```Discovering New UnitsJenkins, Kay                   James Shields Elementary School                               523-6097                           Objective:

With frequent work with Cuisenaire Rods the students will develop a
systematic usage for creating new units or in finding a common
denominator, in use with the skills for adding and subtracting unlike
denominators.

Apparatus needed:

A set of Cuisenaire rods for each student or groups of two.

Recommended strategy:

Before students can develop a system for determining common
denominators, a Hierarchy of Skills must be accomplished.  The student
must develop an understanding of basic concepts metacognitively.  The
goal is to use our entire mathematical schema.  The desired sequence
is:
1.  they must know the meaning of addition and subtraction
2.  they must master all basic concepts (addition, subtraction,
multiplication and division)
3.  they must understand common fractions as related to physical
models (fraction strips, shapes divided into parts, number lines)
4.  they must be familiar with prime and composite numbers (use of a
Hundreds chart is helpful)
5.  they should be able to add and subtract common fractions or like
denominators

Children upon completion of these skills can develop the knowledge
that equivalent fractions will yield a common denominator.

Children can be led to discover the relationship between numerical
values of the rods and multiples that can be derived, thus leading
them to the goal of developing new units not included in rods, i.e.,
12, 15, 18, 21, etc.

Activities:

Review common fractions in everyday use.  (Show each fraction with
your rods.)  1. What fractional part is a stick of butter to a whole
package?  (After each question students will display the appropriate
fraction with any color combination.)  2. I have six eggs.  What
fractional part of the carton do I have?  3. In a mixture of equal
portions, we need 1 part Hydrogen and 1 part Chloride.  What is the
fraction to name each part?  4. Marci's father was asked to design a
playroom for 3 and 4 year olds.  The room needs areas for play-rest-
music-and lunch.  The room should be divided into how many equal
parts?  Write the fraction; show the fraction.  What color rods did
you use to show each fraction above?  Did everyone use the same colors
each time?  (No.)  This begins the development of alternative color
combinations.

With work children can be led to include their knowledge of multiples
of the set of counting numbers, as well as their multiplication facts
or tables of factors (2x3=6; 5x2=10; 3x3=9, etc.).

To add fractions with uncommon denominators such as 1/3 + 5/6 our
denominators need to contain the same unit.  My unit needs to be at
least 6 lengths long.  Ask for a relationship between the numbers 3
and 6.  Children can readily equate 3x2=6.  Display a dark green rod,
make a train of white rods as long as the dark green unit rod.  How
many white rods did you use? What color rod is 1/3 of dark green?
Red, yes.  How many red rods did you use?  Show one red rod and five
white rods.  What is the answer?  You can't tell immediately because
we need to change everything to one color before we can combine the
two, so we now need to trade our red rod in for how many white rods?
(2).  Now we can add them together.  What is your answer?  (7/6 is
correct, write on board).  We call 7/6 an improper fraction because
the numerator is greater than the denominator, therefore we need to
trade up now, I'm trading six white rods in for one dark green rod.
What fraction is left?  Students can easily see the answer to this
question as 1/6 and the answer to the problem as a mixed number of 1
1/6.

Conclusion:

Problems of this nature can be explored by the teacher with the
student's furthering their understanding of the idea of trading up and
down to find common denominators.  This same procedure should be
followed with mixed numbers for adding or subtracting.  Children
should be given the opportunity to develop their own strategies for
finding a common denominator. A list of 5 to 10 problems with unlike
denominators can now be placed on the board for their further
investigation with the teacher giving aid wherever necessary.
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