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Year 8 Mathematics http://www.mathsisfun.com/fraction s.html Fractions Learning Intentions • Learning Intentions – Understand that fractions indicate that a whole is divided into parts – Understand the terms numerator and denominator – Be able to compare and order fractions with different denominators – Be able to add simple fractions and mixed numbers – Be able to subtract simple fractions and mixed numbers What is a Fraction • A fraction is a quantity that has been divided into parts • For example: A Yorkie bar has seven chunks. If you are given two then you have of the bar. • The denominator indicate the number of parts the bar has been divided into • The numerator indicates the number of parts that you have, Equivalent Fractions • Shade one fifth of each of the following diagrams and then write the fraction. Equivalent Fractions • An equivalent fraction is a fraction written using different numbers, but is the same size as the original. • For example, we can write the fraction one fifth in a number of different ways: 1 2 3 4 5 6 5 10 15 20 25 30 • All these fractions are the same if you multiply the numerator and denominator by the same number. Simplifying Fractions • In the previous example the fraction can be written in a number of ways. • We should always try to write a fraction with the smallest denominator. • This is called simplifying the fraction. • Simplify the following fractions: 8 18 24 32 2 14 12 60 Comparing Fractions • Sometime we need to compare fractions to find out which is the largest. • To do this we need to change the fractions so that they have the same denominator. Comparing fractions • Which fraction is the larger 3 or 7 ? 5 11 • To compare these fractions, we need to make the denominators the same. We will make them both 55. • We multiply the numerator and denominator of the first fraction by 11. • We multiply the numerator and denominator of the first fraction by 5. 33 35 or • We now can compare the fractions! 55 55 Order! Order! • To place fractions in order we need to express them with the same denominator • ASCENDING order places the fractions from smallest to largest • DESCENDING order places the fractions from largest to smallest Example • Places the following fractions in ascending order: • The common denominator (the LCM of the denominators) is 20 so we need to express all the fractions out of 20. • Putting them in order gives: • And putting them in their original form: 3 1 7 17 3 5 2 10 20 4 12 10 14 17 15 20 20 20 20 20 10 12 14 15 17 20 20 20 20 20 1 3 7 3 17 2 5 10 4 20 Adding Fractions • If you were given 3 chunks of Yorkie and your friend was given 2, what fraction of the bar would you have? 3 2 • We need to calculate: 7 7 5 • The answer is 7 • We haven’t changed the number of chunks in the bar, so the denominator stays the same. • We have more chunks so the numerator increases. Adding Fractions • When adding fractions, we add the numerators and keep the denominators the same. • Add the following fractions Subtracting Fractions • If you were given 3 chunks of Yorkie and you eat 2, what fraction of the bar would you have? 3 2 • We need to calculate: 7 7 1 • The answer is 7 • We haven’t changed the number of chunks in the bar, so the denominator stays the same. • We have less chunks so the numerator decreases. Subtracting Fractions • When subtracting fractions, we subtract the numerators and keep the denominators the same. • Subtract the following fractions Different Denominators • When adding fractions with different denominators we need to change the fractions into equivalent fractions with the same denominator • Add the following fractions Mix it up • What happens if you have three chunks of Yorkie and your friend has five. How much do you have now? 3 5 • We need to add 7 7 8 • Which gives 7 • This means that we have the 7 chunks making one whole bar and one chunk left over! • We write this as 1 1 7 It’s Improper • The number 8 is called an IMPROPER fraction 7 because the numerator is larger than the denominator. 1 • The number 1 is called a MIXED number 7 because it consists of a fraction and a whole number. • At the end of a sum, we always change improper fractions into mixed numbers. Improper Fractions • Convert the following improper fractions into mixed numbers: Going backwards • Sometimes we need to change mixed numbers into improper fractions. 3 • For example, convert 2 into an improper 7 fraction. • Let’s try drawing a diagram. • Now divide the two wholes into sevenths, 3 17 2 • = 7 7 Calculate: • Convert each of the following fractions into improper fractions: Adding them up • When adding mixed numbers we need to change the numbers into improper fractions first. 3 5 • Calculate 1 2 5 7 Take it Away • When subtracting mixed numbers, we also need to change mixed numbers into improper fractions. 3 1 • Calculate: 3 1 7 5 Decimals • Decimals are used for numbers less than 1. • We extend our number system to the right. Hundreds Tens units tenths hundredths thousandths 3 2 1 5 9 7 Place Value • Write the value of the 5 in each of the following numbers: – 35.327 – 432.456 – 3.578 – 267.435 Place Value • Write each of the following numbers as decimals – One thousand two hundred and six tenths – Three units and fifty tenths – Forty and three hundred and two thousandths – Four units and three fifths – Seven hundred and sixteen fiftyths Adding and Subtracting • When adding and subtracting decimals the decimal point must not move. • Add or subtract the numbers in the columns as normal • Calculate the following: 2.21 + 7.994 13.65 + 7.0026 + 0.6 Adding and Subtracting • Calculate the following: 13.334 – 6.4 8 – 0.645 Multiplying by 10, 100, etc • When multiplying by 10, 100, 1000, etc we move the number to the left by the number of zeros. • Calculate: – 0.36 x 1000 – 0.0054 x 100 – 4.506 x 10 Dividing by 10, 100, etc • When dividing by 10, 100, 1000, etc we move the number to the right by the number of zeros. • Calculate: – 7.5 ÷ 1000 – 1.574 ÷ 100 – 0.0238 ÷ 10 Multiplying Decimals • When multiplying decimals – remove the decimal points – multiply the numbers – put the decimal point back – there must be the same number of decimals in the answer as there were in the question Multiplying Decimals • Calculate: – 51.36 x 6 – 4.5 x 8.2 – 0.46 x 30.5 Dividing Decimals • When dividing decimals – Change the sum into an equivalent sum by multiplying each number by 10, 100, 1000 so that the divisor has no decimal places – Use short or long division – Do not move the decimal point Dividing Decimals • Calculate: – 0.475 ÷ 5 – 142.5 ÷ 2.5 – 45.44 ÷ 0.32 Converting Fractions to Decimals • To convert a fraction to a decimal we need to use short or long division. • Convert the following to decimals: – 7 8 – 32 40 Converting Decimals to Fractions • To convert a decimal to a fraction, write the number as a fraction out of 10, 100, 1000, etc • Convert the following decimals to fractions: – 0.48 – 1.357