5 Quadratics and Higher Degree Polynomials EA1N 966 05 01 indd 103 2 16 08 5 54 46 PM EA1N 966 05 01 indd 104 2 16 08 5 54 48 PM Finding Roots Maximum Rectangle Area You will need Maximum Rectangle Area tns Quadratic functions can model relationships other than projectile motion In this activity you will find an equation relating the area of a rectangle to its width You will also look at real world meanings for the x intercepts and the vertex of a parabola MAKE A CONJECTURE Suppose you have 24 meters of fencing material and you want to use it to enclose a rectangular space for your vegetable garden Q1 What dimensions should you use for your garden to have the largest area possible for your vegetables INVESTIGATE 1 Open the TI Nspire document Maximum Rectangle Area tns on your handheld and go to page 1 2 For this problem the handheld rounds the lengths and widths to whole numbers 2 You should see a rectangle with a fixed perimeter of 24 centimeters Drag vertex C or D to see different dimensions of the rectangle Record the lengths and widths on page 1 3 Get at least eight different rectangles It is okay to have widths that are greater than their corresponding lengths 3 Calculate the area of each rectangle Q2 From the table what is your guess for the largest area 4 Go to page 1 4 to see a scatter plot of your widths areas data Q3 From the scatter plot view has your guess changed If so what your new guess Exploring Algebra 1 with TI Nspire 5 Quadratics and Higher Degree Polynomials 105 2009 Key Curriculum Press EA1N 966 05 01 indd 105 2 16 08 5 54 48 PM Finding Roots Maximum Rectangle Area continued In the next problem you will use the same rectangle but the lengths widths and areas will be captured automatically when you drag the vertex Try to get some very small and very large values of width 5 Go to page 2 1 and drag vertex C or D around to gather many data points As you drag the data will be captured on page 2 2 6 Go to page 2 3 to see the scatter plot of these widths areas data Q4 Write an expression for lengths in terms of widths You can determine this algebraically or use the scatter plot of wid len data on page 2 4 To find the slope for your expression using two points on the scatter plot double click the x coordinate of the given point and enter a new width The cursor will jump to the nearest point on the scatter plot Q5 Using your expression for the length from Q4 write an equation for the area of the garden in terms of the width 7 Go back to page 2 3 and enter this equation on the scatter plot choose Text from the Actions menu and click in an empty space to open a text box Type your equation using x and y and press Press d to put the text tool away then drag the equation to an axis Press to draw the graph 106 Q6 Trace to find the exact largest area choose Graph Trace from the Trace menu Place a point on the graph you may have to press or to trace the graph instead of the scatter plot Press d to put the trace tool away Drag the point toward the top of the graph until an M for maximum appears What is the maximum value of area At what width does this occur Q7 Locate the points where the graph crosses the x axis by double clicking the y coordinate of the trace point and entering a new one To get the other x intercept move toward it and repeat the process Q8 Explain the meaning of the x intercepts in this situation 5 Quadratics and Higher Degree Polynomials Exploring Algebra 1 with TI Nspire 2009 Key Curriculum Press EA1N 966 05 01 indd 106 2 16 08 5 54 52 PM Finding Roots Maximum Rectangle Area Activity Notes Adapted from Discovering Algebra by Jerald Murdock Ellen Kamischke and Eric Kamischke Objectives Students will write equations that model data from a geometric situation Q2 The guess will be determined by which values students used as widths Sample data Activity Time 25 minutes Materials Maximum Rectangle Area tns Mathematics Prerequisites Students should have some number sense involved with measurement and drawing of rectangles they should be familiar with finding the perimeter and area of a rectangle TI Nspire Prerequisites Students should be able to open and navigate TI Nspire documents graph functions and enter data See the Tip Sheets TI Nspire Skills Students will trace functions Notes This activity can be done in multiple ways depending on the skill level of students and amount of time you allocate Before students open the TI Nspire document you might have them fill out a table of lengths widths and areas by hand If students have trouble with this part they can also draw the rectangles Use 24 cm lengths of string for kinesthetic learners A width or length of zero is not acceptable as a measurement but these are useful values to list Regardless of how students get the data have them find and record the length width and area of several possible rectangles Students could also graph the data by hand instead of using the scatter plot on page 1 4 of the TI Nspire document MAKE A CONJECTURE Q1 There could be a variety of guesses for the maximum area INVESTIGATE 2 When students are dragging the vertex of the rectangle to make it change make sure they don t stop when the width becomes greater than the length If they stop too soon they will only get half of the parabola Instead of referring to the values as strictly length and width you might refer to them as two consecutive sides Q3 This answer might change depending on how close the values are that were chosen from earlier steps 5 The rectangle s dimensions of length and width are shown rounded to the nearest whole integer although the table captures more exact values Q4 length 12 width Some students might benefit from solving the equation 2length 2width 24 for length Make sure that the pattern makes sense to students Q5 area width 12 width or area 12width width2 Q6 The maximum area is 36 cm2 at a width of 6 cm The 6 by 6 rectangle is actually a square Q7 0 0 and 12 0 Q8 The rectangle has no area if the width is 0 cm or 12 cm 3 Students can calculate the areas by hand or by typing the formula into the formula cell for column C Exploring Algebra 1 with TI Nspire 5 Quadratics and Higher Degree Polynomials 107 2009 Key Curriculum Press EA1N 966 05 01 indd 107 2 16 08 5 54 54 PM Finding Roots Maximum Rectangle Area Activity Notes continued DISCUSSION QUESTIONS How will the equation for areas change if the …