(x + y)2 fun? It can be!

Jane BurnsMary Aldridge

University of Northern IowaCedar Falls, Iowa J0613

Why does (x + y)2 remain a meaningless abstraction to many studentsin beginning algebra classes? Can the operation be applied in such a waythat middle school students can attain a concrete understanding of theprinciple involved? We believe that it does not have to be drudgery, nordoes it have to be learned in an abstract manner.The spiralling curriculum is based upon the premise that concepts need

to be introduced to children through the use of concrete materials. Thenthe concepts are revisited during subsequent periods of maturation. Ateach point when a concept is reintroduced, the child grasps it with ahigher level of understanding. The idea is no longer unfamiliar and newvariables can be grasped as interrelationships fall into focus.The impact of Piagetian theory on curricular design generally has been

beneficial to students. However, there is greater variation in transitionfrom concrete-operational levels to formal-operational levels of thinkingamong pre-adolescent and adolescent students than has been reflected inteaching materials and instructional strategies. The introduction of ab-stract concepts need not be delayed to accommodate students withmaturational lag, but should be presented in a more concrete, activity-based form. This will facilitate the development of logical thought proc-esses and the transition into higher intellectual functioning.

Algebraic concepts need to be introduced in the framework of prac-tical applications in order to build a base for abstract reasoning. Craftactivities which appeal to students and relate concepts which involve theperception of the whole through the manipulation of the componentparts can motivate learning.

Patchwork handcraft activities utilizing a variety of materials provideconcrete experiences for abstracting the meaning of (x + y)2. Hands-onactivities that result in a product which will instill pride and a sense of ac-complishment should be part of the middle school curriculum. Complet-ing a group project such as a quilt or pillows for a specific purpose cancontribute to a cohesive social atmosphere in a classroom.Where will the students get the materials needed for their projects?

Much of it can be found in the sewing rooms and shops of both the homeand school.

UNDERSTANDING (x + y)2 == x2 + 2xy + y2Before attempting to embark on a project, a student should have an

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idea of the concept and its application. For the gifted student it is no dif-ficult task. However, some students may need a more concrete overview:

(x + y)2 = (x + y) (x + y)= x(x + y) + y(x + y)= x2 + xy + yx + y2= x2 + 2xy + y2

Most middle school students can apply the commutative property ofmultiplication and can identify xy as the same as yx. The square whichresults from the above operation can be described as follows:

x2 means a square with side x.2xy means two equal rectangles with sides x and y.y2 means a square with side y.

Therefore, the square expressed by x2 + 2xy + y2 consists of two squaresof different sizes and two equal rectangles. (Figure 1) The square couldalso consist of four equal squares, when x = y. (Figure 2)

xy

x2

y2

xy

xy

X2

y2

xy

Figure 1. x > yFigure 2. x = y

THE BASIC DESIGN

Using graph paper, students can fill in squares with different dimen-sions for x and y. Following the discussion of (x + y)2 as stated above,the students can reproduce Figures 1 and 2 on the graph paper. Since (x+ y)2 = x2 + 2xy + y2, there must be a pattern piece for the x2, two forthe xy’s and one for the y2. Students can accurately draw each squarefirst, then cut out the drawings for the pattern pieces.

Different values for x and y will result in squares with more interestingproportions for the pieces. For example, six inches might be assigned to xand two inches to y: x = 3y. (Figure 3)

Practice with construction paper or wallpaper patterns will enablestudents to choose color combinations which are interesting and aesthe-tically pleasing. Graph paper is useful for designing an article needingseveral squares, such as a quilt top, table cover, or article of clothing.

(x + y)1 fun? It can be! 617

xy

X2

x

y2

xy

yFigure 3. x=3y

PATCHWORK ACTIVITIES

Quilt squares can be designed with plain or patterned pieces. Four dif-ferent materials may be used in each block, but three solid-colored orpatterned materials can be arranged so that the sides of like pieces do nottouch. (Figure 4)

Figure 4. Quiltblock design ofthree fabrics

A seam allowance of one-quarter inch should be added to the patternsbefore cutting the fabrics. The quilt blocks can be sewn together with theaid of a sewing machine or by hand. (Figure 5) The student’s cooperativeefforts to finish the quilt will give them a sense of group accomplish-ment. Deciding upon a use for the quilt provides opportunity for groupdecision making.

Variety in patchwork design can be achieved in several ways. Differentprints, fabrics, and textures can be combined to achieve effects pleasingto individual tastes. Alternating blocks with different dimensions for xand y can change the overall effect of a completed article. Care must betaken to keep the outside dimension of each block the same size in orderto fit them together. (Figure 6) Rotating the pattern one-quarter turn cancreate different designs. (Figures 7 and 8) Designs generated by making ahalf turn develop a more interesting pattern and are no more difficult toconstruct.

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Figure 5. Fourquilt blocksjoined together

Figure 6. Fourblocks withalternate designs

Creative students will not be content to limit themselves to (x -+- y) asa pattern base, so let them create their patterns using a variety of mathe-matical expressions for squaring. A more intricate design for a pillow topcan be made by using (x + 3y)2. (Figure 9) One block will be sufficient

Figure 7. Fourblocks with blocks2,3, 4 rotatedclockwise

Figure 8. Fourblocks with blocks2,3,4 rotatedcounter-clockwise

Figure 9. Pillowtop (x+3y)2

(x + y^fun? It can be! 619

for a pillow top if a dimension of ten inches is given to x and two inches isgiven to y. (A seam allowance of one-quarter inch must be added to eachpattern piece)Another patchwork project could be a hostess skirt or apron. Skirts

with patchwork pieces go together surprisingly fast, and the student cancreate a unique garment. The dimensions of each square in the garmentwill depend upon the size of the wearer and the patience of the creator.Patchwork pullover shirts can easily be designed from squares. Some stu-dents may prefer to design and make luncheon cloths, napkins, or place-mats.

FURTHER APPLICATIONS

Pieces of wood of various color and dimension can be combined andGrafted into cutting boards any gourmet cook would be proud to use anddisplay. Bookends, magazine racks, jewel boxes, and table tops are otheruseful accessories which can be constructed in middle school classrooms.The painstaking craftsman will use shorter lengths for both the x and ydimensions of the expression with the result being finer detail in the fin-ished product. Some students may choose wood inlay techniques, mosa-ics from a variety of materials, or a leaded-glass design.Working on a craft design may make the square of (x + y) meaningful

to some students for the first time. Designing and creating a beautiful,useful object insures that this particular mathematical procedure is noteasily forgotten. Enthusiastic students engaging in creative hand-craftedactivities declare that (x + y)2 can indeed be fun.

BIBLIOGRAPHY

ELKIND, D. "The Adolescent: Mental Development." A Sympathetic Understanding of theChild: Birth to Sixteen. Boston: Allyn and Bacon, Inc. 1974, 174-80.

JONES, R. S. and C. K. McEwiN. "Creative Learning Environments for the MiddleSchool." Childhood Education, LVI (January 1980), 174-50.

KARPLUS, E. F., R. KARPLUS and W. WOLLMAN. "Intellectual Development BeyondElementary School IV: Ratio, the Influence of Cognitive Style." School Science andMathematics, LXXIV (October 1974), 476-82.

MILLER, J. P. "Piaget, Kohlberg, and Erickson: Developmental Characteristics forSecondary Education."Adolescence, L (Summer 1978), 237-49.

1981 CONVENTION-PLAN NOW TO ATTEND