Measuring Mixed Numbers

Karen Trout                    Sumner Math and Science Academy
                               4320 W. 5th Avenue
                               Chicago IL 60624
                               (312) 534-6730

Objectives:

   Grade 5
     -Students will discover the need and every day use of mixed numbers.
     -Students will measure different lengths to the nearest fraction of an 
      inch.
     -Students will review the equivalent values of fractions (for example:  
      2/4=1/2).
     -Students will review how to come up with the common denominator and why.
     -Students will add mixed numbers phenominologically and then 
      algorithmically.

Materials Needed:

     -1 pair of scissors per student
     -1 ruler per student
     -5 pieces of tape per student
     -1 worksheet per student
     -1 piece of scratch paper per student
     -Construction paper of different colors
     -Large clear container with graduated cups
     -Measuring cup set
     -Bucket full of milky water (mix in a little corn starch)

Strategy:

     1.  Before class, use your construction paper to make a couple sets of 
fraction bar charts (be sure to include whole, halves, fourths, eighths, and 
sixteenths).  Use different colors for each different fraction length, and make 
your pieces board size.
     2.  If you have a large container that does not have graduated cups, just 
place a strip of tape down one side of the container, then pour a cup of water 
in at a time and mark the level for each cup on the piece of tape.
     3.  Create a worksheet that has about 4 different problems similar to the 
example below.  Draw different length bars that the students will have to 
measure individually, cut out, connect, and then remeasure as a whole unit. 
They will only be measuring the length of these rectangles or bars which should
be at various lengths between 1/2 an inch and 5 inches.  This is a sketchy 
example of what one problem would look like:

     Measure these lengths as accurately as you can, then cut them out, tape
them together and remeasure the length of these two together:          

    `ffffffffffffffffffffffffp              `fffffffffffffffp
    w                        w              w               w
    w                        w      +       w               w   =
    w                        w              w               w
    affffffffffffffffffffffffq              afffffffffffffffq

     4.  Begin the class with this question:  "Pretend you are a zookeeper and 
your responsibility is to feed the baby animals their formula each day.  The 
baby porcupine eats 2 1/2 cups of formula, the baby seal eats 5 1/4 cups, and 
the baby ape eats 7 3/4 cups.  How much baby formula do you need to make to feed 
all three of the baby animals?"
     5.  Allow the class time to reflect on the problem and then explain that 
they are going to need to understand how to add mixed numbers in order to solve 
this problem.  Demonstrate for them briefly on the board how they are going to 
use the worksheet.  They are to measure the length of each bar and write that 
length inside the figure.  Then they are to cut out each bar and connect 
them with tape onto the piece of scratch paper they have been given.  They 
should now have a longer bar which they are to measure and write down the total 
length of the two bars combined. 
     6.  When they have completed this worksheet go back and compare some of 
their answers.  Then using the fraction pieces which you will have taped up on 
the board, show them the different lengths.  If they measured 3 1/2 inches for
one bar, show them 3 1/2 in fraction pieces.  If their second bar was 1 1/4 
inches long, then show them 1 1/4 in fractions.  Demonstrate for them how to add 
those two fractions of different denominators (1/4 and 1/2) by replacing the 1/2
fraction piece with two 1/4 pieces and finding their length to remain the same.  
Using the fraction pieces, continue to visualize for them how we come up with a 
common denominator when adding fractions of unlike denominators.
     7.  When you feel they have grasped this concept, you then go back to your 
original question about feeding the baby animals.  This time have them find the 
common denominator between 1/2 and 1/4 and then add their mixed numbers together 
to get the total amount of baby formula needed.
     8.  Finally, to prove whether or not the answer they got on the baby 
formula was correct, have someone come up and measure out the amount of cups fed 
each baby animal.  Have them take the milky water (water and a little corn 
starch) out of the bucket and carefully pour it into your graduated, clear 
container.  The amount of formula that is now in the container should be the 
same as the amount of cups the class came up with on paper.

Conclusion:

The students visualize adding mixed numbers physically by adding different 
lengths together to the nearest fraction of an inch and by adding cups together.  
Then they discover the simplified algorithmic process which reduces the amount 
of work they need to do.  However now they also understand why they need to add 
mixed numbers and what mixed numbers are.  In conclusion, their performance 
should improve greatly as they comprehend the usefulness of these skills and 
what it is that they are actually doing when they come up with a common 
denominator and add their mixed numbers together.