Equalities And Inequalities

Perry Lemon Shoop School
1460 W. 112th Street
Chicago, IL. 60643


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)


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

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.
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