Mathematics: Modeling Our World Unit 4: THE RIGHT STUFF S U P P L E M E N TAL ACTIVITY AREA PROOFS S4.1 Deductive reasoning can be used to establish area formulas. 1.Many area proofs are based on a fact with which most people agree, namely that the area of a rectangle is the length of its base times the length of its height (or just written as bh). Use the 3 x 6 rectangle in Figure 1 to explain why the area can be found by multiplying the base times the height. page 1 of 2 Figure 1. A 3 x 6 rectangle. 2.The solid lines in Figure 2 form a parallelogram. a) Identify the parallelogram s base and height. b) Use deductive reasoning to explain why the parallelogram uses the same area formula as a rectangle. Figure 2. A parallelogram. 3.Figure 3 is triangle. a) Identify a base and the corresponding height. Figure 3. A triangle. b) Does a triangle have only one base? Explain. Is your answer the same for a parallelogram? 3 4 7
S U P P L E M E N TAL ACTIVITY Unit 4: THE RIGHT STUFF Mathematics: Modeling Our World S4.1 page 2 of 2 AREA PROOFS c) Explain why the area of a triangle is half the area of a parallelogram with the same base and height. 4.Calculating the area of a figure or proving an area formula sometimes involves dissecting the figure into several parts and, perhaps, reassembling the parts in a different configuration. The dissecting and reassembling technique is fine, but it must be done with care. Figure 4 shows a triangle drawn on dot paper and divided into six parts. a) What is the original triangle s area? Figure 4. b)make a copy of the figure on graph or dot paper. Cut out the six pieces and reassemble them so that the two small right triangles at the top of the triangle change positions with the larger right triangles at the bottom. You will have to reorient the other two pieces slightly to complete the triangle. c) What does the area of the six pieces appear to be now? 3 4 8
Mathematics: Modeling Our World Unit 4: THE RIGHT STUFF S U P P L E M E N TAL ACTIVITY CIRCLES AND TANGENTS INDIRECTLY S4.2 Geometers occasionally use a method of pro o f called the indirect method. When you use the i n d i rect method to prove something, you ask what would happen if it weren t true. You use logical reasoning to show that if the thing you are trying to prove weren t true, then something that you a l ready know would be contradicted. Another way to think of indirect reasoning is that you eliminate all the possibilities except one. Usually there are only two to consider. For example, to use indirect reasoning to show that a radius and tangent must be perpendicular, you have to consider only two possibilities: either they are perpendicular or they are not. You begin the indirect proof by asking what would happen if they were not perpendicular. Figure 1 shows a radius and a tangent. What if they are not perpendicular? (Since they look perpendicular, you will have to use your imagination on this one.) Since you are imagining that the radius and tangent are not perpendicular, you will also have to imagine some other line from point A that is perpendicular to the tangent (Figure 2). Like most proofs, indirect proofs use facts that have previously been proved or that are simple enough to be agreed on by everyone. An example is the fact that the shortest distance from a point to a line is along a perpendicular from the point to the line (Figure 3). Many people accept this fact without proof, but it can be proved. 1.What does this fact tell you about the length of the imagined perpendicular AC in Figure 2 and the length of the radius AB? page 1 of 2 Figure 1. A radius and a tangent. Figure 2. AC is the imagined perpendicular to the tangent. 2.What does your answer to Item 1 tell you about the location of point C? Explain. 3.Remember that C must be on the tangent line. What does your answer to Item 2 tell you about this tangent line? Figure 3. The shortest distance from P to the line is along a perpendicular. 3 4 9
S U P P L E M E N TAL ACTIVITY Unit 4: THE RIGHT STUFF Mathematics: Modeling Our World S4.2 page 2 of 2 CIRCLES AND TANGENTS INDIRECTLY 4.Summarize the reasoning you used in Items 1 3. 5.The proof you just completed makes use of the fact that the shortest distance from a point to a line is along a perpendicular. If you are familiar with the Pythagorean formula, show how it can be used to prove this fact. 3 5 0
Mathematics: Modeling Our World Unit 4: THE RIGHT STUFF S U P P L E M E N TAL ACTIVITY CENTS OF SPACE S4.3 In this activity you examine the efficiencies of packaging items of several sizes in the same kind of container, much as is done with melons. 1.Four 2 x 3 in. rectangular regions are shown below. Without actually trying with real coins, predict how many of each kind of coin you can fit inside one of these rectangles. a) pennies: b) nickles: c) dimes: d)quarters: page 1 of 2 2.Now carry out the experiment. Using any arrangement you can think of, without overlapping or mixing types of coins, of course, place as many of each coin in a box as will fit. 3 5 1
S U P P L E M E N TAL ACTIVITY Unit 4: THE RIGHT STUFF Mathematics: Modeling Our World S4.3 page 2 of 2 CENTS OF SPACE 3.Complete the table below. The Fit Number in column two is the number of coins you placed in the rectangle. Measure the diameter of each coin and record the results in the column three. Find the radius and record it in column four. Then to complete columns five and six, calculate the total area filled by the coins, the area of the rectangular container, and the efficiency of each packing. Explain why you think the efficiency results turned out as they did. Summarize the things you discovered. 3 5 2
Mathematics: Modeling Our World Unit 4: THE RIGHT STUFF S U P P L E M E N TAL ACTIVITY THE LEAST SQUARES REGRESSION LINE S4.4 An important part of many mathematical modeling pro b l e m s is fitting a line or other curve whose equation is known to a set of data and using residuals to judge whether the curve is a good fit. Many geometric drawing utilities have a coordinate feature that can tell you the equation of a line that you draw or construct. Figure 1 shows a line and ten data points. The residuals were built by constructing a perpendicular to the x-axis from each data point, constructing the intersection of the perpendicular and the line, constructing a segment from the intersection point to the data point, and hiding the line. The sum of the residual squares was founded by measuring each residual, squaring it, and adding these squares. The equation of the line was calculated by the drawing utility. Each of the data points can be moved by selecting and dragging it to a new position. The line can be moved by selecting either of the points on it and dragging the selected point to a new position. The idea is to adjust the line gradually until the sum of the residual squares is as small as possible. page 1 of 2 Figure 1. 3 5 3
S U P P L E M E N TAL ACTIVITY Unit 4: THE RIGHT STUFF Mathematics: Modeling Our World S4.4 page 2 of 2 THE LEAST SQUARES REGRESSION LINE Your calculator or computer fits a line to data by a process called least squares. That means that no other line can produce a smaller sum for the squares of the residuals. Figure 2 shows squares each having a residual as a side. Geometrically speaking, the least squares line produces the smallest possible total for the areas of these squares. Figure 2. Either design your own drawing utility sketch similar to the one in Figure 1 or use the Geometer s Sketchpad file Least Squares Demo on the Course 2 CD-ROM to adjust the line until you have the smallest possible total for the sum of the squares of the residuals. Record the line s equation. Then enter the coordinates of the data points into a calculator and find the linear regression model for the data. Compare the calculator s model with the equation of the line you found with the drawing utility. Prepare a report on the results. 3 5 4