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Fun with Fractions! Kaitlyn Murray EDCI 270 Spring 2012 Students Teachers Teachers Have the students go through this application with a pencil and paper to try out all the problems to get the right answers! •1. Target Audience • This power point presentation is directed towards upper elementary students that are currently learning all about fractions. These students can be first discovering fractions, or students that already have an understanding, but are building new ideas on top of fractions, such as simple mathematic procedures. •2. Learning Environment • This power point presentation is intended to use in the classroom and as homework. First the students will learn how to go through the presentation for help in the classroom, where the teacher is available to help to enable the students to understand the material at hand. Then, the students can use this presentation for homework to fully understand the material for the specific night’s homework. •3. Objective/Purpose 1. 2. 3. Given a fraction, students will be able to compare other fractions in relation to size with the use of strip diagrams. Given a fraction, students will describe where the fraction falls on the number line between numbers zero and one. Given an addition/subtraction, or multiply/divide fraction problem, students will be able to do the mathematical process through strip diagrams and equations with high accuracy. Students This is Fraction Jackson! He’s here to help you through the lesson! •Students: How to use this application: Click this to go to the HOME page • Click this to find out what this word means • Link • Click these buttons to go the next or the previous slide •First of all…… • You need to understand what a fraction is, how to compare it to the whole unit, and do simple mathematic processes. • LET’S GET STARTED!! • You and your four friends order a pizza. Before the pizza arrives, you want to know how much of the pizza each of you and your four friends will get. • Before solving this problem, you need to know more about FRACTIONS. START •Understanding the meaning of A/B 1. 2. A -------B 3. • The unit, or the whole, is clearly in mind. (What=1?) The denominator (B) tells how many pieces of equal size the unit is cut into (or thought of being cut into) The numerator (A) tells how many such pieces are being considered. There can be different ways that the fraction can look like. Click here to find out! •What a fraction looks like on the computer! • This is what a fraction looks like when we write them down: • AND this is what a fraction looks like on the computer: •Draw strip diagram!! •Now you try! -- ⅔-First draw the whole unit – one long strip • Then, draw how many sections the whole is cut up into (the denominator) • Finally, shade in the number of pieces that are being considered. • Look back at the previous slide for examples!! • Now that you know what a fraction is, it is important to know Equivalent Fractions Lowest Terms and all in Greatest Common Factor terms of fractions to be able to do the mathematical processes! •Greatest Common Factor (GCF) The greatest common factor of two (or more) whole numbers is the largest number that is a factor of the two (or more) numbers. For example: 2, 3, and 6 are all common factors of 12 and 18, and 6 is the GREATEST COMMON FACTOR of 12 and 18. (sometimes called the greatest common divisor) •Common Factors and Lowest Terms A factor of a number divides the number, leaving no remainder. (2 is a factor of 4) A common factor is one that each of the numbers will share and have in common. 12/30 = (6x2)/(15x2) = 6/1; 6/15 = (2x3)/5x3) = 2/5 Because 2 and 5 have no common factors (except 1), we say that 2/5 is the SIMPLEST FORM of 12/30. Alternatively, we say 12/30 in LOWEST TERMS is 2/5 Click here for examples! •EXAMPLES: •Equivalent Fractions 2/3 and 100/150 look different but they are equal EX: 2/3 and 4/6 Strip Diagrams!!! •Now you try! 1. What is the GCF of 45 and 20. a) b) c) 2. 9 5 4 What is 18/24 ‘s equivalent fraction? a) b) c) ¼ ½ ¾ •YES!! 1. The GCF of 45 and 20 IS 5! • Good job!!! •UH-OH!! • Remember that the GCF is the largest factor that both the numbers share. • HINT** write out the factors for both numbers and circle the greatest common factor •YES!! 2. The equivalent fraction to 18/24 is ¾!!! • Good Job!! •UH-OH!!! • Remember to draw out the strip diagrams to compare how much is shaded in with each one to find out which one is the equivalent fraction. • OR find the lowest terms of the fraction to find the equivalent one! •Fraction Jackson! Now that you know more about fractions in lowest terms, and their common factors, it is important to relate them to their decimal values and what category of numbers fractions fall into. Let’s Go! •Relating Fractions and Decimals: Terminating Decimals A fraction in simplest form can be presented with terminating decimals when the denominator has only 2’s and 5’s as factors, because we can always find an equivalent fraction with a denominator that is a power of 10. EXAMPLE: ½ = 0.5 •Relating Fractions and Decimals: Nonterminating Decimals A fraction in simplest form can be presented with a nonterminating repeating decimal if the denominator has factors other than 2’s and 5’s. Try with your calculator what are nonterminating, repeating decimals! EXAMPLE: 1/3 = 0.333333333333333333333333 •How to convert Fractions to decimals The simple method is to divide. A fraction bar is a division symbol, so divide the numerator by the denominator to get the decimal value of that fraction •How to convert Decimals to Fractions Take the decimal, for example, .25, and use that number as the numerator. However many digits are after the decimal, is how many zeros that need to be in the denominator in base 10; in this case it is 100. Now simplify the fraction into the lowest terms to get the answer of ¼. •Numbers! • Real Numbers • Rational Numbers • Whole Numbers • Irrational Numbers •Real Numbers The set of rational numbers together with the set of irrational numbers. EXAMPLES: 2, -3, ½ •Rational Numbers Any fraction or its equivalent terminating or repeating decimal names EXAMPLES: 1, 7/9, 1/3, 9 •Whole Numbers Any of the numbers 0, 1, 2, 3, 4…… etc •Irrational Numbers Nonterminating, nonrepeating, decimals cannot be represented as factors with the whole numbers as the numerator and as the denominators EXAMPLES: ∏(3.14), √2 •Decimals to Fractions Practice Which fraction is closer to the decimal?? .29 • 1/3 • 1/4 .78 • 3/4 • 7/10 .05 •0 • 1/2 •YES! .29 is close to and very similar to 1/3 Good Job! •UH-OH!! ¼ is very close to .29, but think about how to convert fractions to decimals…. HINT** Set the decimal as the numerator and the denominator is a base 10 number…. •YES!! .78 is closer to ¾! Good Job!! •UH-OH!! .78 is close to 7/10, but remember to think about changing decimals to fractions…. •YES!! 0.05 is closer to 0! Good Job!! •UH-OH!!! Although .05 may look like 0.5, it is not. Remember to keep in mind how to convert decimals to fractions…. •Fraction Jackson! Now that we know what the fraction is equivalent to in decimal form, we can now estimate the fractional values and compare them to one another to know which fraction is the largest in a set of numbers! •Estimating Fractional Values • Look at the denominators for the fraction sizes • • A larger denominator means a smaller piece of the whole A smaller denominator means a larger piece of the whole You would rather have a larger slice of pizza than a smaller one when sharing with you and your four friends…. •Fractions Close to Zero 2/11, 11/108, 3/36 *Look at the size of the denominator, imagine the pieces and how many are in the entire whole and how many are being considered. The reason these fractions are close to zero is because there aren’t many pieces being considered in comparison to the whole. •Fractions Close to ½ 17/35, 4/7, 25/50 *Look at the denominator and figure out what is exactly half of it to determine if the fraction is close to ½ •Fractions Close to 1 9/8, 11/12, 99/102 *Look at the denominator and determine how many pieces there are in relation to the numerator. •Practice with Estimating Fractions Look at each fraction and determine if the fraction is close to 0, ½, or 1. A. B. C. D. 11/9 Close to 0, Close to ½, Close to 1 18/36 Close to 0, Close to ½, Close to 1 7/12 Close to 0, Close to ½ , Close to 1 1/25 Close to 0, Close to ½, Close to 1 •YES!!! 11/9 is close to 1 9 goes into 11 one full time. It is an improper fraction •UH-OH!!! Remember to look at the denominator and determine how many pieces the whole is cut up into and consider how many pieces you want from the numerator! •YES!!! 18/36 is equal to ½!! •YES!!! 7/12 is just over ½, so it is closer to ½! •YES!! 1/25 is close to zero!! It is one small piece over a large denominator •Fraction Jackson!! • Now you know what a fraction is, and how to compare it to other sizes to find the common factors, you can now determine the answer to computation with fractions!! You can do it!! •Computing with Fractions: Adding When adding fractions, it is important to find the common denominator by finding the common factors of the denominators of both the fractions in the problem. It is also good to draw strip diagrams. •Draw strip diagram!! •Adding ½ + 1/3 ½+1/3 -Must find a common multiple the denominators share. ½*3/3+1/3*2/2 -In this case, the common multiple is 6. Multiply each fraction to have the common denominator! 3/6+2/6 - Add only the numerators!! =5/6 •Computing with Fractions: Subtracting! Subtracting is very similar to addition in finding a common denominator. ¾-1/3 -Must find the common denominator 9/12-4/12 -Subtract! =5/12 • When using strip diagrams, shade in the whole unit with was the first fraction is and then X out what is being subtracted after there is a common denominator shown on the diagram. The boxes without an X make up the answer. Try it with this problem! •Now you try! Add or subtract the following, using strip diagrams if you need them! • ¾-16/64 • • Check your answer! 3/5+1/2 • Check your answer! •¾-16/64 • 64 is the common denominator!! • When reducing the final fraction, the answer is…. ½! •3/5+1/2 10 is the common denominator of 2 and 5. • When adding, you should have gotten an improper fraction. • When the fraction is reduced, the answer is… =1 1/10 • •Computing with Fractions: Multiplying! THE RULE: multiply the numerators to obtain the numerator of the product and multiply the denominators to find the denominator of the product. *To make multiplication of fractions meaningful, one must understand the referent units for the fractions and what the product actually stands for. •Referent Unit The referent unit is the number that starts on the left. For example, 1/3 of ½ of 1 whole 1/3 is the referent unit 1/3 is multiplied by ½ of the 1 whole unit •Multiplying Fractions by Whole Numbers When multiplying fractions by whole numbers, the denominator of the whole number is 1. Then you simply multiply across and simplify the fraction. Example: 3 * 1/5 3/1 * 1/5 3*1/1*5 = 3/5! • •Multiplying A Fraction by another Fraction The process is still the same when multiplying fractions. Instead of one whole number and a fraction, there are now two fractions. Example: 3/5 * 3*4 Multiply across – 9/20 ANSWER: 9/20! • •Computing with Fraction: Dividing Dividing by a fraction gives the same result as multiplying by the reciprocal (flip the numerator and the denominator – ½ -> 2/1) of the fraction symbolically, when the divisor is a fraction, N ÷ c/d = N * d/c, or if N itself is a fraction, a/b ÷ c/d = a/b * d/c (c/d cannot equal 0) •a/b ÷ c/d = (a*d)/(b*c) Dividend ÷ Divisor = Quotient •Dividing by Unit Fractions a/b ÷ 1/c = a/b ÷ c/1 = ac/b Examples: 11 ÷ 1/3 = 11 * 3/1 = 11*3 / 1 = 33 ½ ÷ 1/3 = ½ * 3/1 = (3*1)/(1*2) = 3/2 = 1 ½ •Now You Try! Multiply or Divide the following fraction problems: • • 2/3 * 5/2 • Check your answer! • Check your answer! 2/3 ÷ 1/6 •2/3 * 5/2 When multiplying across, you should get 10/6. • When you simplify the improper fraction into a mixed number, the answer is… • =1 2/3 •2/3 ÷ 1/6 • When going through the processes of dividing, the reciprocal is 6/1. After the multiplication across, the answer is… =4 •Video!! • Now watch this video to recap how to draw strip diagrams, how to add, subtract, multiply, and divide using them. •Now It’s Time for the Quiz!! I know! Quizzes aren’t cool, but see how you do! You can do it! You and your friends splitting the pizza need you to do well!! Make sure you click the answer to check your work to see how well you do! Once you start the quiz, you cannot go “home” or go onto the next question until you get the answer correct. You must finish it. •1. Quiz A ---------B What is the indicator of how many pieces of the whole are being considered in the fraction? A? B? •A! Right! The numerator tells us how many pieces are being considered in the fraction of the whole! •B…. OH-NO!!! Remember what each part of the fraction is called and what each part stands for of the whole of the fraction. •2. Quiz What is the lowest term for the fraction to the left?? 25 ----------625 A. B. ¼ 1/25 •A. ¼ UH-OH!!! You have the right thinking in that when you see 25, you think of fourths, but think of what the common factor in both the numerator and the denominator to find the lowest tem. •B. 1/25 Good Job!!! You knew to look at the common factors of the numerator and the denominator to simplify the fraction into its lowest term! •3. Quiz 14 ---------27 Is the fraction to the left equal to a terminating or a repeating decimal? A. B. Repeating decimal Terminating decimal •A. Repeating Decimal YES!! You knew that the multiples of the denominator do not include 2 or 5, so that it had to be a repeating decimal. •B. Terminating Decimal UH-OH!! Remember that terminating decimals have only 2’s or 5’s as factors. Look again at the fraction.. •4. Quiz What kind of number is shown to the right? -31/2 A. B. Rational Number Real Number •A. Rational Numbers YES!! Rational Numbers are any fractions or its equivalent terminating or repeating decimal names. •B. Real Number YES!! Real Numbers are the sets of rational numbers together with the set of irrational numbers. •4. Quiz -3 ½ is a Real AND a Rational Number!! You might have been questioning your answer, but you are right in the thinking that -3 ½ fits both of the descriptions of rational and real numbers! GOOD JOB!!! •5. Quiz What fraction is the decimal closest to? 0.29 A. B. 1/3 ¼ •A. 1/3 YES!! That’s right!! 0.29 is close to 0.3333 which is the same as the fraction 1/3 •B. 1/4 UH-OH!! Although 0.29 seems like it would be close to 0.25 which is ¼, but think of a smaller fraction, that has a larger decimal value… •6. Quiz Is this fraction closer to ½ or 1? 67 ------------100 A. B. ½ 1 •A. ½ UH-OH!! Remember to think of what exactly ½ is of the denominator and think of how far away the numerator is from the midpoint of the denominator. It is close to ½, but not as close to 1 is. •B. 1 YES!!! 67 ---------100 is closer to 1 than ½!! GOOD JOB!! •7. Quiz 5 --------8 Add these two fractions together. A. B. 1 ----------3 6/11 11/12 •A. 6/11 UH-OH!!! Remember that in adding or subtracting fractions, there first must be a common denominator between the two original fractions. •B. 11/12 YES!! You knew to find the common denominator of 24 and add the numerators equally 22/24, but when the fraction is simplified, the answer is 11/12! GOOD JOB!! •8. Quiz 3 -----4 Subtract these two fractions. A. B. 1 -------3 5/12 2/1 •A. 5/12 YES!! You knew to find the common denominator of 12 and subtract across to get the answer of 5/12!! GOOD JOB!!! •B. 2/1 UH-OH!! 2/1 is the same as the whole number 2. When subtracting fractions, you cannot get a whole number. Try to remember to find the common denominator of the two fractions. Whatever you do to the denominator, you must do to the numerator. •9. Quiz 2 -------3 Multiply these fractions together. A. B. 5 ---------2 10/6 60/36 •A. 10/6 YES!!! When multiplying fractions, you knew not to find a common denominator, and just simply multiply across the numerators to the get the numerator of the product, and multiply across the denominators to get the denominator of the product. GOOD JOB!!! •B. 60/36 UH-OH!!! When multiplying fractions, there is no need to find a common denominator. Just simply multiply across the numerators to get the numerator of the product and multiply across the denominators to get the denominator of the product. •10. Quiz 6 1/3 ÷ 1/3 Divide these fractions. Use the first fraction as the dividend and the second fraction as the divisor to get the quotient. A. 39/2 B. 19/1 •A. 39/2 YES!! You knew to change the mix number into an improper fraction and multiply that by the reciprocal of the divisor (the second fraction) GOOD JOB!! •B. 19/1 UH-OH!! Remember that you need to change the divisor to its reciprocal, then multiply. Don’t just change the mix number into an improper fraction and divide by 1/3. •Fraction Jackson!! Congratulations!!!! You did it!! Now you can finally do the arithmetic to find out how much pizza you and your four friends will get. •Conclusion! You and your four friends order a pizza. Before the pizza arrives, you want to know how much of the pizza each of you and your friends will get. • • • Remember that the pizza is the whole unit. There are five people (which makes the five the denominator) One person will get 1/5 of the pizza then, to be able to share it with you and your four friends!