ncc ify mt student guide v1.0 [2009]

8/14/2019 NCC IFY MT Student Guide v1.0 [2009]

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FOUNDATION: LEVEL 32009/10

MATHEMATICAL TECHNIQUES

Student Guide


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Modification History

Version Revision Description

V0.1

NCC Education Limited, 2009

All Rights Reserved

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Published by: NCC Education Limited, The Towers, Towers Business Park, Wilmslow Road,Didsbury, Manchester M20 2EZ, UK

Tel: +44 (0) 161 438 6200 Fax: +44 (0) 161 438 6240 Email: [email protected]://www.nccedu.com


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Title Here

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CONTENTS

Module Overview and Objectives....................................................................................... 5

1. Contents......................................................................................................................5

2. Learning Outcomes.....................................................................................................53. Teaching and Learning ...............................................................................................6

3.1 Lectures ...............................................................................................................63.2 Tutorials ...............................................................................................................63.3 Seminars..............................................................................................................63.4 Private Study........................................................................................................6

4. Assessment ................................................................................................................6

Unit 1: Number 1 ..................................................................................................................7

1. Learning Objectives ....................................................................................................72. Timings .......................................................................................................................7

3. Seminar Notes ............................................................................................................84. Private Study...............................................................................................................9

Unit 2: Number 2 ................................................................................................................ 19

1. Learning Objectives ..................................................................................................192. Timings .....................................................................................................................193. Seminar Notes ..........................................................................................................204. Private Study.............................................................................................................35

Unit 3: Algebra ...................................................................................................................41

1. Learning Outcomes...................................................................................................41

2. Timings .....................................................................................................................413. Lecture 6: Student Exercise ...................................................................................... 424. Private Study.............................................................................................................43

Unit 4: Probability ..............................................................................................................53


Unit 5: Shape, Space and Measures ................................................................................61


Unit 6: Data Handling......................................................................................................... 79



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Module Overview

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Module Overview and Objectives

The aim of this module is to develop basic mathematical competence and confidence, so that youcan use mathematics effectively in later years of study and also in the modules that containquantitative materials at this level of study. It is also focused on ensuring a functional level ofmathematics that will be valuable in your everyday life and future career.

1. Contents

Unit One: Number 1

Unit Two: Number 2

Unit Three: Algebra

Unit Four: Probability

Unit Five: Shape, Space and Measure

Unit Six: Data Handling

2. Learning Outcomes

Learning Outcomes

Knowledge andUnderstanding

Demonstrate knowledge and understanding of functional maths skills in:

1. Number

2. Algebra

3. Shape, Space and Measures

4. Probability and Statistics

Intellectual Skills 1. Apply mathematical knowledge to real life problems and situations.

2. Demonstrate mathematical understanding and methods of problemsolving.

3. Demonstrate ability to check calculations and interpret results.Practical Skills 1. Perform written and mental calculations.

2. Demonstrate problem solving skills.

3. Draw graphs and tables.

Transferable Skills 1. Make use of English to describe mathematical thinking and processes.

2. Present mathematical information and solutions in written form.

3. Demonstrate ability to work in groups.

4. Demonstrate independent thinking.


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3. Teaching and Learning

Suggested Learning Hours

Lectures: Tutorial: Seminar: Laboratory: Private Study: Total:

25 15 20 40 100

The teacher-led time for this module is comprised of lectures, tutorials and seminars. Thebreakdown of the hours for each Unit is given in the Unit notes below.

3.1 Lectures

Your teacher will be presenting the basic knowledge required for the unit during this time. Yourteacher will use PowerPoint slides during the lecture time and you will be expected to take notesand answer questions.

3.2 Tutorials

These are designed to deal with the questions arising from the lectures and private study sessions.

3.3 Seminars

During this time, you will be working on various tasks which may involve group work, investigationand independent learning. You will be completing practice exercises to consolidate yourunderstanding of the material covered during the lecture sessions.

3.4 Private Study

In addition to the taught portion of the module, you will also be expected to undertake private study.During this time, you may be undertaking further exercises, preparing for future sessions, revisingcontent already covered to deepen understanding etc. Your teacher will set deadlines for thecompletion of exercises. You will usually need to complete the Private Study exercises prior to thescheduled tutorial for each unit, as there is time allowed during this session for the review ofanswers.

3.5 Textbooks

The following textbook provides supplementary reading for this module and will be available for youto use in your centres library.

Croft, A. and Davison, R (2006). Foundation Maths. 10thEdition. London: Pearson Education.

ISBN-10: 0 131 979213ISBN-13: 978-0131979215

4. Assessment

This module will be assessed by means of an examination. You will be expected to demonstrate that

you have met the learning outcomes for the module by responding to 5 questions. Each question willrequire you to show your calculations as appropriate, as well as giving your final answer.


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Unit 1

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Unit 1: Number 1

1. Learning Objectives

This unit provides an overview of a basic number work to ensure that you are competent in calculatingwith numbers and solving problems. On completion of the unit, you will be able to:

Understand place value and use this to multiply and divide numbers by powers of 10.

Use all 4 operations with large numbers and decimals up to 2 decimal points.

Round numbers and estimate answers to calculations.

Solve written problems using the four rules of number.

2. Timings

Lectures: 5 hours

Tutorials: 2 hours

Seminars: 2 hours

Private Study: 6 hours

You should complete the exercises given below as instructed by your teacher. Some are forcompletion during class time, while others will need to be completed during Private Study time.


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3. Seminar Notes

The seminars in this unit will last for 2 hours.

Activity 1 Group project

Working in groups of 4

1. Reflecting on the work you have covered in this unit, your task is to produce a question sheet ofword problems based on real life situations that use the skills you have covered in this unit.

Your worksheet should include questions on:

Addition

Subtraction

Multiplication

Division Rounding and Estimation

Dont forget to include the use of decimals.

The worksheet should have 10 questions. You can combine more than one area within onequestion. For example, a question could have a part requiring addition followed by anotherneeding multiplication.

2. When the worksheet is complete, pair up with another group. You will then complete the othergroups worksheet individually and pass back for marking. They will receive your worksheet andyou will then mark their answers.

3. Now working together with the other group, evaluate your worksheets together. What workedwell? What could have been improved? What did people find easy? Are there any noticeableareas where you need to do more practice?

4. Report back to the class on your findings.

a. How did you find the exercise?

b. What were your key findings?

c. What would you conclude about the use of maths in real life?


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4. Private Study

You should spend approximately 6 hours on the Private Study for this unit. In addition to completingthe exercises as directed by your teacher, you will need to spend time reviewing your notes andpreparing for future classes.

Exercise 1: Place value and multiplying and dividing by powers of 10

1. Give the value of each underlined figure

a. 16b. 980c. 7160d. 5237e. 1 547 458f. 34 768g. 0.345

h. 1.436i. 145.324

2. Multiply each of these numbers by 10:

a. 4b. 155c. 7009d. 2.3e. 0.34f. 143.65


a. 3b. 54c. 1320d. 30.7e. 2.35f. 0.045


a. 5b. 55c. 2670d. 2.4e. 14.32f. 0.009

5.a. Which number is ten times bigger than fifty three?b. Find the number that is one hundred times bigger then 0.73.c. How many times bigger is 3.4 than 0.034?

6. Calculate:

a. 47.1 x 10 =


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b. 0.985 X 100 =c. 1000 X 209.54 =d. 64.3 X 100 =

7. Find the missing numbers in these calculations

a. 3.43 x ? = 343b. 17.9 x ? = 179c. ? x 0.07 = 7d. ? x 0.07 = 7

8. Divide each of these numbers by 10:

a. 60b. 8500c. 85d. 3

e. 1.3f. 0.07


a. 800b. 9400c. 505 00d. 350e. 9f. 1.34


a. 2000b. 23 000c. 1 340 000d. 950e. 60f. 3.2

11.a. Which number is ten times smaller than five hundred and seventy?b. Which number is one hundred times smaller than 2.3?

c. Which number is one thousand times smaller than six million?

12. Calculate:

a. 26.8 10 =b. 149 100 =c. 268 1000 =d. 0.087 100 =

13. Find the missing numbers in these calculations.

a. 649 ? = 64.9

b. 432 ? = 0.432c. 490 ? = 4.9


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d. 0.03 ? = 0.0003

14. Calculate:

a. 4.2 x 10 =

b. 218.19 10 =c. 4.965 x 1000 =d. 0.86 100 =e. 2.3 1000 =f. 0.098 x 100 =g. 0.67 x 100 =h. 24.76 10 =i. 0.098 10 = j. 1000 x 2.36 =k. 0.098 10 =l. 1000 x 2.36 =m. 237 1000 =

n. 24 100 =

15. Write as powers of 10.

a. 10 x 10 x 10b. 1 100

16. Calculate:

a. 1.28 X 102b. 102c. 45.7 102

d. 0.867 X 103e. 3479 103f. 0.659 X 103g. 45.9 102h. 0.863 X 102i. 371 X 10-1 j. 38.7 10-1k. 0.12 X 10-2l. 102m. 10-3n. 175 X 10-1


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Exercise 2: Addition and Subtraction

1. Do the following calculations in your head and write down your answers.

a. 16 + 18

b. 35 + 17c. 69 + 15d. 35 + 77e. 73 + 68f. 45 22g. 76 32h. 75 38i. 154 73 j. 274 109k. 541 373l. 1053 846m. 0.4 + 0.5

n. 3.6 + 4.5o. 7.25 + 1.3p. 3.3 + 2.84q. 2.05 + 0.6r. 7.9 + 6.3s. 1.4 0.6t. 4.9 2.4u. 6.3 1.9v. 12.6 5.4w. 16.4 5.9x. 23.2 11.8

2. Find answers to the following without using a calculator.

a. 313 + 442 + 124b. 633+ 40 + 78c. 2365 + 456 + 7893d. 347 + 875 + 2349e. 988 + 42.6 + 3.245f. 2340 949g. 6091 3746h. 4321 768i. 23.84 12.72 j. 37.42 15.65k. 67.32 24.75l. 42.7 16.832

3.a. Find the sum of 27, 657 and 4092.b. Find the difference between 3211 and 875.c. Add together 2.75, 34.67 and 56.d. Subtract 5.78 from 12.4.e. Find 45.67 take away 8.5.f. Find the total of 12, 5.6 and 0.85.g. Find 200 minus 9.9

h. Add together 34.5 and 5.67 then take 11.99 from the total.


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4.a. The crowds at the four best attended football matches on one Saturday were:- 75 325, 52

435, 47 302 and 45 389. Find the total number of people who attended these matches.b. How many more people went to the match with the highest attendance than the second

highest?

5.a. Four pieces of wood are laid in a line. What is the total length of the four pieces of wood if

their individual lengths are: 2.36m, 5m, 0.7m, 9.78m?b. What is the difference in length between the longest and shortest piece of wood?

6. Three small bottles contain 45cl, 33cl and 66cl. If they are all poured into a 2litre bottle, how muchof the bottle is still empty?

7.a. Three people weigh 65.8kg, 76.4kg and 55kg. What is their total weight?b. Could they all travel in a lift with a weight limit of 200kg?

Exercise 3: Multiplication


a. 45 X 5b. 25 X 8c. 33 X 8d. 5.3 X 3e. 8.5 X 7

f. 14.2 x 3g. 10.8 x 7

2. Use your answer to part i) to help find the next 2 answers.

a.i. 43 x 4ii. 0.43 x 4iii. 0.43 x 0.4

b.i. 58 x 6ii. 5.8 x 0.6

iii. 0.58 x 0.006

3. Find the answers to the following without using a calculator:

a. 34 X 25b. 63 X 49c. 658 X 47d. 402 X 33e. 747 x 245f. 13.5 x 0.4g. 759 x 0.9h. 6.34 x 0.6

4. Use your answer to part i) to help find the next 2 answers.


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a.i. 64 x 23ii. 6.4 x 23iii. 6.4 x 0.23

b.

i. 752 x 74ii. 7.52 x 74iii. 7.52 x 0.74

5. Find the total length of 5 bricks if each one is 22cm long.

6. Find the total weight of 8 plastic toys which each weigh 8.5 grams.

7. A sheet of paper is 0.9mm thick. Calculate the thickness of a pack of 500 sheets of paper.

8. What is the total weight of 9 bags of rice each weighing 5.45kg?

9. A tin of paint holds 1.75litres. How much paint would 6 tins hold?

Exercise 4: Division


a. 96 6b. 78 3c. 11 2d. 23 4

e. 25 3f. 160 20g. 400 50h. 320 40i. 800 400 j. 4200 600k. 4.6 2l. 2.4 6m. 0.48 4n. 0.56 7o. 8.8 11p. 30 0.5

q. 28 0.7r. 18 0.3s. 420 0.7t. 13.2 0.6

2. Find answers to the following without using a calculator.

a. 255 5b. 352 4c. 492 6d. 261 9e. 200 8f. 137 2g. 119 4


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h. 157 8i. 185 9 j. 197 5k. 180 12l. 256 16

m. 744 31n. 480 32o. 492 42p. 135 12q. 320 21r. 923 23s. 946 29t. 745 51

3. Work out answers to the following and show your working.

a. 6 0.3

b. 5.4 0.9c. 24 0.6d. 7.2 0.008e. 560 0.07f. 19.2 0.02g. 29.4 0.4h. 0.3 0.005i. 22.1 1.3 j. 3.1 0.025

4. Work out answers to the following. Keep going until you have 2 places of decimals, then roundyour answer to 1 decimal place.

a. 38 0.3b. 42 0.7c. 15.7 0.8d. 0.6 0.011e. 0.04 0.015

5. Find answers to the following problems. Show your working.

a. Cheese rolls are sold in packets of 18. Abdul needs 300 cheese rolls for a picnic. Howmany packets does he need to buy? How many more rolls will he get than he needs?

b. Karl is using beads to make necklaces. Each necklace needs 46 beads. He has 1000beads. How many necklaces can he make? How many beads are left over?

c. Sarah is packing 1948 chocolates into cartons. Each carton holds the same number. Shehas enough chocolates to fill 59 cartons. How many chocolates are left over?

d. How many lengths of cable 2.3m long can be cut from a reel of cable 74.6m long?


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Exercise 5: Rounding and Estimation

1. Round the following correct to the nearest 10.

a. 34b. 58c. 234d. 555e. 2589f. 10761

2. Round the following correct to the nearest 100.

a. 345b. 576c. 61

d. 4567e. 11419

3. Round the following correct to the nearest 1000

a. 1567b. 20439c. 537d. 234e. 234432

4. Round the following correct to 1 significant figure.

a. 1. 23 2. 69 3. 324 4. 471 5. 4513

5. Round the following correct to 2 significant figures.

a. 234b. 418c. 4581d. 62 117e. 22 222

6. Round the following correct to 3 significant figures.

a. 4516b. 4881c. 11 562d. 98 128e. 276 387

7. Round correct to the nearest whole number.

a. 23.4b. 92.9c. 25.18

d. 49.93e. 265.48


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8. Round correct to 1 decimal place.

a. 23.573

b. 47.9123c. 0.251432d. 405.283e. 2600.37432

9. Round the following correct to 2 decimal places.

a. 3.50723b. 23.14343c. 0.2187432d. 947.145212e. 0.008

10. Use rounding to estimate the answers to the following calculations. Do not use a calculator.

a. 5.1 x 0.31b. 421 x 0.22c. 0.48 x 0.316d. 3.7 x 0.92e. 133 x 0.49f. 0.031 x 82g. 6.25 x 0.058h. 91.6 x 0.78i. 23 x 497

91 j. 205 x 71

19 x 4.8

Exercise 6: Personal Development

Use the results from the activities in the lectures, seminars and exercises above to record yourpersonal skills, knowledge and understanding in the development planner table below. You shouldspend approximately 1 hour on this activity.

Notes:

1. For Confidence Level completion: 1 = Very Unsure, 5 = Very Confident

2. You should use the final column (Plans to Improve) to list ways you can improve your confidencelevel in this area. This might include further review of your notes, further practice, research in thelibrary/on the internet etc.

3. You might also like to use this action plan to test yourself on the key areas. Without looking atyour notes, see how much you can write about each area. This will also be helpful for examinationpreparation.

This activity only has value if you follow up your action plan.


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You should keep re-visiting, updating and amending this Action Plan throughout (and beyond!) yourstudies on this module.

Number 1 Action Plan

Date:Developing Skill,

Understanding andKnowledge

Confidence Level1-5

Plans to Improve

I understand place value andcan use it to multiply anddivide by powers of 10.

I can use different strategiesto add and subtract mentally

including with decimals.

I can use written methods toadd and subtract includingwith decimals.

I can use mental and writtenmethods to multiply largenumbers and decimals.

I can use mental and writtenmethods to divide largenumbers and decimals.

I can round numbers topowers of 10, decimal placesand significant figures. I canuse rounding to estimateanswers to calculations.


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Unit 2

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Unit 2: Number 2


This unit provides you with an understanding of important number concepts, using ratio and workingwith percentages and fractions. On completion of the unit, you will be able to:

Understand the terms square, cube, root and power.

Know and use the index laws.

Understand and use the terms multiple, factor and prime.

Use the four rules of number with negative numbers.

Find the Highest Common Factor, Lowest Common Multiple and product of prime factors.

Understand and use ratio.

Manipulate fractions and calculate with fractions.

Use and calculate with percentages

Understand and find equivalent fractions, decimals and percentages.

2. Timings

Lectures: 5 hours

Tutorials: 3 hours

Seminars: 4 hours




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3. Seminar Notes

The seminar sessions in this unit will last for 4 hours.

Exercise 1: Indices

Write down the values of each of the following

1.a. 12b. 92c. 122d. 152e. 72f. 132g. 112

h. 142

2.a. 42 + 122b. 52 + 142c. 122 + 72d. 132 + 22e. 152 122f. 112 102g. 142 112h. 152 - 132

3.a. 302b. 502c. 1102d. 2002e. 8002f. 12002g. 70002h. 140002

4.a. 203

b. 50

3

c. 3003d. 40003

5. Use the square numbers up to 225 to work out:

a. 0.12b. 0.52c. 0.82d. 0.92e. 1.52

f. 1.32

g. 1.12

h. 1.42


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6. Use the cube numbers up to 125 to work out:

a. 0.23b. 0.33c. 0.53

7. Write down the positive and negative values of each of the following:

a. 36b. 81c. 121d. 225e. 100f. 144g. 196h. 169

8. Write down the values of each of the following:

a. 38b. 327c. 3125

9. Work out the values using positive square roots:

a. 122 + 225b. 144 33c. 121 + 144d. 52 + 196

e. 225 - 81f. 103 - 169g. 169 + 43h. 225 + 225

10. Use a calculator to show that 56 is greater than 7.4 and less than 7.5

11. Write each of the following in index form:

a. 3x3x3x3x3x3x3b. 5x5x5x5x5x5c. 4x4

d. 2x2x2x3x3e. 2x2x5x5x5x5x5x5f. 2x2x3x3x3x4x4x4x4g. 3x3x4x4x4x4x4x4x4

12. Write each of the following using multiplication signs:

a. 42b. 63c. 71d. 210e. 22 X 33

f. 45 X 63g. 45 X 54


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h. 22 X 33 X 4i. 3 X 43 X 54 j. 24 X 33 X 42 X 5

13. Copy and complete the table.

x y x2

y3

x2+ y

3x

2y

3

a 3 2 9 8 9 + 8 = 17 9 x 8 = 72b 2 3c 10 5d 5 10e 1 4f 4 1

14. Use your knowledge of squares and cubes to work out the value of the letter symbol:

a. 2

a

= 4b. 3b = 27c. 4c = 64d. 10d = 1000e. 5e = 125

15. Find the value of the letter symbol:

a. 2a = 25 x 24b. 3b = 35 x 33c. 5c = 52 x 53 x 54d. 7d = 7 x 72 x 75

e. 2e

= 27

24

f. 3f= 35 33g. 5e = 54 53h. 7h = 75 74i. 2i = 24 x 25 23

Exercise 2: Negative numbers

1. Adding and Subtracting Negative Numbers:

a. 7 - 8 + 5b. 6 8 3c. -5 8 + 3d. 12 5 + 7e. 5 4 8f. 6 - (+7)g. 8 ( -6)h. 6 + ( -6)i. -5 ( -2) j. 8 + ( -9)k. 6 ( + 11) ( -8)l. -4 ( + 8) + ( -7)

m. -1 ( -1) ( -1)n. 6 ( + 7) + ( -6)


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o. 4 + (-5) + ( + 9)

2. Multiplying and Dividing Negative Numbers:

a. 7 x 5

b. -4 x 8c. 11 x -8d. -4 x -8e. 2.5 x 8f. -7 x -5 x 4g. 8 x -6 x 3h. 12 6i. -20 5 j. -35 -5k. 35 -7l. -66 -11 x 7m. 7 x 6 -21

n. -15 x -4 -12o. -72 -12 x 5

Exercise 3: Multiples, Factors and Primes

1. Write down all the factors of:

a. 8b. 18c. 28

d. 23e. 36f. 50g. 64

2. Write down the next five even numbers after:

a. 14b. 36c. 200

3. Write down the next five odd numbers after:

a. 17b. 53c. 311

4. Write the first 5 multiples of:

a. 4b. 7c. 10d. 15e. 50

5. Look at the list of numbers: 9, 12, 15, 17, 18. Write down:


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a. an even numberb. an odd numberc. a multiple of 5d. a factor of 24e. a prime number

6. Look at the list of numbers: 3, 11, 20, 24, 25. Write down:

a. an even numberb. an odd numberc. a multiple of 5d. a factor of 27e. a prime number

7. Find the Highest Common Factor of:

a. 24, 40

b. 50, 75c. 168, 60d. 240, 150e. 242, 176f. 24, 36, 60g. 26, 182, 65

8. Find the Lowest Common Multiple of:

a. 2, 3, 5b. 5, 7, 9c. 2, 6, 10

d. 6, 7, 9e. 2, 5, 7, 11f. 3, 5, 8, 12,g. 6, 7, 9

9. Write down the prime factors of:

a. 8b. 14c. 15d. 25e. 35

10. Write each of these numbers as a product of its prime factors.

a. 10b. 24c. 28d. 30e. 30f. 36g. 60

11. Write these numbers as a product of its prime factors.

a. 112


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b. 125c. 150d. 180e. 184f. 500

12. Write these numbers as a product of its prime factors. Give your answers in index form.

a. 132b. 156c. 250d. 400e. 620f. 900

13. Use the product of prime factors to find the highest common factor.

a. 48, 54b. 30, 65c. 56, 108d. 36, 42, 54e. 16, 24, 60f. 15, 36, 48g. 36, 48, 90

14. Use the product of prime factors to find the lowest common multiple.

a. 12, 16b. 11, 14

c. 8, 18d. 10, 12, 15e. 8, 12, 18f. 7, 9, 15g. 8, 12. 20

Exercise 4: Ratio and Proportion

1. Write each ratio in its simplest form.

a. 3 : 6b. 8 : 4c. 15 : 10d. 4 : 24e. 24 : 40f. 1.2 : 2.4g. 4.8 : 3.6h. 15 : 12 : 9i. 24 : 36 : 48 j. 14 : 21 : 35k. 100 : 20 : 25

2. Divide these amounts into the given ratio.


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a. 21 6 : 1b. 36 5 : 4c. 20Litres 3 : 5d. 48m 2 : 3 : 3e. $100 5 : 2 : 3

3. A bag contains 20 coloured counters: 5 red, 10 green, 1 blue and 4 yellow. Find the proportionof:

a. red countersb. green countersc. blue countersd. counters that are not yellow

4. Three tins of paint cost 10.50. Find the cost of:

a. one tin

b. five tinsc. 15 tins

5. There are 7 soft sweets out of every 9 sweets in a tin. Altogether there are 72 sweets in the tin.

a. What proportion are soft sweets?b. How many in total are soft sweets?c. How many in total are not soft sweets?

6. 40 tea bags cost 71 pence. 80 cost 1.43. Which is better value?

7. In a large hotel, the manager estimates that the proportion of light bulbs not working is 1 out of

30. If 29870 lights are working, estimate the number of lights altogether.

8. 5 miles = 8 km. Which is further, 60 miles or 88 km? Show your working.

Exercise 5: Fractions

1. Copy and complete the following pairs of equivalent fractions.

a.3

4 12=

b.4

5 25=

c.1

6 42=

d.3

7 42=

e.7

10 40=

f.2

3 12

=


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g.6

7 35=

h.5

8 24=

i. 35 45

=

j.4

10 15=

2. Change these improper fractions to mixed numbers.

a.3

2

b.6

5

c.8

4

d.9

7

e.11

2

f.13

4

g.19

3

h.27

5

3. Write these mixed numbers as improper fractions.

a.1

22

b.1

33

c. 516

d.3

34

e.2

67

f.3

210

4. Simplify each of the following fractions.


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a.4

8

b.9

12

c. 46

d.20

25

e.50

70

f.9

27

g.18

30

h.45

50

i.54

63

j.44

50

5. Work out the following. Simplify your answers where possible and write any improper fractionsas mixed numbers.

a. 1 35 5

+ =

b.4 1

7 7 =

c.7 1

12 12+ =

d.8 5

9 9+ =

e.7 3

8 8+ =

f.1 3

1 26 6

+ =

6. Work out the following:

a.2 1

3 6+ =

b.3 1

4 8 =

c.3 1

5 10

+ =


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d.5 1

9 3 =

e.1 3

2 16+ =

f. 4 35 10

=

g.7 5

12 6+ =

h.1 11

4 12+ =

7. Work out the following:

a.1 1

5 4

+ =

b.1 1

2 3 =

c.1 2

8 3+ =

d.2 1

3 2 =

e.1 1

3 10+ =

f.1 1

5 6

=

g.3 1

10 4 =

h.3 1

8 5+ =

i.2 1

5 4+ =

j.3 3

4 5 =

k.1 3

6 8+ =

l.1 1 1

2 3 4+ + =

8. Work these out. Simplify where possible and write improper fractions as mixed numbers.

a.1

613

b.3

516


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c.2

65

d.5

156

e. 1 15 6

=

f.2 1

3 6 =

9. Work these out, cancelling first where possible.

a.5 3

6 4 =

b.1 3

6 8

=

c.5 2

6 5 =

d.5 16

12 25 =

10. Work these out. Simplify and write improper fractions as mixed numbers where possible.

a.1

42

=

b.2

6 5 =

c.2

23

=

d.3

44

=

e.1 1

2 3 =

f.1 2

3 5 =

g. 1 34 5

=

h.3 2

7 5 =

11. Dave makes some waffles. He puts honey on3

5 of them. Then he puts yoghurt on1

4 of thosethat have honey.

a. What fraction of his waffles have honey as well as yoghurt?b. Given that he made 40 waffles, how many have honey on as well as yoghurt?

12. Gail is weaving rugs, which are all identical. In1

4 hour she weaves2

3 of a rug. How many

complete rugs will she weave in 4 hours if she works at the same rate?


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Exercise 6: Percentages

A. Find the given percentage of the quantities.

1. 10%

a. 60b. 130c. 57

2. 30%

d. 40e. 250f. 35

3. 25%

g. 60h. 140i. 30

4. 75%

j. 40k. 120l. 90

5. 5%

m. 60n.o. 220p. 70

6. 2.5%

q. 80r. 140s. 90

7. 17.5%

t. 80u. 180v. 230


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B. Write the first number as a percentage of the second

1.a. 20, 40

b. 14, 56c. 36, 48

2.a. 26, 104b. 34, 85c. 44, 60

3.a. 112, 140b. 14, 100c. 6, 130

4.a. 15, 75b. 10, 15c. 17.5, 35

C. Increase the following by the given percentage:

1.a. 60 by 10%b. 12Kg by 25%c. 450g by 5%

2.a. 545m by 8%b. 34 by 12%c. 75 by 20%

3.a. 340Kg by 15%b. 82 by 75%c. 130g by 95%

D. Decrease the following by the given percentage:

1.a. 70 by 10%b. 20Kg by 25%c. 120m 5%

2.a. 320g by 15%b. 30cm by 8%c. 144 by 30%

3.

a. 150mm by 12%b. 64$ by 5%


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c. 4.3m by 10%

E. Copy and complete the table to find the percentage change:

Start Finish % change

1 50 602 40 603 48 244 80 705 25 206 40 167 150 758 600 400

F. Find answers to the following problems:

1. A car is bought for 6000 and loses 10% of its value each year. What will be its value after 3years?

2. 500 is invested at a rate of 4% which is added at the end of each year. How much is theinvestment worth after 4 years?

Exercise 7: Fractions, Decimals and Percentages

A. Write the following fractions as decimals:

1. 2. 3. 2/54. 15/205. 7/25

B. Write the following decimals as fractions:

1. 0.252. 0.43. 0.654. 0.875. 0.48

C. Write the following fractions as percentages:

1. 2. 4/53. 7/104. 6/205. 37/40

D. Write the following decimals as percentages:

1. 0.6

2. 0.673. 0.04


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4. Private Study

You should spend approximately 6 hours on the Private Study for this unit. In addition to completingthe exercises as directed by your teacher, you will need to spend time reviewing your notes and

preparing for future classes.

Exercise 1

A: Negative Numbers

Calculate:

1. 3 + -5 =2. -5 + 7=3. -6 + -9=4. 4 16=

5. 5 - -3=6. -8 - -12=7. 3 x -6=8. -5 x 7 =9. -36 4 =10. -6 x -4 =11. -72 =12. -63 -7 =

B: Indices

1. What is:

a. 5 squared?b. The square root of 49?c. 4 cubed?d. The cube root of 27?e. Which numbers are missing?

1, __, 9, 16, __, __, 49, 64, __, ___

2. Find the value of the letter.

a. 2a = 24 X 26

b. 5b

= 55

X 58

c. 2c = 212 23d. 3fd= 35 36

e. (39)2

C: Multiples, Factors and Primes

1. Which word multiple or factor goes in each of these?

a. 6 is a of 2.b. 6 is a of 30.c. c) 10 is a .of 60

d. 8 is a ... of 32.e. 8 is a of 32


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f. 36 is a of 9.

2. 1 2 5 25 80 150 10 75 50 100

a. Circle the numbers above that are factors of 25.

b. Underline the numbers above that are multiples of 25.

3. List the multiples of 6 between 10 and 50.

4. What is the Lowest Common Multiple of 7 and 9?

5. List the factors of 45.

6. What is the Highest Common Factor of 45 and 60?

7. Write the following as a product of their prime factors:

a. 120b. 285

8. Find the Highest Common Factor of 120 and 285.

D: Ratio

1. Four apples cost 1.40. How much will five apples cost?

2. Rosie and Sophie share some chocolates in the ratio 3:2. Sophie gets 12 chocolates. Howmany does Rosie get?

3. Danny and Richard share 49 in ratio 4:3. How much is the smaller share?

4. Packets of sausages are sold in two sizes:

A pack of 16 sausages costs 2.35A pack of 4 sausages costs 1.19, but they are buy one, get one free.

a. Which packet is better value?

5. 1250 people watch a school concert. The numbers of teachers, pupils and visitors are in theratio 1:6:3.

a. How many pupils watched the concert?b. How many visitors watched the concert?

6. Making 5 litres of tree green paint uses 3.5litres of yellow paint and 1.5litres of blue paint. Howmany litres of yellow paint and blue paint are needed to make:

a. 10 litres of tree green paint?b. 3 litres of tree green paint?

E: Fractions

Work these out:


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1.1 1

4 3+ =

2.2 1

7 3+ =

3. 2 15 6

+ =

4.1 3

7 5+ =

5.5 3

9 8 =

6.3 1

4 6+ =

7.7 2

8 3 =

8.5 2

8 6 =

Cancel first where you can to work these out.

9.1 2

4 9 =

10.5 2

6 15 =

11.5 3

9 10

=

12.15 4

16 25 =

13. Find the missing fraction.

? 3 2

? 5 5 =

14. In a bag of beads, 15 of the beads are blue. The probability of getting a blue when selecting a

bead at random is3

5

.

How many beads are in the bag altogether?

15. Jake bought a large bar of chocolate. He gave away3

4of the bar and ate

5

7of the rest. There

are 6 squares left. How many squares were in the whole bar?

F: Percentages

1. There are 50 members of a rowing club.

a. There are 32 male members. What percentage is male?

b. The number of members increases by 20% of the original number of members eachyear. Calculate the number of members in the rowing club after two years.


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2. Tara has a weekend job. She earns 4.50 per hour on Saturdays. On Sundays her pay is 50%more than it is on Saturdays.

a. What is her pay on Sundays?b. Last weekend Tara worked for 6 hours on Saturday and for 5 hours on Sunday. How

much did she earn in the two days?

3. The cost of a camera is 250 plus VAT of 15%. Calculate the amount of VAT charged on thecamera.

4. Roy invests 5000 at a rate of interest of 4.5% per annum. How much is his investment worthafter 3 years?

5. An antiques dealer buys a table for 450. He sells the table for 60% more than the price he paidfor it. It costs the dealer 70 to restore the table before he sells it.

a. For how much does the dealer sell the table?

b. Calculate the amount of profit the dealer made.c. Calculate this profit as a percentage of the price the dealer paid for the table.

G: Fractions, Decimals and Percentages

1. Change these decimals to fractions and percentages.

a. 0.5b. 0.2c. 0.45d. 0.12

2. Change these fractions to decimals and percentages.

a.1

4

b.2

25

c.4

5

d.5

8


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Exercise 2: Textbook

Additional explanations and exercises can be found in the supplementary textbook for this modulewhich is available in the library at your Accredited Partner Centre.

Croft, A. and Davison, R (2006). Foundation Maths, 10thEdition. London: Pearson Education.

Consult the textbook to improve your understanding and complete additional practice exercises asinstructed by your teacher.

Indices: Chapter 7. Exercises 7.1, 7.2, 7.3

Negative numbers: Chapter 1. Exercise 1.1

Multiples, Factors and Primes: Chapter 1. Exercises 1.3 and 1.4

Ratio: Chapter 5. Exercise 5.2

Fractions: Chapter 2. Exercises 2.2, 2.3, 2.4

Percentages: Chapter 5. Exercise 5.1

Fractions, Decimals and Percentages: Chapter 3. Exercise 3.1



Notes:

1. For Confidence Level completion: 1 = Very Unsure, 5 = Very Confident

2. You should use the final column (Plans to Improve) to list ways you can improve yourconfidence level in this area. This might include further review of your notes, further practice,research in the library/on the internet etc.

3. You might also like to use this action plan to test yourself on the key areas. Without looking atyour notes, see how much you can write about each area. This will also be helpful forexamination preparation.



Date:

Developing Skill,Understanding and

Knowledge

Confidence Level1-5

Plans to Improve

I understand the termssquare, cube and root andknow the square numbers upto 225.

I know and can use the three


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index laws.

I can calculate with negative

numbers.

I understand the termsmultiple, factor and prime.

I can find the LowestCommon Multiple, Highest

Common Factor and productof prime factors.

I can solve problemsinvolving ratio andproportion.

I can recognise equivalentfractions and simplifyfractions.

I can use the four rules ofnumber with fractions.

I can solve problemsinvolving percentagesincluding percentageincreases, decreases andcompound interest.

I can recognise and findequivalent fractions,decimals and percentages.


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Unit 3

Page 41 of85

Unit 3: Algebra1. Learning Outcomes

This unit provides students with the fundamentals of algebra. On completion of the unit, you will beable to:

Understand the terms expression, formula, equation and function.

Write expressions and formulae using algebra

Substitute values into formulae

Find term-to-term and position-to-term rules for sequences.

Simplify algebraic expressions by collecting like terms and multiplying out brackets.

Factorise algebraic expressions

Solve linear equations

Draw linear graphs

2. Timings

Lectures: 6 hours

Tutorials: 4 hours




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3. Lecture 6: Student Exercise

Linear Graphs Exercise:

1.

a. Draw and label a grid with x-axis from -4 to 4 and the y-axis from -10 to 10.b. Find the coordinates of points that lie on the graph of y = x + 5 at x = -2, x = 0 and x = 2.c. On the grid draw and label the graph of y = x + 5.d. Repeat parts b. and c. for the graphs of:

i. y = x + 3ii. y = x + 1iii. y = x 1iv. y = x 3v. y = x 5

Note: Draw all the graphs on the same grid.e. What do you notice about the lines you have drawn?

2.

a. Draw and label a grid with x-axis from -4 to 4 and the y-axis from -10 to 10.b. Find the coordinates of points that lie on the graph of y = 2x at x = -2, x = 0 and x = 2.c. On the grid draw and label the graph of y = 2xd. Repeat parts b. and c. for the graphs of:

i. y = 3xii. y = 4xiii. y = xiv. y = -2xv. y = 3x


3. a. Draw and label a grid with x-axis from -4 to 4 and the y-axis from -10 to 10.b. Find the coordinates of points that lie on the graph of y = 2x + 1 at x = -2, x = 0 and x = 2.c. On the grid draw and label the graph of y = 2x + 1d. Repeat parts b. and c. for the graphs of:

i. y = 2x + 3ii. y = 2x - 1iii. y = 2x 3iv. y = -x + 3


4. Using your findings, try these questions.a. Where will these graphs cross the y-axis:i. y = x + 4ii. y = 2x 1?

b. Which of these graphs will be steeper: y = 2x + 3 or y = 5x 2?c. What can you say about the different parts of the equation y = 3x - 2?


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4. Private Study


Exercise 1: Expressions, Formulae and Equations

A

Write down the algebraic expressions that say:-

1. 2 more than x2. 6 less than x3. k more than x4. t less than x5. 8 multiplied by x

6. h multiplied by j7. x divided by 48. y divided by t9. a multiplied by a10. g divided by g

B

Writing formulae:

1. There are 3 feet in a yard and the rule F = 3Y connects the number of feet and the number ofyards. Write down rules that connect:

a. The number of centimetres in metres.b. The number of wheels on cars.

2. I am twice as old as my son. I am T years old. My son is S years old. Write down formulaeconnecting T and S to explain the following:

a. How old is my son?b. How old will my son be in 4 years time?

C

Write equations from the following information:

1. Crisps cost 50p per packet. I buy x packets and spend 3.50.2. To find p multiply q by 2 then add 4.3. A man buys a daily paper for d pence and a Sunday paper for 1. The total weekly bill is 3.40.4. Two bags of sweets each contain n sweets. When 4 are eaten there are 30 left.5. Sarah buys n bottles of pop. Each bottle costs 35p and she spends 2.10.

Substituting in a formula:

If x = 3 find the value of:

1. x + 62. 3x 13. x24. 16 4x


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c. What is the clearance when w = 6.2?d. What is the ground clearance when the lorry is empty?e. The lorry cannot be driven if d is less than 30. What is the maximum weight the lorry can

carry?

6. Use the given formula to solve the questions.The formula b = 100 - h gives the boiling point of water b at the height above sea level infeet. 1000

a. Work out b when h = 5000 feetb. What is the boiling point when h = 2000 feet?c. What is the boiling point of water at the top of Mount Everest at 30 000 feet?

Additional exercises are available in the Foundation Maths textbook. If you need more practice, refer topage 58, Section 6.3, Substitution and Formulae and Exercise 6.3 (p.61).

Exercise 3: Sequences

A. Write down the next two terms in the sequence.

1. 7, 12, 17, 22, 272. 1, 5, 25, 125, 6253. 3, 4, 7, 11, 184. 1024, 512, 256, 128, 645. 1, 3, 9, 27, 816. 60, 56, 52, 48, 447. 1000, 100, 10, 1, 0.18. 13, 19, 25, 31, 37

9. 1, 3, 6, 10, 1510. 0.002, 0.02, 0.2, 2, 20

B. Each of these is the formula for the nth term. Find the first four terms of the sequence.

1. n + 12. 2n3. 2n - 14. n + 55. 3n6. 3n + 1

7. 5n - 38. 10n9. 7n - 710. 2 - n

C. Find the nth terms for each of these sequences:

1. 1 2 3 42. 4 6 8 103. 4 8 12 164. 0 2 4 65. 7 11 15 196. 1 7 13 197. 11 21 31 41


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8. 5 8 11 149. 101 201 301 40110. 25 23 21 19

Exercise 4: Simplifying expressions and factorising

A. Simplify the following expressions by collecting like terms:

1. x + x + x + x + x2. y + y + y + z + z3. a x a x a4. a + 2b + 2a + b5. 3m + 2n + n + m6. 5x - 3x + 2y y7. 6p + 2q - 5q - 3p

8. 2b x 3b

B. Simplify where possible:

1. 3a - 4b + 2a - 2b2. 4ab + 2bc - 3ba - bc3. 3ab + 2ac + ad4. 8a3 - 4a2 + 5a3 - 2a2

C. Multiply out the brackets in each of these expressions.

1. 2(a + 4)

2. 4(c + 3)3. 4(e 1)4. 4(5 j)5. -4(2 k)6. 2(3a + 5)7. 3(4e 1)8. -3(1- 5k)9. 3(2x + 3y 1)10. 3(a + 5) + 711. 5(d + 3) + 2d12. 4(2g - 3) + 5g13. a(a + 5)

14. c(c 2)15. e(3e - 1)16. k(3 2k)17. 2a(4a + 3)18. 4(a + 3) + 7(a + 4)19. 4(2c +3) + 3(3c 1)20. 2i(4 -5i) + 5i(3i + 4) + i(i +1)

D. Factorise completely:

1. 18a + 242. 18b - 273. 21d 354. 32f - 24


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5. 32g + 206. 18h 457. Dave is asked to completely factorise 20x + 40. He gives the answer 10(2x + 4). Explain why he is

wrong.8. Melanie is asked to factorise 3x2 + 2x. She gives the answer x(3x + 1). Explain why she is wrong.

E. Factorise completely:

1. a2 + 4a2. 5b2 b3. 4e 3e24. ax + bx5. 2x + 3xy6. 12x2 + 8y2

F.

1. You are given that 6(3x 5) + 3(4x - 5) = 15(ax + b). Find the values ofaand b.2. You are given that 5(4x -5y) 2(3x 2y) = c(2x + dy). Find the values ofcand d.

G. (x + 2)(x + 3) = x(x + 3) + 2(x + 3)

So multiplied out this gives: x2 + 3x + 2x + 6 which simplified is: x2 + 5x + 6Use this information to multiply out:

1. (x + 4)(x + 3)2. (x + 2)(x + 7)3. (x - 3)(x + 5)4. (x + 1)(x 4)

Further exercises and explanations are available in the Foundation Maths textbook, Chapters 10 and11:

Exercises 10.1, 10.2 and 10.3.

Exercise 11.1.

Exercise 11.2: This is an extension exercise on factorising quadratic expressions.

Exercise 5: Equations

1.a. 2a + 4 = 10b. 3t + 5 = 11c. 2p + 3 = 10d. 2r 4 = 8e. 2p 3 = 13f. 2q + 8 = 13g. d + 2 = 8

5h. 4w 1 = 23i. 4y 2 = 6

3 j. 10p 7 = 33


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k. 4f + 2 = 19l. 7y 4 = 38

2.a. 2(n + 3) = 16b. 3(p + 4) = 24

c. 2(4u + 5) = 26d. 3(2r + 1) = 21e. 3(j 2) = 12f. 2(5t + 3) = 26g. 22 = 2(4k + 1)h. 2(2w 7) = 4i. 5(2t 5) = 15 j. 6(2b 1) = 18

3.a. 8p + 4 = 7p + 6b. 12w + 7 = 2w + 27c. 8i + 7 = 4i + 35

d. 3d + 1 = 2d + 10e. 9j + 4 = j + 24f. t + 18 = 6t + 3g. 6h + 1 = 3h + 13h. 8h + 1 = 4h + 15i. 3g + 10 = 4g + 3j. 3r + 20 = 5r + 3

4.a. 4p 5 = 3p + 1b. 6y 2 = 5y + 4c. 8h 4 = 5h + 8d. 6t 7 = t + 3e. 4f 7 = f + 8f. 9i 3 = 5i + 7g. 7k 2 = 3k + 11h. 12j 5 = 8j - 3i. 9t 1 = 7t + 7j. 14j 14 = 4j 24k. 2h 1 = 4 4hl. - i + 7 = 5 3i

5.a. 3(3f + 2) = 2(3f + 6)b. 2(5w + 1) = 4(2w + 3)

c. 4(3w 1) = 2(5w + 7)d. 3(5t 2) = 3(3t + 6)e. 9t 1 = 2(4t + 1)f. 8(3d 2) = 2(7d + 5)

6.a. 3(2t 1) + 7 = 5t + 9b. 5(3e 1) 9 = 16c. 9(y 1) = 4(2y 3)d. 3(2d 3) = 2(2d 3)e. 5(4j 3) = 14j + 9f. 2(4f 1) + 5(f 2) = 6(2f + 1)g. 8(2w 1) 2(7w 5) = 10

h. 4(2f + 5) = 8 3(f + 7)i. 2 2(k 3) = 4k 4(k 3)


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j. 5d + 6(d + 3) = d 2(d 3)7.

a. 4x + 2 = x + 24 2

b. x + 6 = x + 3

5 2c. 2x + 1 = x + 3

4 7d. _ 2 = _ 6_

2x 1 x + 18.

a. Jose has 3 suitcases of banknotes and another $1200. Miguel has 1 suitcase of banknotesand another $4000. Each suitcase has the same amount of money in it. Jose and Miguelhave the same amount of money each.

i. How many dollars are in each suitcase?ii. How much money do Jose and Miguel have each?

b. Rhys thinks of a number. He multiplies the number by 7. Then he takes off 25. The answer

is 5 more than the number he first thought of. Turn his number puzzle into an equation andsolve it to find the number he first thought of.

c. Imran thinks of a number. He multiplies his number by 7 and takes the result away from120. He gets the same answer if he adds 16 to his original number. What number wasImran thinking of?

d. Lauren and Rebecca are buying CDs. Lauren starts the day with 43 and Rebecca startswith 79. Lauren buys 4 CDs and Rebecca buys 10. Each CD costs the same. After buyingthe CDs they each have the same amount of money left. How much does a CD cost?

Further explanations and exercises are available in the Foundation Maths textbook Chapter 14:

Exercise 14.1

Exercises 14.2 and 14.3 These are extension exercises on simultaneous and quadraticequations and quad

Exercise 6

1. Plot the following lines on one set of axis.a. y = 2x + 1b. y = 5x 2c. y = 4x + 3

Write down the gradient and y-intercept for each graph.

2. Repeat for the following graphs.a. y = 1 2xb. y = 2 3xc. y = -x

3. Calculate the gradient of the lines joining each of these pairs of points:a. (4,0) and (6, 8)b. (-1, 4) and (7, 2)c. (1, 5) and (3, 5)d. (-2, 6) and (0, 4)

e. (2, 10) and (10, 30)f. (-3, 6) and (-1, -2)


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4. Write down the equations of the straight line which:a. has a gradient of 4 and passes through the point (0, -1)b. has a gradient of -2 and passes through the point (0, 5)c. has a gradient of 3 and passes through the origin

d. is perpendicular to the line in part c) and passes through the point (0,2)

5. Find the equation of the line which passes through the points:a. (0, 3) and (2, 7)b. (0, 1) and (4, 3)c. (2, 0) and (0, 6)d. (-3, 15) and (0, 0)e. (0, -5) and (-2, 0)

Further exercises and explanations are available in the Foundation Maths textbook, Chapters 16 - 18:

Exercises 16.2, 17.1, 18.1, 18.2 Extension exercises are also available:

- Composite functions Ex 16.3

- Inverse functions Ex 16.4

- Plotting quadratic functions Ex 17.3

- Solving simultaneous equations graphically Ex.17.6



Notes:

1. For Confidence Level completion: 1 = Very Unsure, 5 = Very Confident2. You should use the final column (Plans to Improve) to list ways you can improve your confidence

level in this area. This might include further review of your notes, further practice, research in thelibrary/on the internet etc.

3. You might also like to use this action plan to test yourself on the key areas. Without looking at

your notes, see how much you can write about each area. This will also be helpful for examinationpreparation.

This activity only has value if you follow up your action plan. Using the remaining self-study time forunit one to begin to develop one or more of the above areas.

Algebra Action Plan

Date:


Knowledge

Confidence Level1-5

Plans to Improve

I understand the the termsexpression, formula,


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Unit 4

Page 53 of85

Unit 4: Probability


On completion of this unit, you will be able to:

Use a probability line to show probabilities.

Calculate probabilities for mutually exclusive events.

Understand relative frequency.

Calculate the probability of 2 or more independent events happening.

Use tree diagrams to calculate conditional probabilities.

2. Timings

Lectures: 1 hour

Tutorials: 1 hour

Seminars: 1 hour




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3. Seminar Notes

This seminar is designed to last for 1 hour.

Exercise 1

Below is a description of a game to play. Discuss in pairs if you think it is fair.

Exercise 2

You need 2 dice.

You are going to conduct an experiment in order to find experimental probabilities and then compareyour findings with the theoretical probabilities.

1.a. What is the highest number you can get when rolling two dice?b. What is the lowest number you can roll with two dice?

Step 1: Throw the two dice together.

Step 2: Record the result in the table below.

Step 3: Repeat the trial 100 times.

Step 4: Complete the frequency column in the table.

Step 5: Calculate the relative frequency for each possible total and complete the column.

Total on theDice

Tally Frequency RelativeFrequency

TheoreticalProbability

1

2

3

4

Coin Game

I flip two coins. If both land heads, youwin.

If they are different, I win.

If they are both tails, we flip again!


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5

6

7

8

9

10

11

12

2. Which total has the highest frequency? Which has the lowest? Why is this?

Youll return to fill in the theoretical probability later.

3. All the possible outcomes when rolling two dice can be shown in a sample space diagram. Seebelow. Complete the sample space diagram to show all the possible outcomes.

4.a. How many possible outcomes are there altogether?

b. What is the probability of getting 12?P(12) =

FirstDie

+

3

8

Second Die


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Use your sample space diagram to find the probability of the following:

c. P(11) =

d. P(11 or 12) =e. P (2) =f. P (5) =g. P (5 or less) =h. P (3) =i. P (10) =j. P (3, 4 or 5) =k. P (7) =l. P (10 or more) =m. P (multiple of 5) =n. P (square number) =

5. Using your results from your experiment and the information in your sample space diagram,complete the table below.

Total on thedice

ExperimentalProbability

TheoreticalProbability

2

3

4

5

6

7

8

9

10

11

12


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4. Private Study


Exercise 1

A: The Probability Scale

1. For each of the following, choose the word that best describes the probability of each outcomehappening.

Impossible Unlikely Even Chance Likely Certain

a. The sun will never shine again.b. It will snow in Scotland next year.c. The roll of a dice will show an odd number.d. The next baby to be born will be less than 100 Kg.e. It will snow in London in July.f. The colour of a card picked from a pack of playing cards will be red.g. December will be the next month after November.h. Your maths teacher will be on TV tonight.

2. Draw a probability scale using the words from Question 1. Mark with an arrow the probability ofeach of the following outcomes. Label the arrows with the correct letters.

a. A spun coin will come down on tails.b. This evening the sun will set in the West.c. You will score more than 10% in your next maths test.

B: Simple Probabilities

1. A fair 5 sided spinner is numbered 1 to 5. Jane spins the spinner once.a. Find the probability that the spinner will land on the number 4.b. Find the probability that the spinner will land on an even number.

2. Six coloured counters are in a bag; 3 are red, 2 are green, 1 is blue. Write down the colour of the

counter which is:a. Most likely to be chosen.b. Least likely to be chosen.

Find the probability that the chosen counter will be:a. Redb. Greenc. Blue

3. Lisa has 5 cards numbered 1 to 5. Lisa shuffles the cards and chooses one at random. What isthe probability that the card will be:

a. a 4?

b. an even number?c. an odd number?


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d. a number bigger than 3?e. a number less than 10?f. a 6?

4. One card is picked from a regular pack of cards. What is the probability of each of the followingevents happening?

a. Picking a Jackb. Picking a 10c. Picking a red cardd. Picking a 10 or a Jacke. Picking a Jack or a red Card.f. Picking a red jack

C: Relative Frequency

1. In a statistical experiment, Brendan throws a die 600 times. The table shows the results.

Number on Dice 1 2 3 4 5 6Frequency 48 120 180 96 54 102

a. Estimate the probability that he will throw a 2.b. Estimate the probability that he will throw an even number.c. Brendan throws the dice another 200 times. Estimate the number of times it will land on a

6.

2. In a bag there are 30 balls; 15 are red, 5 yellow, 5 green and 5 blue. A ball is taken out at random

and then replaced. This is repeated 300 times. How many times would I expect to get:a. a red ball?b. a yellow or blue ball?c. a ball that is not blue?d. a pink ball?

D: Tree Diagrams

1. A coin is tossed twice. Draw a tree diagram to show all the outcomes. Use your tree diagram towork out the probability of getting:-

a. two headsb. a head and a tail

c. at least one tail

2. On my way to work, I drive through two sets of traffic lights. The probability of the first set beinggreen is 0.3 and the probability of the second set being green is 0.5. Draw a tree diagram to showall the possible outcomes. Find the probability that:

a. both sets are greenb. both sets are redc. a least one set is red

3. A bag contains 10 balls, 7 red and 3 black. A ball is selected but not replaced and then anotherball is selected. Draw a tree diagram to illustrate the possible outcomes.

a. What is the probability that the first ball is a red?

b. What is the probability that the second ball is red if the first ball was red?c. What is the probability that both balls are red?


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d. What is the probability that there is a ball of each colour?e. What is the probability that at least on ball is red?

Exercise 2: Textbook

Additional explanations and exercises can be found in the supplementary textbook for this modulewhich is available in the library at your Accredited Partner Centre.

Croft, A. and Davison, R (2006). Foundation Maths, 10thEdition. London: Pearson Education.

Consult the textbook to improve your understanding and complete additional practice exercises asinstructed by your teacher.

Probability: Chapter 35. Exercises 35.2, 35.3, 35.4, Test and assignment exercises 35


Use the results from the activities in the lectures, seminars and exercises above to record yourpersonal skills, knowledge and understanding in the development planner table below. You shouldspend approximately half an hour on this activity.

Notes:



3. You might also like to use this action plan to test yourself on the key areas. Without looking atyour notes, see how much you can write about each area. This will also be helpful for examinationpreparation.

This activity only has value if you follow up your action plan. Use the remaining self-study time for thisunit to begin to develop one or more of the above areas.

Probability Action Plan

Date:


Knowledge

Confidence Level1-5

Plans to Improve

I understand the languageassociated with probability.

I can use a probability scaleto represent probabilties.


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I can calculate simpleprobabilties where theoutcomes are equally likely.

I can calculate probabilities ofmutually exclusive events.

I understand and can use

relative frequency.

I know and can use the ORrule and the AND rule.

I can calculate the probabilityof two or more successiveindependent eventsoccurring.

I can use tree diagrams torepresent the probability ofsuccessive events occurring.

I can calculate conditionalprobabilities.


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Unit 5

Page 61 of85

Unit 5: Shape, Space and Measures1. Learning Objectives

On completion of this unit, you will be able to:

Identify the metric measures used to measure length, weight and capacity.

Convert between metric measures.

Demonstrate knowledge of 2D and 3D shape names and properties.

Use angle facts to find unknown angles

Calculate the perimeter and area of 2D shapes and the volume of 3D shapes.

Demonstrate knowledge of Pythagoras Theorem

Use Pythagoras Theorem to solve problems

2. Timings

Lectures: 4 hours

Seminars: 4 hours

Tutorials: 2 hours




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3. Seminar Notes

The seminar sessions for this unit will last for 4 hours.

Exercise 1: Triangles

There are 4 types of triangle:

Scalene

Isosceles

Equilateral

Right angled

1. Investigate the properties of each triangle angles, sides, symmetry.

2. Using your information, name each of these triangles.


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Exercise 2: Quadrilaterals

The names, pictures and properties of the most common quadrilaterals are given below, but they areall mixed up.

Match each name with the correct picture and description.

Parallelogram

All sides are equal.

All angles are equal.

Opposite sides are parallel

Diagonals bisect each other at right angles

Four lines of symmetry

Rotational symmetry of order 4

Rhombus

Two pairs of equal sides

Two pairs of equal angles

Opposite sides are parallel.

Diagonals bisect each other.

No lines of symmetry


Rectangle

All sides are equal.

Opposite angles are equal.

Opposite sides are parallel

Diagonals bisect each other at right angles. Diagonals bisect the angles.

Two lines of symmetry


Trapezium

Two pairs of equal adjacent sides.

The longer diagonal bisects its shorterdiagonal at right angles.

One pair of equal angles

One line of symmetry

Square

One pair of parallel sides.

Interior angles at the ends of each parallelside add up to 180.


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Kite

Two pairs of equal sides

Four right angles

Opposite sides parallel

Diagonals bisect each other Two lines of symmetry


Exercise 3: 3D shapes

All 3-D shapes have faces, vertices and edges.

(Note: vertices is the plural of vertex.)

1. Look at the shapes in the table. Find pictures of the shapes and sketch them in.

2. Complete the numbers of faces, vertices and edges. Remember that there are hidden faces,vertices and edges.

3. For each shape, can you find the connection between the following properties?

a. The number of faces, F

b. The number of vertices, V

c. The number of edges, E

4. Find pictures of a cylinder, cone and sphere. What do they all have in common?

Shape Name Number of faces(F)

Number of vertices(V)

Number of edges(E)

Cuboid

Square-basedPyramid


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Triangular-based pyramid(tetrahedron)

Octahedron

Triangularprism

Hexagonalprism

Hexagon-based pyramid

Decahedron

Exercise 4

Change the following quantities to the dimensions shown

A. Length

1. 40mm = ?cm

2. 600cm = ?m3. 2000m = ?Km4. 45mm = ?cm5. 2759cm = ?m6. 5690m = ?Km7. 3.4 cm = ?mm8. 7.8m = ?cm9. 9.87Km = ?m

B. Weight

1. 4000mg = ?g

2. 6000g = ?Kg3. 4700mg = ?g


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4. 1387g = ?Kg5. 3.45g = ?mg .6. 0.86Kg = ?g

C. Capacity

1. 70ml = ?cl2. 500cl = ?l3. 3400ml = ?l4. 6.7l = ?ml5. 4.5cl = ?ml .6. 6.78l = ?cl


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Page 67 of{85

Exercise 5:

1. Estimate the size of the following angles.

2. Find the size of the angles marked x.

3. Find the size of the unknown angles.

4. Find the value of x in each picture.

a. b. d.

e.

c.

f.

d)

b.

a.

b. c. d.a.

c.

b. c.

a.


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5. Find the missing angles.

6. Find the value of the each letter.

a. b. c.


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7. Find the size of the lettered angles.

8. Find the missing angles in these polygons.

a.b.

c.

d.

e)

a. b.

c.

e.

d.


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9. A regular polygon is one with all its sides the same length and all interior angles equal. For aregular hexagon give:

a. the sum of the interior angles

b. the size of one interior anglec. the sum of the exterior anglesd. the size of one exterior angle.

10. A shape has 20 sides. Work out:a. the size of one exterior angleb. the size of one interior angle

11. A regular shape has an exterior angle of 20. How many sides does it have?

12. A regular shape has an interior angle of 150. How many sides does it have?

13. A polygon has an exterior angle of 44. Explain why it cannot be a regular polygon.

Exercise 6:

1. Copy and complete the following table for rectangles a to h.

Length Width Perimeter Areaa 7cm 3cmb 5cm 4cmc 4cm 12cm

d 5cm 16cme 6cm 18cm2f 7cm 28cm2g 2cm 14cmh 5cm 35cm2

2. Find the area and perimeter of these shapes.

1.2.3.4.5.

3. Find the area of these triangles.

c.

a.b.

b.a.


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4. Calculate the area of these parallelograms.

5. Calculate the area of these trapeziums.

6. Find the circumference and area of these circles. Give your answers to 1dp.a. Circle radius 6cmb. Circle radius 4.2cmc. Circle diameter 14cmd. Circle diameter 8.8cm

7. Find the diameter of a circle with a circumference of 56.52cm.

8. Find the radius of a circle with a circumference of 125.6cm.

9. Calculate the volume of a cylinder radius 5cm and length 4cm.

10. Calculate the volume of a cylinder radius 8mm and height 2.4cm. Give your answer in cm to 2dp.

11. Find:a. the volume of this cuboid

b. the surface area of the cuboid.

12. The volume of a cube is 343cm3. Find the total surface area.

a. b. c.

a. b. c.


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13. The volume of this cuboid is 336cm3. What is the value of x?

14. Calculate the volume of these prisms.

a.

b.

c.


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Exercise 7

1. Find the length of x in each of these triangles.

a. b. c.

d. e.f.

g. h.


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2. ABCD is a rectangle. Calculate the length of the diagonal AC.

3. ABCD is a rectangle. Calculate the length from B to the mid-point of AC.

4. Calculate the length of the diagonal of a square with sides 8cm long.

5. A ship sails 80km west, then 100km south. How far is the ship from its starting point?

6. A plane flies 200km north. It then flies west. The plane is 250km from its starting point. How farwest did the plane fly?

7. The sides AB and BC are in the ratio 1:2. AC is 10m. Calculate the length AB.

8. A square has a diagonal length of 10cm. Calculate the area of the square.


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4. Private Study

You should spend approximately 5 hours on the Private Study for this unit. In addition to completingthe exercises as directed by your teacher, you will need to spend time reviewing your notes and

preparing for future classes.

Exercise 1

1. Name the quadrilateral described by the following statements:a. Four equal sides, four equal anglesb. Four equal anglesc. Four equal sidesd. Two pairs of equal length sides adjacent to each other

2. Name the triangle described by the following statements:a. 2 equal angles

b. 3 equal anglesc. No equal angles

3. How many faces do the following 3D shapes have?a. cubeb. square based pyramidc. triangular prism

Exercise 2: Metric Measures

A. Put the following in order, starting with the largest.

1. 36cm, 1.45m, 0.49m, 254mm, 105cm2. 3760cm, 0.02km, 3.65km, 492m, 2150m3. 92mm, 15cm, 3.5m, 0.19m, 0.006km4. 670cm, 6.5m, 6755mm, 0.007km, 76cm5. 1.01m, 1001mm, 0.01001km, 110cm, 1.101m

B. Put the following in order, starting with the largest.

1. 300g, 0.5kg, 0.07kg, 67g, 892g, 1.04kg, 0.985kg2. 0.643kg, 6340mg, 640g, 0.06kg, 63g3. 1.59kg, 3879g, 657g, 89 000mg, 0.0043t4. 0.005t, 6.7kg, 34 200g, 27kg, 0.36t5. 35cl, 0.3l, 320ml, 30ml, 3.1l, 3200ml

Exercise 3: Angles

1. Three angles on a straight line are given in each case. Find the value of the letter in each case.

a. 52, 61 and a

b. x + 10, 10x and x + 20

2. A triangle has angles of 5x, 4x and 3x. Find x.


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3. Calculate the values of a and b in the diagram below.

4. ABCD is a parallelogram. Calculate the vales of h and j.

5. The sum of 4 angles of a pentagon is 482. What is the size of the fifth angle?

6. A polygon has 22 sides. Calculate the sum of the interior angles.

7. A regular polygon has 12 sides. Calculate the size of each exterior angle of the polygon.

8. Each interior angle of a regular polygon is 162. How many sides does this polygon have?

Exercise 4: Area, Perimeter and Volume

1. A bathroom wall measures 2m by 1.5m.a. Work out the area in cm.b. A tile measures 10cm by 10cm. Work out its area in cm.c. How many tiles are needed to cover the wall?

2. A triangle of base 10cm and perpendicular height 8.5cm is cut from a rectangle of length 14cmand width 12cm. What area of the rectangle is left?

3. Mr Greens lawn is the shape of a parallelogram with base 6m and perpendicular height 3.5m. MrGreen sows each square metre with 40 grams of grass seed. The seed costs 1.05 for 140grams. Calculate the cost of the grass seed.


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4. A door mat is the shape of a semicircle of diameter 75cm. Calculate the perimeter of the mat.Give your answer to the nearest centimetre.

5. A circular flower bed of radius 1.2m is surrounded by a path 55cm wide. Calculate the area of the

path in square metres. Give your answer to 1dp.

Exercise 5: Pythagoras Theorem

1. Find the length of the hypotenuse of these right angled triangles where the two shorter sidesmeasure:

a. 15cm and 8cmb. 3.2cm and 5.4cm

2. Find the length of the right angled triangle that has hypotenuse 10cm and height 2cm.

3. A rectangle has length of 11.4cm and width 7cm. Calculate the length of the diagonal.

4. An equilateral triangle has sides of length 10cm. Calculate the vertical height of the triangle.

5. A ladder 8.4m long rests against a wall. The foot of the ladder is 1.7m away from the wall. Howhigh up the wall does the ladder reach?



Notes:



3. You might also like to use this action plan to test yourself on the key areas. Without looking at

your notes, see how much you can write about each area. This will also be helpful for examinationpreparation.



Date:


Knowledge

Confidence Level1-5

Plans to Improve


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Unit 6

Page 79 of{85

Unit 6: Data Handling1. Learning Objectives

On completion of the unit, students will be able to:

Recognise the difference between discrete and continuous data.

Represent discrete data using bar graphs, stem and leaf diagrams, and pie charts.

Represent continuous data using frequency polygons and histograms.

Calculate the mean, median, mode and range for a set of data including grouped frequencydistributions and finding the estimated mean.

Draw scatter graphs and recognise correlation.

Carry out a survey and draw conclusions by using the tools listed above.

2. Timings

Lectures: 4 hours

Tutorials: 3 hours

Seminars: 9 hours




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3. Seminar Notes

The seminar sessions for this unit will last for 9 hours.

Exercise 1

A: Data Collection

A questionnaire is a set of questions. Questionnaires are often used in a survey to collect data onpeoples opinions on a particular topic. The questions need to be thought about carefully to ensurethey are appropriate and relevant. Here is some advice on writing suitable questions:

Avoid open questions where there is no restriction on possible answers.

Use closed questions which give a choice of answer. Use a response section containing tickboxes. This makes analysing the results much easier.

Make sure every answer can only go in one box. Avoid leading questions that encourage a particular answer. These are biased questions

and make a questionnaire invalid.

Using the information above, work in pairs to discuss and answer these questions.

1. A group of college caterers decide to ask students these questions:a. What is your age?b. We think it would be great to offer Chinese food in the college canteen. Do you agree?

i. Criticise each question.ii. Write a better version for each question.

2. For each of the following questions, say whether it is suitable, leading or biased. For eachquestion that is not suitable, write a new question.

a. Smoking kills people and should be totally banned. What do you think?b. How far do you travel to school?

Less than 3 milesMore than 3 miles

c. What is your favourite type of music?3.

a. Design a questionnaire for the public to be used by the owner of a new restaurant. It needsto have 5-8 questions. Include questions about menus, cost, opening hours etc.

b. Redesign the questionnaire to give the owner a very positive set of responses.

B: Discrete Data

1. There are 60 admissions to a hospital one day. The table shows the number of each type ofadmission. Draw a pie chart to represent the data.

Type of Admission Frequency

Medical 18

Surgical 12

Children 6

Geriatric 24

2. The information gives the weights of 30 girls and 30 boys.


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Girls 40, 66, 68, 48, 51, 54, 56, 57, 58, 59, 49, 47, 65, 67, 66, 71, 64, 62, 59, 54, 52, 53, 64,47, 72, 49, 57, 58, 59, 63

Boys 61, 57, 58, 61, 64, 65, 66, 62, 66, 68, 70, 71, 72, 76, 56, 57, 80, 59, 66, 70, 72, 77, 81,

75, 74, 60, 66, 69, 74, 73

a. Draw a back to back stem and leaf diagram to represent the data.b. A back to back stem and leaf diagram has the stems in the middle. The girls are on the

left, the boys on the right of the stems.c. Find the range of each set of data.d. Find the median of each set of data.e. Use your findings to compare the data.

3. The frequency table shows information about the number of certificates awarded to each studentin a class last month.

Number ofcertificates

0 1 2 3 4

Frequency 3 7 3 9 8

a. Write down the modal number of certificates awarded.b. Work out the range of the number of certificates awarded.c. How many students were in the class?d. Work out the total number of certificates awarded.e. Work out the mean number of certificates awarded to students.f. Work out the median number of certificates awarded to students.

4. Fifty 20 year old males and fifty 20 year old females were interviewed about how they spent their

incomes. The percentage of income that each group saved is shown in the tables.

Males% saved 0-10 11-20 21-30 31-40 41-50 51-60Frequency 2 0 19 10 11 8

Females% saved 0-10 11-20 21-30 31-40 41-50 51-60 61-70Frequency 3 3 14 12 9 6 3

a. Draw a double bar chart to compare the data. You need one set of axes and to put thebars in pairs (i.e. male and female next to each other).

b. Write about what you notice about the two sets of data.

C: Continuous Data

1. Mrs Wilson wants to sell her herd of dairy cows. A buyer will need to know the herds averagedaily yield of milk. The daily milk yield, p litres, is monitored over 5 weeks. The table shows theresults of this survey.

Milk yieldP litres

140p


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b. Calculate an estimated mean daily milk yield.c. Which is the more suitable average for the buyer to use? Give a reason for your answer.

2. A class of 30 children were asked to estimate a minute. Their teacher recorded the times they

actually said. The table shows the results.

Time(seconds)

20


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Money how they earn it, how they spend it. Who spends more: girls or boys?

Travel how far away do they live? How do they travel? How much does it cost?

Food what do they eat? How often do they eat out? How much do they spend?

Remember to include data that is numerical so you can analyse it using statistics.

Writing a hypothesis:

A hypothesis is a statement you are going to test. The plural is hypotheses.

e.g. The boys in this class are taller than the girls.

This is a hypothesis. You then create a survey to collect the data to test whether this data is true.Your conclusions would include a statement about whether or not your hypothesis is true.

2. Write a questionnaire for your survey. Remember to think carefully about your questions and

ncc ify mt student guide v1.0 [2009]

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