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1 Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.2 TASK 2.13.2: FIND IT YET? Solutions Suppose we are given two points in a plane, A = (a, b) and B = (c, d). 1. By changing the first coordinate in point A from “a” to “t”, locate a new point C = (t, b) such that the slope of the line through the new point and B is half that of the line through A and B. What do you notice about the difference in the first coordinates of C and B and the difference in the first coordinates of A and B? How does the y-intercept of the line AB relate to the y-intercept of the line EB? d!b . We want to find a point c!a C = (t, b) so the slope of the line through C and B is d!b c!t . Since we want to halve the First note that the slope of the line through A and B is 1 d !b 1 1 1 original slope, we get d!b . This reduces to and so c ! t = 2 c ! a . = c!t = c!t 2 c!a 2 c!a Thus t = 2a ! c and C = (2a-c, b). Participants should notice that the difference between the x-values of points C and B is twice the difference between the x-values of points A and B. ( ) ( ) ( ) We must now find the y-intercept of the lines AB and CB. Starting with the line AB, let I be the y-intercept. Then d!b y= x+I c!a Substitute the coordinates in point B for x and y. d!b d= (c) + I c!a d!b I=d! (c) c!a Now for line CB, let I’ be the y-intercept. Then 1 " d ! b% x+ I' 2 $# c ! a '& Substitute the coordinates in point B for x and y. y= d= 1 " d ! b% (c) + I ' 2 $# c ! a '& 1 " d ! b% I'= d ! $ (c) 2 # c ! a '& November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 2 Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.2 Notice that in I’, we are subtracting from d half the amount that is being subtracted from d in I. 2. By changing the first coordinate in point A from “a” in A to “q”, locate a new point D = (q, b) such that the slope of the line through the new point and B is one-third of that of the line through A and B. What do you notice about the difference in the first coordinates of D and B and the difference in the first coordinates of A and B? This time we want to find the point D = (q, b) so that the slope of the line through D and B is 1 d !b one-third that of the line through A and B. So we want d!b . This reduces to c!q = 3 c!a 1 1 1 . Thus c ! q = 3 c ! a and q = 3a ! 2c . Therefore D = (3a-2c, b). = c!q 3 c!a Participants should note that the difference in the x-values is three times that of the difference of the x-values of the points A and B. ( ) ( ) ( ) We must now find the y-intercept of the lines AB and DB. From before, we know that the y-intercept of AB is d!b I=d! (c) c!a Now for line DB, let I’ be the y-intercept. Then 1 " d ! b% y= $ x+ I' 3 # c ! a '& Substitute the coordinates in point B for x and y. 1 " d ! b% d= $ (c) + I ' 3 # c ! a '& 1 " d ! b% I'= d ! $ (c) 3 # c ! a '& Notice that in I’, we are subtracting from d one-third of the amount that is being subtracted from d in I. 3. By changing the first coordinate in point A from “a” to “v”, locate a new point E = (v, b) such that the slope of the line through the new point and B is one-fourth of that of the line through A and B. What do you notice about the difference in the first coordinates of E and B and the difference in the first coordinates of A and B? November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 3 Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.2 This time we want to find the point E = (v, b) so that the slope of the line through E and B is 1 d !b one-fourth that of the line through A and B. So we want d!b . This reduces to c!v = 4 c!a 1 1 1 . Thus c ! v = 4(c ! a) and v = 4a ! 3c . Therefore E = (4a – 3c, b). = c!v 4 c!a Participants should note that the difference in the x-values is four times the difference of the x-values of the points A and B. ( ) ( ) We must now find the y-intercept of the lines AB and EB. From before, we know that the yintercept of AB is d!b I=d! (c) c!a Now for line EB, let I’ be the y-intercept. Then 1 " d ! b% x+ I' 4 $# c ! a '& Substitute the coordinates in point B for x and y. y= d= 1 " d ! b% (c) + I ' 4 $# c ! a '& 1 " d ! b% I'= d ! $ (c) 4 # c ! a '& Notice that in I’, we are subtracting from d one-fourth of the amount that is being subtracted from d in I. 4. Let F be a point created by changing only the first coordinate of A. Make a conjecture about the location of a point F such that the slope of the line through F and B is 1/n of the slope of the line through A and B where n is a natural number. Explain your reasons for choosing this point and then show that it is the correct choice. Participants may conjecture that based on the previous exercises, the difference in the xvalues of the points F and B is n times that of the points A and B. Thus the difference will be n*(c-a). So the point will be F = na ! (n ! 1)c,b . To show that this is true, we can ( ) calculate the slope of the line between F and B. d!b d!b d!b 1 " d ! b% m= = = = *$ !na + nc n c ! a n # c ! a '& c ! na ! (n ! 1)c ( ) ( ) Thus the slope of the line between F and B is 1/n times that of the line between A and B. November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 4 Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.2 5. Make a conjecture about the y-intercept of the line FB. Explain your reasons for choosing this value and then show that your choice was correct. Participants should conjecture that in the y-intercept of FB, we are subtracting from d, 1/n times the amount that is being subtracted from d in the y-intercept of AB. To show this, we must find the y-intercept of the lines AB and DB. From before, we know that the y-intercept of AB is d!b I=d! (c) c!a Now for line FB, let I’ be the y-intercept. Then 1 " d ! b% x+ I' n $# c ! a '& Substitute the coordinates in point B for x and y. y= d= 1 " d ! b% (c) + I ' n $# c ! a '& I'= d ! 1 " d ! b% (c) n $# c ! a '& Notice that in I’, we are subtracting from d, 1/n times the amount that is being subtracted from d in I. 6. What pattern did you notice in the y-intercepts as the slope was changed? (If you didn’t notice any pattern, go back and make sure that you were always consistent about which point you used to find the equation of the line. Try always using the point B.) Explain what created the pattern. If the point B is on the y-axis, what changes? " d ! b% The y-intercept of the original line is I = d ! $ (c) . # c ! a '& The y-intercept of the new line that was created by changing the slope by a fraction (say 1/k) " 1 % " d ! b% ends up being I = d ! $ ' $ (c) . This is because of the fact that the y-intercept is # k & # c ! a '& exactly y2 ! mx2 , where B = (x2, y2). This point is consistent throughout the process. All that is changing is the value of m which is being multiplied by 1/k. If B is on the y-axis, then B is also the y-intercept of the new line. We can see that because B = (0, d) making I, the intercept of the new line, equal to d. November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 5 Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.2 Teacher notes In this activity, it is essential that the participants always use the point B as the point used to determine the y-intercept. Failure to do so will make it much harder to observe the relationship that is sought. November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 6 Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.2 TASK 2.13.2: FIND IT YET? Suppose we are given two points in a plane, A = (a, b) and B = (c, d). 1. By changing the first coordinate in point A from “a” to “t”, locate a new point C = (t, b) such that the slope of the line through the new point and B is half that of the line through A and B. What do you notice about the difference in the first coordinates of C and B and the difference in the first coordinates of A and B? How does the y-intercept of the line AB relate to the y-intercept of the line CB? 2. By changing the first coordinate in point A from “a” to “q”, locate a new point D = (q, b) such that the slope of the line through the new point and B is one-third of that of the line through A and B. What do you notice about the difference in the first coordinates of D and B and the difference in the first coordinates of A and B? How does the y-intercept of the line AB relate to the y-intercept of the line DB? 3. By changing the first coordinate in point A from “a” in A to “v”, locate a new point E = (v, b) such that the slope of the line through the new point and B is one-fourth of that of the line through A and B. What do you notice about the difference in the first coordinates of E and B and the difference in the first coordinates of A and B? How does the y-intercept of the line AB relate to the y-intercept of the line EB? 4. Let F be a point created by changing only the first coordinate of A. Make a conjecture about the location of point F given that the slope of the line through F and B is 1/n of the slope of the line through A and B where n is a natural number. Explain your reasons for choosing this point and then show that it is the correct choice. 5. Make a conjecture about the y-intercept of the line FB. Explain your reasons for choosing this value and then show that your choice was correct. 6. What pattern did you notice in the y-intercepts as the slope was changed? (If you didn’t notice any pattern, go back and make sure that you were always consistent about which point you used to find the equation of the line. Try always using the point B.) Explain what created the pattern. If the point B is on the y-axis, what changes? November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.