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Common Denominators Focus on… After this lesson, you will be able to... find a common denominator for a set of fractions compare and order positive fractions Jasmin and Tyler collect trading cards. Jasmin has collected __ of a set. 1 3 Tyler has collected 1 of a set. They want to know who has more cards. __ 4 Jasmin and Tyler need to compare the fractions. It is easier to compare fractions when the denominators are the same. So, Jasmin and Tyler need to find a common denominator. How can you determine a common denominator? • coloured pencils 1. Fold a piece of paper into 3 equal parts. Shade 1 of the paper red. __ 3 2. Fold the same piece of paper into 4 equal parts the other way. 230 MHR • Chapter 7 3. a) How many equal parts is the paper divided into? b) Count how many parts you shaded red. Name an equivalent 1 fraction for __ using your answer to part a) as the denominator. 3 4. Fold a different piece of paper into 4 equal parts. Shade 1 of the __ 4 paper blue. 5. Fold the piece of paper into 3 equal parts the other way. 6. Count how many parts you shaded blue. Name an equivalent 1 fraction for __. 4 Reflect on Your Findings 7. a) What is the relationship between the denominators 3 and 4, and the denominator 12? common b) What is one method for determining a common denominator ? denominator • a common multiple of the denominators of a Example: Determine a Common Denominator set of fractions • a common a) 2 Determine a common denominator for __ and __.1 denominator for __1 3 2 4 1 is 12 because a and __ b) Determine equivalent fractions for 2 and 1 using the common __ __ 6 3 2 common multiple of denominator from a). 4 and 6 is 12 Solution Method 1: Use Paper Folding or Diagrams a) Divide a rectangle into 3 equal parts. Either fold a piece of paper, or draw a rectangle. Fold the paper or divide the rectangle into 2 equal parts the other way. There are 6 parts in the rectangle. 2 A common denominator for __ and 1 is 6. __ 3 2 b) Shade 2 of the rectangle red. __ 3 4 of the 6 parts are red. 2 4 __ = __ 3 6 Turn the paper over, or draw another rectangle and divide it as in step a). Shade 1 of this rectangle blue. __ 2 3 of the 6 parts are blue. 1 3 __ = __ 2 6 7.1 Common Denominators • MHR 231 Method 2: Use Multiples 1 a) The denominator of __ is 2. You can use divisibility 2 rules to find multiples. multiple Multiples of 2 are 2, 4, 6, 8, 10, 12, … 6 is divisible by both 2 and 3. • the product of a given number and a natural The denominator of 2 is 3. __ So, multiples of both 2 and 3 3 will be multiples of 6: number like 1, 2, 3, 6, 12, 18, … Multiples of 3 are 3, 6, 9, 12, 15, … and so on • for example, some The ﬁrst multiple divisible by both 2 and 3 is 6. multiples of 3 are 3, 6, 9, 12, and 15 A common denominator is 6. You could use any multiple of 6 as the common denominator, but the first multiple is often better to use. The denominator will be a smaller number, which is easier to work with. b) Write equivalent fractions using 6 as the denominator. ×3 ×2 To determine equivalent fractions, __ = 3 1 __ 2 4 __ = __ multiply the numerator and denominator 2 6 3 6 by the same number. This process does ×3 ×2 not change the value of the fraction. Check: 2=4 __ __ Strategies 3 6 Use pattern blocks. Model It Refer to page xvi. = 1 – 3 – 2 6 = 2 – 4 – 3 6 Determine a common denominator for each pair of fractions. Then use the common denominator to write equivalent fractions. Show two different methods. 1 a) __ and 3 __ 5 1 b) __ and __ 3 4 8 6 232 MHR • Chapter 7 • You can use paper folding, diagrams, or multiples to determine a common denominator. Paper Folding or Diagrams 5 of the 10 parts are blue. 6 of the 10 parts are red. 1 5 __ = ___ 6 3 = ___ __ 2 10 5 10 Multiples The denominator of 1 is 2. Multiples of 2 are 2, 4, 6, 8, 10, … __ 2 The denominator of __3 is 5. Multiples of 5 are 5, 10, 15, 20, … 5 A common denominator is 10. • To write fractions with a common ×5 ×2 denominator, determine equivalent fractions. 1 5 __ = ___ 6 3 = ___ __ 2 10 5 10 ×5 ×2 1. Tina wanted to ﬁnd a common denominator and equivalent fractions for 3 and 2. This is what she did: __ __ 5 3 a) Was she correct? If not, what was her error? b) Draw diagrams to show what she should have done. c) Discuss your diagrams with a classmate. 2. Ian says, “A common denominator for __ and 5 is 12.” Meko 3 __ 4 6 says, “I think it is 10.” Do you agree with Ian or Meko? Why? 3. How can you use multiples to ﬁnd a common denominator for 1 __ 3 the fractions __, 2, and __? 2 5 4 7.1 Common Denominators • MHR 233 7. Use a diagram to determine a common denominator for each pair of fractions. For help with #4 to #9, refer to the Example on Then write equivalent fractions using the pages 231–232. common denominator. 4. Use the folded papers shown to determine a) 3 and __ __ 5 3 1 1 b) __ and __ c) __ and __ 1 8 3 6 4 5 2 a common denominator and equivalent fractions for each pair of fractions. 8. Use multiples to determine a common a) denominator for each set of fractions. Then write equivalent fractions using the common denominator. a) 1 and __ __ 1 1 5 1 5 2 b) __ and __ c) __, __, and ___ 2 5 3 4 8 6 12 1 __ 2 __ 4 3 9. Using multiples, determine a common b) denominator for each set of fractions. Then use the common denominator to write equivalent fractions. a) 3 and __ __ 1 1 1 2 7 1 b) __ and __ c) __, __, and ___ 8 4 6 4 5 3 10 1 __ 3 __ 2 4 5. Look at the diagrams to determine a common denominator and equivalent fractions for each pair of fractions. 10. Determine a common denominator for a) each pair of fractions. Which is the larger fraction in each pair? 3 13 a) __, ___ 5 36 b) __, ___ 4 16 7 49 1 __ 3 __ 11 3 c) ___, ___ 12 4 d) ___, __ 3 5 30 10 27 9 b) 11. Draw a Venn diagram like the one shown to list common denominators that are less 1 1 than 50 for __ and __. 5 __ 1 __ 4 6 6 4 Multiples Multiples 6. Draw a diagram to determine a common of 4 of 6 denominator for each pair of fractions. Then use the common denominator to write equivalent fractions. 1 a) __ and 1 __ 2 b) __ and 1 __ c) 1 and __ __ 2 2 3 3 5 6 5 234 MHR • Chapter 7 12. Fill in the blanks to make equivalent 16. 5 ___ of a schoolyard is taken up by grass. 12 fractions. 7 ___ is the track. The rest is pavement. 1 a) __ = __ = ___ = ___ = ___ = ___ = ___ 18 4 8 12 16 20 24 28 a) What common denominator could be b) 1 2 3 4 5 7 11 __ = __ = __ = __ = __ = __ = ___ used to compare these fractions? 5 24 ___ 6 3 48 9 b) Does the grass or the track take up c) ___ = 12 = __ = __ = ___ = __ 56 more space? 10 5 d) 30 = 15 = ___ = __ = ___ = ___ ___ ___ 48 96 32 13. Fill in each blank with a numerator to 17. a) Copy the shapes. For each shape, make the statement true. Provide as many answers as possible. Use diagrams to show colour in 3. __ 8 how you determined your answers. 1 3 a) __ < __ < __ 4 2 4 1 5 b) __ < __ < __ 3 6 6 2 4 c) __ < ___ < __ 5 10 5 14. Determine a common denominator for b) Which shapes were more difﬁcult to the set of fractions. Use the common colour in? Which were easier? Explain. denominator to write an equivalent c) Imagine you are using paper folding to fraction for each fraction. Then list the determine a common denominator for fractions in order from least to greatest. 3 and __. Which of the shapes would it __ 2 8 5 __ 1 5 2 3 1 1, __, __, __, __, __ be possible for you to use? Show the 3 4 6 3 4 2 work by drawing the fold lines on the 15. The ancient Greeks thought of numbers shapes. as being represented by rectangles. They d) Compare your drawings with a would have made a rectangle like this to classmate’s. represent 6: 18. Write as many different proper fractions in lowest terms as you can that have denominators from 2 to 9 and numerators that are positive numbers. a) How could this rectangle be used to ﬁnd 1 a common denominator for __ and __?1 19. Which of the following fractions is closest 2 3 3 Explain. to ___? 10 b) Use a rectangle to ﬁnd a common 9 A 1 21 __ B ____ C ___ D 2 __ denominator for 3 and __. __ 1 4 100 40 5 4 7 7.1 Common Denominators • MHR 235 20. You have three beakers that are the same 21. The table shows the fraction of the total size. __ of beaker 1 contains oil. 1 of 2 __ number of students at Maple Leaf 3 4 Elementary School that are in each grade. beaker 2 contains water. Beaker 3 is empty. When you pour the liquids into Kindergarten 7 ___ beaker 3, the level of the combined liquids 40 corresponds exactly to one of the Grade 1 3 ___ markings on the side of beaker 3. Which 20 of the following beakers is beaker 3? Grade 2 11 ___ 72 A B 5 Grade 3 ___ 36 Grade 4 26 ____ 180 Grade 5 17 ____ 180 Grade 6 13 ___ C D 90 a) Which grade has the greatest number of students? b) Which grade has the least number of students? c) Which two grades have the same number of students? d) If there are 54 students in grade 1, what is the total number of students in the school? MATH LINK 1 – 8 a) Determine a common denominator for the fractions 1 in the Eye of Horus. Show your work. – 4 1 1 – –– b) Use this common denominator to determine an 2 16 equivalent fraction for each part in the eye. 1 1 –– –– 64 32 236 MHR • Chapter 7

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posted: | 3/25/2011 |

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