The Math Projects Journal

Transcription

1 PROJECT OBJECTIVE The House Painter lesson series offers students firm acquisition of the skills involved in adding, subtracting and multipling polnomials. The House Painter lessons accomplish this b offering students a contet in which the can think about polnomials and their meaning and then be required to create abstract representations of the patterns found within that contet. This lesson also offers opportunit to review graphs of linear equations, to introduce graphing quadratics b plotting points, and to reinforce writing equations in function notation. THE SCENARIO The contet offered here is a house painter who must paint the interior and eterior walls of tract homes. These homes have somewhat similar models Concepts Multipling polnomials Time: 2 Hours Materials Graph paper, Student Handout Preparation None that ma var in sie. The house painter's (student's) job is to discover a mathematical pattern within each model, and then generate a formula (polnomial) to represent that pattern. In each case, the formulas need to be simplified b adding, subtracting or multipling the polnomial, depending on the lesson. LESSON PLANS Da One: Distributing One Term Have students read the scenario for the House Painter. Point out that the painter needs to find the total area of both the shaded and unshaded portions. The main principle behind which students need to understand about multipling binomials is the distributive propert, particularl the concept of distributing both terms of the first quantit. Therefore, the lesson starts b revisiting the distributive propert. The rectangular wall is portioned into two smaller rectangles. The area of the entire wall can be found b calculating the product of its outer dimensions. It can also be found b the sum of the two smaller areas. These epressions must be equal since the total area remains constant. In other words, 8( + 9) 8() + (9). Therefore, the students can SEE how the "8" is distributed to both the "" and the "9." The first two walls further eplore this idea. #1: Assign each student one of the first four values {1..4}. This can be done b one of two methods. The first method is to assign values b groups. In other words, each member of one group can draw the diagram 1, each member of another group can be assigned 2, etc. The second method is to assign values to individuals within each group. For eample, one student in the group draws the diagram for 1, a second student draws a diagram for 2, etc. Assigning individuall makes comparing the results easier, but for lower performing classes, assigning b groups makes it easier for students to get the correct answer. Each student reproduces the given diagram on graph paper according to the dimensions given. For eample, if Sall is assigned 1, then she should draw, on graph paper, a wall that is units high with a left portion that is 1 unit wide () and a right portion that is 12 units wide. Sall should record her calculations in the first line of the chart as both a product and sum: (1 + 12) and (1) + (12). She echanges data with the other groups/members and confirms their answers. Once these instances

2 PROJECT are completed and compared, the group discusses a strateg to figure how much paint is required for walls with values of, and, WITHOUT drawing the diagrams. The grand finale is to complete the final line the abstract generaliation for the total area in terms of written as a product and as a sum. The ke here for the students is seeing that the total width of the wall is: ( + 12). The group then determines whether or not the believe that epression for the product equals the epression for the sum. #2: Here the students determine if this would be true for ANY values. The are to generate unique eamples within their group, complete the chart and compare data just as before. The should see on the final line of the chart that ( + ) +. Da Two - Distributing Two Terms Once the students see that the distributive propert involves distributing a value to both terms of a quantit, epand that principle to multipling binomials. We use the same contet and area model as in the first two walls, but the rectangles have four partitions instead of two. The students are still asked to calculate the area both as a product and a sum. Have the students read the eample and then determine if the area product and the area sum are equivalent. In other words, is the following true? ( + 4)(6 + ) (6) + () + 4(6) + 4(). Of course the must be since the area remains constant, but is their a simpler mathematical method other than drawing the diagrams? Yes, we distribute the both the first two terms to both of the second two terms! To see if this applies when variables are involved in the epression, investigate the net two walls. #3: Have the students create the various instances again. This reinforces the idea that represents a "domain" of numbers, rather than a single unknown value. This is also helpful because students often have difficult seeing that the height and width of the wall are ( + 3) and ( + ) respectivel, instead of 3 and. Having the students SEE the height and width of their instance helps eliminate this misconception. The students will quickl see that, es indeed, ( + 3)( + ) , because the can SEE the four portions. Now have the students look at the two shaded areas which are to be painted. Where are these areas represented in the polnomial? The are the two linear terms that are to be simplified: What does this 8 now sa about our amount of paint? What does the 2 represent? The 1? #4: Determine if idea of distributing both of the first two terms to both of the second two terms would be true for ANY values. The students should generate unique eamples within their group, complete the chart and compare data just as before. The final line of the chart should be ( + )(w + ) w + + w +. SOLUTIONS 1. Total ( + 12) 170 ( + 12) 220 ( + 12) () + (12) () + (12) () + (12) ( + 3)( + ) 180 ( + 3)( + ) 19 ( + 3)( + ) () + () + 3() () + () + 3() + 3() ()

3 You are a house painter and have been hired to paint the interior walls of the houses within a specific tract. The tract has several models that var in sie with the same design and floorplan. You need to determine the area of each model in the tract so ou can purchase the correct amount of each kind of paint. You, being the most mathematicall-minded painter in town, decide to develop formulas for each tpe of wall in the tract. The interior walls require two tpes of paint: enamel paint for kitchens and bathrooms, and flat paint for the other rooms. Sometimes, ou have to paint part of one wall with enamel, while painting the other part with flat. You remembered doing that for another job recentl. The kitchen wall was 8' ', and the famil room was 8' 9'. Your calculations of that wall revealed something interesting. You noticed that if ou calculated the wall as a product b multipling its outer dimensions, ou got the same answer as when ou calculated the area as a sum of the two separate rooms. As shown below: ' 9' 8' 80 ft 2 72 ft 2 Famil Room Product: 8( + 9) 8(19) 12 Sum: 8() + 8(9) You wonder if this pattern is true for paint jobs that ou will do in the future. So for each model shown: a) On graph paper, draw an instance of the diagram. Each group member should draw a different instance. b) Record the instances from our group in the chart provided and complete all but the last line of the chart. c) Generalie our results in the last line (), and determine if, indeed, the area calculated as a product equals the area calculated as a sum. 12' 1. The height of the wall is ft. The width of the kitchen portion varies from house to house. The width of the famil room (shaded) is 12 ft. ' Total

4 2. The height of the wall, and the width of both the kitchen and the famil room var. Famil Room What do our results sa about the epression a(b + c)? You net job just got a little more comple. One of the interior walls has four sections. One section is a window (no paint), two sections are drwall (fl paint), and one section is wood paneled (stain). The window is ' 6' and the panel is 4' ' as shown below. 6' ' ' 4' Product: ( + 4)(6 + ) Sum: (6) + () + 4(6) + 4() You wonder if the pattern that ou discovered with the kitchen famil room job, applies here also. Find out!

5 ' 3. The window is a square of varing dimension. The panel is 3' '. 3' The dimensions of the window and the wood panel all var. Panel Panel w What do our results sa about the epression (a + b)(c + d)?