GCSE Mathematics Revision Guide

Transcription

1 2013 GCSE Mathematics Revision Guide Foundation Tier Arwel Baker 12/19/2013

2 Contents Number... 3 Factors... 3 Multiples... 3 Prime Factors... 3 Prime Numbers... 4 Square Numbers... 4 Cube Numbers... 4 Fractions... 5 Simplifying Fractions... 5 Adding/Subtracting Fractions... 5 Multiplying Fractions... 6 Dividing Fractions... 6 Decimals, Percentages & Fractions... 7 Decimals to Percentages... 7 Percentages to Decimals Percentages to Fractions Fractions to Percentages Fraction of a number... 8 Percentage of a number (With a calculator)... 9 Percentage of a number (Without a calculator)... 9 Ratios Rounding Rounding to the nearest whole number / ten / hundred etc Estimation Money Currency Exchange Units Probability Probability Basics Harder Probability P a g e

3 Probability AND / OR rules Two Events Statistics Mean, Mode, Median and Range Averages from a frequency table Charts Bar Charts Pictograms Scatter Graph Pie charts Frequency Diagram Frequency Polygon Algebra Simplifying Expressions Adding/Subtracting Simplifying Expressions Multiplying/Dividing Using letters Substitution Expanding Brackets Solving Equations Trial & Improvement Shape & Space Perimeter Area Area of a Rectangle Area of a Triangle Area of a Parallelogram Area of a Trapezium Area of a Circle Circumference of a Circle Volume Angles Pythagoras P a g e

4 Number Factors Factors are numbers that multiply into a number. Find all the factors of 24. You need to think of all numbers that multiply to 24. We have 1 24, 2 12, 3 8, 4 6. Therefore the factors of 24 are : 1, 2, 3, 4, 6, 8, 12, 24. Multiples Multiples are numbers in the times-tables. Find the first 5 multiples of 7. Multiples of 7, are numbers in the 7 times tables. So: 7, 14, 21, 28, 35. Prime Factors The question will likely ask you to write a number as a product of its prime factors. These are factors that are prime only. (See below to revise what prime numbers are). Exam Question Write 54 as a product of its prime factors. We split 54 into two numbers that multiply to 54, if one of these numbers are prime, we circle it. If the number is not prime, then we split again and carry on: 54 Once numbers are circled and you can t continue. STOP The final step it to write the answer like: = = P a g e

5 Prime Numbers Prime numbers are number which only have two factors (1 and the number itself). Here are the first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 Which of the following are prime numbers: 8, 4, 12, 11, 17, 21. Remember you need to answer this question WITHOUT using the list of prime numbers at the top of the page. You need to think about how many factors these numbers have. 8 is not prime as its factors are 1, 2, 4, 8 (more than two). 4 is not prime as its factors are 1, 2, 4 (more than two). 12 is not prime as its factors are 1, 2, 3, 4, 6, 12 (more than two). 11 is prime as its factors are 1 and 11 (only two) 17 is prime as its factors are 1 and 17 (only two) 21 is not prime as its factors are 1, 3, 7, 21 (more than two). So the prime numbers in the list are: 11 and 17. Square Numbers Square numbers come from squaring any number. Here are the first few square numbers and how to calculate them: 1 (1 2 = 1) 25 (5 2 = 25) 4 (2 2 = 4) 36 (6 2 = 36) 9 (3 2 = 9) and so on, remember 3 2 means (4 2 = 16) Cube Numbers Cube Square numbers come from cubing any number. Here are the first few cube numbers and how to calculate them: 1 (1 3 = 1) 64 (4 3 = 64) 8 (2 3 = 8) 125 (5 3 = 125) 27 (3 3 = 27) and so on, remember 2 3 means P a g e

6 Fractions Simplifying Fractions Any answers you produce which are fractions you must simplify the fraction if possible. In order to do this, you must divide both the top (numerator) and bottom (denominator) by the SAME number. Simplify We can divide both 18 and 24 by 2. Which would give us. But we can simplify this again; we can divide 9 and 12 by 3. Which would give us. Always check if you can simplify any further However, we could have reached the answer quickly if we divided both 18 and 24 by 6. Adding/Subtracting Fractions They key to adding (or subtracting) fractions is to get the bottom (denominator) of the fraction, of both fractions, the same. That is, we need the fractions to have a common denominator. Calculate The bottom of these fractions are different, so we will need to change one (or sometimes both), to get them the same. In this question, we will multiply (both top and bottom) by 2. This will give us:. This has changed my sum to:. The bottoms of the fractions are now the same. I now add the two top number 1+6=7, but I leave the bottom the same!! My final answer is :. Exam Question 2 Calculate In this question, we need to change both fractions. We will multiply by 3 and by 2. This will give us: We now have the same bottom, so we just need to add the two top number to get an answer of 5 P a g e

7 Multiplying Fractions Multiplying fractions is much easier than adding or subtracting, we multiply the two top number together and multiply the two bottom. Calculate We multiply the two top ( 1 5) and the two bottom (3 7). Which gives us: Remember; try to cancel this fraction like we ve done on page 4. In this question however, we can t. Dividing Fractions Dividing fractions is similar to multiplying fractions. We flip the second fraction and change the divide sign to a multiply. We then carry out the sum as we ve done above. Calculate We flip the second fraction and change the divide sign to multiply. This gives us a new sum: We can then simplify this fraction by dividing both top and bottom by 4, our final answer is 6 P a g e

8 Decimals, Percentages & Fractions Decimals to Percentages To convert a decimal to a percentage, multiply the decimal by 100. Exam Question Convert 0.78 to a percentage. Multiply 0.78 by 100,. The answer is therefore 78%. Percentages to Decimals. To convert a percentage to a decimal, divide the percentage by 100. Exam Question Convert 13% to a decimal. Divide 13 by 100,. The answer is therefore 0.13 Percentages to Fractions. To convert a percentage to a fraction, remember that a percentage is out of 100. Exam Question Convert 18% to a fraction. 18% means 18 out of 100. Write this as a fraction: Remember, you then need to try and simplify this, we can divide both number by 2. So the final answer is: 7 P a g e

9 Fractions to Percentages. To convert a fraction to a percentage, remember that a percentage is out of 100, so you need to change the fraction so the bottom (denominator) is 100. Convert to a percentage. We need to multiply the fraction by 5 in order to have the bottom as 100. This will give us: which is 15%. Exam Question 2 Write, 0.65 and 68% in ascending order. We need to change them so they are all of the same type, we will change them to percentages is 65% (multiply by 100). Ascending mean small to big, therefore: 0.65, and 68% (Notice we ve written them in the original form). Fraction of a number To find a fraction of a number, remember to divide by the bottom of the fraction and then multiply the answer by the top (in any order). Calculate of 30. We take 30, divide it by 5( 30 5), which gives us 6, then multiply the 6 by 3, (6 3), which is 18. Final answer is P a g e

10 Percentage of a number (With a calculator) The easiest way to find a percentage of a number is to use the actual percentage button on your calculator. (We will call this method 1). The other way is to multiply the percentage by the amount and then divide by 100. (This will be method 2). Calculate 37% of 180. Method 1: In your calculator you enter 37% 180 (Instead of of we use multiply). Method 2: We multiply the percentage by the amount and divide by 100: The final answer is Percentage of a number (Without a calculator) The key to finding a percentage without a calculator is to remember how to calculate 10% of a number. To calculate 10% we divide by 10. From this we can calculate 20% (by multiplying 10% by 2), 5% (by halving 10%), 30% (by multiply 10% by 3) and son. Calculate a) 20% of 180. b) 15% of 80. a) We firstly calculate 10% of 180. To do this we divide 180 by 10, which is = 18. We want 20% so we multiply 10% by 2, which is 18 2 = 36. Final answer is 36. b) 10% of 80 is = 8. From this we can calculate 5% by dividing by 2, which is 8 2 = 4. 15% is 10% + 5%, which mean = 12. Final answer is P a g e

11 Ratios To calculate ratios, remember the following steps: 1. Find out how many parts 2. Find the value of 1 part. 3. Calculate the total. 4. Check the numbers add up! Alex, John and Kerry want to share 1,800 using the ratio 3:5:1. How much does each person get? The first note to make is that the order of the names, corresponds to the order in the numbers. Which means: Alex receives 3 parts. John receives 5 parts. Kerry receives 1 part. Step 1: the total parts is: = 9 Step 2: We want to share 1,800 and there are 9 parts. To find the value of 1 part, we calculate = 200. Step 3: We calculate the total each person gets. Alex: One part is worth 200 and we said he receives 3 parts. So his total is: = 600. John: One part is worth 200 and we said he receives 5 parts. So his total is: = 1000 Kerry: One part is worth 200 and we said she receives 1 part. So her total is: = 200 Step 4: If we add their totals: = 1800 this should ALWAYS match how much money we were sharing. If it doesn t you need to check your work carefully for mistakes. 10 P a g e

12 Rounding. Remember: 1 Decimal place means one number after the decimal point. 2 Decimal places means two numbers after the decimal point and so on. When you are rounding you need to locate the cut off point and look at the number after it to decided whether to round up or down. When you are rounding to 1 decimal place you will need to look at the number after it (this will be the 2 nd number after the decimal point) and decide whether to round up the digit or leave it. When you are rounding to 2 decimal places you will need to look at the number after it (this will be the 3 rd number after the decimal point) and decided whether to round up the digit or leave it. When you are rounding to 3 decimal places you will need to look at the number after it (this will be the 4 th number after the decimal point) and decided whether to round up the digit or leave it. And so on, see the examples below for help. Round 7.38 to 1 decimal place. The 3 is the cut off point We need to look at the number after the cut off point (in this case the 2 nd number after the decimal point) As this number is an 8 we must round 3 up to 4 no other numbers change! So our answer is: 7.4 Exam Question 2 Round to 2 decimal place The 4 is the cut off point. We need to look at the number after the cut off point (in this case the 3 rd number after the decimal point) As this number is a 2 we do not change 4. So our answer is: P a g e

13 Exam Question 3 Round to 3 decimal place The 8 is the cut off point. We need to look at the number after the cut off point (in this case the 4 th number after the decimal point) As this number is a 5 we need to round 8 up to 9. So our answer is: Rounding to the nearest whole number / ten / hundred etc. When you are asked to round a number to the nearest ten, think about which ten (0, 10, 20, 30, 40 ) is the number is closest to. When you are asked to round a number to the nearest hundred, think about which hundred (0, 100, 200, 300, 400 ) is the number is closest to. When you are asked to round a number to the nearest whole number, think about which whole number (0, 1, 2, 3, 4 ) is the number is closest to. Round 86.3 to a) the nearest ten b) the nearest whole number c) nearest hundred. a) 86.3 is between 80 and 90, but is closer to 90. So the answer is 90. b) 86.3 is between 86 and 87, but closer to 86. So the answer is 86. c) 86.3 is between 0 and 100, but closer to 100. So the answer is P a g e

14 Estimation When you are asked to estimate a sum you will need to round all the numbers to nicer numbers you can work with. For example you would round 87 to 90, 7.6 to 8. Estimate We are going to round these numbers first. 6.7 will round to will round to 10 (you could keep it at 12 if you like), and 1.2 will round to 1. The new sum is therefore:. To calculate this we calculate 7 10 = 70, then divide by = 70. Final answer is 70. (Remember BODMAS) Exam Question2 Estimate ( ) We are going to round these numbers first will round to 3, 9.2 will round to 9 and I will round 0.48 to 0.5 (half), as I can t divide by 0! The new sum is therefore: ( ), 18 multiplied by a half is 9. So final answer is P a g e

15 Money It is vital that you read these questions carefully and understand what the question is about and what they want you to do. Here are some pointers you might want to remember: VAT: VAT is currently at 20% (will say in the question). VAT is added on top of the price of an item. Bills: If you have an old reading and a new reading (or a reading at the start and end of month), to find the number of units used, you subtract the new from the old (or end from the start). Money: Remember you need to round any money to 2 decimal place, for example A value such as is not correct. We would round this to CR/Credit: If you see this on a bill, it means you need to subtract this amount (as you may have already paid some of the bill). Currency Exchange You will always be given the exchange rate in the question. To help you with this type of question, you will need to work out how to convert from one currency to the other. See the example below. The exchange rate for to is currently 1 = To help you convert from pounds to euros and vice-versa I would do the following: 1.19 = Therefore, to go from pounds to euros I would multiply 1.19 by If I wanted to convert euros to pounds I would divide by Similarly, they could have given the exchange rate as being: 1 = This time I would write: 0.84 = P a g e

16 Knowing that 1 = Find a) 350 in euros. B) 350 in pounds. I start by writing the diagram down: = a) Therefore, to convert 350 in euros I will need to mulyiply by b) Now to convert 350 into pounds, I will need to divide by Note the calculator gives an answer of (or 6 with a dot on top). We always round money to 2 decimal place. Therefore Exam Question 2 Bill converted 700 into dollars. a) How much is this in dollars? b) Whilst over in USA he spent $600. How much pound does he have left? The current exchange rate is 1 = $1.60. I start by writing $ a) Therefore we convert 700 into dollars by multiplying by b) This question can be done in a number of ways. The quickest way is to convert $600 into pounds: We had 700 to start, spend 375, we therefore have: 15 P a g e left

17 Units Some common metric units are: cm, m, g, kg, litres. Imperial units are miles, pounds, pints and inches. You are expected to convert between different metric units. For example cm to m, kg to g etc. Remember: 1 km = 1000 m 1 m = 100 cm 1 cm = 10 mm 1 kg = 1000g 1 litre = 1000ml You are also required to convert between metric and imperial units. It is key that you remember : 1.6 km 1 kg 1 litre 2.5 cm Metric 1 mile 2.2 lb 1.75 pints 1 inch Imperial 1 ft = 12 inches 1 gallon = 8 pints Convert: a) 35 miles into km. b) pints into litres. c) 750mm into feet. Throughout this question you can draw diagrams like we did for the currency, here is one example: mile = 1.6km 1.6 a) We will use the diagram above to change 35 miles into km. b) You can draw a similar diagram for litres and pints. The calculation is: c) We begin by converting mm into cm: We now convert cm into inches: Finally, we convert inches into feet: 16 P a g e

18 Probability Probability Basics Remember all probabilities are between 0 and 1. You will be expected to use words to describe a probability, and you will be expected to calculate probability READ THE QUESTION CAREFULLY! Do not describe the probability as a word if it asks you to calculate. Here is a probability line with numbers and words. Unlikely Impossible 0 Likely Definite Even Chance 1/4 1/2 3/4 1 Mainly we will be writing probability as fraction or decimal. Remember: Probability is calculated by: For instance, think of a pack of cards. If we wanted the probability of choosing a King, then a success would be classed as picking a card with a king of it. Therefore the number of successes would be 4 (King of diamond, hearts, clubs and spades). The total outcomes is the number of cards (in this case) there are in total, which for this example is 52. Therefore the probability of selecting a king is 4/52. A bag contains 6 blue balls, 5 red balls and 9 green balls. Find the probability of picking out a green ball. Count the total number of balls: = 20 balls. As we want a green ball, we need to look at how many green balls there are, which in this example is 9. Therefore, the probability is: Remember, you can also try to simplify the fractions. 17 P a g e

19 Exam Question 2 What is the probability of rolling an even number on a 6-sided dice? There are 6 sides on a dice, this is the total. We want an even number, so we count how many even numbers on a dice (2, 4, 6), which is 3. So the probability is: which can be simplified to. Harder Probability You need to remember that the probability of an event happening and not happening always add to 1. Which means if you know the probability of an event happening, you can calculate the probability of an event not happening. The probability a person has blue eyes is 0.6, what is the probability that a person does NOT have blue eyes. The probability of a person having blue eyes and not having blue eyes MUST add up to 1. As the probability of having blue eyes is 0.6, the probability of a person not having blue eyes is whats left. The calculation is: = 0.4 Therefore the probability of not having blue eyes in 0.4 Exam Question 2 The probability a winning a game in the fairground is. What is the probability of not winning a game in the fairground. The probability of not winning is: 18 P a g e

20 Probability AND / OR rules When dealing with probability, you can convert AND into a multiply and OR into an add. The probability of having blue eyes in 0.6, green eyes 0.1 and any other colour 0.3. What is the probability of having blue or green eyes? The probability of blue eyes is 0.6, the probability of green eyes is 0.1. Because the question asks blue or green, we convert it to : Blue + Green. Replacing these by the probabilities, we get: = 0.7. Therefore the final answer is 0.7. Two Events A typical exam question will be to list all the outcomes when you have two events. For example throwing 2 dice and adding/multiplying/subtracting the result from one die to the result of the other die. They will then go on to ask about probabilities concerning this situation use your list of all outcomes to help here! Two 6-sided dice has thrown and the scores are added together. a) Complete the table below to show all the possible outcomes. Dice Dice b) What is the probability of scoring a total of 11? c) A player pays 5 to play. A player wins if they score 4 or less, and they receive 10 if they win. If 300 play, how much profit would you expect the company to make? 19 P a g e

21 a) The table shows that you have 1-6 on dice 1 and 1-6 on dice 2. We roll the dice and add the two results. Dice 2 Dice 1 b) We now have a table of all the possible outcomes (the numbers not in bold). There are 2 possibilities for 11, and there are 36 possibilities in total. Therefore the probability of gaining a score of 11 is 2/36. c) We need to first find the probability of winning. We need to find how many outcomes are 4 or less. Looking at the table above, we have 6. Therefore the probability of winning is 6/36. If 300 people play we expect 6/36 of 300 to win. Calculating this 300 play, paying 5 each, that s a total of 50 winning 10, gives a total of 500 They would therefore make a profit of = P a g e

22 Statistics Mean, Mode, Median and Range There is always a question on the exam about one or more of these four. Mode The most is the most common, or most popular, value it is possible to have more than one mode. Median The find the median, place the values in order and then find the middle value. (The biggest mistake here is students forget to place the values in order first!!!!). Range The range is the biggest value subtract the smallest value. Mean To calculate the mean, add all the values together then divide by the amount of values. Calculate a) Mode b) Median c) Range d) Mean for: a) The value that appears the most (most common) is 5. b) First put the values in order: The middle value is 5 c) The biggest value is 10, the smallest is 2. So we calculate 10 2 = 8. d) =45 (Add all values together) 45 9 = 5 (Divide by 9 as there are 9 values). 21 P a g e

23 Averages from a frequency table. Near to the end of your paper, you might be asked to calculate the mode or mean from a frequency table. To calculate the mode, just remember the mode is the most popular value (the value with the highest frequency). To calculate the median, you need to find were the middle value would be. In order to do this: 1. Find the total frequency(for example 30). 2. Then find half of this value (for example 15). 3. We now count down the frequency table until we find the 15 th value. This is your median. To calculate the mean, learn the following steps: 1. Add an extra column to the table. 2. Multiply the 1 st column by the 2 nd column (answer in the 3 rd column) 3. Calculate the total of the 2 nd column, then the 3 rd column. 4. Divide the total of the 3 rd column by the total of the 2 nd column. To calculate the range, subtract the smallest value from the largest (not the largest frequency!!). Calculate a) Mode b) Range c) Mediand) Mean for: No of Pets Frequency a) The highest frequency is 10, which corresponds to 3 pets. So the mode is 3 pets. b) The largest value is 3pets and the smallest is 1pet, therefore the range is 3-1 = 2 pets. c) We have a total of 22 values. Half of this would be 11. We now need to find where the 11 th value would be. The first 5 have 1 pet, the next 7 have 2 pets (this gives us a total of 5+7=12). Therefore the 11 th value would be in 2 pets. The median is therefore 2 pets. 22 P a g e

24 d) Following our steps, I will first extend the table, then I will calculate Pets multiplied by the frequency. No of Pets Frequency Pets Frequency = = = 30 3 rd step is to calculate the total of the 2 nd column (frequency) and the 3 rd colum. Total 2 nd = = 21 Total 3 rd = = 49 Final step (Total 3 rd ) (Total 2 nd ), which means = 2.2 Exam Question 2 Calculate a) Mode b) Mean for: Weight Frequency 30 w < w < w < Do not be scared by 30 w < 40, just think of this as meaning, weights between 30 and 40. a) The highest frequency is 18, which corresponds to the group 40 w < 50. b) The difference in this example, is that the Weights are in groups. You will carry out the calculation like the previous example, but you will use the midpoint of each group. The midpoint of 30 and 40 is 35. So we multiply 35 by 8 and so on. Weight Frequency Weight Frequency 30 w < = w < = w < = rd step is to calculate the total of the 2 nd column (frequency) and the 3 rd column. Total 2 nd = = 38 Total 3 rd = = 1750 Final step (Total 3 rd ) (Total 2 nd ), which means = P a g e

25 Bar Charts Charts Remember your categories (like colour, type of pet, etc) are placed along the horizontal line (bottom line), while the frequency is placed on the vertical line (along the side). The frequency must go up in equal steps, (1, 2, 3, 4, 5) or (2, 4, 6, 8, 10) etc. NOT (1, 3, 4, 8, 12). Here is an example of a good bar chart: Pictograms Pictograms need a key to tell you what one picture represents. For instance = 4 houses. If you want to draw 2 houses you draw half a square. If you want to draw 1 house, you draw a quarter of the square. And so on. Here is an example of a good pictogram: 24 P a g e

26 Scatter Graph Here is an example of a scatter graph. The question you will likely be asked is to plot the point this is not as easy as it seems, make sure you have practice on it before the exam! They are also likely to ask to draw a line of best fit this is one straight line using a ruler that goes through the centre of the cluster. Do NOT play join the dots!!!!! For the example above it would look like this: They might also ask what type of correlation this is. Because the line is moving up (looking left to right), then it s called correlation. a Positive If the line was moving down (looking left to right), then it s called a Negative correlation. 25 P a g e

27 Pie charts A common question is to draw a pie chart from a frequency table. The key to drawing a pie chart is to calculate the angle that you need to draw for each section. In order to do this try and remember this formula: The angle to draw is given by: Frequency Total 360 Exam Question Create a pie chart for the following data : Type of Frequency Vehicle Ford 40 Honda 20 Volvo 30 Method 1 We calculate the total first: = 90 We now need to calculate the angle we need to draw for each vehicle, using the formula Ford: = 160 Honda: = 80 Volvo: = 120. Frequency Total 360 To draw the pie chart, draw a line from the centre of the circle to the edge (usually straight up), and then measure the angle for Ford and draw the line, then measure the angle for Honda (from the new line) and so on. Ford Volvo Honda 26 P a g e

28 Method 2 Method 1 is useful if you have a calculator and if you are good are remembering formulae! In method 2 we change the values such that the frequency total is always 360. For example if we have a frequency total of 180, we multiply each frequency by 2 (because = 360). If we have a frequency total of 45, we multiply each frequency by 8 (because 45 8 = 360). If we have a frequency total of 36, we multiply each frequency by 10. The new frequency gives us the angle we need to draw. Exam Question Create a pie chart for the following data : Type of Frequency Vehicle Ford 40 Honda 20 Volvo 30 We calculate the total first: = 90. Using method 2 here, we need to change the total to 360. We will therefore multiply each frequency by 4 (because 90 4 = 360). This gives us: Type of Frequency Vehicle Ford 160 Honda 80 Volvo 120 Therefore the angle for Ford will be 160, 80 for Honda and 120 for Volvo. See the previous page for the pie chart. asure the angle for Honda (from the new line) and so on. 27 P a g e

29 Frequency Diagram A frequency diagram is a type of bar chart, when the data is grouped. As always, the frequency is placed along the side axis. The bars start at the start of the group and end at the end of the group. Notice how the numbers on the bottom axis have been placed (not like a bar chart). Create a frequency diagram for: Time (Mins) Number of People (Frequency) 30 t < t < t < t < t < t < t < Remember 30 t < 35 just means a time between 30 and 35. The first bar will be drawn from 30 to 35. The biggest mistake from students is the labelling of the bottom axis time. 28 P a g e

30 Frequency Polygon A frequency polygon is similar to a frequency diagram, but instead of bars we use points like the scatter graph and we join the dots Exam Question Create a frequency diagram for: Salary ( ) Frequency s < s < s < s < s < Remember s < 15000, just means between and We only need to plot one point for the group but we have two values (10000 and 15000). What we do in this case is we use the middle of the group, which for the firs group will be The biggest mistake from students is the labelling of the bottom axis time. 29 P a g e

31 Algebra Simplifying Expressions Adding/Subtracting It is vital that you remember the basics to simplifying algebraic expressions. We will deal with the adding and subtracting first. a + a = 2a (think apple and apple gives me two apples). NOT 2a 2!! 4b + 3b = 7b (think 4bats and 3bats gives me 7bats). Not 7b 2!!! 8c 3c = 5c (think 8carrots, remove 3carrots gives me 5carrots) We will now look at the harder questions (2 marks): Simplify 2a + 5a + 7b 5b We do NOT add or subtract different letters. We add/subtract the letters separately, as such: 2a + 5a = 7a and 7b 5b = 2b Combine these two to give the final answer: 7a + 2b (a add because the 2 is positive). Exam Question 2 Simplify 6g + 3f + 2g 5f To make the question easier, we rearrange to collect terms this means placing the same letters together. We must remember that the sign in front of the term must stick with it. For example, the minus will stick with 5f. 6g + 2g + 3f 5f 6g + 2g = 8g and 3f 5f = -2f Combine these to give the final answer of: 8g - 2f. (a subtract because the 2 is negative). 30 P a g e

32 Exam Question 3 Simplify 6a 6b + 2b 3a + 3b Again we will collect terms to get: 6a 3a 6b + 2b + 3b 6a - 3a = 3a and -6b + 2b + 3b = -b (remember b means -1b) Combine these to give the final answer of: 3a b Simplifying Expressions Multiplying/Dividing We will now look at how to simplify expression that are multiplying or dividing. a a = a2 b b b = b3 a b = ab (all we can do to simplify here is to put the letters together) 3 2 b = 6b (the answer is NOT 32b, you must multiply all numbers together) 6a 3b = 18ab (multiply numbers together, 6 3 = 18, then place a and b at the end) 5a 8a = 40a2 (5 8 is 40, and a a is a2) a b= Using letters You need to be able to setup an algebraic expression from reading a sentence. Remember these four key points: 4 more than 6 less than can be written as: can be written as: 4 times bigger than can be written as: shared by 4 can be written as: 31 P a g e

33 Charles buys 7 pencils. Write an expression for the total cost, if each pencil costs pence. Charles is buying 7 lots of pencils for pence each. So the total price is. Do not expect your answer to be just a number, as the question is asking to write an expression, this means letters and numbers. Exam Question 2 Jennifer its 5ft tall. John is inches taller than Jennifer. Write an expression for Johns height in terms of. Jennifer is 5ft tall then John is inches more than that. So his height is Substitution Substitution is the process of replacing numbers with letters and calculating the value of the whole expression. You will need to remember 5a means 5 a, and a2 means a a. Calculate the value of 2a + 3 when a = 5. We remember 2a means 2 a and we replace a by 5. Which gives us: Calculating this gives us an answer of 13. Exam Question 2 Calculate the value of 3a + 5b when a = 4 and b = 2. Replacing the letters with the right numbers gives us: - Remember to calculate this we need to calculate the multiplication sums first. Which gives us: - Final answer is 22 The most common mistake is students putting their answer as 22ab. The answers to these should have no letters! 32 P a g e

34 Expanding Brackets Expanding brackets can also be called multiplying out brackets. You need to remember to multiply the term outside the brackets by each term inside individually. Expand ( ). As the 6 is stuck to the bracket, it means the 6 is multiplied by the bracket. We first multiply 6 by the which gives us. We secondly multiply 6 by the 3 which gives us 18. Combine ours answer to get our final answer: I cannot simplify this so this is my final answer. The most common mistakes is multiplying 6 by and forgetting to multiply 6 by 3! Be careful! Exam Question 2 Expand ( ). We firstly multiply by 5, which gives us We then multiply by, which gives us.. Combine these and we get the final answer of Again, we cannot simplify this answer! Exam Question 3 Expand ( ) We firstly multiply 3a by 4, which gives us 12a. We then multiply 3a by b, which gives us 3ab. We do NOT need to do anything with the 6a as this is not inside the bracket. 33 P a g e

35 Combining everything together, we have We are able to simplify this by bringing the 12a and 6a together. Our final answer is therefore Exam Question 4 Expand ( ) ( ) We treat this question as two separate brackets then combine the answer. We will expand 6(a+b) first, then 3(b-a). Expanding ( ) will give us Expanding ( ) will give us Combining these will give us, which we can simplify. Our final answer for this question is Solving Equations Solving an equation means to find the value for x (or whatever letter). In order to do this our first step is usually to rearrange the equation in order to get the letters on one side of the equation and the numbers on the other. Finally we want to x = some value. If you move anything from one side to the other, the sign must change. A plus becomes a minus. A minus becomes a plus. Multiply becomes a divide. A divide becomes a multiply. Solve We rearrange this equation so we only have x on the left and the numbers on the right. We therefore move the +13 (notice I move the sign in front on the number also. This gives me: Remember to change the plus to a minus! When you reach this stage you have got a value for x. 34 P a g e

36 Exam Question 2 Solve We rearrange this equation so we only have the 3x on the left. We therefore move the -6 onto the right, which gives us: Remember the -6 changes to a +6. We now need to get x by itself. As 3x means multiplying by 3, we need to divide the other side by 3. Calculating the right hand side gives us: This is our final answer. Exam Question 3 Solve We rearrange this equation by moving the -x onto the left and the +3 onto the right. Which gives: Remember to change signs. We then simplify both sides: We now divide the 18 by the 6 in order to get x by itself. Calculating this to give us the final answer: Exam Question 4 Solve ( ) In this question we need to firstly expand the bracket on the right and carry on as normal: Here the bracket is expanded. We now move +x onto the left and +6 onto the right. Now simplify. This is our final answer. 35 P a g e

37 Trial & Improvement Trial and Improvement is a method for estimating solutions for harder equations. The question will give you an idea of what x is between, we use this information and guess the value for x. We then substitute this into the equation and see how close we are. We then change the value of x and see if we can get closer. Use the method of trial and improvement to find a solution for, between and to 1 decimal place. We are told our solution for x is between 3 and 4 and the solution needs to be to 1 decimal place, so we will be trying numbers such as 3.1, 3.2, 3.2 etc. When I substitute a value of x into the left hand side, I want the answer to come out as close to the right hand side, which in this question is 0, as possible. To help I draw a table: We first choose a value for x = 3.5 and calculate Value for x High / Low Calculation for = = = 0.69 Low Low High Our value of as 3.5 gave us an answer of -0.75, which is too low (as we want to get close to 0). We therefore choose a higher value for and calculate the same sum. When was 3.6 we got which was too low again, so we tried a higher value for. When was 3.7 we got 0.69 which was too high. We now decided which value of In this example our final answer is 36 P a g e gave us the closer value to 0.

38 Shape & Space Perimeter There is no formula for the perimeter of a shape. Instead you need to remember perimeter is the distance around the shape. Therefore to calculate the perimeter of a shape you add all the sides together. Calculate the perimeter of: 6 cm 14 cm We start by labelling the length of each side. Even though only two sides are labelled, we have enough to label the other sides: 14 cm 6 cm 6 cm 14 cm Remembering perimeter is given by adding all the sides together we have: (Always remember to write the units!) 37 P a g e

39 Exam Question 2 Calculate the perimeter of: 6 cm 14 cm 10 cm 5 cm We start by labelling the length of each side. Even though only two sides are labelled, we have enough to label the other sides: 15 cm 6 cm 14 cm 10 cm 8 cm 5 cm Area If you are asked to calculate the area of a shape which is drawn onto squared paper, all you need to do is count the squares. However, if you are asked to estimate an area of shape you count any square which is half or more as 1. Some question will not be on squared paper and you will need to calculate the area. In order to do this you will need to remember the formulae for each shape. 38 P a g e

40 Area of a Rectangle Height Length Area of a Rectangle = Height Length 3m Find the area of the following rectangle: 5m You should firstly rewrite the formula: We then substitute the values for height and length. Then calculate. Remember the units! Area of a Triangle Height Base Area of a Triangle = Height Base 2 You need to remember that the height is measured straight up (perpendicular) from the height. Such as this triangle: Height Base Area of a Triangle = Height Base 2 39 P a g e

41 Find the area of the following triangle: 7 mm 10 mm You should firstly rewrite the formula: We then substitute the values for height and base. Then calculate. Remember the units! Area of a Parallelogram Height Base Area of a Parallelogram = Height Base Remember, like the triangle, the height is measure perpendicular from the base. 40 P a g e

42 Find the area of the following parallelogram: 9 cm 8 cm 4 cm You should firstly rewrite the formula: We then substitute the values for height and base. Then calculate. Remember units! Remember with this question the height is 8cm. The measurement 9cm is not important, we therefore don t need it to calculate the area. Area of a Trapezium The formula for the area of a trapezium is given to you in the exam 2nd page of the exam booklet. Here it is: a h b ( a represents the length of the side shown in the diagram. b represents the length of the side shown in the diagram. h represents the height of the shape. 41 P a g e )

43 Find the area of the following trapezium: 5 cm 10 cm 7 cm In this question, a is the 5cm, b is the 7cm and the height, h, is 10 cm. We now use the formula: ( ) - Substituting the values in. ( ) Calculating the brackets first, because of bodmas. Then calculate, remember, multiplying by a half is the same as halving. Area of a Circle The area of a circle is given by: r r stands for the radius. Remember: Radius is half the diameter. 42 P a g e

44 Circumference of a Circle The cirumference of a circle is given by: d d stands for the diameter. Remember: Diameter is double the radius. 3cm Calculate the a) area b) circumference of the following circle: 1 We first need to make a note that the measurement given is a radius of 3 cm. a) The formula for the area of the circle is given by: To use the formula we need the radius, which is 3cm. b) The formula for circumference is: To use the formula we need the diameter. We have a radius of 3cm, which means the diameter is 6cm. 43 P a g e

45 Volume Again, you will need to learn some formulae in order to calculate the volume of a 3-D shape Cube/Cuboid The volume of a cube/cuboid is given by: Height Length Width 44 P a g e

46 Calculate the volume of the following cuboid. 10 cm 2 cm 5 cm This cuboid has a height of 10cm, width of 5cm and a length of 2cm. Prism The volume of a prism is given in the exam paper, on the 2nd page, the formula is: The important fact to remember, is that you need to calculate the area of the front then multiply be the length of the shape. 45 P a g e

47 Calculate the volume of the following cylinder: Area = 5cm 2 15 cm Notice the area of the cross-section is 5 cm, and the length of the shape is 15 cm. Therefore using the formula: Calculate the volume of the following cylinder: 2 cm 10 cm Notice the area of the cross-section is not given here. However we are given the radius of the circle is 2 cm and the length of the shape is 10cm. To use the formula we need the area of the circle, which is given by: We can now calculate the volume: 46 P a g e

48 Angles You need to remember the following facts about angles: The angles on a straight line add up to 180. The angles in a full turn add up to 360. The angles inside a triangle add up to 180. The angles inside a quadrilateral add up to 360. The exterior angles on any shape always add up to 360. Calculate the missing angles: a) b) x 80 y a) The angles here are on a straight line and therefore must add up to 180. In order to calculate the missing angle x, we take 70 away from 180. b) The angles here make a full turn, therefore must add up to 360. To calculate the missing angle, y, we take the 200 and 80 away from 360. Exam Question 2 Calculate the missing angles: a) b) b 70 a a) b) Angles here all add up to 180 as they are inside the triangle. The square in the corner represents a right angle which is 90 Angles here all add up to 360 as they are inside a quadrilateral (four sided shape). 47 P a g e

49 Pythagoras Pythagoras is used when you know the measurements of two sides in a right angled triangle and you need to know the third. Here is the formula: h a b a, b represent the length of the sides attached to the right angle. h represents the length of the other side (opposite of the right angle), called the hypotenuse. Calculate the length of the missing side: 5cm 12cm In this question a is the 5cm and b is 12cm, therefore we are calculating the other side, h. using the formula: We now substitute the values in. We now calculate 52 to be 25 and 122 to be 144. Calculate 25 plus 144. You now need to remember the inverse of squaring is square rooting, as we want h on its own. Calculating the square root of 169 for the final answer: The length of the missing side, is, P a g e

50 Exam Question 2 Calculate the length of the missing side: 5cm 4cm You need to be careful here because the 4 cm is the b and the 5 cm is the h. We are calculating a here. We still use the same formula, but look at how it is different from the example above. Substituting the values in. Calculating the 42 and 52. We need to solve or a, therefore move the 16 over. Calculating the right hand side. Now square rooting. 49 P a g e