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FACTS YOU MUST KNOW! USEFUL MEASUREMENTS DISTANCE/LENGTH: WEIGHT: KILOMETRE (km) = 1,000m KILOGRAM (kg) = 1,000g METRE (m) = 100cm GRAM (g) = 1,000mg CENTIMETRE (cm) = 10mm MILLIGRAM (mg) MILLIMETRE (mm) MONEY: VOLUME: LITRE (l) = 1,000ml CENTILITRE (cl) = 10ml MILLILITRE (ml) One Pound = 100 pennies £1 = 100p 50p = £0.50 pound 25p = £0.25 pound 75p = £0.75 pound 10 x 10p = £1 20p x 5 = £1 10% of one pound is 10p 50% of one pound is 50p What other things do I need to know? Even and odd numbers Follow these simple rules for adding even and odd numbers: Odd + Even = Odd Division, or The Odd + Odd = Even Fraction Problem Even + Even = Even As you can see, there are rules that Follow these simple rules for subtracting even and odd numbers: Even - Even = Even Even - Odd = Odd Odd - Odd = Even tell what happens when you add, subtract, or multiply even and odd numbers. In any of these operations, you will always get a particular kind of whole number. But when you divide numbers, something tricky can happen—you might be left with a fraction. Fractions are not even numbers or odd numbers, Follow these simple rules for multiplying because they are not whole numbers. even and odd numbers: They are only parts of numbers, and can Even x Even = Even Even x Odd = Even Odd x Odd = Odd be written in different ways. For example, you can't say that the fraction 1/3 is odd because the denominator is an odd number. You could just as well write that same fraction as 2/6, in which the denominator is an even number. The terms “even number” and “odd number” are only used for whole numbers and their opposites. Square Numbers A square number is the result when you multiply a number by itself (49 is a square number because 7 X 7= 49) Square Root The sum could look like this 6 x 6 = A square root goes the other way: Or it could look like this 62 I know… All square numbers 100 or less: 1, 2, 4, 9, 16, 25, 36, 49, 64, 81, 100 3 squared is 9, so a square root of 9 is 3 Here are a few examples of how you arrive at a square number: 1 Squared = 12 = 1 × 1 The symbol for a square root is this: √ = 1 2 Squared = 22 = 2 × 2 = 4 3 Squared = 32 = 3 × 3 = 9 4 Squared = 42 = 4 × 4 = 16 5 Squared = 52 = 5 × 5 = 25 6 Squared = 62 = 6 × 6 = 36 The sum can look like this: √9 = 3 A square root of a number is a value that can be multiplied by itself to give the original number. A square root of 9 is 3, because when 3 is multiplied by itself you get 9. Remembering the square numbers and square roots will be no problem if you know your times tables! Dividing by 10, 100 and 1,000 Being able to divide by 10, 100 and 1,000 is useful when you want to convert between units. Here are some rules and examples, starting with pounds and pence. Divide by 100 to change pence into pounds. Example: 225 p = 225 ÷ 100 = £2.25 Divide by 1,000 to change grams into kilograms. Example: 1,500 g = 1,500 ÷ 1,000 = 1.500 kg = 1.5 kg Divide by 1,000 to change millilitres into litres. Example: Equivalent decimals, fractions and percentages I know… Equivalent decimals, fractions and percentages: 1 whole = 1 = 100% 3/4 = 0.75 = 75% 1/2 = 0.5 = 50% 1/4 = 0.25 = 25% 1/5 = 0.2 = 20% 750 ml = 750 ÷ 1,000 = 0.750 l = 0.75 l Divide by 1,000 to change metres into kilometres. Example: 750 m = 750 ÷ 1,000 = 0.750 km = 0.75 km Divide by 100 to change centimetres into metres. Example: 902 cm = 902 ÷ 100 = 9.02 m Prime Numbers You need to know that a prime number can only by divided by itself and 1 (13 is a prime number because is only in the 1 and 13 times tables) I Know… prime numbers less than 100: 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Factors In arithmetic, a factor is a whole number that divides exactly into another whole number. For example, what are the factors of 12? Have a go at using multiplication facts to get an answer of 12 in different ways. You can write your numbers in any order you like for a multiplication, so: 2 × 6 is the same as 6 × 2 Here’s another way to find the factors of 48: write your first pair of factors with a reasonable space between them, then move on to the next pair until you have them all. 1 × 12 is the same as 12 × 1 3 × 4 is the same as 4 × 3 Therefore, the full list of factors of 12 is: 1, 2, 3, 4, 6, and 12. When you get to the 6/8 pair, you Now try to find the factors of 48. Start with 1 and pair off your can stop because 7 is not a factor and you already have 8 in your list. numbers: Some numbers have many factors, so 1 × 48, 2 × 24, 3 × 16, 4 × 12 and 6 × organised way or you may miss some. 8 all make 48 Write the list in order: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 it’s a good idea to work in an Don’t forget to include 1 and the number itself in your list. Practising your times tables and really feeling confident with them will help you to understand how factors work! Know your times tables! Multiplication 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 54 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 Table 10 Practice every day to keep your maths skills sharp and to help you solve maths problems. Simplifying Fractions To simplify a fraction, divide the top and bottom by the highest number that can divide into both numbers exactly. What does it mean? Simplifying (or reducing) fractions means to make the fraction as simple as possible. Why say four-eighths (4/8) when you really mean half (1/2)? 4 2 /8 (FourEighths) /4 1 /2 (Two(One-Half) Quarters) Example: Simplify the fraction 24 /108: ÷2 24 108 = ÷2 12 54 ÷2 ÷3 6 = 27 ÷2 = 2 9 ÷3 Method 2 Divide both the top and bottom of the fraction by the Greatest Common Factor (You have to work it out first!). Example: Simplify the fraction 8 /12: How do I Simplify a Fraction? There are two ways to simplify a fraction: Method 1 Try dividing both the top and bottom of the fraction until you can't go any further (try dividing by 2,3,5,7,... etc). 1. The largest number that goes exactly into both 8 and 12 is 4, so the Greatest Common Factor is 4. 2. Divide both top and bottom by 4: ÷4 8 12 2 = 3 ÷4 And the answer is: /3 2 How do we convert a fraction to a decimal manually? Decimals and place value Follow these easy steps: 1: Find a number that you can multiply by the bottom of the fraction to make it into either 10, 100 or 1,000. (Or any 1s followed by zeros.) 2: Multiply both top and bottom numbers by that amount. 3: Then write down just the top number, putting the decimal point in the right place (one place from the right hand side for every zero in the bottom number.) Example 1: Express 3/4 as a decimal 1: We can multiply 4 by 25 to become 100 2: Multiply top and bottom by 25 ×25 3 4 = 75 100 ×25 3: Write down 75 with the decimal point 2 spaces from the right (because 100 has 2 zeros); Answer = 0.75 The chart above shows place value. The decimal point always stays in the same place! Use this to remind you and to help you when you are working out problems involving decimal numbers. Think of decimal numbers as MONEY! DATA HANDLING Mean (average) – To find the mean of a set of data, add up all the numbers in the list and then divide your answer by the number of numbers in the list, e.g. 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 = 135 135 ÷ 9 = 15 15 is the mean (average). Median – To find the median, order the numbers in the list from smallest to largest. The median is the middle number, e.g. 5, 3, 2, 8, 6, 9, 3 Ordered - 2, 3, 3, 5, 6, 8, 9 Mode – To find the mode, find the number that occurs most often in the list, e.g. 2, 8, 25, 26, 3, 26, 25, 9, 26 2, 3, 8 and 9 occur once 25 occurs twice 26 occurs 3 times 5 is the median 26 is the mode If there are two middle numbers in the list, add them together and divide by 2 to find the median, e.g. 4, 9, 23, 16, 8, 45, 12, 14 Ordered - 4, 8, 9, 12, 14, 16, 23, 45 Add together the two middle numbers, 12 and 14 12 + 14 = 26 ÷ 2 = 13 13 is the median Range – To find the range of a set of data, find the lowest number in the list and subtract it from the highest number in the list. e.g. 56, 94, 36, 19, 66, 12 12 is the lowest number in the list 94 is the highest number in the list 94 – 12 = 82 82 is the range of the set of data Shapes Basic information about shapes is something you need to know! How to calculate the perimeter of a shape Square All sides equal Opposite sides parallel All angles 90° Four lines of symmetry Perimeter is the distance around a two-dimensional shape. Rectangle Opposite sides equal Opposite sides parallel All angles 90° Two lines of symmetry Perimeter = l + w + l + w Parallelogram Opposite sides equal Opposite sides parallel Opposite angles equal Rhombus All sides equal Opposite sides parallel Opposite angles equal Two lines of symmetry Example: the perimeter of this rectangle is 7+3+7+3 = 20 How to calculate the area of a shape Example: What is the area of this rectangle? Kite Two pairs of equal, adjacent sides One pair of opposite equal angles One line of symmetry Trapezium One pair of parallel sides Isosceles trapezium One pair of parallel sides Base angles equal Non-parallel sides equal One line of symmetry The formula is: Area = w × h w = width h = height We know w = 5 and h = 3, so: Area = 5 × 3 = 15 All about triangles Angles of shapes Try to remember the total interior angles for the following shapes: A circle has = 360º A triangle has = 180º A square has = 360º A pentagon has = 540º A hexagon has = 720º Identifying angles An acute angle is = LESS THAN 90º A right angle is = EXACTLY 90º An obtuse angle is = BETWEEN 90º AND 180º A straight line is = EXACTLY 180º A reflex angle is = BETWEEN 180º AND 360º A complete turn is = EXACTLY 360º Have a look at the chart below to see what these angles actually look like Ratio and Proportion Ratio is a way of comparing amounts of something. It shows how much bigger one thing is than another. For example: use 1 measure of screen wash to 10 measures of water use 1 shovel of cement to 3 shovels of sand use 3 parts of blue paint to 1 part of white paint Ratio is the number of parts to a mix. For example, the paint mix is 4 parts, with 3 parts blue and 1 part white. The order in which a ratio is stated is important. For example, the ratio of screen wash to water is 1:10. This means that for every 1 measure of screen wash there are 10 measures of water. Mixing paint in the ratio 3:1 (3 parts blue paint to 1 part white paint) means 3 + 1 = 4 parts in all. Ratio is a way in which quantities can be divided or shared. Example: Share £20 between 2 people in a ratio of 3:1. A ratio of 3 + 1 = 4 parts, so the money needs to be divided into 4 parts. 20 ÷ 4 = £5 If 1 person is getting 3 parts they will have 3 × 5 = £15 The other person will have 1 part, £5. Simplest form: ratios can be simplified by finding common factors. Direct proportion: ratios are in direct proportion when they increase or decrease in the same ratio. Equivalent ratios: this is when both sides of a ratio can be multiplied or divided by the same number to give an equivalent ratio. Example There are 15 males and 12 females in a group. What is the ratio of males to females? Give your example in its 3 parts blue paint to 1 part white paint = 3 4 blue paint to 1 4 white paint. If the mix is in the right proportions we can say that it is in the correct ratio. simplest form. So the ratio of males to females is 15:12. However, both sides of the ratio can be divided by 3. Dividing 15 and 12 by 3 gives 5:4. 5:4 is the ratio in its simplest form. 5:4 and 15:12 are equivalent ratios. Direct Proportion Understanding proportion can help in making all kinds of calculations. It helps to work out the value or amount of quantities that are either bigger or smaller than the one about which you have information. Here are some examples: Example: if you know the cost of 3 packets of batteries is £6, can you work out the cost of 5 packets? £2.00 You need to get used to certain symbols and terms which sometimes mean the same thing: + Plus, add, increase, together, more, and - Subtract, minus, take away, fewer, decrease, reduce, difference X Multiply, times, product % Divide, share, divisible by £6.00 £2.00 Basic maths symbols = Equals, total < Less then £2.00 £10.00 To solve this problem you need to know the cost of 1 packet. If 3 packets cost £6, then divide £6 by 3 to find the price of 1 packet. 6 ÷ 3 = 2 Now you know that the batteries cost £2 each, to work out the cost of 5 packets you multiply £2 by 5. 2 × 5 = 10 So 5 packets of batteries cost £10. > More than If you find the less and more than symbols tricky to remember, remind yourself of this rhyme: Time facts What MUST we know and remember? Minutes, seconds and hours There are 60 seconds in one minute There are 60 minutes in one hour Days, months and years There are 7 days in the week: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday A fortnight is TWO weeks There are 24 hours in one day There are 52 weeks in one year AM = ANTE MERIDIEM = morning There are twelve months in one year, each with a different number of days. Learn this simple rhyme to help you to remember: PM = POST MERIDIEM = afternoon 12 hour and 24 hour digital times Thirty days have September, April, June, and November; February has 28 alone, All the rest have 31; Except leap year, that's the time, When February's days are 29 You could also count on your knuckles: When working out how to convert 12hr to 24hr time, if the time is in the afternoon (after midday/12 noon/PM) just add 12! There are 365 days in one year, except in a leap year where there are 366 A leap year occurs every four years A decade is 10 years A century is 100 years A millennium is 1,000 years Problem Solving Tips When you first look at a number problem it’s important that you read the question first and then work out if you need to add, subtract, multiply or divide. Try to find the Step 3 Carry out the > calculations easiest way of working out the problem - use and answer mental or the problem. written Here’s a method you can run through in methods. your head to help solve problems: Make sure you do all the steps Read and needed to What is it Step 1 > answer the about? understand the What are you problem. being asked to do? question. Have you Step 4 answered the Check that Will a diagram your answer help? works. question? > Use estimation to see if your answer is about Do you need to Step 2 Work out what calculations you need to do. > add, subtract, multiply, divide? right. Can you use a different method to Have you done check your a similar answer (eg problem? working Underline any key words to help you decide. backwards, or using the inverse sum)? Some TOP TIPS not to forget! Change them all into the same units to see which is smallest! Put them all as the same units so you can see which is earliest! Get them all as fractions, decimals or percentages