it fact d 1985 e 9988 f 8899 1 a 1111 b 2222 c 3333 Independent practice a 24 24 0 120 24 × 5 = 120 680 340 68 × 5 = 340 d 4444 e 5555 f 6666 12 0 0 600 120 × 5 = 600 5000 25 0 0 500 × 5 = 2500 1 a 54 321 b 65 432 c 76 543 b 68 12 4 00 6200 124 0 × 5 = 6200 g 7777 h 8888 i 9999 d 87 654 e 98 765 f 56 789 c 120 j 10 000 k 11 111 g 45 678 h 34 567 i 23 456 d 500 2 Look for the strategies used to solve the j 12 345 problem. One simple solution is to subtract e 124 0 2 9 875 432 2 345 789 = 7 529 643 123 from 99 999 four times, making the 4 addition 99 507 + 123 + 123 + 123 + 123. 3 a 3268 b 12 619 c 22 656 First Then divide double ÷5 by 10 it Division fact 3 a d 34 579 e 375 777 f 676 068 42 35 g 749 h 36 4 9 i 320 054 52 90 Countr y Paved Unpaved Total 120 roads 2 265 256 6 430 366 a 420 84 420 ÷ 5 = 8 4 1 779 639 3 383 344 70 350 ÷ 5 = 70 354 864 1 870 661 j 65 622 10 4 520 ÷ 5 = 10 4 0 951 220 18 0 900 ÷ 5 = 180 258 000 1 18 3 000 b 350 24 0 1200 ÷ 5 = 24 0 6 663 666 292 USA 4 165 110 626 700 1 0 42 300 473 679 810 6 41 1 655 515 1 751 868 Extended practice India 1 603 705 c 520 1 a 193 635 b 126 296 c 191 790 China 1 515 797 d 900 2 a Teacher to check; using any three digits France 951 220 e 12 0 0 will result in the answer 1089. Japan 925 000 b The answer is still 1089. 5 Spain 659 629 First Multiplication × 10 fact × 30 15 0 Then × 3 220 Canada 415 600 330 15 0 0 Australia 336 962 a 15 2300 450 15 × 30 = 450 660 22 × 30 = 660 Unit 1: Topic 7 990 33 × 30 = 990 4500 150 × 30 = 4500 Brazil 96 353 b 22 6900 230 × 30 = 6900 c 33 b India, Canada and Brazil Guided practice c USA and China d 15 0 1 d China and Canada (981 564 km) e 230 × 10 10 0 10 0 0 10 0 0 0 Extended practice a 29 290 2900 29 000 290 000 1 947 34 4 2 562 996 b 124 124 0 12 4 00 124 000 1 24 0 000 c 638 6380 63 800 638 000 6 380 000 3 The correct answer is 453 487. Teacher to check students’ methods of conrming the d $1.25 $12.5 0 $125 $125 0 $12 500 correct answer. e 75 0 75 0 0 75 000 750 000 7 500 000 4 2 335 358 000 000 km 158 OX FOR D U N I V E RSI T Y PR E S S

6 4 a true b false c true Then Multiplication Unit 1: Topic 9 × 10 fact × 30 First × 3 15 × 30 = 450 22 × 30 = 660 33 × 30 = 990 d false e true f false 150 × 30 = 4500 230 × 30 = 6900 g false h true a 15 45 450 66 660 99 990 Guided practice 450 4500 b 22 690 6900 Independent practice 1 a 69 b 4 42 c 94 c 33 1 a –2 + 4 = 2 b 2 – 3 = –1 d 110 e 4 321 f 1201 d 15 0 c 4 – 7 = –3 d – 6 + 5 = –1 g 934 h 4 8 41 i 4322 e –3 – 5 = – 8 f –8 + 8 = 0 e 230 j 4322 k 12 343 l 54 322 g – 8 + 10 = 2 h 7 – 11 = – 4 7 a 600 b 880 c 1250 2 a 215 b 582 c 358 i –7 + 15 = 8 j 6 – 13 = –7 d 170 0 e 840 f 5000 d 2686 e 659 f 348 2 a –1 b –1 c –4 g 120 0 h 14 40 i 570 Independent practice d –5 e –10 f – 60 j $72 k $90 l 416 1 a 2 b 2 3 a – 60, – 50, – 40, – 30, –20, –10, 0, 10, 20, 2 c e 4 r2 or 4 9 r2 or 9 g 3 5 h m 208 n 62 a c e 3 d 1 30, 40, 50 g 9 r3 or 9 8 r1 or 8 4 8 8 $87.40 5 5 b –25, –20, –15, –10, – 5, 0, 5, 10, 15, 20, 8 r5 or 8 f 8 r5 or 8 9 7 Extended practice 25, 30 3 1 9 r3 or 9 or 9 9 3 c –28, –24, –20, –16, –12, – 8, – 4, 0, 4, 8, 1 4 2 Halve it Add 9 r4 or 9 or 9 12, 16 to nd the two ×5 answers Multiplication 6 3 fact × 15 × 10 3 b 2 d – 35, –28, –21, –14, –7, 0, 7, 14, 21, 28, 116 r3 or 116 90 r2 or 90 4 3 5 2 35, 42 32 r5 or 32 d 148 r2 or 148 6 5 a 12 120 60 18 0 12 × 15 = 180 e – 63, – 54, – 45, – 36, –27, –18, – 9, 0, 9, b 32 320 16 0 480 32 × 15 = 4 80 1 1 858 r1 or 858 f 187 r1 or 187 3 9 18, 27, 36 c 41 410 205 615 41 × 15 = 615 1 h 2 694 r1 or 694 331 r2 or 331 4 a&b Tuesday 2ºC 6 7 d 86 860 4 30 12 9 0 86 × 15 = 1290 3 a 62 and one left over (Possible Sunday 1ºC justication: they would not be able to Wednesday 0ºC e 422 4220 2110 6330 422 × 15 = 6330 split a marble in half.) Saturday –1ºC 1 2 b 7 (Possible justication: they would Monday –2 º C 2 Add Multiplication probably share the extra donut between Friday –3ºC the two fact × 13 × 10 ×3 answers them.) Thursday – 4ºC 4 a $26.50 b $18.50 c $11.50 c Thursday d 6ºC a 15 15 0 45 195 15 × 13 = 195 36 15 6 12 × 13 = 156 69 299 23 × 13 = 299 d $18.25 e $16.50 f $21.50 5 93 403 31 × 13 = 4 03 b 12 120 75 325 25 × 13 = 325 –5 –4 –3 –2 –1 0 1 2 3 4 5 5 a 148.75 b 125.6 c 63.25 c 23 230 A T H d 136.8 e 336.75 f 1231.5 d 31 310 g 2865.5 h 2319.6 i 6523.25 e 25 25 0 Extended practice 6 a 36 b $ 36.25 3 a 250 0 b 6300 c 4 1 a Helsinki, Montreal, Quebec, Moscow Extended practice d 30 e 8800 f $1.80 b Berlin 1 a 291.33 b 124.25 c 83.14 g $68 h $0.9 0 c – 5ºC d 42.33 e 80.4 4 f 1828.33 4 a 60 b 3600 d Montreal and Sydney (– 6 º and 27 º), g 348.6 h 1226.14 i 11 494.75 Helsinki and Acapulco (– 3 º and 30 º), j 14 321.33 k 5095.2 l 10 807.22 Quebec and Melbourne (–7 º and 26 º) 2 a2 b2 c4 d3 e2 f5 g5 2 Unit 1: Topic 8 INTERNATIONAL BIG BANK 3 It erupts 20 times a day. Look for the Date Paid in Paid out Balance $ $ $ strategy that the student uses, e.g. division Guided practice by estimation or 365 × 10 = 3650, double 3650 = 7300. 3 May 10 0 10 0 4 May 30 –20 1 a 162 b 325 9 May 45 14 May 10 2 a 508 b 981 c 630 31 May 120 – 40 50 d 916 e 819 0 f 9415 5 g 8512 h 7285 i 9042 Unit 1: Topic 10 Independent practice Guided practice 3 a $ –100 1 1 a 70 0 b 540 c 1080 b Teacher to check, e.g. it would be d 18 40 e 2010 f 2040 morethan $90 because there would be g 3040 h 9840 i 10 980 4 −4 −3 −1 0 3 5 interest payable on the amount owing. j 55 280 k 186 060 l 208 720 2 $ 4 380 2 3 1 966 080 5 Unit 1: Topic 11 4 4 a 552 b 805 c 980 3 s quar e 2 black dot 5 a 888 b 1053 c 1092 1 Guided practice 0 triangle Extended practice −1 blue dot 1 −2 s tar −3 Multiplication Base number and exponent −4 1 a 725 b 1134 c 741 −5 a 2×2×2×2×2 5 2 d 1419 e 1125 f 2368 b 4×4×4 3 4 g 3198 h 11 178 i 18 612 2 a 11 725 days b 83 700 sneezes c 8×8×8×8 4 8 d 5×5×5×5×5 5 5 3 – 5, – 3, –1, 0, 2, 4 e 7×7×7×7×7×7 6 7 f 10 × 10 × 10 × 10 4 10 OX FOR D U N I V E RSI T Y PR E S S 159

2 Base number Number of times the base number is Multiplication Value of the number Unit 2: Topic 1 and exponent used in a multiplication a three times 3×3×3 27 Guided practice b 3 four times 2×2×2×2 32 c 3 three times 5×5×5 125 1 a 1 b 3 d t wo times 6×6 36 c one -sixth, e 4 t wo times 9×9 81 three -fths, 2 three times 10 × 10 × 10 10 0 0 6 5 f 3 Number fact 7 d 2 5 seven-tenths, √4 = 2 two -thirds, 2 √36 = 6 10 3 6 √9 = 3 √64 = 8 2 a 4 1 b Student shades 4 stars. 2 4 9 The square number or is between 16 3 3 and 4 10 6 and 7 9 4 and 5 7 and 8 3 a b c 10 d 3 Student draws a smiley face at the mark. 10 7 10 6 Student draws a triangle at the mark. 3 10 a Star ting number What number multiplied by Square root of the b itself makes the number? star ting number c 4 Independent practice d 6 3 3 4 5 6 8 4 2×2=4 1 a 6 8 10 12 36 6 × 6 = 36 b 1 Any fraction that is equivalent to 2 7 8 etc. 16 e.g. 14 9 3×3=9 2 The answers below are ones that students 64 8 × 8 = 64 can identify from the fraction wall. However, 2 4 there are other possibilities, such as = 10 20 4 Star ting number Which two square numbers is What are their square 1 2 1 it between? roots? 5 12 4 a a b c b c d 10 e 8 d 12 10 10 9 and 16 √9 = 3 and √16 = 4 1 2 4 3 f Any or all of and a 6 12 42 36 and 49 √36 = 6 and √49 = 7 3 2 b 8 c 6 3 10 8 20 16 and 25 √16 = 4 and √25 = 5 6 8 3 9 12 4 d e f 52 49 and 6 4 √49 = 7 and √6 4 = 8 4 There are several possible correct responses. One possibility is shown. Check Independent practice that the equivalent fractions are correct andthat the shading matches the fractions. 7 5 1 a 2 =2×2×2×2×2×2×2= b 5 = 5 × 5 × 5 × 5 × 5 = 3125 6 3 a = 6 5 8 4 c 3 = 3 × 3 × 3 × 3 × 3 × 3 = 729 d 4 = 4 × 4 × 4 × 4 × 4 = 1024 e 4 7 = 7 × 7 × 7 × 7 = 2401 5 4 5 5 3 5 2 a 8 : 9 = 6561 and 8 = 32 768 b 3 : 5 = 125 and 3 = 243 6 6 3 a 5 = 15 625 b 10 = 1000000 4 Star ting The approximate square root Actual square root (to two Number fact 8 4 number is between decimal places) b = 10 5 5 2 and 3 2 . 24 √5 = 2.24 40 6 and 7 6.32 √4 0 = 6.32 5 Teacher to check shading and to decide how 14 3 and 4 3.74 √14 = 3.74 accurate students need to be with regard to 30 5 and 6 5.4 8 √30 = 5.4 8 dividing the shapes. 99 9 and 10 9.95 √99 = 9.95 6 a 3 (or any equivalent to 1 b 12 c ) 4 3 3 (or any equivalent to ) 4 4 1 2 Most likely answers are and , but 6 12 students could choose other equivalent fractions. Extended practice because this could be confused with 3 × 3 8 × 3 × 3 × 3. d Student draws a star at the mark. 12 1 a 5^2 = 25 Extended practice 3 × 3 (rst time) = 9, × 3 (second time) = b 3^4 = 81 27, × 3 (third time) = 81, × 3 (fourth time) 1 a ÷4 b ÷3 c ÷2 c 10^4 = 10 000 4 4 = 243. 3 = 81. If dealt with correctly, 3 is d ÷3 e ÷5 f ÷4 d 1^10 = 1 seen as the base number (4) being used in a g ×2 h ×2 i ×4 2 Teachers may wish to discuss with students multiplication 4 times: 3 × 3 × 3 × 3 = 81. j ×2 k ×6 l ×4 an appropriate number of decimal places 3 Advanced student should, therefore, see 6 2 Teacher to check method. Below are that are needed. -3 as the opposite of 6 in which 1 is divided -1 probable solutions, but students may a 8 = 1 ÷ 8 = 0.125 by 6 three times (1 ÷ 6 ÷ 6 ÷ 6). Therefore, 4 -2 provide different answers, such as for 2d: 10 b 8 = 1 ÷ 8 ÷ 8 = 0.015625 3 6 can be written as 1 multiplied by 6 three -1 a 2 b 3 c 1 3 4 2 c 4 = 1 ÷ 4 = 0.25 times (1 × 6 × 6 × 6). -2 2 1 4 5 7 5 d 4 = 1 ÷ 4 = 0.25 ÷ 4 = 0.0625 3 d e f a 6 = 1 × 6 × 6 × 6 = 216 (compare with -2 e 10 = 1 ÷ 10 = 0.1 ÷ 10 = 0.01 6 × 6 × 6 = 216) 1 4 g -3 4 f 10 = 1 ÷ 10 = 0.1 ÷ 10 = b 4 = 1 × 4 × 4 × 4 × 4 = 256 (compare 1 4 1 2 5 3 3 a b c 0.01 ÷ 10 = 0.001 with 4 × 4 × 4 × 4 = 256) d 1 e f 4 3 5 3 This is a useful discussion point. For square 4 Some trial and error should lead students to 1 6 numbers, the method of saying “multiplied see that the base number always remains by itself” works. However, for higher the same when the exponent is 1. The exponents, such as the power of 4, it is not method in question 3 is a simple way to see 4 1 this. For example, 5 = 1 x 5 = 5 and so on. correct to say, “3 = 3 times itself 4 times” 160 OX FOR D U N I V E RSI T Y PR E S S

c 45 squares blue d 10 squares green b 1.405 kg c 9.478 kg d heart e Unit 2: Topic 2 Unshaded amount is 1 e Teacher to check but the right must be or 0.25. 4 0.07kg heavier and the two masses 4 0.045, 0.145, 0.415, 0.45, 0.451 Guided practice must be close (e.g. right: 0.58 kg, left: 5 3 4 4 2 8 1 a 0.75 0.51 kg) 0.1 1 a (or ) b =1 0.30 2 c 0.0 9 a 6 3 7 7 0.4 05 0.25 (0) 1 0.0 9 9 f 0.992 kg g 0.943 kg 10 0.01 6 22 1 8 4 b =1 (or 1 ) d (or ) 4 84 2 10 5 3 5 3 2 1 3 8 4 10 + = b – = c 8 4 4 d 9 10 0 Independent practice 405 Unit 2: Topic 5 10 0 0 5 8 4 4 e 8 5 1 a 5 b (or ) c 7 2 d 7 e f 9 3 10 5 4 5 10 5 6 (or ) 25 0 7 10 0 0 12 6 f 8 9 5 10 7 6 g h 1 (or ) i 99 Guided practice 8 10 0 0 10 g 4 1 6 2 6 1 a (or ) b or c or 1 1 a 396 b 39.6 d e f 10 0 8 2 9 3 12 2 h 1 2 6 3 or 4 7 10 5 2 a 8540 b 85.4(0) 4 3 1 9 g h or 6 6 2 Improper Mixed number Decimal 3 a 2982 b 298.2 fraction 4 1 7 3 1 1.75 10 1.3 a or b c or 1.25 4 a 18 438 b 18 4.38 d e 9 f 4.5 8 2 10 6 2 2.75 1.25 7 5 8 9 a 7 3 4 1 5 a 172 b 17.2 4 a 1 b 3 c 4 1 13 3 6 a 171 b 17.1 d 8 e 10 f 3 10 5 or 1 9 1 12 10 4 b 6 1 or 12 2 12 5 25 7 a 82 b 8.2 10 0 4 1 4 3 7 c 1 10 0 + Student shows that 9 = + = 3 9 9 9 450 50 8 a 204 b 20.4 10 0 12 4 1 12 3 1 d 4 (or equivalent) 10 0 a =1 or 1 b =1 or 1 8 8 2 9 9 3 275 75 Independent practice 10 0 16 4 1 6 2 1 e 2 (or equivalent) 12 c =1 or 1 d =1 or 1 10 0 12 3 4 4 2 125 0 25 0 1 a 4 4.1 b 85.6 c 63(.0) 10 0 0 16 6 3 9 3 1 10 =1 or 1 f 1 (or equivalent) 10 5 e f =1 or 1 10 0 0 6 6 2 d 91.5 e 8.6 4 f 10.36 8 4 1 g =1 h =1 8 3 3 g 8.1 h 7.71 i 51.8 Extended practice 4 1 1 a 1 or 1 b 1 j 864.2 k 7.725 l 95.13 9 8 2 1 a 0.1 b 0.25 c 0.7 d 1 e 3 f 1 2 1 2 1 4 10 0 10 0 0 c 1 or 1 d 2 or 2 2 a 5.3 b 12.1 c 12.4 10 5 4 2 6 3 9 e 2 or 2 f 3 d 19.5 e 1.05 f 2.37 10 8 4 2 a 0.2 b 0.125 c 0.75 2 6 3 g 3.19 h 12.1 i 14.38 g 1 h 2 or 2 d 0.375 e 0.8 f 0.875 9 8 4 j 12.47 k 2.57 l 0.527 9 1 11 1 a or 1 b or 1 3 0.33 recurring 4 0.16 8 8 10 10 3 a 174.4 b 327.6 c 36.95 3 1 3 c 2 or 2 d 3 5 Teacher to check rounding. To 3 decimal 8 6 2 d 128.01 e 324.24 f 29.824 5 1 4 places, 0.1428 becomes 0.143 e 2 or 2 f 4 9 10 2 g 722 (.0) h 558.45 i 40.175 7 8 4 11 8 12 a b 1 or 1 c 10 5 4 a $2 b $4 c $7.95 d $19.9 0 d 5 e 1 f 5 5 a $1 b $1.60 c $3 d $1.25 6 12 1 4 Unit 2: Topic 4 Extended practice Extended practice 1 Students should convert all fractions to Look for the strategies the student uses to 2 solve these problems. Discuss whether mental 5 4 3 6 18 Guided practice strategies would be more appropriate in some cases. eighteenths: + + + = =1 18 18 18 18 18 1 5 5 1 a 4166 b 41.66 1 2 1 a b c 2 a 45.2 b 4.37 c 29.12 d 2 e 12 f 8 g 3 3 1 1 5 1 10 4 2 7 d 52.3 e 1.75 f 26.97 3 1 $ 8 4.55 12 3 Answers will vary. For example, Independent practice 2 60 cm (0.6 m) 1 1 5 1 14 1 + + + = =1 1 a 6.02 b 9.36 c 63.936 3 $62.50 4 3 12 6 12 6 d 50.1 e 1.55 f 7.593 4 a Item Cost g 2.21 h 9.9 i 17.415 Sof t drink (1.25 L) $ 6.75 Unit 2: Topic 3 2 a $171.14 b $80.05 Juice (300 mL) $5.0 4 ($5.05) 3 a 54.91 m b 2.287 kg Potato crisps (50 g) $16.20 Guided practice 4 d 8.253 seconds Chocolate (150 g) $9.86 ($9.85) 1 a 3 b 69 5 33.92 m c or 0.03 or 0.69 10 0 10 0 Melon $ 3.8 4 ($ 3.85) 20 2 ( ) or 0.20 (0.2) (Discuss the 6 10 kg 10 0 10 Pies (4 in a pack) (Two packs of four pies are needed for 6 connection with students.) students.) $16.08 ($16.10) 7 8.61 m 2 a 2 b 8 c 125 Extended practice d 20 0 e 75 f9 b $57.80 if items are rounded or $57.77 1 Answers will vary, e.g. 0.2 + 4.62 + 4.36 = g 99 h 999 i1 without rounding. This would then be 9.18 or 0.9 + 4.92 + 3.36 = 9.18 j 10 k 10 0 l 250 rounded to $57.75. 2 a Independent practice Organ Mass c 64 cents (round to 65 cents) skin 10.886 kg d $57.80 ÷ 6 = $9.65 (rounded) liver 1.56 kg 1 Student shades: brain 1.4 08 kg lungs 1.09 kg hear t 0.315 kg $57.77 ÷ 6 = $9.65 (rounded) kidneys 0.29 kg a 5 squares b 35 squares spleen 0.17 kg pancreas 0.098 kg e $57.80 × 4 = $231.20 c 33 squares d 90 squares $57.77 × 4 = $231.08 (= $231.10 rounded) 2 a True b False c True d False e True f True g False h True i True j False k False l True 3 Student shades: a 15 squares red b 5 squares yellow OX FOR D U N I V E RSI T Y PR E S S 161

3 2500 (Five jumps would get to one. 50 jumps 3 The population of Victoria was approximately Unit 2: Topic 6 1 would get to 10, 500 jumps would get to 100 5 million people in 2009. 2 so 2500 jumps would get to 500.) 4 15% of the sheep in the Top 10 sheep Guided practice 4 a 600 b 750 c 10 0 0 countries are in Australia. 1 a 450 b 45 d 250 (150 ÷ 0.6) 5 The Great Sandy Desert covers one- 2 a 740 b 74 5 a $132.948 (rounded = $132.95) twentieth of Australia. 3 a 3750 b 375 b $13 294.80 6 About 13% of the world’s threatened animal c $1329.48 (rounded price = $1329.50) species are in Australia. 4 a 629 0 b 629 5 a 35 b 3.5 6 a 74 b 7.4 Unit 3: Topic 1 Unit 2: Topic 7 7 a 87 b 8.7 8 a 93 b 9.3 Guided practice Guided practice 9 a 326 b 23.5 1 a 8:4 b 3:6 c 9:3 1 a 9 b c 78.92 d 652 c , 0.9, 9% d 10 0 e f 2 a 2:1 b 1:2 c 3:1 10 a 2.35 b 4.275 99 , 0.99, 99% 10 0 d 1:4 e 2:1 f 5:1 80 8 4 c 0.35 d 0.02 ( or ), 0.8, 80% 10 0 10 5 g 1:3 25 1 ( ), 0.25, 25% Independent practice 10 0 4 50 1 ( ), 0.5, 50% 1 a 35 b 350 10 0 2 Independent practice 75 3 ( ), 0.75, 75% 2 a 67 b 670 10 0 4 1 a 9 b 15 c 24 3 a 53.8 b 538 2 a 0.02, 2%, student shades 2 squares 2 a 6 b 10 c 20 20 1 4 a 40.9 b 409 b ( ), 20%, student shades 20 squares c 10 0 5 3 a 1:3:5 b 1:3:2 c 2:3:1 35 7 5 a 0.45 b 0.045 ( ), 0.35, student shades 35 squares 10 0 20 4 Students should recognise that the ratio 6 a 0.79 b 0.079 d 0.7, 70%, student shades 70 squares of 3:1:2 means that there should be 3 blue squares for each yellow square and every 7 a 5.45 b 0.5 45 Independent practice 2green squares. The simplest solution is to 8 a 6.27 b 0.627 1 follow this pattern. However, students may Fraction Decimal Percentage 0.15 15% 9 a 245 b 1737 0.22 22% 0.6 6 0% 0.0 9 9% choose to colour any 12 squares blue, any 0.9 9 0% a 15 0.5 3 5 3% 4squares yellow and any 8 squares green. 0.5 5 0% 10 a 34.161 b 0.0 01 (or equivalent) 0.25 25% 10 0 0.0 4 4% 0.75 75% 11 a 130 0 b 26 0 0 c 3570 b 22 0.2 20% 5 a Look for students who use appropriate (or equivalent) 10 0 d 1270 e 15 470 f 72 950 6 vocabulary, such as, “For every yellow c (or equivalent) 10 bead there are 3 red beads and 4 blue g 96 300 h 25 400 d 9 10 0 beads.” By this stage, students should 12 a 0.4 32 b 0.529 c 0.8 41 9 be beyond simply counting the number 10 e d 0.697 e 1.485 f 3.028 of beads in each colour. g 10.4 36 h 99.999 f 53 10 0 b 1:3:4 13 g 1 × 10 × 10 0 × 10 0 0 (or equivalent) 2 6 a&b Students might see that they can h 1 substitute the yellow beads in 4 a 1.7 17 170 170 0 229.5 2295 22 950 30.2 302 3020 4 question 5 for green and continue 4 4.2 4 42 4 420 b 22.95 5 7.9 3 579.3 5 79 3 i (or equivalent) 215.78 2157.8 21 578 10 0 3 3 0.0 8 3 3 0 0.8 33 008 2 9 0.05 2 9 0 0.5 29 005 the pattern. In this case, they could c 3.0 2 3 j describe the pattern in the following 4 d 4.42 1 5 way: “For every green bead there k e 5.79 3 are 3 red beads and 4 blue beads.” 2 a True b True c False The ratio would be 1:3:4. f 21.578 d True e False f False Students requiring a greater g 3 3.0 0 8 g False h True i True challenge can be encouraged to use h 2 9.0 05 1 6 a different ratio, such as 1:2:5, with 3 a 20%, , 0.3 b 0.07, , 69% c 4 e 10 3 green beads, 6 red beads and 15 14 ÷ 10 ÷ 10 0 ÷ 10 0 0 2 d 1 7.4 0.74 0.074 0.7 0.07 0.0 07 , 17%, 0.2 4%, 0.14, blue beads. 1.8 0.18 0.018 10 0 4 32.5 3.25 0.325 2 9 6.7 29.67 2.9 6 7 1 3 3 6 8.2 36.82 3.6 8 2 a 74 14 5 6.2 14 5.6 2 14.5 6 2 10%, , 0.5 f , 39%, 0.395 7520.8 75 2.0 8 75.2 0 8 10 5 7 Flour Milk Eggs Number of pancakes b 7 4 Matching sets are: c 18 1 • 5%, 0.05, 120 g 250 mL 1 8 20 24 0 g 16 d 325 480 g 32 8 720 g 48 • 8%, 0.08 and 60 g 500 mL 2 3 e 2967 10 0 8 • 80%, 0.8 and 10 1000 mL or 1L 4 f 3682 5 Percentages: 10%, 30%, 60%, 75%, 95% 1 1 3 9 1.5 L 6 4 125 mL g 14 562 Fractions: (or equivalents) 1 10 2 4 10 h 75 208 6 a 5% b 22% c 4 4% d 59% e 72% f 99% 8 a Students who have had experience Extended practice 7 Student draws smiley face and arrow if nding the highest common factor 1 225 × 4 ÷ 1000 pointing to the point approximately mid-way of a set of numbers will use this skill between 0.8 and 0.9. 2 a 0.936 (312 × 3 = 936. to work out the ratio of the animals Divide 936 by 1000 = 0.936) 8 a Triangle: 30% b Star: 40% as 3:8:1:2. O thers will probably use a b 9.36 (312 × 3 = 936. c Circle: 70% d Hexagon: 90% process of trial and error. Divide 936 by 100 = 9.36) b Having worked out the ratio, students Extended practice c 0.609 (203 × 3 = 609. should see that Zoe has 4 ducks, 1 2% of the world’s cattle is in Australia. Divide 609 by 1000 = 0.609) 6 sheep, 16 goats and 2 horses. d 8.004 (4002 × 2 = 8004. 2 4 of Australian mammal species are found Divide 8004 by 1000 = 8.004) 5 nowhere else in the world. 162 OX FOR D U N I V E RSI T Y PR E S S

Extended practice 5 Extended practice Position 1 2 3 4 5 6 7 8 9 10 1 a 3 b 75% c 0.75 Number 2 5 8 11 14 17 20 23 26 29 1 Teachers may wish to discuss the use of 4 a formula in order to simplify a rule before 2 Students may need to revisit the procedure 6 Position 1 2 3 4 5 6 7 8 9 10 students start this work. for nding a fraction of a quantity where the numerator is larger than 1. Number 1 4 9 16 25 36 49 64 81 10 0 a 32 b 40 c 80 d 20 0 a 2 oranges, 8 apples 2 a 10 Rule: Teacher to check, e.g. to nd the b n = 2 + t × 2 or n = t × 2 + 2 (because, number, you square the position number or b 5 oranges, 20 apples to nd the number, you multiply the position number by itself. apart from the two end tables, only two c 10 oranges, 40 apples sides of each rectangle can be used). d 7 oranges, 28 apples Note, the above formulae follow the 7 Depending on the experience and ability 3 a There are ve “portions” and each pattern of the other formulae in this topic. levels of the students, teachers may portion is one-fth. One-fth of 45 However, other formulae are possible, wish to discuss and work through the pieces of fruit is 9 pieces of fruit such as n = 6 + (t – 2) × 2. Teacher to ow chart and model an example to (45 ÷ 5 = 9). So, 3 x 9 = 27 oranges decide whether discussion on this is show how the chart works. The division and 2 x 9 =18 apples. appropriate. examples allow for various pathways b There are seven “portions” and each c a) 18; b) 22; c) 42; d) 102 through the ow chart. portion is one-seventh. One-seventh of 3 a 7 b 9 c 12 d 22 a 114 r2 b 137 c 86 r1 56 pieces of fruit is 8 pieces of fruit 4 a Teacher to check formula, e.g. n = t + 2 8 Teachers may wish to use this activity as a (56 ÷ 7 = 8). So, 3 x 8 = 24 oranges and or n = 2 + t (because, apart from the cooperative group activity. The ow chart is 4 x 8 = 32 apples. two end tables, only one side of each a basic one. A successfully completed chart c There are four “portions” and each triangle can be used). See also note in may look like this: portion is one- quarter. One- quarter question 2b. of 32 pieces of fruit is 8 pieces of fruit b 22 (32 ÷ 4 = 8). So, 1 x 8 = 8 oranges and How to tell if a numb er is divisible 3 x 8 apples = 24 apples. by 3 Unit 4: Topic 2 d There are eight “portions” and each portion is one- eighth. One- eighth of 72 pieces of fruit is 9 pieces of fruit Add the digit s together Guided practice (72 ÷ 8 = 9). So, 3 x 9 = 27 oranges 1 17 and 5 x 9 apples = 45 apples. 2 a 7 b 10 c 21 Is the d 6 e 4 f8 answer Unit 4: Topic 1 g 14 h 10 divisible by 3? Y ES NO 3 a 12 b 12 c 6 Guided practice d 1 e 21 f 29 4 2 The number g 12 h 36 is not divisible 1 a 4 × 5 = 20 b 6 × 4 = 24 The number is divisible by 3. by 3. 4 a 15 b 10 c 8 2 Position 1 2 3 4 5 6 7 8 9 d 2 e 10 0 f 18 Number 10 9.5 9 8.5 8 7.5 7 6.5 6 g 20 h 30 Rule: Subtract 0.5 3 a Yes (because 24 is divisible by 4). Independent practice b Yes (because 16 is divisible by 4). c No (because 42 is not divisible by 4). 1 a 5×2=2+8 b 5 × 5 = 30 – 5 c 24 ÷ 2 = 4 × 3 e d 20 + 7 = (4 + 5) × 3 1 of 6 + 5 = 24 ÷ 3 4 Teacher to check, e.g. Is the last digit even? 2 2 Problem Split the problem to Solve the problem Answer make it simpler Independent practice 1 Teacher to check, e.g. because only e.g. 27 × 3 = (20 × 3) + (7 × 3) = 60 + 21 = 81 3 sticks are needed for each square after the a 23 × 4 = (20 × 4) + (3 × 4) = 80 + 12 = 92 rst one. b 19 × 7 = (10 × 7) + (9 × 7) = 70 + 63 = 13 3 2 Number of sticks c 48 × 5 = (4 0 × 5) + (8 × 5) = 200 + 40 = 24 0 a 1 + 4 × 3 = 12 × 1 = 13 d 37 × 6 = (30 × 6) + (7 × 6) = 180 + 42 = 222 b 1 + 6 × 3 = 1 + 18 = 19 e 29 × 5 = (20 × 5) + (9 × 5) = 100 + 45 = 14 5 c 1 + 8 × 3 = 1 + 24 = 25 f 43 × 7 = (4 0 × 7) + (3 × 7) = 280 + 21 = 301 3 True g 54 × 9 = (50 × 9) + (4 × 9) = 450 + 36 = 486 4 Teacher to check, e.g. 3 a You use 3 sticks for every triangle. Problem Change the order to Solve the problem Answer make it simpler b You start with 1 stick and then use 2 sticks for every triangle or start e.g. 20 × 17 × 5 = 20 × 5 × 17 = 100 × 17 = 170 0 with 3 sticks for the rst triangle a 20 × 13 × 5 = 20 × 5 × 13 = 100 × 13 = 13 0 0 and then use 2 for every other b 25 × 14 × 4 = 25 × 4 × 14 = 100 × 14 = 14 00 triangle. c 5 × 19 × 2 = 5 × 2 × 19 = 10 × 19 = 19 0 c You use 6 sticks for every hexagon. d You start with 1 stick and then d 25 × 7 × 4 = 25 × 4 × 7 = 100 × 7 = 70 0 use 5 sticks for every hexagon or e 60 × 12 × 5 = 60 × 5 × 12 = 300 × 12 = 3600 start with 6 sticks for the rst f 5 × 18 × 2 = 5 × 2 × 18 = 10 × 18 = 18 0 hexagon and then use 5 for every g 25 × 7 × 8 = 25 × 8 × 7 = 200 × 7 = 14 00 other hexagon. OX FOR D U N I V E RSI T Y PR E S S 163

4 c Teacher may choose to discuss optical Problem Use opposites Find the value of ◊ Check by writing the equation ◊ = 35 – 15 ◊ = 54 ÷ 6 illusions. This particular optical illusion ◊ = 6 – 1.5 e.g. ◊ + 15 = 35 ◊ = 10 × 4 ◊ = 20 20 = 35 – 15 ◊ = 45 ÷ 10 ◊ =9 ◊ = 3.5 × 10 4.5 9 × 6 = 54 is called the Müller-Lyer illusion and ◊ = 1.5 × 4 a ◊ × 6 = 54 ◊ = 725 ÷ 100 40 4.5 + 1.5 = 6 there are various theories as to why we 4.5 1 35 perceive the lines as different in length. 6 of 4 0 = 10 b ◊ + 1.5 = 6 7.25 4 6 Allow for a tolerance of + / – 4 mm per c 4.5 × 10 = 45 d 1 rectangle and 3 mm for the triangle. Allow of ◊ = 10 35 ÷ 10 = 3.5 equivalent lengths. 4 6 ÷ 4 = 1.5 ◊ × 10 = 45 7.25 × 100 = 725 e ◊ ÷ 10 = 3.5 a 10 cm b 11.6 cm c 9.2 cm d 6.3 cm f ◊ ÷ 4 = 1.5 7 Teacher to check. g ◊ × 100 = 725 Extended practice 5 Problem Possible substitutes for ◊ Check 2 1 Tyrannosaurus Rex 12.8 m 2 5 × 3 = 25 × 3 = 75 6800 mm 3 e.g. 2 4 5 6 7 9 × 3 + 5 = 32 0.8 3 m 6 ◊ × 3 = 75 8 9 10 11 54 ÷ 9 – 5 = 1 290 cm 5 9 10 11 12 2 × 5 + 5 = 15 590 mm 7 2 3 15 ÷ 10 – 1.5 = 0 Iguanodon 3700 cm 1 5 10 4 5 24 × 10 – 12 = 228 3500 mm 4 a ◊ × 3 + 5 = 32 12 14 15 20 38 ÷ 2 = 16 + 3 50 cm 8 35 36 16 18 (5 + 2.5) × 10 = 75 1.5 2 37 38 Microraptor 2.5 b 54 ÷ ◊ – 5 = 1 3 Homalocephale c 2 × ◊ + 5 = 15 Saltopus d 15 ÷ ◊ – 1.5 = 0 Puer tasaurus e 24 × 10 – ◊ = 228 Dromiceiomimus f 2 Micropachycephalosaurus ◊ ÷2=4 +3 g (5 + ◊) × 10 = 25 × 3 2 Answers will vary. Look for students who come up with a plausible suggestion such Extended practice 3 5 cm 50 mm 420 mm 90 mm as a dog or a cat. 32 mm 1 a ◊ × 3 – 4 = 11, ◊ × 3 = 11 + 4, ◊ = 15 ÷ 42 cm 75 mm 125 mm 124 mm 3 Puertasaurus 99 mm 3, ◊ = 5 9 cm 4 Answers will vary, e.g. about 26 b ◊ × 10 – 15 = 19, ◊ × 10 = 19 + 15, ◊ × 3.2 cm (37 ÷ 1.4 = 26.43). Look for students who 10 = 34, ◊ = 34 ÷ 10, ◊ = 3.4 7.5 cm can make a reasonable estimate of the 2 a 16 height of Year 6 students and use this to 12.5 cm b Answers may vary, e.g. (10 + 2) × 4 – 2 come up with a plausible response. 12.4 cm = 46 or 10 + 2 × 4 – 2 = 16 5 Answers will vary. Look for students who 9.9 cm c Answer will depend on the calculator are able to make a reasonable estimate of used. Most basic calculators will not 4 Teacher to check and possibly ask student their own height and to accurately calculate be programmed to follow the order of to justify answers. the difference between their given height operations and will give the answer 46. and that of the microraptor. a cm or mm b cm d Teacher to check, e.g. (10 + 2) × 4 – 2 c mm d m 6 Teacher to check and decide on level = 46, 10 + 2 × (4 – 2) = 14, (10 + 2) × of accuracy that is required. Looking at e m or cm f km (4 – 2) = 24 problem-solving strategies may be seen to 3 Answers will vary, e.g. 4 – 4 + 4 – 4 = 0; (4 ÷ Independent practice be more important than absolute accuracy. 4 – 4) + 4 = 1; 4 ÷ 4 + 4 ÷ 4 = 2; (4 + 4 + 4) ÷ 1 Answers may vary. Teachers could ask 4 = 3; (4 – 4) × 4 + 4 = 4; (4 × 4 + 4) ÷ 4 = 5; students to justify answers. (4 + 4) ÷ 4 + 4 = 6; 4 + 4 – 4 ÷ 4 = 7; 4 – 4 + Unit 5: Topic 2 a The length of a pencil 157 mm 15.7 cm 4 + 4 = 8; 4 ÷ 4 + 4 + 4 = 9 b The height of a Year 6 1.57 m 157 cm student Guided practice Unit 5: Topic 1 c The length of a nger nail 15 mm 1.5 cm 2 157 m 0.157 km 8 cm 1 d The distance around a 2 12 cm 2 school yard Guided practice 3 2 10 cm e The length of a bike ride 1570 m 1.57 km 4 a 2 b 2 c 2 4 cm 1 4 km 4000 m 7000 m 19 000 m 2 2 2 2 6000 m 2 cm 9 cm 18 cm 7500 m mm cm and mm cm 5 a b c 3500 m 4 cm 5 mm 4.5 cm 7 km 4250 m 7 cm 5 mm 7.5 cm 9750 m 8 cm 2 mm 8.2 cm a 45 mm 6 cm 9 mm 6.9 cm 19 km Independent practice b 75 mm 6 km 1 a 2 L: 3 cm, W: 2 cm, A: 6 cm c 82 mm 2 L: 3 cm, W: 5 cm, A: 15 cm 7.5 km b d 69 mm 2 2 2 40 m 63 m 150 m 3.5 km 2 a b c 4.25 km 3 Teacher to check. Note: accuracy in 3 a 2 b 2 c 2 21 m 56 m 24 m measuring is less important than the ability 9.75 km 4 Students’ own answer. Look for students to convert between the units of length. who show an understanding of why the 4 a Teacher to check estimates. Look for formula would not work, e.g. because the 2 1m 100 cm 4 00 cm 550 cm estimates expressed in the correct unit shape is not a rectangle. 250 cm 4m 710 cm 820 cm 156 cm and that are reasonable in comparison 2 2 2 75 cm 20 cm 25 cm 18 cm 5 a b c e 2 5.5 m 16 cm with the length of Line B. 2 16 cm d 2.5 m b Line A: 6.8 cm, Line C: 9.2 cm 6 2 5000 m 7.1 m 5 a Teacher to check estimates. Look for 7 a 2 b 2 20 000 m 40 000 m 8.2 m estimates that are reasonable given the c 2 50 000 m 1.56 m length of Line B. 8 2 30 cm × 21 cm = 630 cm b All lines are 6 cm long. 0.75 m 164 OX FOR D U N I V E RSI T Y PR E S S

5 Teacher to check. Look for students who are Extended practice Unit 5: Topic 4 able to describe the relationship between 2 2 1 a ABCD = 20 cm , ABC = 10 cm the total mass of the pad and the 2 2 b EFGH = 18 cm , EFG = 9 cm c d mass of each sheet, e.g. nd the mass IJKL = 21 2 JKL = 10.5 1 2 Guided practice cm , cm (10 ) 2 of 100 sheets and divide the answer 2 2 MNOP = 16 cm , NOQ = 8 cm by 100. 1 a 5t 5000 kg 7500 kg 2 a 2 1250 kg b 12 cm 2355 kg c 995 kg 6 Check that the total equals 1.85 kg and 1 2 7.5 t cm 7.5 (7 ) also that the masses are appropriate. (For 2 1 2 1.25 t cm 12.5 (12 ) example, one item at 1.84 4 kg and the 2 2.355 t remaining six items at 1 g would not be 0.995 t appropriate.) 7 a 62.5 kg Unit 5: Topic 3 b 3.5 kg 3500 g 4500 g 850 g b Twelve Year 6 students. (12 × 40 kg = 250 g 4.5 kg 3100 g 480 kg, 13 × 40 kg = 520 kg) Guided practice 0.85 kg 8 Check that total equals 1 kg and also that 3 3 0.25 kg the masses are appropriate. (For example, 8 cm 16 cm 1 a b 3 3 one mango at 985 g and the remaining three 12 cm 12 cm c d 3.1 kg at 5 g would not be appropriate.) 3 3 2 a 12 cm , 3, 36 cm c 3 3 5.5 g 5500 mg 3750 mg b 8 cm , 2, 16 cm 1100 mg Extended practice 355 mg 3.75 g 1 mg 3 a 1 Teacher to check. Look for students 3 kL 3000 L 9000 L 3500 L 1.1 g 6250 L who are able to come up with a strategy 9 kL 0.355 g that connects millilitres and grams and 3.5 kL demonstrates their understanding of mass. 0.001 g 6.25 kL It is unlikely for normal primary classroom 2 a– d Multiple possible answers, e.g. equipment to be accurate enough to prove b 2L 2000 mL a: a truck, b: a person, c: our, d: a grain 7000 mL 5750 mL that 1 mL of water has a mass of 1 g. 4500 mL of salt. Look for students who make 7L This could be a useful discussion point for appropriate choices for each unit of mass students. 5.75 L and who can justify their answers. 2 a potato crisps 4.5 L 3 b breakfast cereal has 40 mg more sodium. c 3 kg and kg and kg and g 500 cm fraction decimal 500 mL 3.5 kg 3 kg 500 g (However, students could be asked to 3 225 mL 2.5 kg 2 kg 500 g 225 cm 1L 1 3.25 kg 3 kg 250 g consider the normal serving size.) 1750 mL or 1.75 L 4.7 kg 4 kg 700g 3 a 3 kg 1.9 kg 1 kg 900 g 1000 cm 2 3 1750 cm c 505 mg (2 × 135 mg + 55 mg + 180 mg) 1 b 2 kg 2 d 216 mg ( Total = 2516 mg less 2300 mg 1 c 3 kg = 216 mg) 4 7 d 4 kg 10 Independent practice 9 e 1 kg 3 10 Unit 5: Topic 5 15 cm 1 a 15 b 2 Teacher to check. Look for students who Independent practice are able to nd the correct answer using a Guided practice 3 1 a 1 kg 700 g (or equivalent) reliable strategy, e.g. because 6 cm would t on the top layer and there are two layers b 4 kg 250 g (or equivalent) 1 a 1:20 pm, 1320 the same. c 850 g (or equivalent) (Allow a tolerance b 6:48 pm, 1848 of +/– 10 g) c Clock to show 2:42, 0242 3 Teacher to check, e.g. you multiply the width by the length to nd the number 2 Most likely answers are below. Look for d 11:07 pm, 2307 of cubes that will t on the one layer and students who can explain why they would e Clock to show 10:22, 10:22 pm then multiply that by the height to nd the choose the particular scale to nd the mass f 6:27 am, 0627 volume. ( V = L × W × H) of each item. g Clock to show 10:35, 1035 4 a 3 b 3 c 3 a Scale C h Clock to show 11:59, 11:59 pm 40 cm e 18 cm 48 cm d 3 3 3 b Scale B 48 cm 80 cm f 180 cm Independent practice c Scale A 5 B, C, A, D, F, G, E 1 54 minutes d Scale B or C 6 a 1400 mL b 1500 mL c 1300 mL 2 Train 8221 3 The scale has 50 g increments so the d 1250 mL e 750 mL f 1350 mL pointer should be between 900 and 950. 3 9 minutes Teacher to check that the shading is Teacher to decide on the required level of appropriate. 4 Train 8215 accuracy. 5 Because the train only stops there to pick up Extended practice passengers. 1 The most likely answer is millimetres. 6 55 minutes Students could be asked to justify their 7 responses. Station Time 16 3 0 2 3 0 Southern Cross 16 3 8 30 m × 3 m × 0.15 m = 13.5 m kg 16 5 6 170 4 Footscray 1710 1714 3 Teacher to check. Look for students who 1716 1720 1 1724 50 0 are able to accurately follow the directions Werribee and who make the link between volume Lit tle River and capacity to arrive at a plausible answer. L ara Note: equipment in primary schools is Corio not usually accurate enough to prove that 4 a B, A, C, D 3 Nor th Shore 1 mL water has a volume of exactly 1 cm b A & B (exactly 5 t) Teachers may choose to discuss this with c C & D (5.945 t) Nor th Geelong students. Geelong OX FOR D U N I V E RSI T Y PR E S S 165

8 d Right-angled isosceles triangle. Two f hexagonal pyramid 195 0 1955 sides are the same length. One angle is g square prism 19 6 0 19 6 5 a right angle. 1970 1975 h cone 19 8 0 19 8 5 e Scalene triangle. All sides are different 19 9 0 19 9 5 i triangular pyramid 2000 lengths. All angles are different sizes. Fir s t spac e craf t – Sputnik 1 Independent practice F ir s t animals in space F ir s t humans in space 2 Teacher to check descriptions. For example: F ir s t space w alk F irs t person on t he Moon 1 a rectangular prism Fir s t spac e craf t on Mar s a Square. All the angles are right angles. F ir s t space shu t t le launched b square (-based) pyramid Space shu t t le C hallenger ex ploded F ir s t space pr obe t o Nep t une All the sides are the same length. F ir s t operat ional rover on ano t her b Trapezium. There are two obtuse angles. 2 Teacher to check. N ote: it may be plane t (Mar s) There is one pair of parallel sides. necessar y to discuss the reason for c Rectangle. A ll the angles are right adding tags to some of the faces. T he angles. Two pairs of sides are the page could be enlarged by photocopying. same length. A lternatively, students could copy the nets onto grids of a larger size. d Parallelogram. There are two pairs of parallel sides. There are two obtuse and 3 Teacher to check. N ote: teachers will two acute angles. probably wish to provide ex tra paper e Rhombus. A ll the sides are the same for additional practice and to reassure length. T here are t wo pairs of parallel students that success is not necessarily sides. assured at the rst at tempts in such activities. Look for students who show 3 Teacher to check. Examples of similarities an understanding of the faces and edges and differences that can be observed: of 3 D shapes and are able to accurately Similarities Dif ferences reproduce them on isometric dot paper. a Neither shape The diameters on the circle are Extended practice has any straight all the same length but they are dif ferent on the oval. 1 ?krow waL s’reluE seoD lines. Name segde b They are both One has ve sides and the fo rebmuN other has eight sides. regular shapes. secit rev fo rebmuN c They each have The parallelogram has t wo at least one pair pairs of parallel sides but the secaf of parallel sides trapezium has only one. fo rebmuN 9 1971 (allow 1972 at teacher’s discretion) a rectangular 6 8 12 Yes 8 12 (14 – 12 = 2) Extended practice prism 5 5 d They are One has three pairs of parallel 5 sides but the other has only 4 6 1 a 3 hours b 2 hours 51 minutes one pair. b hexagonal 6 4 18 Yes 8 (20 – 18 = 2) both irregular 10 16 prism c 9 minutes hexagons. c square 8 Yes (-based) (10 – 8 = 2) 2 a 3:26 pm b Clock to show 3:26 e They are both One is regular but the other pyramid pentagons. is not. c 22 minutes f They are both One (the rhombus) has all sides d triangular 9 Yes parallelograms. of equal length but the other (11 – 9 = 2) 3 does not. Depar ts Arrives prism Big Town Small Town e triangular 6 Yes (8 – 6 = 2) g They are both One has obtuse and acute angles and the other has four right angles. pyramid Bus A 12 0 8 15 07 quadrilaterals. Bus B 15 33 18 32 Bus C 195 4 2 25 3 f square prism 12 Yes (14 – 12 = 2) h They are both The rst is a scalene triangle but the second is isosceles. g pentagonal 7 15 Yes The rst shape has a reex 10 (17 – 15 = 2) right-angled angle. The rst is a regular octagon but the second octagon is not. prism The rst shape has 10 sides but triangles. the second shape has 8 sides. h octagonal 24 Yes (26 – 24 = 2) i They both have prism a right angle. Unit 6: Topic 1 j They are both octagons. Guided practice k They both have Unit 7: Topic 1 at least four 1 a regular hexagon reex angles. b irregular quadrilateral Guided practice c regular quadrilateral (square) Extended practice 1 a 80 º, acute b 100 º, obtuse d irregular pentagon 1 a circumference b radius c 35º, acute d 145º, obtuse e regular octagon c diameter 2 Teacher to check that angles of 25º are f regular pentagon 2 a sector b quadrant drawn. (Decide whether to allow a tolerance g irregular triangle c semi- circle of x º.) Look for students who understand h irregular hexagon 3 Teacher to check and to decide on the level how to align the centre of the protractor 2 It has ve equal sides and some of the of accuracy that is appropriate. correctly along the baseline of the angle and angles are the same size. 4 8 who understand which set of numbers on the protractor to read. Independent practice 1 Teacher to check descriptions. For example: Independent practice a Equilateral triangle. A ll the angles are Unit 6: Topic 2 1 (Allow a tolerance of +/– 1º) the same size. A ll the sides are the a 125º b 165º c 99 º d 169 º same length. Guided practice 2 Teacher to check, e.g. The size of the b Isosceles triangle. Two angles are the same size. Two sides are the same “outside angle” is 40 º less than 360 º. 1 a rectangular prism length. b square (-based) pyramid 3 a 330 º b 315º c 215º d 265º c Right-angled scalene triangle. All the c triangular prism 4 a 95º b 112 º c 270 º d 120 º sides are different lengths. One angle is d cylinder e 333 º f 40º g 30 º h 120 º a right angle. e octagonal prism i 45º j 155º 166 OX FOR D U N I V E RSI T Y PR E S S

Extended practice Unit 8: Topic 2 Unit 9 : Topic 1 1 a a = 60 º, b = 180 º b a = 125º, b = 55º, c = 125º c a = 48 º, b = 132º, c = 132º Guided practice Guided practice d a = 50º 1 yellow triangle 1 a 42 b6 2 b = 38 º, c = 142º, d = 38 º 2 a (– 8,– 6) b (4,4) 2 2 e = 38 º, f = 142º, g = 38 º, h = 142º 3 green circle and yellow triangle 3 Accept from $310 to $315. i = 142º, j = 38 º, k = 142º, l = 38 º, 4 True m = 38 º, n = 142º, o = 38 º, p = 142º 4 9 5 a The student draws a line from the origin 3 Teacher to check and decide on level Independent practice point to the yellow triangle. of accuracy that is required. Looking at b (1,–1), (2,–2), (3,– 3) 1 a Teacher to check students’ graphs. problem-solving strategies may be seen to c&d Teacher to check. Look for students b 27 (Red & blue = 50. be more important than absolute accuracy. who demonstrate an understanding of Yellow and purple = 23. 50 – 23 = 27) the quadrant system and who interpret the coordinates correctly. Unit 8: Topic 1 Favo u r i t e c o l o u r s f o r Ye a r 6 Independent practice Key = 4 people 1 a (3,5) b (– 4,5) c (– 4,1) d (3,1) Guided practice 2 (– 6,5) → (–7,3) 1 a reection b translation c rotation 3 Starting point students’ choice. Endpoint 2 Teacher to check patterns. Look for must be the same as the starting point, e.g. students who are able to correctly identify (– 4,1) → (3,1) → (3,5) → (– 4, 5) → (– 4,1) and continue the transformation used in 4 Teacher to check drawing and coordinates. Red Ye ll o w Blue Green P ur p l e each pattern. (It can be an advantage to ask students Independent practice to give each other their plotted points to check that the drawings match the ordered 2 Teacher to check information. Totals must 1 Teacher to check descriptions. For pairs.) Look for students who demonstrate be 6 more than the information from example: an understanding of how ordered pairs question 1. a The hexagon has been translated work and who can use them to accurately horizontally. 3 a Teacher to check graph. T he most describe the points of their gure. appropriate increment for the ver tic al b T he triangle has been rotated 5 axis is 3, as the highest possible bar h o r izo nt all y. y 10 total is 3 2. c The hexagon has been translated b Answers will vary, e.g. a bar graph vertically. 9 because it is easier to work out the d The pentagon has been reected 8 numbers for each bar, or a pictograph vertically. 7 because it looks better. e The triangle has been reected horizontally and vertically. 6 f The arrow has been translated 5 horizontally. The second row is the same as the rst one, but it has been reected 4 horizontally. 3 2 Teacher to check pattern. Look for students 2 who can accurately use the language of 1 transformation to describe the pattern. x 3 Teacher to check patterns. (Shapes are –1 1 2 3 4 5 6 7 8 9 10 –1 coloured to simplify the identication of the patterns.) Examples of possible 6 Teacher to check. descriptions: a T he shape has been reected Extended practice horizontally on the rst row. T he y 1 second row is a ver tic al reection of 10 the rst row. 9 b The shape is rotated through 180 º 8 clockwise on the rst row (or has been 7 J H reected horizontally then vertically D 6 on the rst row). The second row is a 5 A vertical translation of the rst row. G F 4 B c The shape is translated horizontally on 3 the rst row. The second row is a vertical 2 reection of the rst row. L O 1 x 4 Teacher to check pattern. Look for students –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10 –1 who are able to successfully demonstrate their understanding of transformations I –2 C N through the construction of their pattern. –3 K –4 Extended practice –5 1 & 2 Teacher to check. Look for students –6 E who are able to demonstrate prociency with digital technologies to construct a M –7 pattern that draws on their understanding –8 of transformations. –9 –10 OX FOR D U N I V E RSI T Y PR E S S 167

4 a students in the town think (although neither could the newspaper editor!). Graph to show the hourly temperatures at the ski resort 9°C 6 a About $130 to $135 b four c About $42 8°C 7 a sample b Teacher to check e.g. “No, because it 7°C was only based on the views of 200 people”. This issue could be used as a basis for a group or class discussion. 6°C 9 c 180 ( of 200) 10 erutarepmeT Extended practice 5°C 1 a sample b secondary 2 a Any number over 50 4°C b 49% c Teacher to check, e.g. the public has a 3°C right to know how many people were surveyed and what type of people they were. 2°C 3 a Teacher to check. Look for students who show an understanding of data collection 1°C and the importance of accurately representing data and its sources; e.g. the people surveyed (all students) were 0°C not a fair representation of public opinion. 0700 0800 0900 1000 1100 1200 1300 1400 1500 1600 1700 b Teacher to check. Look for students Time who show an understanding of how data collection inuences results and c an apply this to the situation, e.g. b Teacher to check. Look for students on each. For example, the top boys’ name (Jack) was deliberately not included, whereas the top girls’ name was. See more c arr y out a sur vey of 10 0 people of at https://online.justice.vic.gov.au/bdm/ who comment on the information popular-names. various ages and backgrounds. (rather than on the appearance of c Teacher to check , e.g. it is based on the graph). For example, the lowest truth, but it c annot necessarily be temperature was at 7 am, OR the trusted as a fair reection of public temperature remained the same 4 a Any number between 1550 and 1649. opinion. T his is a possible group or between 1200 and 1300 (not “I like the (Actual number was 1596). class discussion point. way the lines go up and down”). b Teacher to check, e.g. there were 4 a Teacher to check. Look for students 5 Teacher to check. Look for the way that more than 1500 more babies with who c an of fer reasoning to the student labels the graph and, if the top name for boys than there were demonstrate their understanding of necessary, whether an appropriate scale has for girls. manipulation of sur vey responses, been used. An appropriate way to present e.g. bec ause it probably inuenced the information would be on a dot plot, a bar the answers that people gave. graph or a pictograph. Students who choose Unit 9 : Topic 2 b Teacher to check. Look for students a line graph have not understood that this is who choose a question that is likely used to show how recorded data changes Guided practice to result in the collection of more over a period of time. accurate data, e.g. Do you think a 1 a secondary b primary Extended practice fast- food restaurant should be opened 2 Some questions may have two possible 1 a Annabelle, Jade and Eva near the high school? answers. Teachers may wish students to b Approximately one-sixteenth justify their responses. Likely answers are: 2 a Numbers for each name a sample b census b Any two numbers around 600 that have c sample or census (depending on size a difference of 14. ( The actual numbers of school) Unit 9 : Topic 3 were 612 named Eva and 598 named d sample Annabelle.) 3 a primary b sample 3 Teacher to check, e.g. Similarities: Half Guided practice of each graph is taken up with one name. Independent practice 1 a Range is 16% to 80% = 64% The proportion of the second most popular 1 Secondary data (the class teacher did the b Range is 75 cm to 150 = 75 cm name to the most popular name is about survey). 2 a 35% b 76 the same on each graph (half). There are 5 1 2 a 5( ) b 1( ) 8 8 3 a 4 4% b 16 two names on each graph that are about as 3 Yes, because the principal wrote that it popular as each other. 4 a 30% b 15 was the majority of the students who were Differences: The fraction of the second surveyed. most popular name is slightly smaller for girls than it was for boys. 4 It could be true (because it is not clear what the other students would have answered). Note for teachers: The information 5 … and could be true. Teachers may wish to was taken from a Victorian government website. The data was deliberately chosen open up a discussion about this as it raises to make the graphs look similar. The names are all from the top 100, but the positions issues about data in the media. Many may of the names in the top 100 are different disagree with the newspaper headline but we cannot be sure what the majority of 168 OX FOR D U N I V E RSI T Y PR E S S

Independent practice Unit 10 : Topic 1 1 Week Seven-day minimum Order Range Mode Median Mean temperatures Guided practice 1 3 º, 6 º, 7 º, 9 º, 7 º, 8 º, 2 º 2 º, 3 º, 6 º, 7 º, 7 º, 8 º, 9 º 7º 7º 7º 6º 2 7º 7º 6º 5º 3 4º 8º 8º 9º 1 Teacher to check. Answers may vary and 4 8º 10 º 8º 9º 1 º, 3 º, 2 º, 9 º, 7 º, 7 º, 6 º 1 º, 2 º, 3 º, 6 º, 7 º, 7 º, 9 º students could be asked to justify their 9 º, 6 º, 8 º, 8 º, 10 º, 7 º, 8 º 6 º, 7 º, 8 º, 8 º, 8 º, 9 º, 10 º answers. Probable answers are: 10 º, 9 º, 10 º, 8 º, 7 º, 3 º, 2 º 2 º, 3 º, 7 º, 8 º, 9 º, 10 º, 10 º a even chance b highly likely c impossible d likely 2 Number set Order Range Median e certain f highly unlikely 8, 2, 6, 4, 10 2, 4, 6, 8, 10 8 6 25, 14, 17, 12, 6, 4 4, 6, 12, 14, 17, 25 g unlikely 12, 8, 2, 6, 2, 5, 21 2, 2, 5, 6, 8, 12, 21 21 13 a 19 6 2 1 b c (1 out of 10) 10 d 3 50% 4 0.3 82, 23, 3, 8, 15, 3, 16, 2 2, 3, 3, 8, 15, 16, 23, 82 80 11.5 or 11½ Independent practice 1 15% 3 Number set Mode Mean 2 Students may choose a fraction, a decimal 8, 2, 6, 4, 10 None 6º a 25, 14, 17, 12, 6, 4 None 13 and a percentage in any order but possible b 12, 8, 2, 6, 2, 5, 21 2 8 c 82, 23, 3, 8, 15, 3, 16, 2 3 19 answers are: d 2 1 a (or ) b 0.4 c 10% 10 5 3 There are 8 out of 10 ways the spinner will not land on green. Answers should be any 8 4 or all of , , 0.8 or 80%. 10 5 4 Teacher to check the appropriateness of student responses. Look for students who demonstrate an understanding of the 4 a This could be done as a ‘think, pair, Extended practice language and application of probability and share’ activity, with students sharing 1 a Yes, technically Sam is correct because who are able to justify their responses. their thoughts before arriving at their 10 occurs more frequently than the other responses. 5 Teacher to check. Sectors should be scores. coloured as follows: The modes should be fairly easy to b Answers will vary, teacher to check. A estimate as (8 hours for Sydney and 6 hours for London). • yellow: 2 sectors • blue: 3 sectors possible answer is that more than half of • green: 2 sectors • white: 2 sectors the scores are less than 10, with two of • red: 1 sector Looking at the high and low points for the scores being very low. each city is likely to result in answers of around 6 or 7 hours of sunshine a day 6 a 0.8 b 7 c 0.07 for Sydney and 3, 4 or 5 hours a day of e 10 sunshine for London. c The median score is 7 out of 10. 3 d 4 4 f 8% 10 4 d The mean score is 6 out of 10. 10 7 Teachers may choose to ask students to 8 2 should be red, 4 should be yellow and reect on whether Sam’s achievement b level is best reected by the mean, median or mode. Sydney London 6 should be blue Mode 8 hours 6 hours 9 A: 25 blue & 75 yellow Mean 88 ÷ 12 = 7.33 50 ÷ 12 = 4.166 B: 60 blue & 40 yellow (rounded to 7 (rounded to 4 hours) hours) 2 a The range is 19 (20 – 1 + 19). C: 90 blue & 10 yellow b 19 c 19 D: 50 blue & 50 yellow d The mean score is 155 ÷ 10 = 15.5 or c This could take the form of a class or 1 Extended practice 15 2 group discussion. Answers will vary. e Teachers may choose to use this task for 1 a 37 b $ 37 Responses will likely revolve around a group discussion about why the mean c Answers may vary, e.g. because the the difculty to accurately estimate the score does not reect Sam’s ability. boss only gives back $36 of the $37. mean without doing a calculation. The score of 1 out of 20 could be for a d $10 0 0 d Sydney. The mode (8) is only slightly variety of reasons ranging from lack of 2 a Answers may vary, e.g. 18 out of 37 is more than the median (7.5) compared to effort to not feeling very well. When London, where the mode (6) is 1.5 hours interpreting data in the real world, an almost the same as 18 out of 36, and 18 1 out of 36 = 2 different to the median (4.5). anomaly (or outlier) is often ignored in b 19 order to give a truer interpretation of the e London. The mean for October to March c Students’ own responses, e.g. because data. Students could be further extended is 16 ÷ 6 = 2.66 hours a day, rounded $37 was collected but the boss only by carrying out a similar activity for their to 3 hours. The mean for April to paid back $36. Look for students who own assessments. September is 34 ÷ 6 = 5.66 hours a day, understand the probability of landing rounded to 6 hours. The difference is an 3 This could be carried out as a group activity. on black and can apply this to supply a average of 3 hours a day less sunshine in Multiple answers are possible. Look for plausible response. the colder months. students who total the four temperatures d $10 000 (108) and subtract this from 203 (7 x 29). f Sydney. The mean for October to The answer of 95 needs to be divided March is 46 ÷ 6 = 7.66 hours a day, rounded to 8 hours. The mean for April appropriately between the three remaining to September is 42 ÷ 6 = 7 hours a day. The difference is 1 hour a day less days. For example, 31 ºC, 32 ºC and 32 ºC, sunshine in the colder months. instead of 93 ºC, 1 ºC and 1 ºC. OX FOR D U N I V E RSI T Y PR E S S 169

5 Probable number of times for each total: Unit 10 : Topic 2 Independent practice 12: 2 11: 4 10: 6 9: 8 1 a 2 8: 10 7: 12 6: 10 5: 8 b one (1 + 1) Guided practice 4: 6 3: 4 2: 2 2 5 Total Ways the dice can land Total number 1 a or 5 out of 6 of ways 6 6 Answers may vary, but likely answers are: of two b Students’ own responses. Look 3 11 6 20 20 20 dice Spinner 1: yellow, blue, red; for students who demonstrate an 1 7 4 12 6+6 1 Spinner 2: yellow, blue, red. 12 12 12 understanding of probability and the fact 11 6 + 5, 5 + 6 2 that, although there is a greater chance Extended practice of not rolling a 6, it is still possible, e.g. 10 6 + 4, 4 + 6, 5 + 5 3 1 a 1 out of 7 b one counter each number has the same chance so 9 6 + 3, 3 + 6, 5 + 4, 4 + 5 4 2–3 Teachers may wish to model this game there is as much chance for 6 as for 8 6 + 2, 2 + 6, 5 + 3, 3 + 5, 4 + 4 5 with students and discuss the implications every other number. 6 + 1, 1 + 6, 5 + 2, 2 + 5, 4 + 3, 6 of the game, which demonstrates why the 3+4 7 2 Answers will vary. This could prove an only sure, long-term winner in a gambling interesting group or class discussion point, 6 5 + 1, 1 + 5, 4 + 2, 2 + 4, 3 + 3 5 situation is the “banker”. It may be with students being asked to justify their 5 4 + 1, 1 + 4, 3 + 2, 2 + 3 4 necessary to allow more than ten rounds of responses. Look for students who can the game to establish a pattern. Teachers 4 3 + 1, 1 + 3, 2 + 2 3 explain why different students obtained will also decide whether to “tweak” the different results using the language of 3 2 + 1, 1 + 2 2 rules so that each player must choose a probability. 2 1+1 1 different number each time. In this case 3 The probability for each number is 6. The the banker’s balance is certain to increase 3 7 (6 out of 36 ways) likelihood of this occurring is probably not by one counter each round. However, if, 4 Allow fractional equivalents of the following: very high given the relatively small number for example, each of the seven students of rolls of the dice. This could prove an a 2 b 3 c 4 d 36 e 36 f 36 g h i “bets” on the same number and the spinner j 5 k 6 5 interesting group or class discussion point 36 36 36 4 3 2 lands on that number, the “banker” will 36 36 36 about what would be likely to happen after, 1 0 36 36 obviously lose. In the long term, however, say, 360 or 3600 rolls of the dice. the probabilities of the game will ensure that the only certain winner is the “banker”. 170 OX FOR D U N I V E RSI T Y PR E S S

Oxford Mathematics Primar y Years Programme is a comprehensive and engaging series for Kindergarten to Year 6. Designed by experienced classroom teachers, it supports sequential acquisition of mathematical skills and concepts, incorporates an inquiry-based approach, and is fully aligned with the understandings and outcomes of the PYP K– 6 mathematics curriculum. Student Book PY P Practice and Master y Book PY P Teacher Book PY P O x ford Ma thema tics O x ford Ma thema tics O x ford Ma thema tics Pr imar y Year s Programme Pr imar y Year s Programme Pr imar y Year s Programme Br ia n Mur r a y Br ia n Mur r a y A n n ie Fac ch i net t i Br ia n Mur r a y The series includes:  Student Books with guided, independent and extended learning activities to help students understand mathematical skills and concepts  Practice and Master y Books (Years 1– 6) with reinforcement activities and real-world problems that allow students to explore and apply their knowledge  Teacher Books with hands-on activities, blackline masters and activity sheets, as well as pre- and post-assessment tests for every topic. Oxford Mathematics Primar y Years Programme supports differentiation in the classroom by helping teachers nd the right pathway for every student, ensuring that each child can access the PYP mathematics curriculum at their own point of need. ISBN 978-0-19-031225-1 9 780190 312251 1 How to get in contact: web www.oxfordprimary.com/pyp email [email protected] tel +44 (0)1536 452620 fax +44 (0)1865 313472