Session 5 Linear Functions and Slope

Transcription

1 Session 5 Linear Functions and Slope Key Terms for This Session Previously Introduced closed-form description [Session 2] recursive description [Session 2] origin [Session 4] New in This Session independent variable dependent variable slope rate linear relationship Introduction and Review In the previous session, you developed proportional reasoning skills by making absolute and relative comparisons, comparing ratios, making scale drawings, and looking at graphs of proportional relationships. In this session, we ll explore linear relationships by looking at lines and slopes. [SEE NOTE 1] We ll also use spreadsheets to aid us in our exploration. If you feel comfortable using a spreadsheet program, feel free to start the lesson now. If you ve never used a spreadsheet before, or if you have but would like a review of the basics, go to page 135 now for a short tutorial. Learning Objectives Try It Online! The spreadsheet tutorial is available online. Go to the Patterns, Functions, and Algebra Web site at and find Session 5, Part A, Spreadsheet Tutorial. In this session, we ll use spreadsheets to explore dynamic dependence and linear relationships. We will: Recognize linear relationships as they are expressed in tables, equations, and graphs Understand that linearity is expressed in constant differences Use spreadsheets to work with tables, equations, and graphs Explore the role of slope and dependent and independent variables in graphs of linear relationships Develop an understanding of rates and how they are related to slopes and equations NOTE 1. In this session, we ll explore the concepts of linear function and slope. Another goal is to examine how the use of computer spreadsheets can affect our understanding of these two ideas. In mathematics, technology is often introduced for its own sake, with little or no consideration for how it can enhance students understanding of mathematical concepts. In this session, spreadsheets will allow us to observe changes in dependent and independent variables for multiple data points. We ll also use spreadsheets to produce graphs that illustrate the connection between rate and slope. NOTE 1 cont d. next page Patterns, Functions, and Algebra 113 Session 5

2 Part A: Linear Relationships in Patterns (35 MINUTES) Finding the Pattern A function expresses a relationship between variables. For example, consider the number of toothpicks needed to make a row of squares. The number of toothpicks needed depends on the number of squares we want to make. If we call the number of toothpicks T and the number of squares S, we could say that T is a function of S. In this case, S is called the independent variable and T the dependent variable the value of T depends upon whatever we determine the value of S to be. [SEE NOTE 2] In this section, we ll explore the dependence of one variable on another with the help of a spreadsheet program. Make a row of squares using toothpicks. The squares are joined at the side. Problem A1. How many toothpicks are needed for 1 square? For 2 squares? For 5 squares? Make a table of these values. Squares Toothpicks NOTE 1, CONT D. To complete the session, you ll need to know how to perform a few basic procedures. If you re not already familiar with spreadsheets, you may want to read the spreadsheet tutorial in the course material. Groups: Those with more computer experience can provide a quick demonstration to others, or pair up with those who have less computer experience. Materials Needed: Computer with spreadsheet program for individuals working alone or for each pair or small group, graph paper, rulers (optional), toothpicks (optional) Review Groups: Discuss the homework, particularly Problem H3 and the differences between the proportional equations and the linear equations. NOTE 2. This section begins with a variation of the toothpick problem from Session 2, allowing us to explore the concept of linearity in a familiar context. Groups: It s likely that people will have varying degrees of comfort and familiarity with spreadsheets, so working in groups may be helpful.work on Problems A1-A3, completing the table and coming up with the rules.you may want to share rules before moving to the computers to work on the rest of Part A. Session Patterns, Functions, and Algebra

3 Part A, cont d. Problem A2. Develop a formula describing the number of toothpicks as a function of the number of squares. [SEE TIP A2, PAGE 139] VIDEO SEGMENT (approximate times: 3:09-4:30): You can find this segment on the session video approximately 3 minutes and 9 seconds after the Annenberg/CPB logo. Zero the counter on your VCR clock when you see the Annenberg/CPB logo. In this video segment, Gina explains her solution to Problem A2, including how she generated a rule for the number of toothpicks in each row of squares.watch the segment after you have completed Problem A2. If you get stuck on the problem, this segment may help you come up with a solution. Does Gina s work involve a closed-form description or a recursive description for the number of toothpicks in each new row? Refer to Session 2, Part E (see page 49) for more information on these two kinds of descriptions. Problem A3. As you add each new square to the row, how many toothpicks are added? This gives you a recursive rule for the number of toothpicks. Patterns, Functions, and Algebra 115 Session 5

4 Part A, cont d. Using a Spreadsheet If at any time you need help, refer to the spreadsheet tutorial, page 135. Problem A4. Open a spreadsheet on your computer and enter your table from Problem A1 into the first two columns. Each cell in the first column will contain the number of squares in a row. The cell next to it in the second column will contain the number of toothpicks needed to build that number of squares. [SEE NOTE 3] Problem A5. Use your spreadsheet program to create a graph of the numbers in these two columns. [SEE TIP A5, PAGE 139] VIDEO SEGMENT (approximate times: 7:23-8:07): You can find this segment on the session video approximately 7 minutes and 23 seconds after the Annenberg/CPB logo. Zero the counter on your VCR clock when you see the Annenberg/CPB logo. In this video segment, Professor Cossey asks whether or not the points on Problem A5 s graph should be connected with a line. Watch the segment after you ve completed Problem A5. In what situations should the points on a graph be connected? What does connecting two points on a line imply about the points in between? NOTE 3. The first use of the spreadsheet in Problem A4 requires only the entry of data.to draw the graph for Problem A5, select the data, and then select the graphing tool (this is called the Chart Wizard if you re using Excel). The type of graph or chart for representing a functional relationship is an XY graph, or scatterplot. Some of the scatterplots will display only points, like this: Others will connect the points with a line: Groups: After creating the graphs, pause to discuss which graph was chosen to represent the function. Some may say that the points should not be connected because, in this context, the function makes sense only for whole numbers. For example, it may not make sense to talk about how many squares 4.5 toothpicks can make. It is often useful, however, to connect the points on a graph like this so that we can see the shape of the function (in this case, a straight line) more clearly. This is often done by indicating the actual data points by large dots or diamonds and then connecting them with a narrower line. Session Patterns, Functions, and Algebra

5 Part A, cont d. Different types of function rules can be entered into a spreadsheet. [SEE NOTE 4] Example: Closed-Form Rule The formula y = 3x + 7 expresses the dependent variable y as a function of the independent variable x. Suppose that the values of x appear in the first column, beginning at cell A2, and the corresponding values of y are placed in the third column beginning with C2. The formula would first be entered in C2 as =3*A2+1, and then the fill down command would be used to complete the remaining values of y. Example: Recursive Rule Suppose you have a recursive rule that says The first output is 10; to get any other output, subtract 2 from the output before it. You would enter the first output into cell C2.Then you could enter the formula =C2-2 in C3, and use the fill down command for the remaining cells. Problem A6. Enter your formula from Problem A2 into the third column of your spreadsheet. Compare the values computed by the spreadsheet program using your formula with those you entered by hand. Are they the same? Problem A7. Enter your recursive rule (from Problem A3) into the fourth column of your spreadsheet. Are the values computed by the spreadsheet program the same as those you entered by hand? You can extend your formula to more values by filling down the output column. You will need to either enter the input numbers by hand or enter a formula that increases numbers in that column by one each time. See the spreadsheet tutorial for more information on how to do this. Problem A8. Use the spreadsheet program to create a graph of the input/output columns for either of your rules. NOTE 4. An important feature of spreadsheet programs is the ability to transfer formulas from one cell to another, updating cell references appropriately. This is called filling, and it is done most easily in Excel by selecting Fill and then Down from the Edit menu. Some may also know how to do this by dragging the formula from one cell to another.this method also works, but is hard to learn at first. Once all four columns are filled, the spreadsheet should look something like this: It s worth noting some pitfalls in working with spreadsheets: Forgetting the = at the beginning of a formula Beginning to type data before selecting a cell Expressing the cell coordinates in the wrong order the letter for the column must precede the number for the row Forgetting to use the * for multiplication End Part A by printing copies of the toothpick graphs or copying the graphs onto graph paper. Patterns, Functions, and Algebra 117 Session 5

6 Part B: Slope (40 MINUTES) Thinking About Slope Slope is an important concept in mathematics, and in Part B we ll explore how it is used to solve problems. [SEE NOTE 5] Write and Reflect Problem B1. Take a minute to think about what you already know about slope. What does it mean? Where is it used? You may be familiar with the idea of slope as a measure of steepness. The formula for slope is usually described as: slope = (change in y) / (change in x) The slope of a line is often described as a ratio of rise/run. Another way to think of slope is as the amount that the dependent variable changes for each increase by 1 in the independent variable. In other words, as x changes by 1, what happens to y? Look at the four graphs on page 119. For each graph, select four pairs of points, and calculate the slope of the line between each pair of points. Remember that slope = (change in y) / (change in x). As you calculate the slopes for each of the graphs, ask yourself why the slope between pairs of points would change or why it would stay the same. [SEE NOTE 6] Try It Online! These problems can be explored online as an Interactive Activity. Go to the Patterns, Functions, and Algebra Web site at and find Session 5, Part B, Thinking About Slope. NOTE 5. Groups: Discuss what everyone already knows about slope. Go over the definition of slope in the course materials, and then work on Problems B1-B4. Graph paper is needed for this exercise. NOTE 6. While working on this activity, choose different pairs of points. For each pair, it s important that the difference between the y coordinates and the difference between the x coordinates are computed by considering the pairs in the same order.that is, for points (x 1, y 1 ) and (x 2, y 2 ), the slope can be computed as either: (y 1 - y 2 ) / (x 1 - x 2 ) or (y 2 - y 1 ) / (x 2 - x 1 ). Note that students sometimes confuse this by computing (y 2 - y 1 ) / (x 1 - x 2 ). Session Patterns, Functions, and Algebra

7 Part B, cont d. Graph A Graph B Graph C Graph D Problem B2. What happened when you tried to find the ratio of rise/run for Graph D, a curved object? Patterns, Functions, and Algebra 119 Session 5

8 Part B, cont d. The drawing at left shows a cable attached to a wall. Problem B3. Calculate the ratio rise/run for each pair of points: [SEE NOTE 7] Points P and Q Points P and R Points Q and R Problem B4. Describe the difference between the rise/run ratios for the graph in Problem B3 and the ratios for the graph of a line. [SEE NOTE 8] Comparing Slopes Here and on the following pages are graphs of nine different lines. Look at the equation and the slope of each line. What changes when the slope becomes positive or negative, and when the slope is larger or smaller than 1? Try It Online! These problems can be explored online as an Interactive Activity. Go to the Patterns, Functions, and Algebra Web site at and find Session 5, Part B, Comparing Slopes. NOTE 7. For Problem B3, consider measuring directly on the picture or tracing the picture onto graph paper to compute the ratios. NOTE 8. Think about the main points for this part of the session: The rise/run for a line is the same between any pair of points on the line; lines are the only kind of graph for which this is true. Problem B3 is taken from IMPACT Mathematics Course 3, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p. 26. Session Patterns, Functions, and Algebra

9 Part B, cont d. Patterns, Functions, and Algebra 121 Session 5

10 Part B, cont d. Session Patterns, Functions, and Algebra

11 Part B, cont d. Problem B5. Consider this line. What are the coordinates of the points R and S? Problem B6. Find the slope of the line through points R and S. Problems B5 and B6 are taken from IMPACT Mathematics Course 3, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p. 27. Patterns, Functions, and Algebra 123 Session 5

12 Part B, cont d. Problem B7. Many people describe slope as a measure of the steepness of a line. Look at the two graphs below. Which line has larger slope? Which line appears to be steeper? Explain what is happening. [SEE NOTE 9] [SEE TIP B7, PAGE 139] VIDEO SEGMENT (approximate times: 11:32-14:21): You can find this segment on the session video approximately 11 minutes and 32 seconds after the Annenberg/CPB logo. Zero the counter on your VCR clock when you see the Annenberg/CPB logo. In this video segment, the onscreen participants discuss methods of comparing the slope of two lines. Watch this segment after you have completed Problem B7 and compare your methods of comparison with those of the onscreen participants. If two graphs are drawn with the same scale, is it true that the graph that appears steeper has the larger slope? Do the following problems on graph paper. Problem B8. On one graph, create three different lines with slope 2. Describe what is the same and what is different about these graphs. [SEE NOTE 10] Problem B9. On one graph, create three different lines with negative slope. Describe what is the same and what is different about these graphs. NOTE 9. Problem B7 focuses on the relationships between slope, scale, and the appearance of a line s graph. It s an important idea that slope only relates to steepness if you are comparing two lines with the same scale. NOTE 10. In Problem B8, notice that the lines are all parallel and therefore do not intersect. You may want to talk briefly about what you know about parallel lines, as we will return to this idea in future sessions. Session Patterns, Functions, and Algebra

13 Part B, cont d. Slopes and Architecture Architects and carpenters use rise and run to describe and build staircases. The picture below shows the rise and run for each step and the total rise and run for the staircase. Take It Further Problem B10. An architect is designing a staircase for a house with a difference of 10 feet between floors. The staircase has 18 steps and a total run of 14 feet. What is the ratio (total rise/total run) of the staircase? Problem B11. Find the rise and run for each step (what we ll call the step rise and run) in inches. What is the ratio (step rise/step run)? Explain your answer. Problems B10-B13 are taken from IMPACT Mathematics Course 3, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000). Patterns, Functions, and Algebra 125 Session 5

14 Part B, cont d. Problem B12. Find a nearby staircase and measure the rise and run of one step. How could you use this to estimate the total rise and run of the staircase? Problem B13. Design a staircase with a total rise of 14 feet, a step rise between 6 and 8 inches, and a sum of step rise and step run between 17 and 18 inches. All steps should have the same rise and the same run. Your answer should include: a. the number of steps b. the height of each step c. the run of each step d. the total run e. the ratio (step rise/step run) f. the ratio (total rise/total run) for the staircase. [SEE TIP B13, PAGE 139] VIDEO SEGMENT (approximate times: 21:08-25:30): You can find this segment on the session video approximately 21 minutes and 8 seconds after the Annenberg/CPB logo. Zero the counter on your VCR clock when you see the Annenberg/CPB logo. In this video segment, taken from the real world example at the end of the Session 5 video, master carpenter Norm Abram describes the importance of slope in construction. Watch this segment after completing Part B. In what other professions is slope used on a regular basis? Session Patterns, Functions, and Algebra

15 Part C: Rates (40 MINUTES) A rate describes how much one variable changes with respect to another. Rates are often used to describe relationships between time and distance. When an object or person moves at a constant rate, the relationship between distance and time is linear. [SEE NOTE 11] Problem C1. Achilles runs at a constant rate of 9 miles per hour. [SEE NOTE 12] a. Write an equation describing the relationship between the distance Achilles covers and the time he runs. b. How far will Achilles travel in 1.5 hours? c. If you graphed the relationship between the distance Achilles covers and the time he runs, what would the graph look like? d. Enter your equation in a spreadsheet and use the spreadsheet to draw the graph. Is the graph what you expected? Explain why or why not. [SEE TIP C1, PAGE 139] NOTE 11. The exercises in this part of the session link the concept of direct variation, covered in Session 4, to the idea of problem solving, which will be covered in Session 6. NOTE 12. Groups: Work in pairs on Problems C1 and C2. Consider sketching the situations or drawing tables to help come up with the equations for Achilles and the tortoise. After generating graphs for these situations, compare the graphs with the toothpick graph developed in Part A. In the case of Achilles and the tortoise, non-integer values make sense, so there is no problem with drawing connected lines between points. The graph should look like this: Patterns, Functions, and Algebra 127 Session 5

16 Part C, cont d. Problem C2. Achilles is going to race against a tortoise, who moves at only 1 mile per hour. To make the race fair, the tortoise gets a head start of 32 miles. a. Write an equation describing the relationship between distance and time for the tortoise. b. Enter the tortoise s equation into a spreadsheet. c. How long will it take for Achilles to catch up to the tortoise? VIDEO SEGMENT (approximate times: 17:56-19:07): You can find this segment on the session video approximately 17 minutes and 56 seconds after the Annenberg/CPB logo. Zero the counter on your VCR clock when you see the Annenberg/CPB logo. In this video segment, participants use a spreadsheet program to answer Problem C2. Watch this segment after you have completed Problem C2. If you get stuck on the problem, you can watch the video segment to help you. Could the participants have answered Problem C2 using a recursive rule? Problem C3. Make a single graph that shows the progress of Achilles and the tortoise. Where do the two lines cross? Session Patterns, Functions, and Algebra

17 Part C, cont d. Problem C4. What is the relationship between the points where the lines cross and Achilles passing the tortoise? Problem C5. Which of the two lines in Problem C3 represents a proportional relationship? How do you know? [SEE TIP C5, PAGE 139] Problem C6. Suppose that two people were traveling a distance of 100 miles at the same speed, and the first person got a head start of 25 miles.when would you expect them to be at the same point? What does this tell you about their distance graphs? [SEE NOTE 13] NOTE 13. Problem C6 addresses the idea of parallel lines.you might want to refer to the graphs drawn in Part B. Ask if lines with the same slope ever intersect. Students and teachers often miss the essential connection between solving linear equations and finding the intersection of lines. The intersection of two lines happens at an (x, y) pair that satisfies both linear equations. At this point in each equation, the xs are equal and the ys are equal. To find where this happens, set the ys equal to each other and solve each equation for x. When trying to solve an equation like 5x = 5x + 25 by the usual method of doing the same thing to both sides, students end up with the equation 0 = 25 and don t know how to interpret it.the point is that the two lines are parallel, so they never intersect, which means there is no solution to the equation. And the equation 0 = 25 is never true. Solving equations is covered more completely in the next session, but if there is time, think about some of these ideas now. Patterns, Functions, and Algebra 129 Session 5

18 Part D: Putting It Together (35 MINUTES) Linear functions have come up in many situations so far in this session. In this section, we ll consolidate some of our ideas about linear functions and look for connections between them. [SEE NOTE 14] Below are several input/output tables. [SEE NOTE 15] For each table: Find a closed form rule for taking an input and finding the correct output. Find a recursive rule for going from one output to the next. Graph the pairs of numbers in the table. Determine if the rule describes a linear function or not. If it is a linear function, find the slope. Problem D1. Input Output NOTE 14. The point of this section is to make connections between the different situations in which we ve seen linear functions. NOTE 15. Take 15 to 20 minutes to work on Problems D1-D7.Then think about the connections in the different representations, paying particular attention to instances where linear functions are different from other kinds of functions. In Problem D6, the recursive rule for the function y = 1/x can be quite challenging. The easiest way to describe it is to use your input in the rule: Some people consider a rule recursive only if it truly depends on previous outputs, with no reference to the input. In summary: Closed forms for linear functions look like y = ax + b, where a and b are some numbers, x is the independent variable, and y is the dependent variable. Recursive rules for linear functions add a constant value from one output to the next. This constant is the same as the value of a in the formula y = ax + b. Graphs of linear functions look like lines. The slopes of the lines are the same as the difference between successive outputs, and the same as the value of a in the y = ax + b formula. Groups: Discuss the statements above. Session Patterns, Functions, and Algebra

19 Part D, cont d. Problem D2. Input Output [SEE TIP D2, PAGE 139] Problem D3. Input Output Problem D4. Input Output Patterns, Functions, and Algebra 131 Session 5

20 Part D, cont d. Problem D5. Input Output Problem D6. Input Output /2 3 1/3 4 1/4 5 1/5 Problem D7. Input Output Session Patterns, Functions, and Algebra

21 Part D, cont d. Write and Reflect Problem D8. What are the characteristics of a linear function? How can you tell that a function is linear if you are given: a closed-form rule for a function? a recursive rule for a function? a description of a situation? a table? a graph? Part E. Thinking About Technology (10 MINUTES) In mathematics, technology can be used to learn new concepts, apply known mathematics, avoid tedious computation, and gather information. It can also be exciting to learn new technology in and of itself. In this session, you used spreadsheet technology to look at the concept of linearity. [SEE NOTE 16] Write and Reflect Problem E1. How has the use of technology in this session affected your approach to the material? In what ways did it help? Are there ways in which it may have hindered your learning process? Problem E2. Did the technology help you think mathematically? NOTE 16. Leave some time to think about technology. As technology is introduced in classrooms, it is important to reflect on how it can facilitate or interfere with the learning process. We ve all heard the horror stories about the college freshman multiplying a number by 10 on a calculator or the clerk who can t make change without the help of the cash register! Patterns, Functions, and Algebra 133 Session 5

22 Homework Problem H1. In the Achilles and the tortoise problems in Part C, Achilles runs at a constant rate of 9 miles per hour, and the tortoise moves at 1 mile per hour. Suppose that the speeds of Achilles and the tortoise are unchanged but Achilles catches up to the tortoise in 1.5 hours. How much of a head start did the tortoise get? [SEE TIP H1, PAGE 139] Problem H2. The tortoise has taken some turtle speedup potion and can now walk at 2 miles per hour. If Achilles still runs at 9 miles per hour and catches up to the tortoise in 3 hours, how much of a head start did the tortoise get? Problem H3. Here s a trick that master carpenter Norm Abram might use when building supports for roofs. He knows he ll need evenly spaced supports along the roof. He carefully measures what length he needs for the first one, and finds that it s 12 feet.then he measures what he ll need for the second, and finds it is 9 feet. He calls to his assistant: Don t measure the others, just make them 6 and 3 feet long! Why does Norm s trick work? Problem H4. You ve worked with undoing functions. Take a moment to think about undoing a linear function. If given the formula d = 3t + 2 for distance traveled in terms of time, what would you do to express time in terms of distance? When undoing a linear function, will the result always be a new function? If so, will the new function always be a linear function? Session Patterns, Functions, and Algebra

23 Spreadsheet Tutorial Because of technology, spreadsheets have progressed from a handwritten accounting practice to a powerful tool in mathematics and modeling. A spreadsheet is a grid of cells, where each cell has a column label and a row label. This section is designed for those who don t have much, or any, experience using spreadsheets. If you are familiar with spreadsheets and spreadsheet software, please proceed with the session. Cells For the purposes of this tutorial, the column heading for a cell will be lettered, starting with A, and the row heading for a cell will be numbered, starting with 1. For example, the cell in the second column and third row would be cell B3. It is a common error to reverse these labels. While a cell can only contain one piece of information, there are several types of information it can hold. To enter information into a cell, simply select that cell by clicking on it. A box will surround the selected cell at any time, and you can move to any surrounding cell by using the keyboard s arrow keys. A cell can contain text, such as a word or a title, or numbers in many different formats (whole number, decimal, dollars and cents, among others). A cell can also contain a mathematical computation; this is typically done by placing an equal sign ( = ) in the cell followed by the computation, like this: When performing computations, the equal sign is very important; without it, a spreadsheet program may interpret the computation as text. Computers use certain characters for each operation, as follows. Addition: use +. Typing =6+7 in a cell will display 13. Subtraction: use -. Typing =8-3 in a cell will display 5. Multiplication: use * (shift-8).typing =5*6 will display 30. Do not use the letter x for multiplication on a spreadsheet. Division: use / (forward slash, to right of period). Typing =12/3 will display 4. Exponentiation: use ^ (shift-6). Typing =4^3 will display 64. Important: Do not type the quotation marks used in the examples! They are not part of any computation or formula. If you type quotation marks, the spreadsheet program will automatically interpret what you type as text, and will not perform any calculations. Even though the result of the calculation is displayed, the spreadsheet remembers the details of the calculation. If you click on a cell, you can see the details of the calculation on screen. Patterns, Functions, and Algebra 135 Session 5: Spreadsheet Tutorial

24 Spreadsheet Tutorial, cont d. Formulas The power of a spreadsheet lies in its use of formulas. Each cell in a spreadsheet is able to contain its own formula. A formula in one cell can perform a calculation using values placed or calculated in other cells. A demonstration here will be helpful in understanding this. Suppose you wanted to make a list of consecutive numbers. You could simply enter the value of each number in the cells, or you could create a formula to accomplish the same goal. First, type the value 1 in cell A1. In cell A2, type =A1+1 instead of typing the value. The use of A1 is a reference to cell A1, and the calculation will use whatever value was stored in cell A1. Since that value was 1, cell A2 will display 2. This can be continued in any other cell of the spreadsheet. Continue the list by entering into cell A3 =A2+1, and entering into cell A4 =A3+1. You should now see the numbers 1, 2, 3, and 4 listed in these cells. Dynamic Change Most spreadsheet programs include a feature that updates all values in the spreadsheet whenever one is changed. In the last example, cell A2 was defined to be 1 more than cell A1, cell A3 was defined as 1 more than cell A2, and cell A4 was defined as 1 more than cell A3. Change the value in cell A1 to 23 and observe what happens. Once a formula is entered, many values can be tested easily and quickly using this feature. Users can create a spreadsheet to model population growth, then instantly judge the long-term effects of a change. At home, this feature allows quick updating to a budget or a retirement plan. In many professions, the ability to use spreadsheets has become as important as the ability to use word processing software. Filling Cells With Formulas In the above example, you typed the same kind of formula several times. Cell A2: =A1+1 Cell A3: =A2+1 Cell A4: =A3+1 All of these formulas can be interpreted as Take the value in the cell above, and add 1. Nearly all spreadsheet programs allow the user to fill a formula over a group of cells, and the formula will adjust to the new location. To do this, the user must first highlight (select) a group of cells. Session 5: Spreadsheet Tutorial 136 Patterns, Functions, and Algebra

25 Spreadsheet Tutorial, cont d. Click on cell A4 and hold down the mouse button. Now, drag the mouse down to cell A10.The cells from A5 to A10 should become darkened, while cell A4 remains selected. Now, find the command Fill Down and select it. In a menu-driven spreadsheet, the Fill command is probably located in the Edit menu. The pattern you started in cells A1 through A4 should be continued through cell A10. More importantly, the formulas used in cells A2 through A4 should be continued, so that changing the value in cell A1 will change the entire spreadsheet. Change cell A1 back to the number 1 before proceeding. Closed-Form and Recursive Rules In Session 2, Part A (see page 36), we first encountered tables which could be described by several different rules. A closed-form rule is a rule that describes how to take an input and directly determine an output. For example, the rule take the input, multiply by 4, then add 2 is a closed-form rule. In a spreadsheet, a closed-form rule can be expressed in two columns: the left-hand column of inputs, and the right-hand column of outputs. The rule just described could be expressed in a spreadsheet by entering the following formula into cell B1: =A1*4+2 This formula takes the value in A1, multiplies it by 4, and then adds 2.To produce the information in other cells, fill down the formula from B1 to B10. In B2 you should have =A2*4+2, and so on. A recursive rule is a rule that describes how to proceed from one input to the next. A recursive rule for the table of outputs we just created might be The first output is 6; to get any other output, take the last output and add 4. Think about how you could enter this rule into a spreadsheet in column C before reading the solution tip below. Tip: Enter the number 6 in cell C1, then enter =C1+4 in cell C2. Finally, fill down the formula from cell C2 to C10. Important: Do not fill down from cell C1, since it does not contain a formula! If you filled down from cell C1, each entry in the column would be 6. Closed-form and recursive rules can work in tandem. There are many situations where one type of rule more easily describes a situation. We will be using both types of rules, so make sure you are comfortable using them both. Patterns, Functions, and Algebra 137 Session 5: Spreadsheet Tutorial

26 Spreadsheet Tutorial, cont d. Graphs While different spreadsheets create charts and graphs in their own way, most have an automatic graphing tool included. To create a graph, first highlight the cells you wish to use in the graph. On your spreadsheet, highlight cells A1 through B10 by clicking on A1, holding down the mouse button while you move the mouse, and releasing on cell B10. Then find the graphing tool, which may be an icon at the top of the screen that looks like a chart. Selecting this will bring up a window asking what type of chart you would like to use; this choice will depend on the situation, but for this example, select an XY (Scatter), with data points connected by lines. Be sure to select a graph that places the first column of information on the horizontal axis and the second column on the vertical axis. If you have done this correctly, the graph should be a straight line containing 10 points. Most graphing tools allow you to preview the graph before it is finalized, and you may need to try several options before the graph looks the way you want it to. If you have trouble with this feature, you should consult the help function or user manuals for your spreadsheet program. Session 5: Spreadsheet Tutorial 138 Patterns, Functions, and Algebra

27 Tips Part A: Linear Relationships in Patterns TIP A2. As in Session 2 (see page 41), try to develop the formula based on the context of the toothpick squares. TIP A5. If you need more help with this, see the portion of the spreadsheet tutorial on graphs, page 138. Part B: Slope TIP B7. Is there anything about the graphs that makes comparison more difficult? TIP B13. There are many different solutions to this problem. Remember that the total of the step rise and the step run must be between 17 and 18 inches. Part C: Rates TIP C1. Would a closed-form or a recursive rule be easier to work with in this situation? TIP C5. If you need more help with proportional relationships, refer to Session 4, Part C, page 100. Part D: Putting It Together TIP D2. The recursive rule here includes n, the input number. Homework TIP H1. Using a spreadsheet can help solve this problem. Patterns, Functions, and Algebra 139 Session 5: Tips

28 Solutions Part A: Linear Relationships in Patterns Problem A1. The 1st square requires 4 toothpicks. The 2nd square requires 3 more, bringing the total to 7. Each additional square requires 3 more toothpicks. Here is the completed table: Squares Toothpicks Problem A2. The formula is T = 3S + 1. Problem A3. Three toothpicks are added each time, so a recursive description is that the nth square will require 3 more toothpicks than the one before it. A formula would be written T n = T n Problem A4. Problem A5. Problem A6. If both rules describe the pattern correctly, the values computed by the spreadsheet should be the same as the values in the table. Note that the spreadsheet information in the columns is not the same; the third column contains formulas, and the screen shows the result of these formulas. Problem A7. Again, the values should be the same, even though the formulas are not. Problem A8. This should be the same graph as the one you found in Problem A5. (See above.) Session 5: Solutions 140 Patterns, Functions, and Algebra

29 Solutions, cont d. Part B: Slope Problem B1. On a basic level, slope measures vertical change over some horizontal distance a road can slope up or slope down. The slope of a line describes how much vertical change (change in y) there is per horizontal change (change in x). Problem B2. On a line, the ratio of rise to run is always constant; on a curved object, this value is constantly changing. So, the slope of a curved object changes depending on the points selected, while the slope of a line is always constant. Problem B3. From P to Q, the ratio is roughly 0.6. From P to R, the ratio is roughly 1.3. From Q to R, the ratio is roughly 4. Problem B4. The ratios for the graph of a line would be constant throughout. For example, a line connecting P to R would have a rise/run ratio of 1.3, regardless of where a new point was located on it. This is different from the graph used in Problem B3, which has a slope that varies depending on which two points are used. Problem B5. The coordinates are R(1, 6) and S(4, 12). Problem B6. The slope is (change in y) / (change in x). The change in y is 12-6 = 6, and the change in x is 4-1 = 3.Therefore, the slope is 6 / 3 = 2. Note that the line may not appear to have slope 2, since the vertical axis is labeled by twos, while the horizontal axis is labeled by ones. Problem B7. Graph A has a larger slope, even though graph B appears steeper. This appearance is caused by the different scale used in the two graphs. If it were placed on the other scale, the line in graph B would appear nearly flat! Problem B8. All the lines have the same slope at all points. None of the lines intersect, and they are always the same distance from each other they are all parallel lines. Problem B9. A line with negative slope travels down and to the right ( decreasing ). Lines with negative slope may still intersect one another. Any line with negative slope will travel through the top left and bottom right quadrants of the graph paper. Problem B10. The total rise / total run is 10 / 14, which is roughly Problem B11. To calculate step rise and run, divide both rise and run by 18.The step rise is 10 / 18 feet (6 2 / 3 inches) and the step run is 14 / 18 feet (9 1 / 3 inches). The ratio of step rise / step run is (10 / 18) / (14 / 18), which again equals The step rise and step run are in proportion to the total rise and total run. Problem B12. The total rise and run can be estimated accurately by multiplying the rise and run of a single step by the total number of steps, because each step has roughly the same rise and run. Problem B13. There are a lot of answers here! Number of steps: between 21 and 28 Height of each step: between 6 and 8 inches Run of each step: between 9 and 12 inches Total run: between 189 and 336 inches Ratio of step rise to step run: between 6/12 and 8/9 Ratio of total rise to total run: between 6/12 and 8/9 Regardless of what you did, the ratios of step rise to step run and total rise to total run should be identical. Patterns, Functions, and Algebra 141 Session 5: Solutions

30 Solutions, cont d. Part C: Rates Problem C1. a. The equation A = 9h describes the distance in miles that Achilles travels, in terms of time measured in hours. A stands for the distance run by Achilles, while h stands for hours. b. For this problem, h = 1 1 / 2, so A = 9 (1 1 / 2 ) = 13 1 / 2 miles. c. The graph should be a line through the origin, since the relationship is a proportional one. d. The graph is a line through the origin. You should find that the line has a constant slope, or rate of change, of 9 miles per hour. Problem C2. a. The equation is T = h There is an invisible 1 in front of h, since the tortoise runs at 1 mile per hour. b. As in Problem C1, you should find a constant rate; this time, the rate is 1 mile per hour. c. A comparison of the spreadsheets finds that after exactly 4 hours, both Achilles and the tortoise are 36 miles from the start. Problem C3. According to the graphs, the intersection of the two lines occurs at the point (4, 36). Note that the independent variable (on the horizontal axis) represents time, and the dependent variable (on the vertical axis) represents distance. Session 5: Solutions 142 Patterns, Functions, and Algebra

31 Solutions, cont d. Problem C4. At the time h = 4, the distance for both Achilles and the tortoise is 36 miles. Since the graph represents both travelers positions over time, the two are at the same point at h = 4. It s at this time that Achilles overtakes the tortoise. Problem C5. Achilles graph is the proportional relationship because it is the graph of a line passing through (0, 0). Problem C6. Because the two people are traveling at the same speed, the person with the 25-mile lead will keep that lead, at exactly 25 miles, for the entire race. The two distance graphs will never intersect, no matter how long the race is, which suggests that the graphs will be parallel. So, linear graphs with the same rate of change will be parallel. Part D: Putting It Together In the following problems, solutions refer to the input variable as x, and the output variable as y. Problem D1. The closed-form rule is y = x - 1. The recursive rule is y n = y n-1 + 1, since the outputs grow by 1 each time. This is a linear function, according to its graph, and the slope is 1. Problem D2. The closed-form rule is y = x 2. The recursive rule is harder to formulate for this one: it is y n = y n-1 + (2n - 1). The key here is finding the pattern in the differences between each term. This is not a linear function. Problem D3. The closed-form rule is y = 2x + 1. The recursive rule is y n = y n Outputs grow by 2 each time. This is a linear function, and the slope is 2. Problem D4. The closed-form rule is y = -x + 10, or y = 10 - x (both are the same). The recursive rule is y n = y n-1-1. Outputs drop by 1 each time. This is a linear function, and the slope is -1. Problem D5. The closed-form rule is y = 5x. The recursive rule is y n = y n Outputs grow by 5 each time. This is a linear function, and the slope is 5. Problem D6. The closed-form rule is y = 1 / x. The recursive rule is very difficult. Two possible answers are: 1 / y n = 1 / y n-1 + 1, and y n = y n / (n)(n - 1). This is not a linear function. Notice that the rate of change is not constant. Problem D7. The closed-form rule is y = -7. The recursive rule is y n = y n-1 because every term is the same as the last. This is a linear function, according to the graph, and the slope is 0 (which means it is a horizontal line). Problem D8. If there is a closed-form rule for a function, and the function is linear, it will be in the form y = Mx + B, where M and B can be any real number positive, negative, or 0. Note Problems D5 and D7, in which one of the two values is 0. If there is a recursive rule given, it should be in the form y n = y n-1 + M, where M is the slope of the line. If a situation is described, it should involve a constant rate of change, such as a constant speed of a car, the constant slope of a ramp, or the constant price of gasoline per gallon. Patterns, Functions, and Algebra 143 Session 5: Solutions

32 Solutions, cont d. If a table is given, the rate of change (change in output, divided by change in input) should always be the same number. If inputs are a sequence of numbers (like 1, 2, 3, 4, 5), the outputs should also form a sequence (3, 5, 7, 9, 11; 5, 10, 15, 20, 25). If a graph is given, it should be a straight line (a linear function). Homework Problem H1. Earlier we found that Achilles ran 13 1 / 2 miles in 1 1 / 2 hours. Therefore, we have to find a way to get the tortoise to the 13 1 / 2 -mile mark after 1 1 / 2 hours.the tortoise walks at 1 mile per hour, so it can walk 1 1 / 2 miles in that time. The remaining distance must be its head start: 13 1 / / 2 = 12 miles. Problem H2. Use the same logic as in Problem H1. Achilles runs 27 miles in the 3 hours, therefore the tortoise needs a head start that will get it to the 27-mile mark after 3 hours. Because it walks at 2 miles per hour, it can walk 6 miles in 3 hours, so the head start is 27-6 = 21 miles. An algebraic equation could also be used for this problem. Problem H3. It works because the roof is a straight line, and therefore it has a constant rate of change. By carefully measuring the 1st and 2nd support, the carpenter has calculated a rate of change: (change in height of support) / (distance between supports). Since this rate is constant, and the distance between supports stays the same, the change in the support s height must also be constant. This is identical to predicting the next number in the output of a linear function; in this case, the output drops by 3 for every new support. Problem H4. It can be done by solving the algebra equation for the other variable, using the technique of undoing that was first used in Session 3. For the equation d = 3t + 2, start by subtracting 2 from each side to produce d - 2 = 3t. Then divide both sides by 3, so that the equation is (d - 2) / 3 = t. If a linear function can be undone, the result will always be a new, linear function. The only linear functions which cannot be undone are constant functions like y = -7. See Session 5, Problem D7 (page 132) and Session 3, Problems E9-E12 (page 75). Session 5: Solutions 144 Patterns, Functions, and Algebra