week 10

2. Study Graphs A, B, C, and D (below).  Each graph shows the hare's movement in subsequent races. 

Graph D  Where does the hare start the race in relation to the starting line?  Finish the race in relation to the starting line? The hare starts the race at the end of the line and finishes at the beginning.  Write a story about the race the Hare ran. How fast is the hare moving on each time interval compared to his speed on the other time intervals? What direction is the Hare moving (toward the starting line? Away from the starting line? Staying still?) The hare moves in the direction to the beginning with constant speed, then moves three times slower for the same distance, and then moves with twice the speed of the first time interval to the beginning of line. During the race between the tortoise and the hare, the fox recorded the speed at which both the tortoise and the hare were traveling.  The fox observed that the tortoise plodded along at a rate of 20 meters per minute throughout the 1000-meter race. 3. a. Complete the table below indicating the distance traveled by the tortoise in the time elapsed. Time elapsed in minutes, t Distance traveled in meters, D 0 0 1 20 2 40 3 60 4 80 5 100 6 120 7 140

b. What does this number represent in terms of the tortoise's movement during the race? This is speed c. How can you find the distance the tortoise traveled directly from the number of minutes that have elapsed since the start of the race? d. Let t represent the elapsed time in minutes and D represent the distance in meters traveled by the tortoise.  Write an equation showing the relationship between t and D.   D= 20t___________________ e. Replace t in the equation in problem 6d with some values in the table.  What values did you get for D from the equation?  f. Did the equation values for D match those in your table? Yes g. Should they match? Yes h. How does your answer to problem 6a relate to the equation you found in problem 6d? What is the connection between how far the tortoise can travel in one minute and

, yes the hare would have won e. How far could the hare have traveled if he continued at this rate for the entire 50.5 minutes? f. Find an equation relating the hare's elapsed time and distance traveled for 2 minutes to 49.5 minutes.  Keep in mind that he is not sitting on the starting line at the beginning of this time interval but he is sitting still. g. What is the slope of the equation you found in problem 7b?  Why does this slope make sense? The slope is 0. The object is not moving, and the coordinate does not change. h. What is the slope of the line the shows the hare's movement during the late minute of the race? The slope is 500, since the speed is 500 8. a. Why do you think the examples in this assignment are called linear relationships? They are represented by line segments on graph b. Think about the examples above, what are some general properties of linear relationships? The slope of the line is constant. For the equal periods of time the coordinate changes equally.

b. The hare ran at a steady rate of 250 meters per minute throughout the race.  He stopped only after he crossed the Finish Line.  On the same graph as in 9a, draw a graph of the hare's distance from the starting line over the time it took him to complete the race. Include that picture below:

f. Who would win the race?  Explain. 

c. How are the graphs related? The graphs are parallel 11. Study the table and the graphs. a. For each graph and corresponding data set, find an equation to model the graph and data. Equation for the tortoise starting at the starting line: Equation for the tortoise starting 500 meters from the starting line: Equation for the tortoise starting 100 meters before the finish line: b. What do the equations have in common?  Explain. They have common slope which is the speed 20 m/s c. What is different in each of the equations?  The y-intercept, which is the starting position

d. Why? By the condition, the tortoise starts in different positions e. How does Tortoise's head start show up in each equation? It is y-intercept f. Why does this make sense? This is the position on start, when t=0 g. You have studied the idea of a "head start" in linear relationships in previous mathematics classes.  What is the mathematical name for the "head start" value in a linear equation? y-intercept 12. a. Study the table. (Don't complete the tables yet. You’ll be given tables to complete later.) Table A Table B x y Another way to write y x y Another way to write y 0 0 0 0 5 5 1 3 0 + 3 or 0 + 3( 1 ) 1 8 5 + 3 or 5 + 3( 1 ) 2 6 0 + 3 + 3 or 0 + 3( 2 ) 2 11 5 + 3 or 5 + 3( 2 ) 3 3 4 4

5 5 6 6 27 27 108 108 b. What patterns do you see in what is in Table A?  3 is added to the previous value c. What patterns do you see in what is in Table B? 3 is added to the previous value  d. How are the tables similar?  Explain. The same value is added e. How are the tables different?  Explain. The starting value is different f. Graph the data in both tables on the same pair of axes.  Include the picture of the graph.

g. What do you notice? 

The lines are parallel h. Does your observation seem reasonable?  Yes i. Explain. The same value is added, which means slopes are the same 13. a. In the column labeled  Another way to write y,  show how the y-values in the table build from the y-value when x = 0.  This is the starting y-value.  Do not simplify at any step.  Continue extending the table.  Pay attention to the x-values. Table A Table B x y Another way to write y x y Another way to write y 0 0 0 0 5 5 1 3 0 + 3 or 0 + 3( 1 ) 1 8 5 + 3 or 5 + 3( 1 ) 2 6 0 + 3 + 3 or 0 + 3( 2 ) 2 11 5 + 3 or 5 + 3( 2 ) 3 9 0+3+3+3 or 0+3(3) 3 14 5+3+3+3 or 5+3(3) 4 12 0+3+3+3+3 or 0+4(3) 4 17 5+3+3+3+3 or 5+4(3) 5 15 0+3+3+3+3+3 or 0+5(3) 5 20 5+3+3+3+3+3 or 5+5(3) 6 18 0+3+3+3+3+3+3 or 0+6(3) 6 23 5+3+3+3+3+3+3 or 5+6(3) 27 81 0+3+...+3 or 0+27(3) 27 32 5+3+...+3 or 5+27(3) 108 324 0+3+...+3 or 0+108(3) 108 113 5+3+...+3 or 5+108(3) b. What patterns are evident from your work? Table A: Add 3 the number of times in cell x

Table B: Add 3 the number of times in cell x and then add 5 c. Write an equation in the form y = mx + b for each table. Table A:  Table B: d. How does patterns you noticed in the tables relate to the equation? Add 3 x times means 3x 14a. Complete the Tables C and D below. Find an equation for each table. In the column, Another way to write y,  show how the y-values in the table build from the y- value when x = 0.   This is the starting y-value.  Note how Table D doesn’t start with x = 0. How would you find the y-value when x = 0 in this case? Do not simplify at any step.  Continue extending the table.  Pay attention to the x-values. Table C Table D X y Another way to write y x y Another way to write y 0 9 9-0(2) 1 6 4+1(2) 1 7 9-1(2) 2 8 4+2(2)

2 5 9-2(2) 3 10 4+3(2) 3 3 9-3(2) 4 12 4+4(2) 4 1 9-4(2) 5 14 4+5(2) 5 -1 9-5(2) 6 16 4+6(2) 6 -3 9-6(2) 7 18 4+7(2) Equation for Table C: Equation for Table D: b. Are you tired of filling out the column  Another way to write y?  Now that you've experienced the constant growth of a linear relationship, you can simplify the process.  Study Table E.  Notice how the process below simplifies the work you did in problems 1a and 6.  In this process you find the difference between two consecutive y-values instead of writing each y-value in terms of the starting y-value.  Table E Table F x Y Change in y X y Change in y 0 7 0 15 4 -3 1 11 1 12 4 -3

2 15 2 9 4 -3 3 19 3 6 4 -3 4 23 4 3 4 -3 5 27 5 0 4 -3 6 31 6 -3 c. Find an equation to fit the Table E.  How do you know your equation is correct? Change in y defines the slope, starting point defines the y-itercept d. Find an equation to fit the Table F.  How do you know your equation is correct? Change in y defines the slope, starting point defines the y-itercept 15. Reflections: Below are more graphs showing the hare running a race. The y-axis is the hare’s distance from the starting line. The x-axis is time.

Graph A How would the hare need to move in relation to the starting line to create this graph?  Is the graph possible to create in real life? This is movement with constant speed from the beginning to the end. It is possible

Graph B How would the hare need to move in relation to the starting line to create this graph?  Is the graph possible to create in real life? This is movement with constant speed from the end to the beginning. It is possible Graph C How would the hare need to move in relation to the starting line to create this graph?  Is the graph possible to create in real life? It does not move. It is possible. Graph D How would the hare need to move in relation to the starting line to create this graph?  Is the graph possible to create in real life? It moves from the beginning towards end and then returns back with slightly higher speed. It is possible. Graph E How would the hare need to move in relation to the starting line to create this graph?  Is the graph possible to create in real life? It moves from the end towards beginning and then returns back with slightly lower speed. It is possible. Graph F

How would the hare need to move in relation to the starting line to create this graph?  Is the graph possible to create in real life? It moves from the beginning towards end, then stays, and then moves in the same direction with slightly lower speed. It is possible. Graph G How would the hare need to move in relation to the starting line to create this graph?  Is the graph possible to create in real life? It stays at the start, then moves forward, then stays again, and then return back with slightly higher speed. It is possible. Graph H How would the hare need to move in relation to the starting line to create this graph?  Is the graph possible to create in real life? It stays, then moves forward in no time, the stays, moves forward in no time, repeats three more times. It is impossible. Write your answers to the above questions in one file (recommended .doc/.docx or PDF; not pages).   Make your work neat and easy to follow.   Answer any questions that require written work with complete sentences.   Include pictures or illustrations where requested.