Holy Cross High School

Transcription

1 Holy Cross High School Numeracy Across Learning A Guide for Parents/Carers and Staff explaining how topics involving numbers are taught within Holy Cross High School

2 Introduction All teachers now have a responsibility for promoting the development of Numeracy. With Curriculum for Excellence comes an increased emphasis upon Numeracy for all young people so it is important that Numeracy skills are revisited and consolidated throughout schooling. This information booklet has been produced to inform staff and young people how the Numeracy experiences and outcomes of Curriculum for Excellence are taught within the Maths Department at Holy Cross High School. It includes the Numeracy skills useful in subjects other than Mathematics. It is important that we (all staff at Holy Cross High School) deliver a consistent approach to our young people. Young people always have difficulties with transferable skills and if we can deliver consistent approaches of Numeracy across the school we will be helping our young people become more successful learners. It is hoped that parents/carers will also use the booklet and by being given an insight into the way number topics are being taught in the school it will easier for them to help the young people with their homework, and as a result improve their progress. Your feedback on this booklet would be appreciated. Mathematics Department Holy Cross High School Note: Each topic starts by displaying the outcomes for levels 2, and 4. Remember that the fourth level is for the majority of young people to reach by the end of S. In Holy Cross High School we have some young people working at each of these three levels. Maths Department 2 Holy Cross High School

3 Page Topic 4 Number and Number Process Estimating and Rounding 8 Fractions, decimal fractions and percentages 24 Money 0 Time 5 Measurement 40 Data and Analysis 48 Ideas of Chance and uncertainty 49 Mathematical Dictionary 5 Appendix Four Operations of Fractions 5 Appendix 2 Calculator Skills 54 Appendix Prefixes and their meaning Maths Department Holy Cross High School

4 Number and Number Process Second Level Third Level Fourth Level I have extended the range of whole numbers I can work with and having explored how decimal fractions are constructed can explain the link between a digit, its place and its value. MNU 2-02a Having determined which calculations are needed, I can solve problems involving whole numbers using a range of methods sharing my approaches and solutions with others. MNU 2-0a I have explored the contexts in which problems involving decimal fractions occur and can solve related problems using a variety of methods. MNU 2-0b I can show how my understanding of how the number line extends to include numbers less than zero and have investigated how these numbers occur and how they can be used. MNU 2-04a I can use a variety of methods to solve number problems in familiar contexts, clearly communicating my process and solutions. i.e + - x MNU -0a I can continue to recall number facts (eg multiplication tables) quickly. MNU -0b I can use my understanding of numbers less than zero (eg negative numbers) to solve simple problems in context. MNU -04a Having recognised similarities between new problems I have solved before, I can carry out the necessary calculations to solve problems set in unfamiliar contexts. MNU 4-0a When young people come to secondary school they have to cope with many different subjects and have a lot of new interests but it is still important that they practise their basic number work which may be reinforced as it was in primary school. Everyone should know their tables particularly as they move up the school. The six, seven, eight and nine times tables are very important and can be practised at home. The eleven and twelve times tables should also be reinforced. Primary School learning about place value is often forgotten and can be reinforced at home. Millions M Hundreds of Thousands HTh Tens of Thousands TTh Thousands Th Remember Hundreds H Tens T Units U Decimal Point tenths t hundredths h From the table above the: stands for Hundreds i.e stands for 5 Tens i.e stands for 6 Units i.e. 6 7 stands for 7 tenths i.e. 0 7 or stands for 8 hundredths i.e or 8 00 Reading and writing large numbers should be encouraged when appropriate.,678,02 reads Three million, six hundred and seventy eight thousand, and twenty three. It is also commonplace to use a small space instead of commas when writing large numbers. Maths Department 4 Holy Cross High School

5 ADDITION Young people are encouraged to set out their working neatly. It is important that they learn to line up whole numbers and decimals as a large number of them have difficulties with this and end up with the wrong answer. Example Calculate Note: When adding or subtracting whole numbers or decimals, figures with the same place value must be in line with each other. Zeros can be added in to help pupils line up and consequently answer question correctly. Example 2 How much does it cost altogether for a book costing 6.68 and a maths set at 2.4? Note: We put the carry on figure underneath the line. It would cost 9. altogether. Note: Communicate your answer using words. Maths Department 5 Holy Cross High School

6 SUBTRACTION The method for subtraction is called DECOMPOSITION (or exchange). We DO NOT borrow and pay back. Example What is the difference between 6.79 and.85? Note: Communicate your final answer using appropriate units. The difference in price is We also expect pupils to carry out subtraction mentally. o Counting on: To solve 4 27, count on from 27 until you reach 4 o Breaking up the number being subtracted: e.g. to solve 4 27, subtract 20 then subtract 7 MULTIPLICATION We all must encourage our young people to learn their times tables so that they are able to recite them confidently and improve their skills in multiplication. Maths Department 6 Holy Cross High School

7 MULTIPLYING by 0, 00, 000 etc When young people are multiplying by 0, 00 or 000 we teach them to move the digits, 2, or places to the left as illustrated in the examples below. Example Multiply 54 by 0 Example 2 Multiply 50 6 by 00 Th H T U Th H T U t x 0 = x 00 = However, with some classes it is easier to explain this idea by moving the decimal point. MULTIPLICATION by MULTIPLES of 0, 00, 000 etc Example To multiply by 0 multiply by 0 then by. 5 x 0 = 5 x 0 x = 50 x = 050 Example 2 To multiply by 600 multiply by 00 then by x 600 = 46 x 00 x 6 = x 6 = These rules also apply when multiplying decimal numbers. Example 2 6 x 20 Example x 500 = 8 4 x 00 x 5 = 2 6 x 0 x 2 = 840 x 5 = 2 6 x 2 = = 47 2 Maths Department 7 Holy Cross High School

8 It is important to emphasise the setting out of a multiplication. Example Example 2 A packet of crisps weighs 26 7 grams. What is the weight of 8 packets? Note: Always communicate your final answer using appropriate units. 8 packets of crisps weigh 2 6 grams DIVISION When young people are dividing by 0, 00 or 000 we teach them to move the digits, 2, or places to the right as illustrated in the examples below. DIVIDING by 0, 00, 000 etc When dividing 0, 00, 000 we move the point one, two or three places to the left. Th H T U Th H T U t = = Again with some classes it is easier to explain this idea by moving the decimal point. Maths Department 8 Holy Cross High School

9 DIVISION by MULTIPLES of 0, 00, 000 etc Example To divide by 0 divide by 0 then by = = 05 = 5 Example 2 To divide by 600 divide by 00 then by = = = 46 These rules apply when dividing decimal numbers. Example = = = 2 6 Example = = 92 5 = 8 4 It is important to emphasise the setting out of a division. Example There are 92 pupils in a year group. They are shared equally between 8 classes. How many pupils are in each class? ³2 There are 24 pupils in each class. Example 2 Divide 24 by 5 4 r 4 5 2²4 Warning 4 r 4 is NOT the same as 4 4 Maths Department 9 Holy Cross High School

10 Example Tony is paid for working 7 hours. How much does he earn each hour? When dividing a decimal number by a whole number, the decimal points must stay in line. Note: Communicate your final answer using appropriate units. Tony earns 6.42 each hour. Example 4 A jug contains 2 2 litres of juice. If it is poured evenly into 8 glasses, how much juice is in each glass? Each glass contains litres Zeros are added until there is no remainder or the required number of decimal places is reached. KNOWING which OPERATION to use. Young people often have difficulty interpreting what to do in a question. The tables below will help your son/daughter understand the vocabulary that can be used in problems and what operation to carry out. These lists are not exhaustive. ADDITION SUBTRACTION + - add Subtract the sum of take away the total of the difference between altogether how many more the value of how many less how much how much left MULTIPLICATION DIVISION X multiply Divide times Share product how many per altogether how much each (you can sometimes multiply instead of doing lots of additions) Maths Department 0 Holy Cross High School

11 NEGATIVE NUMBERS Young people should: Be able to recognise negative numbers in real life: Temperature o Bank balances Floors in a building o Stock market losses Golf scores o Sea level Know the position of negative numbers on a number line and be able to put negative and positive numbers in order. Be able to read temperatures from a thermometer. Be able to add and subtract negative and positive numbers, for example: o (- 2) + 5 = o 7-0 = - o (- 4) - 6 = -0 o 5 + (- 9) = - 4 * Remember - adding a negative is the same as subtracting * Be able to solve problems involving negative numbers, for example: o One morning in Hamilton the temperature was - C. In Aberdeen it was 5 C colder. What was the temperature in Aberdeen? Since it was colder, the temperature in Aberdeen was 5 C less than - C so the calculation is: (-) - 5 = - 6 C o My bank balance at the end of last month was (- 400). The next day my salary of 00 was paid into my account. What was my new balance? The starting balance was (- 400) and 00 was added so the calculation is: (- 400) + 00 = 700 Maths Department Holy Cross High School

12 ORDER of CALCULATION (BODMAS) Consider this: What is the answer to x 8? Is it 7 x 8 = 56 or = 42? The correct answer is 42. Calculations which have more than one operation need to be done in a particular order. The order can be remembered by using the mnemonic BODMAS. The BODMAS rule tells us the order in which the operations should be carried out. BODMAS represents (B)rackets (O)f (D)ivide (M)ultiply (A)dd (S)ubtract Scientific calculators use this rule, some basic calculators may not, so care must be taken in their use. Example Division first = 5 2 then subtraction. = Example 2 (9 + 5) x 6 Brackets first = 4 x 6 then multiplication. = 84 Example (5 2) Brackets first = then division = then add. = 20 Maths Department 2 Holy Cross High School

13 Estimating and Rounding Second Level Third Level Fourth Level I can round a number using an appropriate degree of accuracy, having taken into account the context of the problem. MNU -0a I can use my knowledge of rounding to routinely estimate the answer to a problem, then after calculating, decide if my answer is reasonable sharing my solution with others. MNU 2-0a ESTIMATING Having investigated the practical impact of inaccuracy and error, I can use my knowledge of tolerance when choosing the required degree of accuracy to make real-life calculations. MNU 4-0a For every calculation we perform, we should really carry out a rough check in order to satisfy ourselves that the result is reasonably accurate. Example How much money would I need to be able to buy 8 fudges at 9p each? As an approximation we could easily find the cost of 20 fudges costing 20p each i.e. 20 x 20 = 400p = 4.00 This answer is obviously too high (the actual answer is.42), but it does give us a rough idea of how much we should expect to pay. Example 2 What would be the approximate weight of 48 packets of crisps each weighing 2 5g? We can roughly interpret this problem as being 50 packets each weighing 0g i.e. 50 x 0 = 500g = 5kg Young people can practise estimating sensibly and getting the feel of large and small weights, heights and distances and using money in a practical way. Maths Department Holy Cross High School

14 ROUNDING Examples - When using large or small numbers it is useful to round numbers to give an approximation. a) Round 4 8 cm to the nearest cm. 4 8 cm is between 4 cm and 5 cm It is nearer 5 cm. So, we say that 4 8 cm = 5 cm (to the nearest cm). b) Round 2 cm to the nearest cm. 2 cm is between 2 cm and cm It is nearer to 2 cm. So, we say that 2 cm = 2 cm (to the nearest cm). c) Round 8 5cm to the nearest cm. 8 5 cm is between 8 cm and 9 cm. When the number is half way between, we always round up to the higher number. You would say that 8 5 cm is rounded up to 9 cm (to the nearest cm). Maths Department 4 Holy Cross High School

15 Reminders These rules apply to all units of measurement e.g. a) Round 2 kg to the nearest kilogram (kg). 2 kg is between kg and 4 kg. It is nearer to kg. So, 2 kg kg (to the nearest kg). b) Round 4 9 m to the nearest metre (m). 4 9 m is between 4 m and 5 m. It is nearer to 5 m. So 4 9 m 5 m (to the nearest m). c) Round 6 to the nearest whole number. 6 is between 6 and 7. It is nearer to 6. So, 6 6 (to the nearest whole number). The rules for rounding are: If the digit after the one you are rounding to is a 0,, 2, or 4, the last digit stays the same. Otherwise if the digit is a 5, 6, 7, 8 or 9, you have to add on (i.e. round up) to the last digit. The above rule for rounding works in all cases. Examples ) 27 (rounded to the nearest ten) 0 2) 5 64 (rounded to the nearest hundred) ) 84 (rounded to the nearest ten) 840 4) 95 (rounded to the nearest thousand) ) 2 5 (rounded to decimal place) 2 4 Maths Department 5 Holy Cross High School

16 DECIMALS 74 has 2 decimal places because it has 2 digits to the right of the decimal point. The number of digits to the right of the decimal point is the number of decimal places. e.g has decimal places; 9 8 has one decimal place. Reminders When you write an amount of money in pounds, you use 2 digits after the pound to show the pence. e.g..65 means and 65 pence Example Round.968 to the nearest penny. When you round off an amount like.968 you look at the pence..968 Any number after the pence means a bit more. Since the 8 is bigger than 5 we round up to 97 pence. This is the pence..968 =.97 (to the nearest penny). Example 2 Round 2.59 to the nearest penny This is the pence. Any number after the pence means a bit more. Since the is less than 5 we leave it as 59 pence. (We can ignore the other digits) = 2.59 (to the nearest penny) Maths Department 6 Holy Cross High School

17 SIGNIFICANT FIGURES Sometimes a number has far too many figures in it for practical use. This can be overcome by reducing the number to a certain number of significant figures, e.g. John won,467,809 in the lottery. It would be much more useful and practical to say John has won.5 million. A digit in a number is significant if it gives some sense of quantity and accuracy. Zeros can be complicated when do we count them? When do we leave them out? When zeros are used to determine the position of the decimal point or place value then they are NOT significant. Examples. 8 rounded to sig fig 40 (Zero is here for place value.) rounded to 2 sig figs (Zeros are here for place value.) rounded to sig fig (Zeros for position of decimal point and place value.) rounded to sig figs (Zeros for position of decimal point and place value. The 0 between the 5 and 6 is significant.) Maths Department 7 Holy Cross High School

18 Fractions, Decimal Fractions and Percentages Second Level Third Level Fourth Level I have investigated the everyday contexts in which simple fractions, percentages or decimal fractions are used and can carry out the necessary calculations to solve related problems. MNU 2-07a I can show the equivalent forms for simple fractions, decimal fractions and percentages and can choose my preferred form when solving a problem, explaining my choice of method. MNU 2-07b FRACTIONS I can solve problems by carrying out calculations with a wide range of fractions, decimal fractions, decimal fractions and percentages, using my answers to make comparisons and informed choices for real life situations. MNU -07a I can show quantities that are related can be increased or decreased proportionally and apply this to solve problems in everyday contexts. MNU -08a I can choose the most appropriate form of fractions, decimal fractions and percentages to use when making calculations mentally, in written form or using technology, then use my solutions to make comparisons, decisions and choices. MNU 4-07a Using proportion, I can calculate the change in quantity caused by a change in a related quantity and solve real-life problems. MNU 4-08a Everyone should be able to calculate fractions of different amounts. They are taught (and encouraged) to divide by the denominator (bottom number) and multiply by the numerator (top number). Example of 2 4 (2 ) Example 2 of 70 4 (70 5) 5 Example of 2 9 (2 7 ) Example 4 of 76 2 (76 4) 7 4 Young people should be able to give fractions in simplified form. Fractions can be simplified by dividing the top and bottom number by the same common number. You can also find equivalent fractions by multiplying the top and bottom by any number. Example 5 Simplify a) Top and bottom can be divided by 5 here. 5 is known as the Highest Common Factor. b) Top and bottom can be divided by here. is known as the Highest Common Factor. Maths Department 8 Holy Cross High School

19 Example 6 Multiply the top and bottom number of the fraction by a simple number to create a new equivalent fraction. a) b) PERCENTAGES It should be known that % means out of 00. Percentages without using a calculator. Every percentage can be written as a fraction or a decimal. The following should be learned to facilitate calculations without a calculator % 00 50% 50 50% % % % % % 0 0 0% 00 % % %, 0%, 40%, 60% etc can be calculated by: Finding 0% and then multiplying. eg 0% = 0% x, 70% = 0% x 7 2%, %, 4%, 6% etc can be calculated by: Finding % and then multiplying. eg 4% = % x 4, 9% = % x 9 % 2 66 % 2 75% 4 20% 5 Note: It is easier for some pupils to find 0% and then multiply by 2 to find 20%. 5% can also be found by finding 0% and dividing by 2. To calculate 7 5% we would show pupils to find 0%, (divide by 2) to find 5%, (divide by 2 again) to find 2 5% and then add up. (0% + 5% + 2 5% = 7 5%) Maths Department 9 Holy Cross High School

20 Examples (no calculator) a) 2 66 %of 6 2 of6 24 (6 2) b) 0%of500 cm (0% 0% ) of cm c) 9%of 720 ml (9% % 9) of ml PERCENTAGES using a CALCULATOR. To calculate percentages using a calculator we always get pupils to change the percentage into a decimal by dividing by 00 and then multiply. We do NOT use the percentage button on a calculator. Examples (with calculator) a) 4% of 60 (4 00) b) 68 5% of 500 ( ) Note: When dealing with money problems always give answers correct to 2 decimal places. FRACTIONS PERCENTAGES To change a fraction into a percentage we change to a decimal first by dividing and then multiply by 00. Example Sandra scored 24 out of 0 in her Maths test. Calculate her percentage. Make a fraction: Change to a percentage: 00 = (24 0) x 00 = 80% 0 Sandra scored 80% in her Maths test Maths Department 20 Holy Cross High School

21 PERCENTAGE INCREASE or DECREASE To find a percentage increase or decrease you first of all find the increase or decrease and then express it as a fraction of the original amount and then multiply by 00 to change into a percentage. Example A jacket cost 25. In a sale it is reduced to 85. Calculate the percentage decrease. Decrease = = Fraction of original price = Percentage Decrease = 00 2% 25 Example 2 Matthew bought a flat for Three years later he sold it for What was his percentage profit? (i.e. percentage increase) Increase = = Fraction of original price = Percentage Profit = 00 = 9 09% Maths Department 2 Holy Cross High School

22 PERCENTAGES in ACTION Example Last year a painting cost This year the painting increased in value by 2%. How much is the painting worth now? Old Price = Increase (2% of 2400) = 288 (2 00 x = ) New Price ( ) = Example 2 John the joiner buys a new tool kit costing 860. He receives a trade discount of 40%. How much does John have to pay for his new tool kit? Old Price = 860 Decrease (40% of 860) = 744 (40 00 x 860 = ) New Price ( ) = 6 Maths Department 22 Holy Cross High School

23 RATIO A ratio is used to compare two or more related quantities. The compared to is replaced with two dots : For example 2 boys compared to 8 girls can be written as 2:8. To simplify ratios, you divide both parts of the ratio by the highest common factor. For example 2:8 = 2: as you divide both sides by 6. (The same method as simplifying a fraction). Example Find the ratio of to in simplest form. : 0:6 (divide both sides by 2) = 5: SHARING a QUANTITY in a GIVEN RATIO To share a quantity in a given ratio young people are instructed to add up the total parts of the ratio, e.g. 2: total 5 parts. They are then instructed to work out one part by dividing the quantity by the total number of parts. You are then able to work out how the quantity is shared by multiplying the values of one part with the ratio values. Example is shared in the ratio :7 between Bob and Andy. How much does each receive? + 7 = 0 parts 0 parts = part = = So Bob gets x = and Andy gets 7 x = Maths Department 2 Holy Cross High School

24 Money Second level Third Level Fourth Level When considering how to spend my money. I can source, compare and contrast different contracts and services, discuss their advantages and disadvantages, and explain which offer best value to me. MNU -09a I can budget effectively, making use of technology and other methods, to manage money and plan for future expenses. MNU -09b I can manage money, compare costs from different retailers, and determine what I can afford to buy. MNU 2-09a I understand the costs, benefits and risks of using bank cards to purchase goods and obtain cash and realise that budgeting is important. MNU 2-09b I can use the terms profit and loss in buying and selling activities and can make simple calculations for this. MNU 2-09c I can discuss and illustrate the facts I need to consider when determining what I can afford, in order to manage credit and debt and lead a responsible lifestyle. MNU 4-09a I can source information on earnings and deductions and use it when making calculations to determine net income. MNU 4-09b I can research, compare and contrast a range of personal finance products and, after making calculations, explain my preferred choices. MNU 4-09c BEST BUYS Young people are encouraged to use unit amounts (i.e. find the value/cost of ) to decide which is the better value for money. Example The same brand of coffee is sold in two different sized jars as shown. Which jar represents the better value for money? o Find the cost per gram for both jars. 00g costs 86p so =.86p per gram. 250g costs 247p so = 0.988p per gram. o Since the large jar costs less per gram it is better value for money. Maths Department 24 Holy Cross High School

25 WAGES and SALARIES Young people learn that people earn money in all sorts of ways, e.g. hourly, weekly, monthly or yearly (salary). Remember: 52 weeks per year, 2 months in a year and annual means yearly. Example Isobel gets paid 9760 per annum. What is her weekly wage? = 80 Example 2 Duncan is a chef. His wage last week was 249 for working 0 hours. a) Calculate his hourly rate of pay. Hourly rate = = 8.0 b) This week he worked 8 hours. How much did he earn? OVERTIME This week he earned 8 x 8.0 = 5.40 In some jobs the rate of pay is higher for people working at night, weekends or holidays. o Double time is the normal rate x 2 o Time and a half is the normal rate x 5 Example Stuart is a long distance lorry driver with a basic wage of 4.50 per hour. His overtime pay is paid at double time. Calculate what he gets for 7 hours overtime. Overtime = 7 x (2 x 4.50) = 20 Example 2 Janet works in a petrol station, getting 6 per hour. Her overtime rate is time and a half. Calculate her total pay for a week in which she works 4 hours plus 5 hours overtime. Basic wage = 4 x 6 = 204 Overtime = 5 x ( 5 x 6) = 45 Total pay = = 249 Maths Department 25 Holy Cross High School

26 GROSS PAY, NET PAY and DEDUCTIONS Gross pay is the amount that an employer pays you. Deductions are taken from your gross pay and include things like:- o Superannuation a type of extra pension paid to you when you retire. o National Insurance (NI) to pay for loss of earnings if you are sick / unemployed. o Income Tax paid to the government to pay for education, health, transport etc. Net pay is the amount that you take home after deductions are made. Net Pay = Gross pay Deductions Example Blair has a gross pay of per annum. He pays in deductions. a) Calculate his annual net pay. Net pay = = 2 08 b) Calculate his monthly take home pay. Monthly pay = = 759 INCOME TAX Income tax calculation is a difficult and sometimes very confusing process. The Inland Revenue (H.M.R.C.) do not calculate your bill purely on your gross income. Instead they give you allowances and relief on part of your income. The allowances change after we have a budget. Taxable Income = Gross Pay Allowances. Maths Department 26 Holy Cross High School

27 COMMISSION Some people, particularly salespersons, receive a lower basic wage, but boost their earnings by adding on a percentage of their total sales this is called commission. Example Sally sells kitchens. She earns % commission on each kitchen she sells. How much is she paid for selling a 000 kitchen? Commission = % of 000 = 00 x 000 = 90 HIRE PURCHASE Hire Purchase is a way of paying for a product over a period of time. This is useful as people can get a product and pay it off over time. Hire purchase works as follows:- o A deposit is paid and the product can be taken by the customer. o The customer pays weekly or monthly instalments until the product is fully paid. o When an item is bought through hire purchase, it usually ends up costing more than it would have if the product was paid for outright. This extra money is sometimes called interest. In today s economic climate, however, many companies allow goods to be paid in instalments without adding anything extra this is called Interest Free Credit. Example The cash price for a sofa is 00. To pay for the sofa through hire purchase a 5% deposit has to be paid then 2 monthly instalments of 90. a) How much will the deposit be? Deposit = 5% of 00 = 5 00 x 00 = 65 b) How much would be paid for all 2 instalments? Instalments = 2 x 90 = 080 c) What is the total hire purchase price of the sofa? HP Price = = 245 d) How much more is this than the cash price? Extra = = 45 Maths Department 27 Holy Cross High School

28 FOREIGN EXCHANGE The rate of exchange for each currency will normally be given by an amount per and it changes daily with the stock market. Great Britain uses the pound (GBP) as its currency. Many European countries use the euro ( ). Foreign Money = Number of Pounds x Exchange Rate Number of Pounds = Foreign Money Exchange Rate In November 20 the exchange rate was:.5 Example Robert goes on holiday to Paris and takes 600 spending money with him. Using the exchange rate above how many euros would he get? Euros = 600 x 5 = 690 Example 2 Jim returns from a school trip to Germany with 85. Use the exchange rate above to find out how many pounds he will get back. Pounds = 85 5 = 7.9 VALUE ADDED TAX (VAT) The government also raises money by charging V.A.T. Most items that you purchase include VAT (currently 20%) although there are some items which are VAT free. Example Find the total cost of a car costing VAT. Vat = 20% of = x (or /5 of 7800) = 560 Total = = 9 60 Three quick methods. o To find VAT only x 0 2 o To find the price including the VAT x 2 o To find the price without VAT 2 Maths Department 28 Holy Cross High School

29 INSURANCE Questions on insurance usually involve reading values from tables and performing calculations. There are many different types of insurance: People take out Building Insurance policies to protect themselves against fire damage, storm damage, burst pipes etc. Household Content cover protects us from e.g. theft or accidental damage to the living room carpet. The payment made each year is called a Premium and Insurance Companies usually quote the cost of insuring contents for each 000 of value. Some people take out Life Insurance policies (Whole Life) so that when they die their loved ones are left with some money. There is also an endowment policy scheme where people place savings to earn interest and take out the money later in life. If you drive, you are required by law to have your car insured in case of accident or theft. The cost of your motor insurance depends on:- o The make of your car and engine size. o Where you live o Your age A No Claims Discount is money deducted from your annual premium the longer you are able to drive without making a claim. People are obliged to take out Travel Insurance mainly to cover against their holiday being cancelled or delayed, their luggage being lost on the journey or to pay for medical attention if they are ill while on holiday. The cost of your insurance depends on where you are going and for how long. Children under 2 years normally pay 50% less and under s go free. Maths Department 29 Holy Cross High School

30 Time Second Level Third Level Fourth Level I can use and interpret electronic and paper -based timetables and schedules to plan events, and make time calculations as part of my planning. MNU 2-0a I can carry out practical tasks and investigations involving times events and can explain which unit of time would be most appropriate to use. MNU 2-0b Using simple time periods I can give a good estimate of how long a journey should take, based on my knowledge of the link between time, speed and distance. MNU 2-0c Using simple time periods, I can work out how long a journey will take, the speed travelled at or distance covered, using my knowledge of the link between time, speed and distance. MNU -0a I can research, compare and contrast aspects of time and time management as they impact on me. MNU 4-0a I can use the link between time, speed and distance to carry out related calculations. MNU 4-0b UNITS of TIME century = 00 years decade = 0 years year = 2 months = 52 weeks = 65 days (66 in a leap year) week = 7 days day = 24 hours hour = 60 minutes minute = 60 seconds 0 days have September, April, June and November. All the rest have thirty-one except February alone, which has 28 days clear and 29 in each leap year. 2 HOUR CLOCK Uses a.m for morning, p.m for afternoon/evening Midday = noon = 2.00 p.m Midnight = 2.00 a.m The digits should have a point between the hours and minutes so 9.20 a.m is twenty past nine in the morning Maths Department 0 Holy Cross High School

31 24 HOUR CLOCK Has to have four numbers, doesn t have a point, no a.m/p.m 2 blocks of 2 numbers, first block for hours, second block for minutes Hours bigger than 2 indicate p.m Midday = 200 Midnight = is twenty past nine in the morning 220 is twenty past nine in the evening Example Change from 2 hour clock into 24 hour clock a) 6.0 a.m = 060 b) 0.5 p.m = 225 c) five to nine in the morning = 0855 d) five past seven in the evening = 905 Example 2 Change from 24 hour clock into 2 hour clock a) 075 = 7.5 a.m b) 205 = 8.5 p.m c) 000 = 2.0 a.m TIME INTERVALS A number line can help when calculating time intervals. The easiest way of finding how long something lasts is by counting on. Example How long is it from 0755 to 0948? (5mins) + (hr) + (48 mins) Total time = hr 5 minutes Maths Department Holy Cross High School

32 CHANGING UNITS To change decimals/fractions of hour: multiply by 60. Young people often make mistakes with the following: they think 2 5 hrs is 2 hours 5 minutes instead of 2 hours 0 minutes or 25 hrs is hour 25 minutes instead of hour 5 minutes. To change minutes to a decimal of an hour you divide by 60. You should learn that: 2 hour = 0 5 hour = 0 minutes, hour = 0 25 hour = 5 minutes, 4 4 hour = 0 75 hour = 45 minutes,. hour = 0 hour = 20 minutes, * 2. hour = 0 6hour = 40 minutes, hour = 0 2 hour = 2 minutes 5. * (Note:0 means 0 it is called a recurring decimal). Example Change 0 8 hour into minutes. 0 8 hour = 0 8 x 60 = 48 minutes Example 2 Change 27 minutes into hours. 27 min = = 0 45 hour Time is a life skill which everyone uses every day of their life. Parent/carers can encourage their children to use time calculations in the home or planning journeys by looking at timetables. Maths Department 2 Holy Cross High School

33 SPEED, DISTANCE and TIME The following three formulae are used to calculate Speed, Distance and Time. D S T S D T D T S These formulas can be easily remembered by putting the letters D, S and T in alphabetical order into a triangle as follows. To help you work out the formula, place your finger over the letter you want to find and the position of and the remaining letters leave the formula you require. Example Alison jogs at an average speed of 6km/h for hours. What distance does she jog? S = 6 km/hr T = hours D =? D = S x T = 6 x = 8km Example 2 A hot air balloon travelled 25 kilometres at an average speed of 0km/hr. For how long was the balloon in the air? D = 25 km S = 0 km/hr T =? T = S D 25 = 25 hours = 2 hours 0 minutes 0 Maths Department Holy Cross High School

34 Example George can walk to the office in 0 minutes. The distance from his house to his work is 2 5 miles. Work out George s average speed in miles per hour. T = 0 mins D = 2 5 miles S =? S = T D = = 5 mph Note: When doing Speed, Distance and Time calculations it is important that the units correspond. For example, if the speed is in km/hr and the time is in minutes, you will get the wrong answer unless you change the unit of time into hours. Maths Department 4 Holy Cross High School

35 Measurement Second Level Third Level Fourth Level I can use my knowledge of the sizes of familiar objects or places to assist me when making an estimate of measure. MNU 2-a I can use the common units of measure, convert between related units of the metric system and carry out calculations when solving problems. MNU 2-b I can explain how different methods can be used to find the perimeter and area of a simple 2D shape or volume of a simple D shape. MNU 2-c I can solve practical problems by applying my knowledge of measure, choosing the appropriate units and degree of accuracy for the task and using formula to calculate area or volume when required. MNU -a CHOOSING UNITS of MEASUREMENT I can apply my knowledge and understanding of measure to everyday problems and tasks and appreciate the practical importance of accuracy when making calculations. MNU 4-a It is important that we are able to pick out an appropriate unit of measure for different objects. Here are a few examples. LENGTH a doorway is about 2 m high and about m wide a small ruler is about 5 cm long a DVD is about mm thick WEIGHT a bag of sugar weighs kg (or 000g) a small bag of crisps weighs about 0 g a medium sized apple weighs about 50 g an average man weighs about 75 kg VOLUME (CAPACITY) a can of coke holds 0 ml a medicine spoon holds 5 ml a bucket holds about 0 litres of water fresh orange juice usually sold in litre cartons Maths Department 5 Holy Cross High School

36 Young people can practise these skills at home while helping out in the kitchen or discussing DIY projects in the home. Young people can be made aware at home of metric and imperial weights and measures. They should also be aware of their weight and height in both metric and imperial measures. UNITS of MEASUREMENT The following table should be learned by everyone. Conversion between units is an essential skill when solving practical problems. Length Volume (Capacity) Weight 0 mm = cm 000 ml = litre 000 mg = g 00 cm = m 00 cl = litre 000 g = kg 000 m = km 000 cm = litre 000 kg = tonne CONVERTING UNITS If changing from small units to large units (for example, g to kg), you divide. If changing from large units to small units (for example, km to m), you multiply. The diagram below will help you convert between metric lengths Km m cm mm X 000 X 00 X 0 Maths Department 6 Holy Cross High School

37 PERIMETER The total distance around the outside edge of a shape is called the perimeter. The units in the perimeter calculation should be the same. Example Calculate the perimeter of the shape below. P = = 20 cm 6cm 5 cm 5 cm 4cm AREA Area is defined as the space inside a 2D shape. Again like perimeter before you perform any calculations you have to check that the units are the same. The area of a rectangle is given by A = l x b (length times breadth). The area of a triangle is given by A = ½ x b x h ( half times the base times the height). Note that the base and the height of a triangle must be perpendicular (at right angles to each other). Example Calculate the area of the rectangle. mm A = l x b A = 8 x A = 24 mm 2 8 mm Example 2 Calculate the area of the square. The length and breadth are the same. A = l x l A = 5 x 5 A = 25 m 2 5 m Maths Department 7 Holy Cross High School

38 Example Calculate the area of the triangle. 4 cm A = ½ x b x h A = ½ x 7 x 4 A = 4 cm 2 7 cm Note : More complicated shapes can be split up into separate shapes. Example 4 Calculate the area of the shape below. Area = l x B A = 4 x 2 2 A = m 2m Area 2 = ½ x b x h A = ½ x 4 x ( 2 2 2) A = 2 4m Total area = = 0 8 m 2 VOLUME The volume of a shape is simply the amount of space it takes up and is three dimensional. A small cube measuring cm by cm by cm has a volume of cubic centimetre or cm³. This space is equivalent to ml of liquid. The volume of a cuboid is calculated by multiplying the length by the breadth by the height, the formula is V = l x b x h. Again you should always check that the units are the same throughout. Maths Department 8 Holy Cross High School

39 Example Calculate the volume of the cuboid. V = l x b x h cm 9 cm 2 cm V = 9 x 2 x V = 54 cm Example 2 Calculate the volume of the cube with side 2 metres. The length, breadth and height are all the same. 2 m mm V = l x l x l V = 2 x 2 x 2 V = 8 m When using Area or Volume Formulae young people are expected to: Write down the formula Substitute appropriate values Calculate answers with appropriate units [as shown in the above examples] Maths Department 9 Holy Cross High School

40 Data and Analysis Second Level Third Level Fourth Level Having discussed the variety of ways and range of media used to present data, I can interpret and draw conclusions from the information displayed, recognising that the presentation may be misleading. MNU 2-20a I have carried out investigations and surveys, devising a variety of methods to gather information and have worked with others to collate, organise and communicate the results in an appropriate way. MNU 2-20b I can work collaboratively, making appropriate use of technology, to source information presented in a range of ways, interpret what it conveys and discuss whether I believe the information to be robust, vague or misleading. MNU -20a I can evaluate and interpret raw and graphical data using a variety of methods, comment on relationships I observe within the data and communicate my findings to others. MNU 4-20a Nowadays young people are encouraged to draw graphs using software packages, but from time to time they have to draw graphs by themselves. The following list is a guide to help them. For all graphs that are drawn we expect that a sharpened pencil and ruler are used at all times. The following pages illustrate examples of the various types of graphs young people are expected to be able to construct. Maths Department 40 Holy Cross High School

41 Number of pupils BAR GRAPHS Bar graphs are often used to display data so that the information is easier to interpret. The horizontal axis should show the categories and the vertical axis the frequency. The graph should have a title and each axis must be labelled. Example The table below shows the homework marks for class 4B. Mark Number of Pupils (frequency) Here is the BAR GRAPH that illustrates the data. Homework marks for class 4B 0 graph title axis label there should be an equal space between the bars Frequency is 0 labelled on the lines on the vertical axis Mark all the bars are the same width axis label bars are labelled in the centre (either numbers or words) Note: A Bar Graph should have spaces of EQUAL WIDTH between the bars. When there are no spaces, the graph is called a HISTOGRAM. Maths Department 4 Holy Cross High School

42 Weight(kg) LINE GRAPHS Line graphs are used to show trends in data. The trend of a graph is a general description of it. Line graphs consist of a series of points which are plotted then joined by a line. The graph should have a title and each axis must be labelled. Example The table shows Heather s weight over 4 weeks are she follows an exercise programme. Week number Weight(kg) Here is the LINE GRAPH that illustrates the data. 85 x axis label x x x x x Heather s Weight x x x x x x graph title x x on the lines Week Notes axis label on the lines Since the numbers are quite high we can break the vertical axis. Please note in Science the vertical axis is never broken. The TREND of the graph is that Heather s weight is decreasing. Maths Department 42 Holy Cross High School

43 Height SCATTERGRAPHS A scattergraph is used to display the relationship between two variables. A pattern may appear on the graph. This is called the CORRELATION. This can be either POSITIVE or NEGATIVE. Example The table below shows the height and arm span of a group of first year boys. Arm Span (cm) Height (cm) The SCATTERGRAPH below is produced from the data. The information is plotted as a series of points - similar to a line graph but without joining the points up. 70 S Boys 65 x x x x x x x 45 x x x 40 5 x x x Arm Span The graph shows a general trend as the arm span increases so does the height. This graph shows a POSITIVE correlation. The line drawn is called the LINE of BEST FIT. We normally instruct the young people to draw a line by having the same number of points above it as below it. The line can be used to provide estimates. For example, a boy with arm span 50cm would be expected to have a height of around 52cm. The method of finding this is shown on the diagram. Maths Department 4 Holy Cross High School

44 PIE CHARTS INTERPRETING PIE CHARTS A pie chart can be used to display information. Each sector (slice) of the chart represents a different category. The size of each category can be worked out as a fraction of the total using the number of divisions or by measuring angles. Example 0 people were asked the colour of their eyes. The results are shown in the pie chart. HAZEL BROWN BLUE GREEN How many people had brown eyes? The pie chart is divided up into ten parts so people with brown eyes represent 0 2 of the total. 2 0 of0 6 so 6 people had brown eyes. If no divisions are marked we can work out the fraction by measuring the angle of each sector. The angle of the brown sector is 72 o. 72 So the number of pupils with brown eyes = x 0 = 6 people 60 Note: If you find a value for each sector this should add up to 0 pupils. Maths Department 44 Holy Cross High School

45 PIE CHARTS DRAWING PIE CHARTS In a pie chart the size of the angle for each sector is calculated as a fraction of 60 o. Example In a survey about television programmes a group of people were asked about their favourite soap. Their answers are given in the table below. Draw a pie chart to illustrate the information. Soap Number of People Eastenders 28 Coronation Street 24 Emmerdale 0 Hollyoaks 2 None 6 The angles of the pie chart are calculated as follows: Eastenders Coronation Street Emmerdale Hollyoaks None NONE Check that the total is 60 o. HOLLYOAKS EASTENDERS EMMERDALE CORONATION STREET Remember to label all the sections of the pie chart Note: when the total frequency divides into 60 without leaving a remainder it is useful to calculate the size of the angle for and then multiply up from there. Example If the total frequency is 60 then each one would have 6 o (60 60) so 2 would be represented by 6 x 2 = 72 o. Maths Department 45 Holy Cross High School

46 AVERAGES To give information about a set of data, the average value may be given. There are three methods of finding the average value. They are the MEAN, the MEDIAN and the MODE. MEAN The mean is found by adding all the data together and dividing by the number of values. i.e. sumof allvalues mean = thenumberofvaluesused MEDIAN The median is the middle value when all the data is written in numerical order from smallest to largest. If there are two middle values the median is half-way between these values. MODE The mode is the value that occurs most often. RANGE The range of a set of data is a measure of spread. Range = highest value - lowest value Example A class scored the following marks for their homework assignment Find the mean, median, mode and range Mean = So mean is 7 to decimal place Median first write data in order Median = 7 Maths Department 46 Holy Cross High School

47 Mode since 7 is the most frequent mark we say the mode is 7. [If more than one number occurs most often we say there is no mode] Depending on the data you have, at times it can be misleading and it is important that you use the appropriate average. The mean is useful when a typical value is wanted. Be careful not to use the mean if there are extreme values. The median is a useful average to use if there are extreme values. The mode is useful when the most common value is needed. The range is used to help us decide how spread out the data is. The range is calculated as follows. Range = Highest Number - Lowest Number In the above example the range would be 0 4 = 6 When the range is small that means that your data is close together or consistent. If the range is large then your data is well spread out and has extreme values. Maths Department 47 Holy Cross High School

48 Ideas of Chance and Uncertainty Second Level Third Level Fourth Level I can conduct simple experiments involving chance and can communicate my predictions and findings using the vocabulary of probability. MNU 2-22a I can find the probability of a simple event happening and explain why the consequences of the event, as well as its probability, should be considered when making choices. MNU -22a By applying my understanding of probability, I can determine how many times I expect an event to occur, and use this information to make predictions, risk assessment, informed choices and decisions. MNU 4-22a Probability is a measure of how likely or unlikely an event is to happen. It is measured between 0 and and can be shown as a fraction or a decimal. 0 ½ Impossible 50/50 Certain Chance To find the probability of an event, we use: Probability (event) = number of favourable outcomes total number of possible outcomes Example What is the probability of picking a black counter from a bag containing 5 red, blue and 2 black counters? Number of favourable outcomes = 2 (number of black counters) Number of possible outcomes = = 0 (total number of counters) P(black) = Always leave your fraction in its simplest form. Example 2 If a fair die is thrown 00 times, approximately how many 5 s are likely to be obtained? P (5) 6 We multiply 00 by 6 since 5 is expected fives 6 of the time. Young people should be familiar with packs of cards and sets of dominoes playing these games at home will help with many aspects of numeracy. Maths Department 48 Holy Cross High School

49 Mathematical Dictionary (key words) a.m. Ante meridiem. Any time in the morning between midnight and 2 noon. Approximate An estimated answer, often obtained by rounding number to nearest 0, 00 or 000. Axis A line along the base or edge of a graph. Plural - Axes Calculate Find the answer to a problem. It does NOT mean that you should use a calculator! Interest Money paid on a savings account in the bank. Data A collection of information may include facts, numbers or measurements. Denominator The bottom line of a fraction. Digit A number Discount The amount an item is reduced by. Equivalent fractions Estimate Evaluate Even Factor Frequency Greater than (>) Gross pay Increase Least Less than (<) Maximum Mean Median Minimum Mode Multiple Fractions which have the same value. E.g. 2 6 and 2 are equivalent fractions. To make an approximate or rough answer often by rounding. To work out the answer to. A number that is divisible by 2. Even numbers end with 0, 2, 4, 6 or 8. A number which divides exactly into another number leaving no remainder. E.g. the factors of 5 are,, 5, 5 How often something happens. In a set of data it is the number of times a number or a category occurs. Is bigger or more than. Example: 0 is greater than 6 is written 0 > 6. Pay before deductions. The amount by which something has gone up. The lowest or minimum value in a group. Is smaller than or lower than. Example: 5 is less than 2 is written 5 < 2. The largest or highest number in a group. A measure of the average of a set of numbers. A measure of average the middle number in an ordered set of data ordered from the lowest to the highest. The smallest or lowest number in a group. Another type of average the most frequent number or category. A number which can be divided by a particular number leaving no remainder. Example: some of the multiples of 4 are 4, 8, 2, 6, 48, 72 The stations (or answers) in the times tables. Maths Department 49 Holy Cross High School