1
Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.4
TASK 2.13.4: ONE MORE LOOK
Solutions
Suppose we are given two points in a plane, A = (a, b) and B = (c, d).
1. By changing the second coordinate in A from “b” to “m”, locate a new point C = (a, m) 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?
First note that the slope of the line through A and B is
C = (a, m) so the slope of the line through C and B is
original slope, we get
d!m
=
c! a
d!b
. We want to find a point
c!a
d!m
c! a
. Since we want to halve the
1 d !b
. This reduces to 2 d ! m = d ! b and so
2 c!a
( )
(
)
1
(b + d) and C = (a, 12 (b + d)). Participants should notice that the difference between
2
the x-values of points C and B is twice the difference between the x-values of points A and
B.
m=
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 of 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 of point B for x and y.
y=
d=
1 " d ! b%
(c) + I '
2 $# c ! a '&
November 23, 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 2.13.4
1 " d ! b%
I'= d ! $
(c)
2 # c ! a '&
Notice that in I’, we are subtracting from d half the amount that is being subtracted from d in
I.
2. By changing the second coordinate in A from “b” to “p”, locate a new point D = (a, p) 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 a point D = (a, p) so the slope of the line through D and B is
Since we want to take a third of the original slope, we get
d! p
=
c! a
d! p
c! a
.
1 d !b
. This reduces to
3 c!a
( )
1
3 d ! p = d ! b and so p = (b + 2d) and C = (a, 13 (b + 2d)). Participants should notice
3
that the difference between the x-values of points C and B is three times the difference
between the x-values of points A and B.
(
)
From before, we know that the y-intercept of the line through AB is
" d ! b%
I = d !$
(c)
# c ! a '&
Now for line CB, let I’ be the y-intercept. Then
1 " d ! b%
y= $
x+ I'
3 # c ! a '&
Substitute the coordinates of 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 the amount that is being subtracted from
d in I.
3. By changing the second coordinate in A from “b” to “r”, locate a new point E = (a, r) such
that the slope of the line through the new point and B is one-fourth the slope of the line
November 23, 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 2.13.4
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?
This time, we want to find a point E = (a, r) so the slope of the line through E and B is
Since we want to take one fourth of the original slope, we get
d !r
=
c! a
d !r
c! a
.
1 d !b
. This reduces to
4 c!a
( )
1
(b + 3d) and C=(a, 14 (b + 3d)). Participants should notice that
4
the difference between the x-values of points C and B is four times the difference between
the x-values of points A and B.
(
)
4 d ! r = d ! b and so r =
From before, we know that the y-intercept of the line through AB is
d!b
I=d!
(c)
c!a
Now for line CB, let I’ be the y-intercept. Then
1 " d ! b%
x+ I'
4 $# c ! a '&
Substitute the coordinates of 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 the amount that is being subtracted
from d in I.
4. Let F be a point created by changing only the second 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.
Participants may conjecture that based on the previous exercises, the difference in the xvalues of the points F and B is 1/n times that of the points A and B. Thus the difference will
"
%
1
b + (n ! 1)d ' . To show that this is true, we
be 1/n*(b-d). So the point will be F = $ a,
n
#
&
can calculate the slope of the line between F and B.
(
)
November 23, 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 2.13.4
"1
% " nd ! b ! nd + d %
d ! $ (b + (n ! 1)d)' $
'& 1 " d ! b %
n
#n
& #
m=
=
= $
c!a
c!a
n # c ! a '&
Thus the slope of the line between F and B is n times that of the line between A and B.
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, 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 of 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 '&
November 23, 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 2.13.4
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) makes I, the intercept of the new line, equal to d.
November 23, 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 2.13.4
Teaching 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 23, 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.
7
Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.4
TASK 2.13.4: ONE MORE LOOK
Note: In this activity, we are specifically looking for the pattern that occurs naturally. The
“Big Idea” is what’s important here.
Suppose we are given two points in a plane, A = (a, b) and B = (c, d).
1. By changing the second coordinate in A from “b” to “m”, locate a new point C = (a, m) 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 second coordinate in A from “b” to “p”, locate a new point D = (a, p) 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 second coordinate in A from “b” to “r”, locate a new point E = (a, r) 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 second 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.
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 23, 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.