Discovering New Units

Jenkins, 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.