Equalities And Inequalities

Perry Lemon                    Shoop School
10024 S. Eggleston             1460 W. 112th Street
Chicago, IL. 60628             Chicago, IL. 60643
(312)-238-4532                 (312)-535-2715

Objectives:

1.  to involve students in discovery
2.  to teach partitioning and equivalency
3.  to teach the order of fractions

Materials needed:

Overhead projector (teacher)
Tower of bars (class)
Rulers (class)
Markers: Bingo chips, candy, etc. (class)
Activity cards (one per group of three)

Strategy:

    The tower of bars is a model for fractions.  The whole bar at the top 
represents the unit 1 and the bars below it illustrate fractions with 
denominators from 2 through 12.  Various fractions can be illustrated by 
placing markers on the tower of bars, such as three markers on the seventh 
bars to represent 3/7 and so on. 
 
    Place a transparency of the tower of bars on a overhead projector.  Discuss 
numerator and denominator using tower of bars.  Stress the number of equal parts 
determine the denominator and the numerator tells how many equal parts are being 
considered.  Divide class into groups of three.  Each individual will have a 
copy of the tower of bars, a ruler or straight edge and markers.  One student 
will have the responsibility of demonstrating the model by placing markers on 
the overhead projector.  Another would write the fraction on the chalkboard and 
the third student would lead discussion of observations and/or explanations 
regarding the model.  Activity card 1, for example, would be a practice to 
reinforce the part-whole interpretation of a fraction, such as, 3/5 indicates 
that a whole has been partitioned into 5 equal parts and 3 of those are being 
considered. 

  Equalities: every second row on the tower of bars has a line down the center. 
That is, a line appears down the center of the halves bar, the fourths bar, the 
sixths bar and so on.  It may be helpful to color these lines on a transparency. 
On another activity card students would place a ruler on the line down the 
center and use markers to show the patterns on every second line.  Then, write 
the fractions seen using the equal sign.  The corresponding numerical pattern is 
1/2 = 2/4 = 3/6 = ... Every third row has vertical lines that line up with those 
on the thirds bar.  These lines can be colored a second color.  Similar 
observations can be made for every fourth bar, every fifth bar and so on.  The 
corresponding numerical patterns are 1/3 = 2/6 = 3/9 = 4/12 ... and 2/3 = 4/6 = 
6/9 = 8/12.


   To help see the equality pattern, markers can be placed on bars and a ruler 
or the edge of a piece of paper can be used to match up vertical lines. 
Additional activity cards can be devised to make this discovery. 

  Inequalities: The first part of each bar as you move down the left side of 
tower of bars represent a unit fraction and these parts become smaller and 
smaller.  Similarly, looking down the right side of the tower of bars shows that 
1/2 < 2/3 < 3/4 < 4/5 .... These fractions get closer and closer to one. 
Activity cards can be devised to show this and other patterns of inequalities. 

   The lesson may be concluded by summarizing concepts discovered from the use 
of the activity cards and the vocabulary review.  Suggested terms are: 
numerator, denominator, equivalent fractions, unit fractions; symbols: <  is 
less than, > is greater than, = is equal to.