Understanding Functional Skills Maths level 2

Transcription

1 Understanding Functional Skills Maths level 2 Workbook 7 - Find area, perimeter and volume of common shapes

2 INTRODUCTION TO THE MATHEMATICS FUNCTIONAL SKILLS QUALIFICATION AT LEVEL 2 In order to meet the assessment criteria for mathematics at level 2 you will be required to demonstrate your ability to represent, analyse and interpret, using number, geometry and statistics plus a selection of other skills in the coverage and range within functional contexts. Represent Analyse Interpret This means that you will need to: Understand routine and non routine problems in familiar and unfamiliar contexts and situations. Identify the situation or problems and identify the necessary mathematical methods needed to solve them. Choose from a range of mathematics to find solutions. This means that you will need to: Apply a range of mathematics to find solutions. Use appropriate checking procedures and evaluate their effectiveness at each stage. This means that you will need to: Interpret and communicate solutions to multi-stage practical problems in familiar and unfamiliar contexts and situations. Draw conclusions and provide mathematical justifications. Functional Skills in mathematics at level 2 has 12 areas of coverage and range : Understand and use positive and negative numbers of any size in practical contexts Carry out calculations with numbers of any size in practical contexts Understand, use and calculate ratio and proportion including problems involving scale Understand and use equivalences between fractions, decimals and percentages Understand and use simple formulae and equations involving one- or two-step operations Recognise and use 2D representations of 3D objects Find area, perimeter and volume of common shapes Use, convert and calculate using metric and, where appropriate, imperial measures. Collect and represent discrete and continuous data, using information and communication technology (ICT) where appropriate Use and interpret statistical measures, tables and diagrams for discrete and continuous data, using information and communication technology (ICT) where appropriate Use statistical methods to investigate situations Use probability to assess the likelihood of an outcome Page 2

3 You will be required to sit a formal test in order to assess your skills. You will be given 1 1/2 hours to complete the test, in which there will be a total of 40 possible marks to achieve. The 40 marks will be broken down into different tasks or a series of questions. Over 75% of the questions will require open-response answers. Open response assessment is defined as: Task-based assessment based on real-life contexts that require learners to apply their skills, knowledge and understanding in order to resolve problem/s or produce effective outcome/s Presenting purposeful tasks and problems, embedded in realistic scenarios but does not prescribe the processes or the methods by which the learner responds Instead of choosing from answers given to you, with this type of question you will need to show the process you have used to obtain your final answer. It is very important to read the question carefully. The way the question is worded will give you valuable clues about how you should answer it. Please note - Calculators are provided for use during the test. Top Tip After each question the maximum number of marks you can obtain through your answer will be displayed. This should give you a clue about how much detail you are expected to show. It is a good idea to try to complete a question, even if you are unsure that you have the correct answer, as you may be awarded some marks for the method you have shown. Page 3

4 Please find below a selection of very useful websites that can provide additional support resources in English. Please take the time to study and review these. Click on a topic you are interested in and you will see the different types of materials or activities that are there to help you with that topic. For each topic, you will find fact sheets, worksheets, quizzes and games. Job skills related resources also available. A good site with adult literacy and numeracy activities including more than 1500 free Functional Skills and Skills for Life resources. These are interactive literacy and numeracy practice materials, designed to supplement teaching. The practice material is generic and has been set in everyday recognisable settings. Interactive quizzes that allows you to develop both English and maths skills. Even has a facility to be used on a mobile device. Page 4

5 FIND AREA, PERIMETER AND VOLUME OF COMMON SHAPES Key definitions: Perimeter Area Volume MSS1/L1.8, MSS1/L2.7 MSS1/L1.9, MSS1/L2.8 MSS1/L1.10, MSS1/L2.9 The perimeter of a shape is the total length of its boundary i.e. the distance around the edge. The area of a shape is a measure of the amount of space it covers. The volume of a 3-D shape is a measure of the amount of space it occupies. Perimeter The perimeter of a shape is the total length of its boundary. For a rectangle, the perimeter, P, is given by the formula: P = 2(l + w) where l is the length and w is the width. w Or alternatively it could be written as 2w + 2l. Whichever way the answer would be the same. l Example: A lawn measuring 5m by 2m would require a boundary fence of: 2 (2 + 5) = 14m 2m 5m Page 5

6 Example: The plan shows the dimensions of an L-shaped sitting room. The house owner wants to buy skirting board to go around the edge of the room, leaving gaps at the doors. What is the shortest length of skirting boards he can buy? 7m 2.5m 90cm 5m Not to scale 90cm Starting at the top of the room you will need to add: 7m 5m 4.5m 2.5m 2.5m +2.5m =24m From this you will need to subtract the lengths of the doors (where no skirting board is required): 90 x 2 = 180cm 24m -1.8m 22.2m Therefore the final answer is 22.2m 4.5m These figures are assumed as they are not directly given on the plan, however they must be incorporated into the calculation. Top Tip: Metric units of perimeter are mm, cm, m and km. Page 6

7 The perimeter of a circle is called a circumference. The diameter of a circle is the distance across the centre. The radius of a circle is the distance from the edge to the centre. Top Tip: Diameter and Radius The diameter is ALWAYS twice the value of the radius in any circle. The circumference, C, of a circle can be written in two ways: C = πd where d = diameter C = 2πr where r = radius π is a special number, equal to the ratio of the circumference of a circle to its diameter. π is an irrational number, which means it cannot be written as a fraction. π is approximately equal to Diameter, d Radius, r Circumference, c Top Tip: Using π in test You will always have a calculator to use when attempting to calculate using the π value in your test. Example: A child s paddling pool measuring 8m in diameter would have a circumference of: C = πd C = π x 8 C = 25.12m 8m Page 7

8 Area The area of a shape is a measure of the amount of space it covers. For a rectangle, the area, A, is given by the formula: A = l x w Where l is the length and w is the width. w l Example: A lawn measuring 5m by 2m would require how much turf? 2m x 5m = 10 m 2 Please note that all AREA measurements are written as ( 2 ) as this defines that the answer is in units squared. 2m 5m Top Tip: Area and squares Remember that if the shape is 2D (i.e. flat) then the area will be always be expressed in squares (2). Metric units of area are mm 2, cm 2, m 2 and km 2. At level 2 Functional skills you will be more likely to be asked to calculate the area of an irregular or composite shape. To calculate this you will need to split the figure into regular shapes and then total up the relevant shape areas to find the overall area. Page 8

9 Example: This kitchen floor is to be covered in cushioned vinyl. 4.5m What area of cushioned vinyl is needed? Firstly, split the L-shaped room into two rectangles A and B. 5m 7m 2.5m Area of A Area of B = = 2.5 (7 4.5) = 22.5 m 2 = = 6.25 m 2 Total area = = 28.75m 2 Answer: m 2 (Please note that diagrams are NOT to scale) Area of a circle For a circle the area, A, is given by the formula: A = π r 2 where r is the radius. Top Tip: Area and circles It is sometimes easier to remember it as: Area = π (radius 2 ) as you must always remember to square (multiply by itself) the radius value before then multiplying this answer by the π value (3.14 ) Example: A child s paddling pool measuring 8m in diameter would have an area of: 8m A = π r 2 A = π x (radius 2 ) A = π x (4 x 4) A = π x 16 A = m 2 Page 9

10 Area of a triangle You may sometimes need to calculate the area of a triangle. For a triangle the area, A, is given by the formula: A = ½ x base x height. Example: Find the area of this triangle: 5cm 6cm A = ½ x base x height A = ½ x 5cm x 6cm A = ½ x 30cm A = 15 cm 2 Top Tip: Area and triangles Calculate the area of a triangle in the same way that you calculate the area of a rectangle i.e. multiply your two values together. Then simply HALF your answer. Page 10

11 Volume The volume of a 3-D shape is a measure of the amount of space it occupies. The volume of a cuboid (3D box shape) is: l w h Where: l = length, w = width, and h = height. h l w Example: A shoe box with the following dimensions would have a volume of: Volume (V) = l w h V = 40 x 32 x 10 V = 1280 x 10 V = V = 12,800 cm 3 40cm 10cm 32cm Please note that all VOLUME measurements are written as ( 3 ) as this defines that the answer is in units cubed. Top Tip: Volumes and cubes Remember that if the shape is 3D (i.e. 3 - dimensional) then the volume will be always be expressed in cubes ( 3 ). Metric units of area are mm 3, cm 3, and m 3

12 Volume of a cylinder The volume, V, of a cylinder (3D cylindrical shape, e.g. tube) is: V = πr 2 h Where r is the base radius and h is the height. Top Tip: Volumes and cylinders It is sometimes easier to remember it as: Volume = π (radius 2 ) x height You must always remember to square (multiply by itself) the radius value before then multiplying this answer by the π value (3.14) before then multiplying that answer by the height. Example: A can of cola has a diameter of 6cm and a height of 12cm. 6cm What is the volume of the can? 12cm Volume (V) = πr 2 h V = π x (3x3) x 12 V = π x 9 x 12 V = x 12 V = cm 3 Remember: the radius is only HALF the diameter - therefore in this case the radius is 3cm. Page 12

13 3D Volume You may need to solve problems in three dimensions. Example: A tin of shoe polish is 8cm in diameter and 2cm high. The shoe polish must be stored in an upright position. How many tins of shoe polish will fit into a carton that is 40cm long by 32cm wide by 10cm high? 8cm The tins are 8cm wide. 2cm 10cm This means that 40 8 = 5 tins will fit along one edge and 32 8 = 4 tins will fit along the other edge. So a total of 5 4 = 20 tins will fit on the bottom of the carton. The carton is 10cm high. Which means that 10 2 = 5 tins will fit, one on top of another. Not drawn to scale Therefore the total number of tins in the carton is 20 5 = 100. Answer: 100 tins 40cm 32cm Top Tip: Volume and problem solving A sketch diagram can help you understand the question A plan is a view from above A view from the front is called a front elevation Page 13

14 QUESTIONS Making cushions 1a. Rebecca makes cushions. She uses squares of fabric 37cm by 37cm. Rebecca buys a roll of fabric which has a width of 112cm. How many squares can she fit across the width of the fabric? Use the box below to show clearly how you get your answer. 1b. Rebecca needs 2 squares of fabric to make each cushion. The roll of fabric is 22m long. How many cushions can Rebecca make out of the roll of fabric? Use the box below to show clearly how you get your answer. Page 14

15 Village hall 2. Scarlett checks the health and safety regulations for the fair. She needs to work out the available floor area in the village hall. The diagram shows the village hall and the stalls at the fair. The available floor is not shaded. 20m 1m 10m Stall Raffle stall Refreshment stall 3m 1m Diagram NOT accurately drawn 2m Work out the available floor area. Use the box below to show clearly how you get your answer. Page 15

16 Bales of hay 3a. Neil is a farmer. He stores bales of hay in a barn. The barn is 14.5m long by 14.5m wide. Each bale of hay is 0.9m long by 0.45m wide by 0.45m high. He places some bales of hay lengthwise on the ground along the back wall of the barn. Back wall of barn. Diagram NOT accurately drawn How many bales of hay can Neil place along the back wall, on the ground? Use the box below to show clearly how you get your answer. Page 16

17 3b. Neil stores 4,000 bales of hay in the barn. He stacks the bales of hay in rows, with no spaces between the bales. Each row is 10 bales high. He leaves a rectangular space on the ground at the front of the barn. Work out the size of the space on the ground at the front of the barn. Use the box below to show clearly how you get your answer. Page 17

18 Energy drinks 4. Chris buys powder to make energy drinks. The tin holds 1.5kg. The instructions on the tin are to make 1 litre of drink use 4-6 scoops of powder. Each scoop holds 25g. Chris uses bottles holding 750ml of energy drink. He wants to make as many bottles as possible from 1 tin of powder. How many bottles of energy drink can he make? Use the box below to show your answer clearly. Page 18

19 Sandpit 5. Janine wants to put sand in the sandpit at the nursery. The sandpit is in a rectangular space with length 4.5m and width 3.5m. Janine wants to fill the sandpit with sand to a depth of 300mm. She finds out on the internet that each bulk bag of sand covers an area of 5m 2 to a depth of 100mm. Volume of a cuboid = length width depth How many bags of sand does Janine need to fill the sandpit? Use the box below to show clearly how you get your answer. Page 19

20 ANSWERS Making cushions 1. In this question you only need to show a straightforward understanding of how to calculate using area. You will be expected to show that you identified the maximum number of squares that can fit across the width of the fabric. To do this: 37cm is the width of one cushion. 37 x 2 = 74cm 37 x 3 = 111cm therefore 3 cushions will just fit across the width of the fabric. Answer = 3 squares 2 marks would be awarded for identifying this accurately. b. This follows on from part (a) and therefore you are expected to use your answer from part (a) to help with this question. You know that 3 squares will fit across the width The length of the fabric is 22m Convert 22m into cm (as this is the units for each cushion) 22m = 2200cm 2200cm 37 = Therefore only 59 full squares can be obtained from the length 3 squares x 59 squares = 177 squares can be cut from the fabric However each cushion requires 2 squares each therefore = 88.5 As you cannot have half a cushion the maximum number must be rounded down to 88 cushions Answer = 88 cushions There are a possible 3 marks for this question if you complete the calculation close to the example above you will gain full marks however marks will be lost if it is not totally accurate or clear to the marker. 1 mark is simply given for converting the 22m into the correct number of cm! Page 20

21 Village hall 2. In this question you need to show a slightly higher level of understanding of how to calculate using area. You will be expected to show the different stages to your calculation and the method for each: Calculate the overall area of the village hall: 20m x 10m = 200m 2 Then start to subtract the areas that are UNAVAILABLE, there are 3 of these: a. The raffle stall this measures 20m x 1m = 20m 2 b. The stalls this measures 1m x 9m (don t be tricked into assuming it is 10m long as one of these square metres is already included in the raffle stall area previously identified in part (a) = 9m 2 c. Refreshment stall this measures 3m x 2m = 6m 2 Now add these 3 totals together: Subtract this figure from the original overall total: Answer = 165m 2 There are a possible 3 marks for this question if you complete the calculation close to the example above you will gain full marks however marks will be lost if it is not totally accurate or clear to the marker. Page 21

22 Bales of hay 3a. In this question you need to show a higher level of understanding of how to calculate using volume. You will be expected to show the different stages to your calculation and the method for each The back wall measures 14.5m in length and each bale of hay is 0.9m Therefore the calculation is = However this needs to be rounded down to the nearest whole bale Answer = 16 bales of hay 2 marks would be awarded for identifying this accurately. 3b. You will be expected to show the different stages to your calculation and the method for each: Firstly, we know from part (a) that Neil can get 16 bales of hay across the barn We are told that these can only go 10 bales high Therefore 16 x 10 = 160 bales on each row Neil wants to store 4,000 bales in total, so we need to find out how many rows this will be: 4, = 25 rows Each bale is 0.45m deep so these 25 rows would need: 0.45m x 25 = 11.25m of space A re-cap is now worthwhile: we now know the space required that will house the 400 bales: 14.5m (width of barn) x 11.25m (depth of hay) The barn measures 14.5m x 14.5m therefore all that is needed next id to calculate the space at the front of the barn: = 3.25m and this is all the way across the barn (14.5m) Therefore 3.25m x 14.5m = m 2 is the amount of ground at the front of the barn Answer = m 2 There are a possible 5 marks for this question. If you complete the calculation close to the example above you will gain full marks, however, marks will be lost if it not totally accurate or clear to the marker. Page 22

23 Energy drinks 4. In this question you need to show a higher level of understanding of how to calculate using volume. You will be expected to show the different stages to your calculation and the method for each: Firstly, identify which unit of measurement you are more comfortable to use, either ml or litres for the juice and kg or g for the powder Next identify the least amount of energy powder you can afford to use and relate this to your chosen unit of measurement (4 scoops per litre) For this example let s use ml for the juice: therefore 750ml in each bottle equates to 4 scoops x 0.75 = 3 scoops Therefore we are going to use 3 scoops for every 750ml bottle of drink produced One scoop = 25g, so 25g x 3 scoops = 75g used per each bottle produced Let s use g for the powder mix: 1.5kg converted to g = 1,500g 1,500g 75g = 20 Therefore the maximum number of bottles Chris can make is 20 Answer = 20 bottles of energy drink maximum There are a possible 5 marks for this question if you complete the calculation close to the example above you will gain full marks however marks will be lost if it is not totally accurate or clear to the marker. Page 23

24 Sandpit 5. In this question you need to show a higher level of understanding of how to calculate using volume. You will be expected to show the different stages to your calculation and the method for each: Firstly calculate the volume of the sandpit that Janine requires to be filled with sand Volume = length x width x depth Volume = 4.5m x 3.5m x 0.3m (300mm converted into m) Volume = x 0.3 Volume = 4.725m 3 Next you need to identify the volume of each bag of sand: 5m 2 x 0.1m (100mm converted into m) = 0.5m 3 This tells us that each bag will fill 0.5m 3 Therefore = 9.45 bags of sand required However you cannot purchase only 0.45 bags therefore Janine will have to purchase 10 bags to fully fit the sandpit to the required level Answer = 10 bags of sand required There are a possible 6 marks for this question if you complete the calculation close to the example above you will gain full marks however marks will be lost if it is not totally accurate or clear to the marker. Page 24

25 Understanding Functional Skills AON WB7 Find area, perimeter and volume of common shapes L2 V4