1. What specialist uses information obtained from bones to help police solve crimes?

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1 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT VIDEO VIEWING GUIDE H What specialist uses information obtained from bones to help police solve crimes? 2.What are some things that can be predicted from the remains of an individual? 3.The length of what bone is sometimes used in an equation to predict an individual s height? 4. Who developed an accurate equation for predicting height? 5. What is the formula for predicting height? 167

3 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT TI-83 INSTRUCTIONS FOR ENTERING DATA AND MAKING A SCATTER PLOT H4.2 Step 1: Clear previously stored functions from your calculator as follows: Press: Y=, position the cursor on the line with a function, and press CLEAR. Step 2: Clear unwanted data lists. Press STAT ENTER. Move the cursor to the header of the unwanted list and press CLEAR. ENTER. Step 3: Enter your data. Use the blue arrow keys to move the cursor to the first row of the first column. Your screen should resemble the one in Figure 1. Figure 1. Blank data table. Step 4: Enter the boys body-length and head-circumference data from Item 6 of Lesson 1, Activity 3 into L1 and L2 as follows. Begin by entering the length numbers in L1. After entering each number, press ENTER until all seven numbers have been entered. Next, press the right arrow key to move your cursor to the first data row of L2. Enter the circumference data into L2. (See Figure 2.) Step 5: Plot your data. Press: 2nd STAT PLOT and then 1 to select Plot 1. Press: ENTER to highlight ON. Adjust the remainder of the settings to match those in Figure 3. Press: ZOOM 9 (for ZoomStat) to view a scatter plot of your data. Notice that ZoomStat has automatically selected a viewing window for your data. Step 6: Add a line to your scatter plot. Press: Y=,.75x, GRAPH. You should now see a graph of the line y = 0.75x in the same viewing screen as your scatter plot. Try changing the coefficient of x until you find a line that more closely matches the pattern of the scatter plot. Figure 2. Data from Lesson 1, Activity 3. Figure 3. The Plot menu.

Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT EXCEL INSTRUCTIONS FOR ENTERING DATA AND MAKING A SCATTER PLOT H4.3 Step 1: Open your spreadsheet.

5 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT EXCEL INSTRUCTIONS FOR ENTERING DATA AND MAKING A SCATTER PLOT H4.3 Step 1: Open your spreadsheet. Label Columns A and B length and head, respectively, as follows. Click on cell A1. Enter the word length and then press the right arrow key to move to cell B1. Label column B and press ENTER. Step 2: Enter the boys body-length and head-circumference data from Item 6 of Lesson 1, Activity 3 into columns A and B. Here s how. Click on cell A2. Enter the number and then press ENTER. Continue this process until all the length data have been entered. Next, click on B2 and enter the data from the second column. (Figure 1.) Step 3: Make a scatter plot of the data. Click on cell A1, and while still holding down on the mouse button, drag to the lower right hand corner of your data (Cell B8). Release the mouse button. Columns A and B, rows 1 8 should now be highlighted. Click on the Chart Wizard on the tool bar at the top.(figure 2) Figure 1. Data from Lesson 1, Activity 3. Next, imagine where you want to place your scatter plot on your worksheet. Click on the upper-right corner of that spot and drag to the lower-left corner. Release the mouse and the Chart dialog box will appear. (It should tell you the cells you have highlighted.) Click on the Next button, then on XY Scatter, Next. Select the form of the scatter plot you would like to make (1 would be a good selection), Next. A scatter plot of your data will appear. Press Next again. This time you will have an opportunity to put a title on your scatter plot and label the axes. When you have finished, click OK. Figure 2. The chart wizard button.

7 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT TI-83 INSTRUCTIONS: CALCULATING SUMS AND RATIOS OF NUMBERS IN TWO LISTS H4.4 Step 1: Clear previously stored lists and functions from your calculator s memory. Step 2: Press STAT ENTER to display the empty lists. Then enter the boys data from Item 6 of Lesson 1, Activity 3. (See Figure 1.) Step 3: Compute the ratios, (Head circumference in L2)/(Length in L1) and place them in L3 as follows: Move your cursor so that it highlights the header for L3. Your screen should resemble the one in Figure 2. Since you want to divide the numbers in L2 by the numbers in L1, press: 2nd L2 / 2nd L1 ENTER The six ratios will appear in L3. Now practice: Place the sums of corresponding numbers in lists L1 and L2 in L4. Then find the differences of corresponding numbers in lists L4 and L3 and place them in L5. Figure 1. Measurements on length (L1) and head circumference (L2). Figure 2. L3 is highlighted.

Step 3: Compute the ratios, (head circumference)/(length) and place them in column C as follows. Click on cell C2, enter the formula: =B2/A2, and press enter.

9 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT EXCEL INSTRUCTIONS: CALCULATING SUMS AND RATIOS OF COLUMNS OF NUMBERS H4.5 Step 1: Open a spreadsheet and enter the boys data from Item 6 of Lesson 1, Activity 3. (See Figure 1.) Step 2: Label cell C1 Ratio. Step 3: Compute the ratios, (head circumference)/(length) and place them in column C as follows. Click on cell C2, enter the formula: =B2/A2, and press enter. The value of B2/A2 will appear in cell C2. (See Figure 2.) Click on cell C2 again and drag to highlight cells C2 to C8. Click on Edit and drag to highlight Fill Down. The seven ratios will appear in column C. Now practice: Place the sums of corresponding numbers in columns A and B in column D. Then find the differences of corresponding numbers in D and in C and place your results in E. Figure 1. Measurements on length and head circumference. Figure 2. Formula entry into C2.

11 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT DIRECTIONS FOR TARGET The program Target presents a screen of ten randomly distributed targets. The goal is to hit as many of the targets as you can with the graph of a function. The more you hit, the better your score. If you hit one target, you get 200 points. If you hit two targets, you get 300 points per target. If you hit three targets, you get 400 points per target. As your graph is drawn, the targets you hit are removed. When the graph is finished, you see your score. Then any targets you didn t hit are put back on the screen, and you take another turn. This continues until all the targets are gone and the game is over. On any turn, you can draw one practice graph. Practice shows you if your graph really hits the targets you hope it will hit. You can take only one practice per turn, however, and your score for that turn is reduced by 200 points. Although you can use almost any kind of function in the program, there are some functions that cause problems. For example, the function 1/x is undefined when x is 0. The function x is undefined for any negative number. The program has a checking feature for these kinds of functions, but it slows the graphing somewhat. Because of the way the program does its graphing and checking, it takes a long time to graph y = 0. If you want to hit targets on the x-axis, it is better to graph y = 0.2 or y = 0.2 instead. Here s a screen-by-screen description of the game. Run the program and ten randomly distributed, square-shaped targets appear on the graphing screen (Figure 1), which uses the window [ 9.4, 9.4] x [ 6.2, 6.2]. After you determine the equation of a function whose graph you think will hit several of the targets, press ENTER. Figure 1. Ten targets. H4.6 page 1 of 2 You are prompted for the function (Figure 2). Do not type the Y= part of the equation, only the expression that follows the equals sign. Figure 2. Function prompt. The program asks whether you want to take a practice shot (Figure 3). If you choose practice, the graph is drawn, but no targets are removed. Also, your score for the current turn is reduced by 200 points. After a practice attempt, the practice function is displayed while you enter your next attempt. You are allowed only one practice per turn. Figure 3.

12 HANDOUT Unit 4: PREDICTION Mathematics: Modeling Our World H4.6 page 2 of 2 The program asks whether you want checking on or off (Figure 4). Choose 1 if the function is undefined anywhere in the interval [ 9.4, 9.4] to avoid errors (e.g., division by 0) that crash the program. You have the option of turning checking off because checking slows graphing. The checking menu does not appear on a practice attempt. Press ENTER again, and the remaining targets (of your original ten) appear (Figure 7). Note that occasionally two targets overlap, which means that one of them is shielding the other. It can be difficult to hit both targets when one shields the other. Can you do it? Figure 4. Figure 7. The function graphs, and any targets that are hit disappear (Figure 5). Note that the individual points of the graph are not connected, which means that very steep graphs sometimes miss targets you expect to hit. Repeat the process until all ten targets are gone and the program ends (Figure 8). Figure 8. Figure 5. When the busy indicator at the upper right of the screen stops, press ENTER to see your score (Figure 6). The display shows the number of hits and points for the current turn. It also shows the number of turns taken and the total points accumulated. (Practice does not count as a turn.) Figure 6.

13 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT TARGET RECORD PAPER H4.7

15 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT CLASS DATA RECORDING SHEET H4.8 Female Name Forearm Height Stride length (cm) length (cm) (cm) Male Name Forearm Height Stride length (cm) length (cm) (cm)

17 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT TI-83 INSTRUCTIONS: CALCULATING PREDICTED VALUES AND ERRORS These instructions will be useful for Lesson 2, Activity 6, Forearmed is Forewarned. Example: Find the errors using the linear model y = 2x + 1 for the data set in Figure 1. H4.9 Recall that prediction errors are defined as Yactual Ypredicted. 1.Enter the data in lists L1 and L2. List L2 contains the Yactual. Then enter y = 2x + 1 as Y1. Now, plot the data and graph the line. If you used ZOOM 9 to set the viewing window, your results should match the graph in Figure 2. 2.Next, use the linear model to store in L3 the predicted value of y for each of the values of x in L1. Here s how to do it: Press STAT ENTER to display the data table. Then use the arrow keys to move the cursor to highlight the header for L3. Press VARS and use the right cursor to highlight Y-VARS, and select Function. Select Y1. (This is where you stored y = 2x + 1, the prediction equation). Finally, substitute the values of L1 into your equation Y1: Press (2nd L1). At this point, your screen should look like Figure 3. Press ENTER and your predicted values will show up in L3. 3.To get the prediction errors of the y = 2x + 1 model, all you need to do is subtract the predicted values in L3 from the actual y-values in L2, that is, L3 L2. Here s how to do it: Use the cursor to highlight L4 above column 4. Press 2nd L2 2nd L3 ENTER The model s prediction errors are now stored in L4. Figure 1. Sample data. Figure 2. Scatter plot and graph of a line. Figure 3. Screen showing formula for L1.

19 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT WHAT S MY LINE? For use with Lesson 3, Individual Work 6. H4.10 ITEM 2 12 y x Figure Scatter plot of strong linear relationship. ITEM y x Figure Scatter plot of linear relationship. ITEM 4 12 y Line 1 Line 2 Line 3 x Figure Lines describing the data in a scatter plot.

21 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT SUPPLEMENTAL INSTRUCTIONS FOR USING EXCEL 4.0 ON MACINTOSH COMPUTERS FOR LESSON 3, ACTIVITY 8. H4.11 page 1 of 2 This worksheet gives additional instructions for Lesson 3, Activity 8. Follow the items and steps in your book and below. For Part II, see your text (pages 63 68) and the additional instructions below. a) See your text. b) See your text. c) Follow Handout H4.3 from Lesson 1, Activity 3, using the new data and labels. However, at Step 3, the highlighted columns should be A, B, and C. Be sure that you select the XY Scatter graph type and that you use column A as x-values for the plot. d) Absolute references in Excel (and many other spreadsheet application programs) are indicated by using a dollar sign ($) before the affected reference. To follow the example in your text, suppose you wish to multiply the value of D2 by 3, add 2, and have the result appear in cell C2. A relative reference formula in Excel would be =3*D2+2. (The equals sign tells Excel that this is a formula, not just a label.) If this formula were copied down from cell C2 to cell C3, it would read =3*D3+2, still referring to the cell to the right. An absolute reference formula in Excel would be =3*$D$2+2. If it were copied from cell C2 down to cell C3, it would remain unchanged. (There are two dollar signs because there are two directions involved in the reference. The $D fixes the column, and the $2 fixes the row. Since you will only be copying down in this activity, the $D is optional; the $2 is not. You might want to experiment with different combinations of dollar signs in formulas, but for this activity you can use the two-dollar-sign form of absolute references.) e) The formula for your line s prediction for the x in A2 is: =$D$2*(A2-$E$2)+$F$2. Explain the use (and absence) of dollar signs in each of the four references. Check that the resulting value agrees with what you get by hand and with Figure 4.43 before continuing. f) Excel provides two methods of getting the predicted-y-values formula from cell C2 to cells C3 C9. The first is called Fill Down. To do this, highlight cells C2 C9, then select Fill Down from the Edit menu. (Newer versions avoid the use of the menu: First click on C2. Highlight the cells C2 C9 by dragging the small dark square that appears at the lower-right corner of C2. Releasing the drag completes the Fill.) The second method is Copy and Paste. For that method, first

22 HANDOUT Unit 4: PREDICTION Mathematics: Modeling Our World H4.11 page 2 of 2 select only cell C2, then choose Copy from the Edit menu. Then highlight cells C3 C9 and choose Paste from Edit. Pressing the escape key completes the process. Do one of these procedures to obtain the remaining predicted y-values. g) To modify your graph so that the predicted values are displayed as a line instead of as isolated points, open the graph by double-clicking anywhere in the graph. Then open the series containing the predicted y-values by double-clicking on any of the points representing a predicted y-value. Set up the line display as Custom and select the style and color of line you want displayed. OK your selections, then return to the spreadsheet by closing the graph window. h) See your text. i) Remember that prediction errors are defined as actual minus predicted. j) See your text. k) See your text. l) See your text. m ) The command for the sum here is =sum(i2:i9). Note the use of the colon (:) to indicate a range of cells. n) See your text. If you are using an average, its formula is =Average(). Put the proper range of cells in the parentheses.

Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT EDITING A GRAPH S SERIES IN EXCEL H4.12 The columns of numbers you use to define a graph are called the graph s series.

23 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT EDITING A GRAPH S SERIES IN EXCEL H4.12 The columns of numbers you use to define a graph are called the graph s series. Each series needs to include all the numbers you wish to graph together, no more, no less. The columns in Lesson 3, Activity 8, Part III are longer than those in Part II. Here s how to change the graph set-up for Part III. First, open the graph by double-clicking anywhere on it. Next, select Edit Series from the Chart menu. You will see a dialog box similar to the one shown Figure 1. Figure 1. Sample Excel screen. Select the first series you wish to edit. In Figure 1, Series1 is selected. Click in the box to the right of X Values and drag to the right until you can see the end of the text in the box. Click at the end of the text to remove any unwanted highlighting, then edit the last number so that it includes the last row of data. For Part III, it should go through A11. Repeat these same steps to revise the Y Values box to go through B11. Click on Define to complete the revision of Series1. Repeat the same steps to revise Series2.

25 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT TI-83 INSTRUCTIONS FOR FINDING THE LEAST-SQUARES LINE H4.13 Here s how to use the TI-83 to calculate the least-squares line and its residuals. 1. Enter the explanatory variable values into L1 and the response variable values into L2. Then make a scatter plot of the data in an appropriate window. 2.To calculate the equation of the least-squares line, first press STAT, CALC, LinReg(ax + b). To complete the command, you need to tell the calculator which list contains the data for the explanatory variable, which list contains the data for the response variable, and where you want the least-squares equation stored. Complete the entry to match the one shown in Figure 1 and then press ENTER. Figure 1. Setting up linear regression. 3.Press Y=. Has the regression equation been stored as Y1? Press GRAPH to view the scatter plot with a graph of the least-squares line. 4. The TI-83 computes the residuals automatically and stores them in a list named RESID whenever you use a built-in regression command. To see the residuals displayed, place the cursor on the header of list L3, press INS, and select RESID from the LIST menu. Your screen should look like Figure 2. Figure 2. Entering the residuals. Press ENTER again and the residuals will appear. How many are positive and how many negative? What is the absolute value of the largest residual? How can you use the features on your calculator to find the sum of the residuals? (Think back to Lesson 2, Activity 5, I Predict That.)

27 Mathematics: Modeling Our World Unit 4: PREDICTION HANDOUT EXCEL 4.0 GUIDANCE FOR LESSON 3, ACTIVITY 9, ITEMS 1 4 H4.14 page 1 of 2 Complete Activity 9 in your text. Use these instructions to assist your work using Excel Data entry. a) Open your saved spreadsheet from Activity 8. b) Enter the powerboat and manatee deaths data in columns A and B. 2. Finding the least-squares equation. a) Clear the formulas for the predicted y-values by highlighting column C and choosing Clear from the Edit menu. b) Move the cursor to the cell in which you have the slope, D2. Enter the formula =SLOPE(B2:B15, A2:A15). Note that the y-values are listed first in the Excel formula. Press Enter to see the result. c) Move to an empty cell nearby, say, D4. Type a new label, y-int. Then move to D5 and enter the formula =INTERCEPT(B2:B15, A2:A15). Press ENTER for the result. (Figure 1.) Figure 1. Computing the intercept of the least-squares line in Excel 4.0.

28 HANDOUT Unit 4: PREDICTION Mathematics: Modeling Our World H4.14 page 2 of 2 3.Graphing the data and line. a) Modify the predicted y-values formula to reflect the changed equation, and fill down to match the data columns (Figure 2). Figure 2. Computing the predicted y-values in Excel 4.0. b) Revise the Series definitions so that all your data are included in the graph. c) Rescale the axes as you like by opening the graph (double-click on the graph), then opening the axes and selecting Scale to enter your modifications. 4. Assessing the model. a) The residuals are in the column labeled errors. Extend your formula down to reflect the number of rows in your data. b) Add a cell (maybe under the cell for SSE) to compute the sum of the residuals. Check the syntax used in computing the SSE in Activity 8, and be sure to include the full list of residuals in the sum. c) To graph the residuals, highlight only the two columns containing your x-values and your residuals; after highlighting the first of these columns, hold the CONTROL key to allow you to highlight the other column. Then use the Chart Wizard to create a new graph as you have in the past. You can answer Item 9 just by changing the data columns. No formulas need to be altered.

Microsoft Excel Using Excel in the Science Classroom

Microsoft Excel Using Excel in the Science Classroom OBJECTIVE Students will take data and use an Excel spreadsheet to manipulate the information. This will include creating graphs, manipulating data,