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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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