6th Grade. Fractions. 1 Vocab Word. Greatest Common Factor. Slide 1 / 302. Slide 2 / 302. Slide 4 / 302. Slide 3 / 302. Slide 6 / 302.

Transcription

1 Slide / 0 Slide / 0 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. th Grade Fractions Click to go to website: Slide / 0 Fractions Unit Topics Greatest Common Factor Least Common Multiple GCF and LCM Word Problems Distribution Fraction Operations Review (+ - x) Fraction Division Fraction Operations Mixed pplication Glossary Common Core Standards:.NS.,.NS. Vocab Word Slide / 0 The charts have parts. Factor whole number that can divide into another number with no remainder. whole number that multiplies with another number to make a third number. Click on the topic to go to that section Its meaning R. (s it is used in the lesson.) Slide / 0 Vocabulary words are identified with a dotted underline. Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number. How many thirds are in whole? How many fifths are in whole? How many ninths are in whole? (Click on the dotted underline.) The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall. Slide / 0 Greatest Common Factor is a factor of Examples/ Counterexamples x and are factors of is not a factor of ack to Link to return to the instructional page. Return to Table of Contents

2 Slide / 0 Interactive Website Slide / 0 (Rows and Columns can be adjusted prior to starting the game) Player chose to earn points. Player finds,,,,,, and earns points. Review of factors, prime and composite numbers Play the Factor Game a few times with a partner. e sure to take turns going first. Find moves that will help you score more points than your partner. e sure to write down strategies or patterns you use or find. nswer the Discussion Questions. Slide 9 / 0 Discussion Questions. Make a table listing all the possible first moves, proper factors, your score and your partner's score. Here's an example: First Move Proper Factors My Score. What number is the best first move? Why? Partner's Score None Lose a Turn 0, Slide 0 / 0 Player chose to earn points. Player finds and are the only available factors and earns points.. On your table, circle all the first moves that only allow your partner to score one point. These numbers have a special name. What are these numbers called? re all these numbers good first moves? Explain.. On your table, draw a triangle around all the first moves that allow your partner to score more than one point. These numbers also have a special name. What are these numbers called? re these numbers good first moves? Explain.. Choosing what number as your first move would make you lose your next turn? Why?. What is the worst first move other than the number you chose in Question? more questions Slide / 0 Party Favors! ctivity You are planning a party and want to give your guests party favors. You have chocolate bars and lollipops. Discussion Questions What is the greatest number of party favors you can make if each bag must have exactly the same number of chocolate bars and exactly the same number of lollipops? You do not want any candy left over. Explain. Could you make a different number of party favors so that the candy is shared equally? If so, describe each possibility. Which possibility allows you to invite the greatest number of guests? Why? Slide / 0 Greatest Common Factor We can use prime factorization to find the greatest common factor (GCF).. Factor the given numbers into primes.. Circle the factors that are common.. Multiply the common factors together to find the greatest common factor. Uh-oh! Your little brother ate of your lollipops. Now what is the greatest number of party favors you can make so that the candy is shared equally? Note to Teacher

3 Slide / 0 Slide / 0 Use prime factorization to find the greatest common factor of and. x x x x x The Greatest Common Factor is x for steps nother way to find Prime Factorization... Use prime factorization to find the greatest common factor of and. x x x x x The Greatest Common Factor is x Slide / 0 Use prime factorization to find the greatest common factor of and 90. Slide / 0 Use prime factorization to find the greatest common factor of and x x x 90 x x x 9 90 x x x 90 x x x GCF is x x GCF is x x Slide / 0 Slide / 0 Use prime factorization to find the greatest common factor of 0 and x x x x x x x Use prime factorization to find the greatest common factor of 0 and x x x x x x x 9 GCF is x x GCF is x x

4 Find the GCF of and. Slide 9 / 0 Find the GCF of and 0. Slide 0 / 0 Slide / 0 Find the GCF of and 0. Find the GCF of and. Slide / 0 Slide / 0 Slide / 0 Find the GCF of and. Relatively Prime: Two or more numbers are relatively prime if their greatest common factor is. Example: and are relatively prime because their GCF is. Name two numbers that are relatively prime.

5 Slide / 0 Slide / 0 and are not relatively prime. Identify at least two numbers that are relatively True False prime to 9. C D Slide / 0 Slide / 0 Name a number that is relatively 9 Name a number that is relatively prime to 0. prime to and. Slide 9 / 0 Slide 0 / 0 0 Find two numbers that are relatively prime. C D 9 Least Common Multiple Return to Table of Contents

6 Slide / 0 Text-to-World Connection (Click for Link to Video Clip). Use what you know about factor pairs to evaluate George anks' mathematical thinking. Is his thinking accurate? What mathematical relationship is he missing?. How many hot dogs came in a pack? uns? Note to Teacher. How many "superfluous" buns did George anks remove from each package? How many packages did he do this to?. How many buns did he want to buy? Was his thinking correct? Did he end up with hot dog buns? Slide / 0 multiple of a whole number is the product of the number and any nonzero whole number. multiple that is shared by two or more numbers is a common multiple. Multiples of :,,,, 0,,,,... Multiples of :,,,, 0,,... The least of the common multiples of two or more numbers is the least common multiple (LCM). The LCM of and is.. Was there a more logical way for him to do this? What was he missing?. What is the significance of the number? Slide / 0 Slide / 0 There are ways to find the LCM:. List the multiples of each number until you find the first one they have in common.. Write the prime factorization of each number. Multiply all factors together. Use common factors only once (in other words, use the highest exponent for a repeated factor). EXMPLE: and Multiples of :,,,, 0 Multiples of :,, LCM Prime Factorization: LCM: Slide / 0 Find the least common multiple of and. Multiples of :,,,,... Multiples of :,,,... LCM: Prime Factorization: 9 LCM: 9 Slide / 0 Find the least common multiple of 0 and. 0 C 0 D 0

7 Slide / 0 Slide / 0 Find the least common multiple Find the least common multiple of and. of 9 and C C D 0 D Slide 9 / 0 Slide 0 / 0 Find the least common multiple Find the least common multiple of and 9. of and C C 0 D D 0 Slide / 0 Slide / 0 Find the LCM of and 0. Find the LCM of and 0.

8 Slide / 0 Find the LCM of and. Slide / 0 9 Find the LCM of and. Slide / 0 0 Find the LCM of and. Slide / 0 Find the GCF of 0 and. Slide / 0 Slide / 0 Interactive Website GCF and LCM Word Problems Uses a venn diagram to find the GCF and LCM for extra practice. Return to Table of Contents

9 Slide 9 / 0 How can you tell if a word problem requires you to use Greatest Common Factor or Least Common Multiple to solve? Slide 0 / 0 GCF Problems Do we have to split things into smaller sections? re we trying to figure out how many people we can invite? re we trying to arrange something into rows or groups? Slide / 0 LCM Problems Do we have an event that is or will be repeating over and over? Will we have to purchase or get multiple items in order to have enough? re we trying to figure out when something will happen again at the same time? Slide / 0 Samantha has two pieces of cloth. One piece is inches wide and the other piece is 90 inches wide. She wants to cut both pieces into strips of equal width that are as wide as possible. How wide should she cut the strips? What is the question: How wide should she cut the strips? Important information: One cloth is inches wide. The other is 90 inches wide. Is this a GCF or LCM problem? Does she need smaller or larger pieces? click This is a GCF problem because we are cutting or "dividing" the pieces of cloth into smaller pieces (factor) of and 90. Slide / 0 ar Modeling Use the greatest common factor to determine the greatest width possible. The greatest common factor represents the greatest width possible not the number of pieces, because all the pieces need to be of equal length. inches Slide / 0 en exercises every days and Isabel every days. en and Isabel both exercised today. How many days will it be until they exercise together again? What is the question: How many days until they exercise together again? Important information: en exercises every days Isabel exercises every days 90 inches Is this a GCF or LCM problem? re they repeating the event over and over or splitting up the days? This click is a LCM problem because they are repeating the event to find out when they will exercise together again. click inches

10 Slide / 0 ar Modeling Use the least common multiple to determine the least amount of days possible. The least common multiple represents the number of days not how many times they will exercise. Slide / 0 Mrs. Evans has 90 crayons and pieces of paper to give to her students. What is the largest number of students she can have in her class so that each student gets an equal number of crayons and an equal number of paper? en exercises in: Days GCF Problem LCM Problem nswer Isabel exercises in: Days Slide (nswer) / 0 Mrs. Evans has 90 crayons and pieces of paper to give to her students. What is the largest number of students she can have in her class so that each student gets an equal number of crayons and an equal number of paper? Slide / 0 Mrs. Evans has 90 crayons and pieces of paper to give to her students. What is the largest number of students she can have in her class so that each student gets an equal number of crayons and an equal number of paper? GCF Problem LCM Problem nswer C D 90 nswer [This object is a pull tab] Slide (nswer) / 0 Mrs. Evans has 90 crayons and pieces of paper to give to her students. What is the largest number of students she can have in her class so that each student gets an equal number of crayons and an equal number of paper? C nswer C Slide / 0 How many crayons and pieces of paper does each student receive if there are students in the class? C D 0 crayons and 0 pieces of paper crayons and pieces of paper crayons and pieces of paper crayons and piece of paper nswer D 90 [This object is a pull tab] Challenge problems are notated with a star.

11 [This object is a pull tab] [This object is a pull tab] [This object is a pull tab] Slide (nswer) / 0 How many crayons and pieces of paper does each student receive if there are students in the class? C D 0 crayons and 0 pieces of paper crayons and pieces of paper crayons and pieces of paper crayons and piece of paper nswer D Slide 9 / 0 Rosa is making a game board that is inches by inches. She wants to use square tiles. What is the largest tile she can use? GCF Problem LCM Problem nswer Challenge problems are notated with a star. Slide 9 (nswer) / 0 Rosa is making a game board that is inches by inches. She wants to use square tiles. What is the largest tile she can use? Slide 0 / 0 Rosa is making a game board that is inches by inches. She wants to use square tiles. What is the largest tile she can use? GCF Problem LCM Problem nswer nswer Slide 0 (nswer) / 0 Rosa is making a game board that is inches by inches. She wants to use square tiles. What is the largest tile she can use? Slide / 0 How many tiles will she need? nswer in. square tiles nswer

12 [This object is a pull tab] object is a pull tab] [This object is a pull tab] Slide (nswer) / 0 Slide / 0 How many tiles will she need? nswer tiles Y00 gave away a $00 bill for every th caller. Every 9th caller received free concert tickets. How many callers must get through before one of them receives both a $00 bill and a concert ticket? GCF Problem nswer LCM Problem Slide (nswer) / 0 Y00 gave away a $00 bill for every th caller. Every 9th caller received free concert tickets. How many callers must get through before one of them receives both a $00 bill and a concert ticket? Slide / 0 9 Y00 gave away a $00 bill for every th caller. Every 9th caller received free concert tickets. How many callers must get through before one of them receives both a $00 bill and a concert ticket? GCF Problem LCM Problem [This nswer C 0 nswer D Slide (nswer) / 0 9 Y00 gave away a $00 bill for every th caller. Every 9th caller received free concert tickets. How many callers must get through before one of them receives both a $00 bill and a concert ticket? C 0 D nswer Slide / 0 0 There are two ferris wheels at the state fair. The children's ferris wheel takes minutes to rotate fully. The bigger ferris wheel takes minutes to rotate fully. Marcia went on the large ferris wheel and her brother Joey went on the children's ferris wheel. If they both start at the bottom, how many minutes will it take for both of them to meet at the bottom at the same time? GCF Problem LCM Problem nswer

13 object is a pull tab] [This object is a pull tab] Slide (nswer) / 0 0 There are two ferris wheels at the state fair. The children's ferris wheel takes minutes to rotate fully. The bigger ferris wheel takes minutes to rotate fully. Marcia went on the large ferris wheel and her brother Joey went on the children's ferris wheel. If they both start at the bottom, how many minutes will it take for both of them to meet at the bottom at the same time? Slide / 0 There are two ferris wheels at the state fair. The children's ferris wheel takes minutes to rotate fully. The bigger ferris wheel takes minutes to rotate fully. Marcia went on the large ferris wheel and her brother Joey went on the children's ferris wheel. If they both start at the bottom, how many minutes will it take for both of them to meet at the bottom at the same time? GCF Problem nswer nswer LCM Problem [This C D 9 Slide (nswer) / 0 Slide / 0 There are two ferris wheels at the state fair. The children's ferris wheel takes minutes to rotate fully. The bigger ferris wheel takes minutes to rotate fully. Marcia went on the large ferris wheel and her brother Joey went on the children's ferris wheel. If they both start at the bottom, how many minutes will it take for both of them to meet at the bottom at the same time? C D 9 nswer C How many rotations will each ferris wheel complete before they meet at the bottom at the same time? (Input the answer for the small ferris wheel.) nswer Slide (nswer) / 0 Slide / 0 How many rotations will each ferris wheel complete before they meet at the bottom at the same time? (Input the answer for the small ferris wheel.) nswer The small ferris wheel will complete rotations and the large ferris wheel will complete rotations in minutes. Sean has -inch pieces of toy train track and Ruth has -inch pieces of train track. How many of each piece would each child need to build tracks that are equal in length? GCF Problem LCM Problem nswer [This object is a pull tab]

14 [This object is a pull tab] [This object is a pull tab] [This object is a pull tab] Slide (nswer) / 0 Slide / 0 Sean has -inch pieces of toy train track and Ruth has -inch pieces of train track. How many of each piece would each child need to build tracks that are equal in length? GCF Problem LCM Problem nswer What is the length of the track that each child will build? nswer Slide (nswer) / 0 Slide 9 / 0 What is the length of the track that each child will build? nswer inches I am planting 0 apple trees and 0 peach trees. I want the same number and type of trees per row. What is the maximum number of trees I can plant per row? GCF Problem LCM Problem nswer Slide 9 (nswer) / 0 Slide 0 / 0 I am planting 0 apple trees and 0 peach trees. I want the same number and type of trees per row. What is the maximum number of trees I can plant per row? GCF Problem LCM Problem nswer Distribution Return to Table of Contents

15 Which is easier to solve? + ( + ) Slide / 0 Do they both have the same answer? You can rewrite an expression by removing a common factor. This is called the Distributive Property. Slide / 0 The Distributive Property allows you to:. Rewrite an expression by factoring out the GCF.. Rewrite an expression by multiplying by the GCF. EXMPLE Rewrite by factoring out the GCF: (9 + ) ( + 9) Rewrite by multiplying by the GCF: ( + ) ( + ) Slide / 0 Slide / 0 Use the Distributive Property to rewrite each expression: Click to Reveal Click to ( + ) ( + ) ( + ) Reveal Click to Reveal In order to rewrite this expression using the Distributive Property, what GCF will you factor? ( Click + to) ( Click + to ) (9 Click + ) to Reveal Reveal Reveal REMEMER you need to factor the GCF (not just any common factor)! Slide / 0 Slide / 0 In order to rewrite this expression using the Distributive Property, what GCF will you factor? In order to rewrite this expression using the Distributive Property, what GCF will you factor? + + 0

16 Slide / 0 Slide / 0 9 In order to rewrite this expression using the Distributive Property, what GCF will you factor? 0 In order to rewrite this expression using the Distributive Property, what GCF will you factor? + + Slide 9 / 0 Slide 0 / 0 Use the distributive property to rewrite this expression: + Use the distributive property to rewrite this expression: + ( + ) (9 + ) C ( + ) D ( + ) ( + ) ( + ) C ( + ) D ( + ) Slide / 0 Slide / 0 Use the distributive property to rewrite this expression: (0 + ) (0 + ) C ( + ) Fraction Operations D ( + 9) Return to Table of Contents

17 Slide / 0 Slide / 0 Let's review what we know about fractions... Discuss in your groups how to do the following and be prepared to share with the rest of the class. dding Fractions.... Rewrite the fractions with a common denominator.. dd the numerators.. Leave the denominator the same.. Simplify your answer. dd Fractions Subtract Fractions Multiply Fractions Click link to go to review page followed by practice problems dding Mixed Numbers.... dd the fractions (see above steps).. dd the whole numbers.. Simplify your answer. (you may need to rename the fraction) Link ack to List Slide / 0 Slide / 0 Find the sum. Find the sum Slide / 0 Slide / 0 Find the sum. Find the sum. + +

18 Slide 9 / 0 Slide 90 / 0 Find the sum Find the sum. + Slide 9 / 0 Slide 9 / 0 0 Find the sum Find the sum. + Slide 9 / 0 Slide 9 / 0 Find the sum. Is the equation below true or false? True False + Click For reminder Don't forget to regroup to the whole number if you end up with the numerator larger than the denominator.

19 Slide 9 / 0 Slide 9 / 0 Find the sum Find the sum. + Slide 9 / 0 Slide 9 / 0 Find the sum. Find the sum. Slide 99 / 0 Slide 00 / 0 Find the sum. 9 Find the sum.

20 Slide 0 / 0 Slide 0 / 0 0 Find the sum. Find the sum. + quick way to find LCDs... Slide 0 / 0 List multiples of the larger denominator and stop when you find a common multiple for the smaller denominator. Ex: and Multiples of :, 0, Slide 0 / 0 Common Denominators nother way to find a common denominator is to multiply the two denominators together. Ex: and x x x x x Ex: and 9 Multiples of 9: 9,,, Slide 0 / 0 Slide 0 / 0 Find the sum. + Find the sum. + 0

21 [This object is a pull tab] Slide 0 / 0 Slide 0 / 0 Find the sum. Find the sum Slide 09 / 0 Slide 0 / 0 Find the sum. Find the sum. + + Slide / 0 Slide (nswer) / 0 Try this... Try this nswer nswer 0

22 object is a pull tab] Slide / 0 Slide (nswer) / 0 Try this... Try this... + nswer + [This nswer Slide / 0 Slide / C 9 C D 0 D Slide / 0 Slide / C 0 C 0 D D 0

23 Slide / 0 Slide / 0 Find the sum. + Find the sum C D Slide 9 / 0 Slide 0 / 0 Find the sum. Find the sum. + Slide / 0 Slide / 0 Find the sum. Find the sum.

24 Slide / 0 Slide / 0 Find the sum. 9 Find the sum. Slide / 0 Slide / 0 0 Find the sum. Subtracting Fractions.... Rewrite the fractions with a common denominator.. Subtract the numerators.. Leave the denominator the same.. Simplify your answer. Subtracting Mixed Numbers.... Subtract the fractions (see above steps..). (you may need to borrow from the whole number). Subtract the whole numbers.. Simplify your answer. (you may need to simplify the fraction) Link ack to List Slide / 0 Slide / 0 Find the difference. Find the difference. 0 0

25 Slide 9 / 0 Slide 0 / 0 Find the difference. Find the difference. Slide / 0 Slide / 0 Find the difference. Find the difference. 9 Slide / 0 Slide / 0 Find the difference. 9 9 Is the equation below true or false? True False 9 9 9

26 Slide / 0 Slide / 0 9 Is the equation below true or false? 90 Find the difference. True 9 False 9 Slide / 0 Slide / 0 9 Find the difference. 9 Find the difference. Slide 9 / 0 Slide 0 / 0 9 Find the difference. 9 Find the difference.

27 Slide / 0 Slide / 0 9 Find the difference. 9 Find the difference. 9 Slide / 0 Slide / 0 9 Find the difference. 9 Find the difference. Slide / 0 Slide / 0 Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number. Regrouping Review When you regroup for subtracting, you take one of your whole numbers and change it into a fraction with the same denominator as the fraction in the mixed number. How many thirds are in whole? How many fifths are in whole? How many ninths are in whole? Don't forget to add the fraction you regrouped from your whole number to the fraction already given in the problem.

28 Slide / 0 Slide / 0 9 Slide 9 / 0 Slide 0 / 0 99 Do you need to regroup in order to complete this problem? 00 Do you need to regroup in order to complete this problem? Yes or No Yes or No Slide / 0 Slide / 0 0 Regroup 0 0 Regroup

29 Slide / 0 Slide / 0 0 Find the difference. 0 Find the difference. C C D D Slide / 0 Slide / 0 0 Find the difference. 0 Find the difference. 0 C D Slide / 0 Slide / 0 0 Find the difference. 0 Find the difference.

30 Slide 9 / 0 Slide 0 / 0 09 Find the difference. 0 Find the difference. Slide / 0 Slide / 0 Find the difference. Slide / 0 Slide / 0 Trey has a piece of rope that is feet long. He cuts off an foot piece of rope and gives it to his sister for a jump rope. How much rope does Trey have left? C dding & Subtracting Fractions with Unlike Denominators pplications D

31 Slide / 0 Slide / 0 The roadrunner of the merican Southwest has a tail nearly as long as its body. What is the total length of a roadrunner with a body measuring feet and a tail measuring feet? Cara uses this recipe for the topping on her blueberry muffins. / cup sugar / cup all-purpose flour / cup butter, cubed / teaspoons ground cinnamon How much more sugar than flour does Cara use for her topping? Slide / 0 Slide / 0 Jared's baseball team played a doubleheader. Holly made dozen bran muffins and During the first game, players ate lb. of dozen zucchini muffins. How many peanuts. During the second game, players ate dozen muffins did she make in all? lb. of peanuts. How many pounds of peanuts did the players eat during both games? Slide 9 / 0 Slide 0 / 0 The Spider roller coaster has a maximum speed of miles per hour. The Silver Star roller coaster has a maximum speed of miles per hour. How much faster is the Spider than the Silver Star? 9 Great Work Construction used cubic yards of concrete for the driveway and cubic yards of concrete for the patio of a new house. What is the total amount of concrete used?

32 Slide / 0 Slide / 0 0 rectangle has a length of cm. and a width of cm. What is its perimeter? n equilateral triangle has a side length of in. What is its perimeter? n equilateral triangle is a triangle whose sides are equal in length. Slide / 0 Slide / 0 Kelly made friendship bracelets. She kept, gave to her friend Michelle. The rest she will sell. What fraction of bracelets will Kelly sell? Henry spent hours playing on Monday and hours playing on Tuesday. How much time did he spend playing on Monday and Tuesday? Slide / 0 Slide / 0 Evaluate the expression if x. Kyle put seven-eighths of a gallon of water into a bucket. Then he put one-sixth of a gallon of liquid cleaner into the bucket. What is the total amount of liquid Kyle put into the bucket?

33 Slide / 0 Slide / 0. Multiply the numerators.. Multiply the denominators.. Simplify your answer. Multiplying Fractions... Click for Interactive Practice From The National Library of Virtual Manipulatives Multiplying Mixed Numbers.... Rewrite the Mixed Number(s) as an improper fraction. (write whole numbers / ). Multiply the fractions.. Simplify your answer. Link ack to List Slide 9 / 0 Slide 0 / 0 x x Slide / 0 Slide / 0 x 9 ( )

34 Slide / 0 Slide / ( ) x x True False Slide / 0 Slide / 0 x x 9 C C 9 9 D D 0 Slide / 0 Slide / 0 x x True False C 0 D

35 Slide 9 / 0 Salad Dressing Recipe Slide 90 / 0 ( ) ( ) C 0 / cup sugar / teaspoon paprika teaspoon dry mustard / teaspoon salt / teaspoon onion powder / cup vegetable oil / cup vinegar D 9 What fraction of a cup of vegetable oil should Julia use to make / of a batch of salad dressing? She needs / of / cup vegetable oil. xof Slide 9 / 0 Carl worked on his math project for / hours. pril worked / times as long on her math project as Carl. For how many hours did pril work on her math project? Tom walks 0 miles he walks in days? Slide 9 / 0 miles each day. What is the total number of x as long as miles each x day for days 0 x 0 x 0 0 Slide 9 / 0 Jared made cups of snack mix for a Slide 9 / 0 Sasha still has of a scarf left to knit. If she party. His guests ate of the mix. How much snack mix did his guests eat? finishes of the remaining part of the scarf cups today, how much does she have left to knit? cups C cups D cups

36 Slide 9 / 0 Slide 9 / 0 9 In Zoe's class, of the students have 0 eth hiked for hours at an average pets. Of the students who have pets, rate of miles per hour. Which is the have rodents. What fraction of the students best estimate of the distance that she in Zoe's class have rodents? hiked? C 9 miles 0 miles D C D miles miles Slide 9 / 0 Slide 9 / 0 Clark's muffin recipe calls for cups of flour for a dozen muffins and cup of flour for the topping. If he makes of the original recipe, how much flour will she use altogether? Fraction Operations Division Return to Table of Contents Recall from th grade: Slide 99 / 0 Modelling Division When we are dividing, we are breaking apart into equal groups. Dividend Divisor Quotient Slide 00 / 0 pplying to Fractions The previous example used whole numbers and grouped the dividend according to the divisor. The same strategy can be applied when dividing with fractions. Use the model below to demonstrate: Note The model below represents: groups of The pink rectangle represents. See how many you can fit in the squares.

37 Slide 0 / 0 Example Use the model below to demonstrate Slide 0 / 0 Evaluate the following problem using the model below: answer Slide 0 / 0 Evaluate the following problem using the model below: answer Slide 0 / 0 Fraction Divided by a Fraction The same strategy we utilized for the previous examples can also be applied when dividing a fraction by another fraction. In this example our division problem is: We need to determine how many 's there are in Slide 0 / 0 Example Use the model below to evaluate: Slide 0 / 0 Evaluate the following problem using the model below: answer

38 Slide 0 / 0 Evaluate the following problem using the model below: answer Slide 0 / 0 Vocabulary Review Complex Fraction: fraction with another fraction in the numerator, denominator or both Reciprocal: The inverse of a number/fraction Original Number Reciprocal Slide 09 / 0 Patterns Do you notice a pattern between the division of fractions and their solution? Slide 0 / 0 If you think about it, we are dividing by a fraction which creates a complex fraction. You need to eliminate the fraction in the denominator in order to solve the problem. Note To do this, multiply the numerator and denominator of the complex fraction by the reciprocal of the denominator (making the denominator ). You can then simplify the fraction by rewriting it without the denominator of and solve the new multiplication problem. Slide / 0 Slide / 0 Example Dividing Fractions lgorithm x x x x Step : Leave the first fraction the same. Step : Multiply the first fraction by the reciprocal of the second fraction. Original Problem Complex Fraction Multiply by Reciprocal Simplify Denominator Rewrite Without Step : Simplify your answer. There are rules that can be applied to fraction division problems to eliminate steps from this lengthy procedure. x x x source -

39 Slide / 0 Some people use the saying " Keep Change Flip" to help them remember the algorithm. Evaluate: Slide / 0 Example Change Keep Flip Changed Kept Flipped Change Keep Flip Changed Kept Flipped x x x x Slide / 0 Slide / 0 Checking Your nswer To check your answer, use your knowledge of fact families. 0 x 0 x True False is of Slide / 0 Slide / 0 0 True False 9 0 C 0

40 Slide 9 / 0 Slide 0 / Slide / 0 Sometimes you can cross simplify prior to multiplying. without cross with cross simplifying simplifying Slide / 0 Can this problem be cross simplified? Yes No Slide / 0 Slide / 0 Can this problem be cross simplified? Can this problem be cross simplified? Yes No Yes No

41 Slide / 0 Slide / 0 Can this problem be cross simplified? Yes No Slide / 0 Slide / 0 Slide 9 / 0 Slide 0 / 0 Dividing Mixed Numbers lgorithm Step : Rewrite the Mixed Number(s) as an improper fraction(s). (write whole numbers / ) Step : Follow the same steps for dividing fractions x

42 Slide / 0 Slide / 0 Example Evaluate: 9 x 0 Slide / 0 Slide / 0 0 Slide / 0 Slide / 0 pplication Problems - Examples Winnie needs pieces of string for a craft project. How many / yd pieces of string can she cut from a piece that is / yd long? x pieces or x pieces

43 Slide / 0 One student brings / yd of ribbon. If students receive an equal length of the ribbon, how much ribbons will each student receive? Slide / 0 Kristen is making a ladder and wants to cut ladder rungs from a ft board. Each rung needs to be / ft long. How many ladder rungs can she cut? x yards of ribbon x rungs Slide 9 / 0 Slide 0 / 0 box weighing 9 / lb contains toy robots weighing / lb apiece. How many toy robots are in the box? 9 Robert bought / pound of grapes and divided them into equal portions. What is the weight of each portion? pounds x robots C D / pounds / pounds / pound Slide / 0 Slide / 0 car travels /0 miles on / One tablespoon is equal to / cup. It is gallons of fuel. Which is the best estimate of the number miles the car travels on one gallon of fuel? also equal to / ounce. recipe uses / cup of flour. How many tablespoons of flour does the recipe use? C D miles miles miles miles C D tablespoons tablespoons tablespoons tablespoons

44 Slide / 0 Slide / 0 bookstore packs books in a box. The There is gallon of distilled water in the total weight of the books is / pounds. If each book has the same weight, what is the weight of one book? / pound / pounds class science supplies. If each pair of students doing an experiment uses gallon of distilled water, there will be C D / pounds / pounds gallon left in the supplies. How many students are doing the experiments? Slide / 0 Slide / 0 Now we will use the rules for adding, subtracting, multiplying and dividing fractions to solve problems. Fraction Operations pplication e sure to read carefully in order to determine what operation needs to be performed. First, write the problem. Next, solve it. Return to Table of Contents Slide / 0 Slide / 0 EXMPLE: EXMPLE How much chocolate will each person get if people share lb of chocolate equally? How many cup servings are in of a cup of yogurt? Each person gets lb of chocolate. There are 9 servings.

45 Slide 9 / 0 Slide 0 / 0 EXMPLE: How wide is a rectangular strip of land with length miles and area square mile? One-third of the students at Finley High play sports. Two-fifths of the students who play sports are girls. Which expression can you evaluate to find the fraction of all students who are girls that play sports? / + / / - / C / x / D / / It is miles wide Slide / 0 Slide / How many cup servings are in cups of milk? How much salt water taffy will each person get if people share lbs? You MUST write the problem and show LL work! You MUST write the problem and show LL work! Slide / 0 Slide / 0 If the area of a rectangle is square units and its width is units, what is the length of the rectangle? recipe calls for cups of flour. If you want to make of the recipe, how many cups of flour should you use? You MUST write the problem and show LL work! You MUST write the problem and show LL work!

46 Slide / 0 Slide / 0 Find the area of a rectangle whose width is length is cm. cm and Working with a partner, write a question that can be solved using this expression: You MUST write the problem and show LL work! Slide / 0 Slide / 0 cupcake recipe calls for lb. of butter. If you wanted to make times the amount, how much butter would you need? Mike is making a bird house. He needs pieces of wood that are in. in length. How long are all the pieces together? Slide 9 / 0 Slide 0 / 0 square has a side length that measures cm. What is its area? Terri reads for hour every night. Shelly reads of the time Terri reads. How long does Shelly read?

47 Slide / 0 Slide / 0 The park playground area is shaped like a triangle. ill measured the lengths. What is the perimeter of the playground? 9 Riley has a piece of ribbon that is in. long. She wants to cut into pieces of equal length. How long will each piece be? ft. ft. ft. Slide / 0 Slide / 0 0 Miranda has cups of cake batter that she wants to separate equally into layer cake pans. How much will she put in each pan? Evaluate the expression if. + Slide / 0 Slide / 0 If is true, which of the following equations must also be true? Emily is cutting a piece of paper that is in. long into equal pieces. How long will each piece be? C D

48 Slide / 0 Slide / 0 Kelly needs cups of milk for every cookies she is baking. How many cups of milk does Kelly need if she is making 9 cookies? There are 00 th grade students and of them play a sport. How many play sports? Slide 9 / 0 Slide 0 / 0 Karla made dozen cupcakes. She and her family ate 0 cupcakes. What fraction of the dozen does she have left? What number will make this expression true? Slide / 0 Slide / 0 Sarah has 0 eggs that she will color. She will color of the eggs pink. How many eggs will be pink? 9 If is true, which of the following equations must also be true? C D

49 Slide / 0 Slide / 0 90 Miranda is baking a cake. The recipe calls for cups of butter, cups of milk, and cups of sugar. How many cups of ingredients is this? Glossary Return to Table of Contents Slide / 0 lgorithm step-by-step process to find a solution. Slide / 0 ar Model diagram that uses bars to show the relationship between two or more numbers. How to... Step : Step : Step : + dd the ones then add the tens It's like a cooking recipe for mathematics. ack to Part Whole Part Part + Part Whole Whole - Part Part Larger mount Smaller mount Difference Large - Small Difference Large - Difference Small Whole Part One part x # of parts Whole ack to Slide / 0 Common Denominator whole number that is a common multiple of all the denominators of a set of fractions. Slide / 0 Complex Fraction fraction whose numerator or denominator or both contain fractions.,,9,,, LCD is + x x x x + x x ack to Must be written as a fraction. ack to

50 x x x factors Slide 9 / 0 Composite Number number that has more than two factors. x ny number with factors other than one and itself is composite. Slide / 0 Distributive Property x Only factors. Multiplying a sum by a number is the same as multiplying each addend in the sum by the same number and then adding the products. ack to Slide 0 / 0 Cross Simplify Used to make operations with fractions easier. Divide the numerator of one fraction and the denominator of another fraction by their GCF. + 0 GCF of and is. + 0 Slide / 0 Dividend The number being divided in a division equation. + 0 ack to ( + ) a(b-c)ab-ac a(b+c)ab+ac Dividend Dividend x(+) (+) (x)+(x) ack to Dividend ack to Divisor The number the dividend is divided by. Slide / 0 Divisor number that divides another number without a remainder. R Divisor Must divide evenly. ack to Slide / 0 Exponent small, raised number that shows how many times the base is used as a factor. Exponent ase " to the second power" x x x x x ack to

51 Slide / 0 Factor whole number that can divide into another number with no remainder. whole number that multiplies with another number to make a third number. Slide / 0 Greatest Common Factor (GCF) The largest number that will divide two or more numbers without a remainder. is a factor of x and are factors of R. is not a factor of ack to :,,,,, :,,,, Common Factors are,, GCF is Using Prime Factorization x x x x x GCF x GCF is and are common factors, but not the greatest common factor. ack to Slide / 0 Improper Fraction fraction whose numerator is greater than its denominator. Slide / 0 Least Common Multiple (LCM) The smallest number that two or more numbers share as a multiple. ll improper fractions are greater than. ack to 9: 9,,,, :, 0, LCM is Using Prime Factorization 9 x x LCM x x LCM is :,,, :, is the LCM, not ack to Slide 9 / 0 Mixed Number number, greater than one, written as a whole number with a fraction. Slide 90 / 0 Multiple The product of two whole numbers is a multiple of each of those numbers. 0. Improper Fraction Decimal Number 9 ack to x is a multiple of. x Factors Product / Multiple x 0 and are factors of 0, not multiples. ack to

52 Slide 9 / 0 Prime Factorization number written as the product of all its prime factors. Slide 9 / 0 Prime Number positive integer that is greater than and has exactly two factors, one and itself. x x or x There is only one for any number. x x x Only prime numbers are included in prime factorizations. ack to Prime #s to 0,,,,,,, 9,, 9 Two is the only even prime number. One is not a prime number, because it has only one factor. ack to Slide 9 / 0 Proper Factor ll of the factors of a number other than one and itself. Slide 9 / 0 Quotient The number that is the result of dividing one number by another. :,,, Proper Factors: and 9:,, 9 Proper Factor: :, The number does not have any proper factors. ack to Quotient Quotient Quotient ack to Slide 9 / 0 Reciprocal One of two numbers whose product is one. Slide 9 / 0 Regroup (Fractions) To write a whole number as a fraction equal to one to assist with subtracting. x Number is the reciprocal of. x Reciprocal r x r ack to + ack to

53 Slide 9 / 0 Relatively Prime Two numbers who only have as a common factor. When the GCF of the numerator and denominator of a fraction is one. Slide 9 / 0 Simplify To remove brackets, unnecessary terms and numbers by performing all possible operations. :,,, :,, Only Common Factor is ll prime numbers are relatively prime to every other number. 9:,, 9 :,,, Common Factors: and ack to (+) () y++y y+ ack to Slide 99 / 0 Slide 00 / 0 ack to ack to Slide 0 / 0 Slide 0 / 0 ack to ack to