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Quilting with High School Students: Creating Polynomial Quilts Why? The concepts of multiplying binomials and factoring trinomials can be difficult for students. While they can understand the distributive property and basic multiplication, doing more complex computations with binomials can be very confusing. Sometimes a visual or hands-on activity can make the concept much clearer. The idea for polynomial quilts came about as a result of doing a teacher workshop using algebra blocks. Moving the blocks around on an overhead led to thinking about quilt designs. Consequently the instructions for the activity evolved as a way of helping algebra students visualize a polynomial. Using the area model builds on the elementary school model of arrays to make sense of multiplication facts and families. This is a familiar model to students, and it is good to review and remind them of the patterns of multiplication from earlier grades. NCTM Standards If you are going to use this activity, you will want to match your district or state standards for polynomials. Perhaps the two that are most prevalent, and most broad, in this activity are:
Another standard included is:
Setting Up the Activity This activity can be done using construction paper, graph paper, or fabric. Sources of fabric can be donations from parents or local quilt and fabric stores. If you would like to expand quilt models, chances are some of your students' parents are quilters and can bring quilts to class. Quilt and fabric stores will sometimes donate time, materials, and demonstrations about basic techniques in quilting, including making some sample blocks. There are numerous sites available on line with historical information about quilt blocks. You can also use sites of quilts to examine mathematical relationships within the geometry of the quilts. Many of the traditional quilt blocks are very mathematical, as well as historical in origin. This activity is designed to look strictly at quilts built on polynomials. As such, you can create on paper or let students work on fabric. One of the best parts of this activity is letting students decide on the dimensions for their variables. For many students, the move to abstract mathematical thinking required in algebra is difficult; the idea of using a variable to represent some unknown quantity is not always an easy one to grasp. Being able to decide on dimensions for variables through this activity helps students understand the abstract use of a variable. Color choices can also aid in understanding the combination of two variables. Students may decide on one color for an "x" and another color for a "y." Choosing a color that is a blend of the two would help expand understanding of an "xy" combination. If "x" is red and "y" is yellow, then orange would be a logical choice for an "xy" combination. Supplies Construction paper in several colors (see note above concerning blending colors) Scissors, ruler and pencil, glue stick, oaktag or some stronger backing material The Basic Activity 1) This is based on the basic multiplication of (x + y)(x + y). Using construction paper in several colors (see note above about color blends), cut two small strips 1 inch by however many inches for the "x" dimension. Any dimension is appropriate. One inch is suggested as a width so that the paper quilts are easy to work with. Label these two "x." 2) Cut another two small strips one inch by however many inches for the "y" dimension. Label these "y." Students have now determined the "value" of their variables. 3) Glue the "x" strips in place on the backing material, one on the top edge and one to the left side. Glue the "y" strips in place on the backing material, next to the "x" strips. Diagram 1
 4) Measure the "x" strip. Students will now cut out a square in the same color as the "x," using the dimension of the "x" as the measure of one side of the square. In using the FOIL method, students have just multiplied the two First terms, x times x, to get an x-squared. This gets glued into place under and next to the "x" on the backing. Diagram 2
 5) Measure the "y" strip. Students will now cut two rectangles in a "blend" of the "x" and "y" colors (if possible). One side of the rectangle will be the measure of "x" and the other side will be the measure of "y." Using the FOIL method, students have just multiplied the Outer and Inner terms (x times y and y times x). Have students wait to glue these into place, as they will want to intuit that every polynomial can be represented as a square or rectangle, and then determine where these pieces fit. Diagram 3
 6) With the "y" measurement, students will now cut out a square in the same color as the "y," using the dimension of the "y" as the measure of one side of the square. In using the FOIL method, students have just multiplied the two Last terms, y times y, to get a y-squared. 7) Now have students determine where all the pieces will fit to create a rectangular model of this polynomial. just like a beginning multiplication table, the pieces need to fit under their specific dimensions. Diagram 4
 8) Glue pieces into place for a completed model.
 The Activity - Part 2 Now students can begin to experiment with other polynomials. If you ask that dimensions remain within a particular size, you will end up with a series of polynomial blocks that can then be assembled into a larger "polynomial quilt." If the sizes differ, then you can use border strips to connect the various blocks into a cohesive whole quilt. Students do not need to glue the initial dimension strips into place, and what works well is two things: you have an easier time connecting pieces into a "quilted whole," and you have great models to use for the next part of the activity. The bottom photos show two additional polynomial quilts: (2x + y)(2x + y) and (2x+2)(y+3).
 The Activity - Part 3 The reverse of multiplying binomials is factoring. if you want to extend this activity, then use the completed blocks to analyze the factoring process. This time you will work from the middle of the block, looking at the numbers of different pieces that make up the block and trying to determine the dimensions of the original pieces. Make up a couple of blocks that do not have the "original pieces" attached. See if students can determine the original dimensions. In the photo of (2x+2)(y + 3), you can see that you have two blocks with dimensions of x and y, so you need an x and y as factors. Students will see a visual way to factor a polynomial. You have a variety of already-completed blocks that can now be looked at and factored. A Caution All these activities work with positive numbers. The primary purpose is to get students to "see" what happens with the algebra. This is easily accomplished using just positive numbers and avoids an artificial use of negative numbers. As students become proficient at multiplying and factoring, then they won't need the visual, and the role of signs can also be addressed more easily. In Conclusion One of the best services we can provide our algebra students is the use of concrete models whenever possible. Teachers have used manipulatives heavily in elementary school, and the use eases off in middle school. It is rare to see manipulative activities used in the high school algebra class. This is one activity that will give a very strong visual model to students and provide another pathway to understanding. Resources Mathematical Quilts, available from Key Curriculum Press, usually as a package with the second book More Mathematical Quilts, ISBN 1-55953-374-9, available from Key Curriculum Press Mathematical Quilts Posters, ISBN 1-55953-413-3, available from key Curriculum Press. These are excellent posters for looking at advanced ideas of mathematics within quilts. Related : Math And Quilting
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