`Equalities And InequalitiesPerry Lemon                    Shoop School                               1460 W. 112th Street                               Chicago, IL. 60643                               (312)-535-2715Objectives:1.  to involve students in discovery2.  to teach partitioning and equivalency3.  to teach the order of fractionsMaterials 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.`