# NF5-12 Flexibility with Equivalent Fractions and Pages

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2 4/8 = /4 = / Exercises: Imagine erasing the broken line. Write the equivalent fraction. d) Answer: /4 = /, 4/8 = /4, 4/8 = /, d) 4/6 = / Using division to write an equivalent fraction. Draw the first picture below on the board: 4 6 = 4 6 = ASK: How many parts do I have to group together to get parts for the whole? () PROMPT: Six divided by what is three? ( because 6 = ) Group the parts and then show this relationship as in the second picture above. ASK: How many parts are shaded now? ( of the total parts are shaded now) Fill in the numerator (). Exercises: Divide to find the missing numerator. 5 = 5 6 = 4 0 = 6 d) = 0 50 Bonus: = 000, 00 Answers:,, 5, d) 9, Bonus: e) 5 COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Sample Solution: 5 = 5 Reducing a fraction to lowest terms. Remind students that they can make an equivalent fraction using bigger numbers by multiplying both the numerator and the denominator by the same number. Write the fraction on the board and ASK: If you have a fraction like 5 /5 (point to it), how can you make an equivalent fraction with smaller numbers, like /5? (divide the numerator and denominator by the same number) 7 5 = 7 5 Exercises: Divide the numerator and denominator by the same number to make an equivalent fraction Answers: /4, 4/5, /, d) /6 8 d) 5 0 Number and Operations Fractions 5- F-49

3 Point out that the number students are dividing by (7 in the example of /5) is a factor of both the numerator and the denominator, so it is a common factor. If necessary, students can repeatedly divide the numerator and the denominator by a common factor until there is no common factor except. Demonstrate doing so to solve the first exercise below, and have students do the remaining ones. Exercises: Divide the numerator and the denominator by a common factor until there is no common factor except d) Sample Answers: 84 6 = 4 6 = 6 9 = 00, 50 = 0 5 = 6 7,, d) 7 9 When solving part, point out how useful it is to know the times tables. If students can recognize 4 and 6 as both being in the 7 times table, then they can recognize 7 as a common factor. When students finish, tell them that the process they just did results in the equivalent fraction that uses the smallest possible numbers, that is, the fraction in lowest terms. This process is called reducing the fraction to lowest terms. Extension You can reduce a fraction to lowest terms using the greatest common factor (GCF). Step : Find the GCF of the numerator and the denominator. (See Extension 4 in NF5-.) Step : Divide the numerator and the denominator by the GCF. Example: To reduce 4 to lowest terms, first find the GCF of 4 40 and 40. Factors of 4:,,, 4, 6, 8,, 4 Factors of 40:,, 4, 5, 8, 0, 0, 40 So the GCF of 4 and 40 is 8. Divide the numerator and the denominator by = 8 5 Reduce to lowest terms: Answers:, 4, 5, d) 5 0 d) 4 60 COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION F-50 Teacher s Guide for AP Book 5.

7 Answers: 5 9,, 0 Students should create lists of equivalent fractions for each fraction being added. They can then use their lists to subtract the same fractions as above. Exercises: Subtract Answers: 7 5, 5, 0 Reviewing the common denominator. Explain that by finding the first fraction in each list with the same denominator, students are finding the number that is a multiple of both denominators. ASK: When else have you used the common denominators of fractions? (when comparing fractions, we found two fractions with the same denominator) Review comparing fractions with different denominators. For example, remind students that to compare / and /5, first they need to find equivalent fractions with common denominators. SAY: We can multiply by 5 to change /, and we can multiply 5 by to change /5: SAY: Because 5 = 5, the denominators are now the same, and / < /5 because 5/5 < 6/5. Emphasize that to find the common denominator, students can multiply the numerator and denominator of the first fraction by the denominator of the second and multiply the numerator and denominator of the second fraction by the denominator of the first, so to find a common denominator students can multiply both denominators together. Adding and subtracting fractions by finding a common denominator. Write on the board: ASK: What is a common denominator of the fractions? (0) How do you know? (it is 5 4) Point out that we have to multiply 5 and by 4 and 4 and by 5 to change both fractions into fractions with the denominator = 0 = 0 Demonstrate again by adding the fractions /6 + /8. Have a volunteer find a common denominator of the fractions. Remind students that a common denominator of the fractions can be found by multiplying the two denominators. COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION F-54 Teacher s Guide for AP Book 5.

8 Extensions. Subtract. i) - ii) 4 - iii) iv) Find the next two subtractions in the pattern. Then predict and check your answers. Then predict: d) Make up a subtraction problem that has the answer /999,000. Answers: i) 6, ii), iii) 0, iv) 0, = = 4 ; = =, The pattern in the answers is to have 56 as the numerator and the common denominator equal to the product of the two denominators. So the answer is /9,900. d) Use the pattern 999 from part. The question , 999 has the answer. 999, 000. A team won / of its games and lost /5 of its games. What fraction of the games did the team tie? Explain how you know. Solution: / + /5 = 0/5 + /5 = /5, /5 = /5. Find the missing number. + 5 = 5 Answers:,, Bonus: = 7 0 Bonus: = COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Number and Operations Fractions 5-4 F-55

9 NF5-5 Adding and Subtracting Fractions Using Pages 6 7 the LCM STANDARDS 5.NF.A., 5.NF.A. Vocabulary common multiple lowest common denominator lowest common multiple (LCM) multiple times table whole number Goals Students will identify multiples of a number, common multiples of two numbers, and the lowest common multiple (LCM) of two numbers (up to ). PRIOR KNOWLEDGE REQUIRED Can skip count by,, 4, or 5 Can multiply -digit numbers Can find equivalent fractions Knows the times table Can reduce fractions to lowest terms Review whole numbers. Remind students that whole numbers are the numbers 0,,,, and so on. Write whole number on the board, and contrast whole numbers with fractions, which represent parts of a whole. Introduce multiples. The multiples of a whole number are the numbers you get by multiplying the number by another whole number. For example, 6 is a multiple of because = Write multiple and multiply on the board. Point out how similar the words are only the last letter of each is different, e versus y. Emphasize that this makes it easy to remember what a multiple is: You get the multiples of a number by multiplying that number by whole numbers. Skip counting to find multiples. List the first five multiples of 4, including 0, on the board as shown. Have students continue the list by writing the next five multiples of 4. (4 5 = 0, 4 6 = 4, 4 7 = 8, 4 8 =, 4 9 = 6) ASK: Is 6 a multiple of 4? (no) How can you tell? (it is not on the list; it is between 4 and 8, which are right next to each other) Point out that we can list all the multiples of 4 by skip counting: 0, 4, 8,. Exercises: Skip count to decide if each number is a multiple of d) Answers: no, yes, yes, d) no Zero is a multiple of any number. Have students write a multiplication equation that proves that 0 is a multiple of. ( 0 = 0) ASK: Which whole numbers is 0 a multiple of? (all of them, because any whole number can be multiplied by 0 to make 0) The multiples of 0 and. ASK: What are the multiples of 0? (only 0 is a multiple of 0) Demonstrate this by skip counting by 0s from 0: 0, 0, 0,. ASK: What are the multiples of? (every whole number is a multiple of ) PROMPT: What numbers do we get when we skip count by s? (all whole numbers) COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION F-56 Teacher s Guide for AP Book 5.

10 Common multiples. Draw two number lines to on the board, one for the multiples of and the other for the multiples of. Mark the multiples of on the first number line with Xs. Point out that you start at 0 and mark every second number. Ask a volunteer to mark the multiples of, starting at 0 and marking every third number. The number lines on the board should now look like this: : : ASK: Which numbers are multiples of both and? (0, 6, and ) Exercises: Draw two number lines on grid paper to to find numbers that are multiples of and 5 to to find numbers that are multiples of and 6 to 6 to find numbers that are multiples of and 5 Tell students that a number that is a multiple of two numbers is called a common multiple of the two numbers. Ask volunteers to say some common multiples of and. (Examples: 0, 6, ) Challenge students to name more common multiples of and. (8, 4, and so on) ASK: Does anyone know the phrase in common? What does it mean if I say two people have something in common? (the two people share a quality, characteristic, like, or dislike) Have students pair up and find something they have in common with each other. (example: we both like reading) In their pairs, students should take turns saying something about themselves until they find something in common. Relate this to the term common multiple : the numbers and both have as a multiple, so that is something the numbers and have in common. That is why is called a common multiple of and. COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Introduce lowest common multiples. Remind students that 0 is a multiple of every number. Because of that, it is useless as a multiple and cannot give us extra information. When we want to list multiples, we are only interested in the multiples that are not 0. Tell students that the smallest common multiple of two numbers not including 0 is called the lowest common multiple of the numbers. SAY: Some people call the lowest common multiple the least common multiple. Exercise: Find the lowest common multiple. Write your answer in sentence form. Example: The lowest common multiple of 6 and 0 is 0. and and 5 and 6 d) and 5 e) 4 and 6 f) 6 and 8 Answers: 6, 0, 6, d) 5, e), f) 4 Tell students that the term lowest common multiple is often written as LCM. Write on the board: lowest common multiple. Number and Operations Fractions 5-5 F-57

12 is the lowest common multiple of and 5? (5) Demonstrate how to use the LCM to add fractions. = 5 5 and 5 = 6 5. So + 5 = = 5. Introduce the lowest common denominator (LCD). ASK: When else have you used the lowest common multiple of the denominators of fractions? (when comparing fractions, we found two fractions with the same denominator) Tell students that, because the lowest common multiple of the denominators is so useful, we have a special name for it: lowest common denominator, or LCD. Exercises: Find the LCD to add or subtract d) 5 - In the following problems, students cannot just multiply the denominators to find the lowest common denominator. Do the first one together. Exercises: Find the LCD to add or subtract. e) 8-6 f) g) Now, in these problems, students will need to change only one denominator. Again, do the first one together. Exercises: Find the LCD to add or subtract. i) j) k) h) l) Answers:, 5, 9 0, d) 0, e) 5 4, f), g) 0, h) 0, 5 i), j) 8 5, k) 9, l) 0 Sample Solutions: e) LCD is 4; /8 = 9/4, /6 = 4/4; 9/4 4/4 = 5/4 COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION i) LCD is 8; /4 = /8; /8 + /8 = 5/8 ACTIVITY Each student will need two dice of one color and two dice of another color. (Alternatively, students can roll the same pair of dice twice.) The student rolls all four dice, adds the results of the dice of the same color, and finds the lowest common multiple of the resulting numbers. For example, if a student rolls and 6 with red dice and and 4 with blue dice, the student has to find the lowest common multiple of 9 (= + 6) and 6 (= + 4). Number and Operations Fractions 5-5 F-59

18 SAY: Just as we can add 0 to the ones in a number to make sure we have enough ones to subtract from, we can add to the fraction part of a mixed number to make sure we have enough in the fraction part to subtract from. Point to the /4 in 5 /4 and SAY: This number is less than because it s the fraction part of a mixed number. Point to the /4 in 9 /4 and SAY: If we regroup to make this more than, we can be sure to have enough to subtract from. Exercises:. Regroup to make the fraction part more than. 6 /5 = 5 + /5 = 5 4/9 = 4 + 4/9 = /7 = + /7 = d) 7 5/8 e) 6/ Bonus: /5. Regroup the first number and then subtract. /5 /5 5 / / 6 /8 5/8 Answers:. 5 8/5, 4 /9, 8/7, d) 6 /8, e) 7/, Bonus: 7/5;. 7/5 /5 = 4/5, /, 4/8 Using the LCD to add and subtract. For some mixed numbers, the fraction parts must be changed so that they have a denominator equal to the LCD of the fractions. Some problems will require regrouping. Do the first one with students. Exercises: Subtract. 5 /4 / 5 /7 + / /4 5/8 d) 4 /5 / e) 4/9 + 5/7 Bonus: 7 4/9 5/7 Answers: 5 /4 / = 5 9/ 4/ = 5/, 8 /, 5/8, d) 4/5, e) 5 0/6, Bonus: 4 46/6 Extensions. Subtract mixed numbers by adding parts and whole portions of the difference on a number line. Example: 9 /4 5 / COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION = 4. Ron made 6 pies for a bake sale. He has 5/8 pies left. How many pies did Ron sell? (4 /8) Each pie had 8 pieces. How many pieces of pie did he sell? (5) Ron sold each piece for \$. How much money did he make? (\$05). Name two fractions that have a sum between and. a sum between / and /4. different denominators and a difference between / and. 4 Number and Operations Fractions 5-6 F-65

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