Know how to use fractions to describe part of something Write an improper fraction as a mixed number Write a mixed number as an improper fraction

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1 . Fractions Know how to use fractions to describe part of something Write an improper fraction as a mixed number Write a mixed number as an improper fraction Key words fraction denominator numerator proper fraction improper fraction mixed number One way to think of a fraction is as a number of parts of a whole. The bottom number in a fraction is the number of parts we divide the whole into. These parts are the same size. The bottom number is called the denominator. The top number in a fraction is the number of these parts we are counting. The top number is called the numerator. These are proper fractions : the denominator is greater than the numerator. So the fraction is made of some of the parts of the whole. These are improper fractions : the denominator is less than the numerator. An improper fraction can be converted to a mixed number part whole number and part fraction. as a mixed number is. or or Example Convert into an improper fraction. 9 Start by finding the number of eighths in wholes by multiplying by. Example Convert into a mixed number. remainder therefore Divide by to find how many wholes can be made from. Example What fraction of a turn does the minute hand of a clock make between 9:0 a.m. and 9:0 a.m.? The time between 9:0 a.m. and 9:0 a.m. is 0 minutes 0 minutes is of a whole turn 0 minutes is whole turn. Maths Connect R

2 Exercise.... a) Look at each shape. Write the fraction which is shaded. i) ii) iii) b) Write the fraction of the unshaded part of each shape. What fraction of a turn does the minute hand of a clock make between: a) 9:0 a.m. and 0: a.m.? b) : a.m. and : a.m.? c) : p.m. and : p.m.? What fraction of a turn does the hour hand of a clock make between: d) : a.m. and : a.m.? e) :0 a.m. and :0 p.m.? f) : p.m. and : p.m.? Convert these mixed numbers into improper fractions: a) b) c) d) e) 0 f) g) h) i) 9 Convert these improper fractions into mixed numbers: j) a) b) c) d) e) f) g) 0 h) i) j) 9 Write these as mixed numbers. The number of: a) cm in mm b) days in 0 hours c) minutes in 0 seconds Write, or between each pair: a) and b) and c) and d) and e) and f) g) and h) 0 and 0 i) and and Copy and complete: a) b) c) d) e) f) Oranges are cut into quarters. At half-time in a hockey match, eleven players and four substitutes each have a quarter of orange. How many oranges do they have between them? How many are left from a bag of ten oranges? Long sticks of dowelling are cut into fifths to make counting sticks. We have enough to make counting sticks. How much dowelling did we have? Investigation Try converting both numbers to a mixed number to compare them. We will call and -digit fractions. Investigate different -digit fractions between and, such as: 0 Work systematically: start searching for fractions with a denominator of, then, then, and so on. Number : Proportion Fractions

3 . Equivalent fractions Recognise sets of equivalent fractions Simplify a fraction by cancelling Reduce a fraction to its lowest terms Key words numerator denominator equivalent fraction lowest term cancel Fractions are equivalent when they have the same value. You can use fraction walls to find sets of equivalent fractions. whole This wall illustrates these sets of equivalent fractions: (,,, ) (,, ) (,, ) You can also use a multiplication square to find sets of equivalent fractions The two highlighted rows show this set of equivalent fractions:,,, ). Expressed in its lowest terms, this fraction is. (, 0 0 Example Find the missing number in 0 The denominator has been multiplied by ( 0) So, the numerator,, must also be multiplied by ( ), giving 0 The numerator is the top number of the fraction, the denominator is the bottom number. Example Write in its lowest terms: Cancel by dividing both the numerator and denominator by. Divide again by. Now the fraction is in its lowest terms. Maths Connect R

4 Exercise.... Copy each pair of equivalent fractions, writing the missing number: a) b) c) d) e) Write the following in their lowest terms: a) 0 b) c) 0 0 d) 0 e) f) 0 g) 9 h) i) j) Write all the equivalent fractions, which have a smaller denominator, to each of these: a) b) c) Four fractions in each set are equivalent. Write down the fraction that is not. a),, 0 0,, 9 b) 9,,,, c),, 0,, 0 0 d),, 0, 0, 0 e) 0, 0, 0, 0 9, 0 f),,,, g),,,, 9 h), 0,,, Who am I? a) I am a fraction equivalent to. My denominator is. b) I am a fraction equivalent to. My numerator is 0. c) I am a fraction equivalent to. My numerator and denominator have a total of. d) I am a fraction made of four digits. My digits are all even. I reduce to in my lowest terms. e) I am a fraction. I reduce to in my lowest terms. My numerator and denominator have a total of 0. Investigations You need a set of 9 digit cards. Draw a large version of this equivalent fractions board. Investigate different ways of placing the cards to create pairs of equivalent fractions. Here is one example: Copy the rectangular grid on the right. Investigate how many different fractions you can show by colouring different sections. Here are two examples: 9 Work systematically. For example: keep the denominator of the right hand fraction fixed, and explore different ways of arranging the cards to make a fraction which is equivalent. Check to see if any of your fractions are equivalent. Equivalent fractions

5 . Key words decimal convert recurring decimal Convert a decimal to a fraction equivalent fraction Convert a simple fraction to a decimal, using division lowest terms Convert a simple fraction to a decimal, using a calculator, and put a set of fractions in order Recognise recurring decimals Comparing fractions To convert a fraction to a decimal, find an equivalent fraction with a denominator of 0 or 00. For example: 0 ; four tenths 0. ; forty two hundredths is four tenths and two hundredths The fractions,, 0 and can be ordered by either: converting each to a decimal by first converting to tenths or hundredths: ; ; ; or: changing them to equivalent fractions with the same denominator and placing them on a number line: In both cases, the order is:,, 0,. Another way of converting a fraction to a decimal, is to divide the numerator by the denominator, for example, is converted by dividing by, to get 0.. The division can also be done using a calculator Converting the fraction to a decimal using division, leads to 0., where s continue indefinitely. The decimal is called a recurring decimal. We say that is recurring. This is written with a dot over the, like this: 0... Example Write < or > between the pair of fractions and 0, converting each one to a decimal first. 0. is less than 0. so 0 Multiply numerator and denominator by. Multiply numerator and denominator by. Example Convert. to a mixed number.. 0, 0 which reduces to Divide numerator and denominator by. 0 Maths Connect R

6 Exercise.... Draw each of the following number lines. Mark the position of each fraction, then write them out in order, smallest first: a) b) c) 0 0 (twentieths) 0 (eighths) 0 (twelfths) Write or between each pair of fractions, by first converting each fraction to a decimal: a) and b) 0 and c) and 9 d) 0 0 and 0 e) 0 and 0 f) and 0 Convert each decimal to a fraction in its lowest terms: a) 0. b) 0. c) 0. d) 0. d). e). f).0 h). Convert these fractions to decimals, using written division: a) 0 b) c) d) Check each answer using a calculator. Write these fractions as recurring decimals. a) b) 9 c) 9 d) e) 9 f) g) 9 h) Put these in order, smallest to largest: a),, 0., 0, 0 0. b) 9, 0, 0, 0., c) 0., 0.,, 0, d) 0.9,, 9 0, 0., 90 e),.,, 0., 0 Who am I? a) I am a proper fraction. My decimal equivalent is 0.. My denominator is 0. b) I am an improper fraction. My decimal equivalent is.. My numerator and denominator have a total of. c) I am a -place decimal equivalent to a fraction with a denominator of 0. Its numerator is the total of my tenths and hundredths digit. Investigation 0 Convert to tenths or hundredths, then cancel. Put the dot over the first decimal that recurs, e.g is written as 0... You need a set of 0 9 digit cards. Draw a large version of this diagram. Investigate different ways of placing the cards to create a fraction and its equivalent decimal. Here are two examples: 0 0 Comparing fractions 9

7 . Adding and subtracting fractions Add two fractions Subtract two fractions Break up a calculation into steps Key words improper fraction equivalent fraction lowest terms mixed number denominator When you add or subtract fractions, it may help to draw a number line. If fractions have the same denominator, simply add the numerators: 0 If the fractions have different denominators, convert them to that have the same denominator: equivalent fractions Example Work out Once you have the answer, change the improper fraction to a mixed number ( ). Reduce the fraction to its lowest terms ( ). Example Work out so Find an equivalent fraction to so that both fractions in the subtraction have the same denominator. Example Work out Convert each to an improper fraction. Change to equivalent fractions with the same denominators ( ). Change back to a mixed number. 0 Maths Connect R

8 Exercise.... Work these out, and express the answer in its lowest terms: a) b) c) d) e) f) 0 0 g) h) i) j) Work out: a) b) 0 c) d) e) f) 0 g) h) i) 0 Find in each of the following: a) b) 0 0 c) d) e) f) g) h) I completed of my book last month, and this week. What fraction of the book needs completing? What is the perimeter of a rectangular room that is m wide and m long? If in every hour period, you spend of the time sleeping, of the time at school, and of the time eating, and half of the remaining time watching television, what fraction of the day is left? Who am I? a) I am a fraction. I am one half more than one third. b) I am a fraction. I am one quarter less than five eighths c) I am an odd number of fifths. When one half is subtracted from me the result is equivalent to 0. Investigations Start with a number, for example. Explore different additions which have this number as an answer. For example: two fractions with the same denominator: or three fractions with the same denominator: or two fractions with different denominators: Here are some pairs of fractions which have a difference of : and and and Investigate pairs of fractions which have a difference of. Repeat for. Start with thirds and write down about five pairs. Then look at sixths, ninths, twelfths, and so on. Adding and subtracting fractions

9 . Fractions of amounts Calculate fractions of numbers, quantities and measures Multiply a fraction by a whole number Key words expressions of 0 kg is the same as 0 kg which is kg. kg kg kg kg To find of kg, start by finding of kg, then multiply by : of kg kg of kg kg. So kg kg kg kg kg kg 0 and of 0 are identical; of and are identical expressions. of can be written in different ways. These include:,,,,, Example Find the missing number in of cm cm. of, so the missing number is of What do you need to multiply by to get? Exercise.... Work out: a) of cm b) of p c) 9 of days d) of 9 minutes e) of m f) of km g) of litres h) of cm i) of p j) 9 of kg k) 0 of 00ml l) 0 0 of 00 Work out: a) 9 b) 0 c) 0 d) 0 e) f) g) h) Maths Connect R

10 For each of the following lists, put the expressions in order, smallest to largest: a) A: 0 B: C: of D: E: b) A: of B: C: D: E: 9 c) A: B: 0 0 C: D: 0 E: 0 of 0 d) A: 9 of B: 90 of 0 C: D: 0 E: 0 Find in each of the following: a) of 9 b) of c) of d) e) 0 f) 0 of 0 99 g) h) 0 i) j) 9 of Copy and complete each multiplication table: can be written as the result of of, of, of 0, and so on Write at least three different similar calculations for a), b), c) 0 At a test match there were 000 spectators. supported the home team, supported the away team, and the remainder were neutral. How many neutral spectators were there? A -litre jug of punch is three-quarters full. One third of this is poured out. How much punch is left in the jug? The minute hand of a clock rotates through of a complete turn. What is this angle in degrees? A complete turn is 0º. Who am I? a) I am a number of days. One quarter of me is one week. b) I am an amount of money. Three-quarters of me is one half of. c) I am a proper fraction. When I am multiplied by 0, the result is. Investigation Start with a whole number, for example. Here are six different expressions, using multiplication, whose answer is : of of Choose your own numbers, and write six different expressions for each. Give an example of a number that has nine (or more) different expressions. Fractions of amounts

11 . Percentages Understand percentage as the number of parts per 00 Convert a percentage to a decimal Convert a percentage to a fraction Recognise the equivalence of percentages, fractions and decimals Calculate a number or amount as a percentage of another Key words per cent percentage hundreths convert equivalent % is shorthand for per cent. A percentage is the number of parts in every 00. 0% means 0 out of every 00 or % means out of every 00 or So % of A means 0 0 of A Percentages can be written as equivalent 0 out of 00 out of 00 fractions or decimals. To write one amount, for example, as a percentage of another, for example, : Write as a fraction then write an equivalent fraction to this with a denominator of 00. multiplying top and bottom by. 00 So, is % of. Percentage Fraction Decimal 90% % % Example Find the value of when 0% 0 0% so and so 0% 0. Write the fraction in its lowest terms. The denominator has been multiplied by, so multiply the numerator by. Example % Write mm as a percentage of 0 cm. Start by making sure that both amounts have the same unit change the 0 cm to 00 mm. Write one number as a fraction of the other. Find an equivalent fraction with a denominator of 00, by dividing top and bottom by. Maths Connect R

12 Exercise.... a) b) c) i) Write each shaded amount as a percentage. ii) Write each unshaded amount as a percentage. Add the percentages for each shape. What do you notice? Write each percentage as i) a decimal, and ii) as a fraction in its lowest terms: a) 0% b) % c) % d) 0% e) 0% f) % g) % h) % i) % j) % Find the value of in each of the following: a) 0% b) 0% 0 c) 0% 0 d) % e) % f) 0% g) 0% h) % i) % 9 j) % Write the first amount as a percentage of the second amount: a) as a percentage of 0 b) 9 as a percentage of c) km as a percentage of 0 km d) kg as a percentage of 00 kg e) 0 m as a percentage of 00 m f). as a percentage of g) 00 m as a percentage of 0 c h) 00 g as a percentage of 0 kg % of the pupils in a school are boys. What fraction of the pupils are girls? Give the answer in its lowest terms. Harry saves money by collecting coins in a bottle. % of the coins are p pieces, 0% of the coins are 0p pieces, and the remainder are 0p pieces. The total value of the 0p pieces is. a) How many 0p pieces are there? b) How many p pieces are there? c) How many 0p pieces are there? d) How much has Harry saved? Who am I? a) I am a percentage. I am one half less than 0.. b) I am a percentage. When I am doubled, I am equivalent to. c) I am a fraction in my lowest terms. I am equivalent to the percentage of cotton in a top which is made from a mixture of polyester and cotton. It is % polyester. Investigation Look back at Example. There are 0 m in c and 000 g in kg. Estimate what percentage of vowels we use are a, e, i, o, or u. Choose a passage from a book. List the first 0 vowels. Write the percentage of each. How do these compare with your estimates? Percentages