1
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.3
TASK 3.1.3: DIAGONAL PATTERNS
Solutions
1. A 3x4 rectangle is sketched on the grid below. Calculate the exact length of the
diagonal. Record in Table I along with the process that you used. What is the
name of the process?
Participants are expected to use Pythagorean’s Theorem.
Table I
Length
Width
Process
Length of Diagonal
3
4
32 + 42
5
6
8
62 + 82 = 100
10
9
12
92 + (12)2 = 81 + 144 = 225
15
12
16
(12)2 + (16)2 = 144 + 256 = 400
20
15
20
(15)2 + (20)2 = 225 + 400 = 625
25
3n
4n
(3n)2 + (4n)2 = 9n2 + 16n2 = 25n2
5n
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 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.3
2. Sketch on the grid the rectangle that has its lower left hand corner at the point
(0, 0) and whose length and width is twice that of the original rectangle R1. Label
this rectangle R2. Calculate the length of the diagonal of R2. Record the
information in the table. How is the length of the diagonal of R2 related to the
length of the diagonal of R1?
The length of the diagonal of R2 is 10, twice the length of R1.
3. Sketch on the grid the rectangle that has its lower left hand corner at the point
(0, 0) and whose length and width is three times that of the original rectangle R1.
Label this rectangle R3. Calculate the length of the diagonal of R3. Record the
information in the table. How is the length of the diagonal of R3 related to the
length of the diagonal of R1?
The length of the diagonal of R3 is 15, three times the length of R1.
4. Repeat (3) for R4 and R5, rectangles that have their lower left hand corner at the
point (0,0) and whose length and width is 4 and 5 times respectively of the
original rectangle R1. How are the lengths of the diagonals of R4 and R5 related
to the length of the diagonal of R1?
The length of the diagonals of R4 and R5 is 20 and 25 respectively, 4 and 5 times the
length of R1, respectively.
5. Describe the function rule that fits this data.
The length of the diagonal of the rectangles defined in this situation satisfy that if the
lengths of the sides of the rectangle are 3n and 4n then the length of the diagonal will be
5n. In equation form, d = 5 n.
6. Are the lengths of the diagonals of the rectangles proportional to the length of
diagonal of R1? Do they form a direct variation?
The diagonals are proportional to the length of the diagonal of R1. The constant of
proportionality is the multiple of the length of the sides. This is also a direct variation.
7. Plot on your grid the line that passes through the point (0, 0) and the upper right
hand corner of the original rectangle R1. What do you notice? What is the
equation of this line? What does the slope represent in this situation? Explain the
relation of the pattern that you found in calculating the slopes of the diagonal with
the slope of this line.
The line passes through the upper right hand corner of all the rectangles. The equation
of the line is y = 43 x. The slope represents that if we start at the upper right hand corner
of one of our rectangles, say Rk, moving to the right 4 units and up 3units will give us the
corner of the next rectangle. Thus we will add 5 units to the length of the diagonal of the
rectangle Rk to get the length of the next rectangle.
8. How does the line that you drew in (7) help you to explain the results that you
found in Table I?
Participants should see that the diagonal is increasing by 5 each time the length and
width is doubled, tripled, etc.
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 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.3
For the geometric argument, an example follows.
Note that R2 is four copies of R1. Drawing the
diagonal for R2, we see that we get two copies of
the diagonal of R1.
Similarly R3 is nine copies of the R1 rectangle
and the diagonal is made up of 3 R1 diagonals.
Thus we can see that as new rectangles are
created as described in this exercise, the diagonal
will be changing by a multiple of 5, the length of
the original rectangle R1.
9. In rectangles defined like those discussed here, if the diagonal of such a rectangle
with a width of 12 is 15, what is the length of the diagonal of a rectangle with
width of 60? Explain how you determine your answer.
Participants can solve this several ways. One way is to recognize and use the
12 60
proportionality of the situation.
implies x = 75. Another way is to recognize
=
15 x
that n (the multiplier) must be 15 since 4*15 = 60. Thus our rule says that the diagonal
will have length 5n = 5*15 = 75.
10. Suppose that the original rectangle R1 had length 4 and width 3. As in the
previous exercises and on the grid below, construct rectangles that are have 2, 3,
and 4 times the length and width of this rectangle. What is the length of the
diagonals of each of the rectangles? How does this compare with the previous
results? What mathematical property explains this?
The diagonals have length 5, 10, 15, and 20 respectively. The length of the diagonals
will exactly match those found in the original 3x4 rectangle. This is a result of the
commutative property of addition of real numbers.
of the idea of symmetry across the line y = x.
a 2 + b2 = b2 + a 2 It also a result
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 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.3
11. How is the line that passes through the upper right hand corner of the new set of
rectangles related to the line you found for the first set?
The line that passes through the corners of the new set of rectangles is the line y = 4/3 x.
The slope of this line is the reciprocal of the original 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.
5
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.3
12. Go back now to the original set of rectangles that you created. Calculate the areas
of rectangles R1 – R5 to complete the table below.
Table II
Length
Width
Process
Area of Rectangle
3
4
3*4
12
6
8
6*8
48
9
12
9*12
108
12
16
12*16
192
15
20
15*20
300
3n
4n
3n*4n
12n2
13. How is the area of R2 related to the area of R1?
Use the drawing of R1 and R2 on your grid to
justify your relationship.
The area of R2 is four times the area of R1. This is
geometrically visible by seeing that there are four
copies of R1 in R2.
14. Calculate the area of R3. How is the area of R3 related to the area of R1? Use
the drawings on your grid to justify your relationship.
The area of R3 is nine times the area of R1. This is geometrically visible by seeing that
there are nine copies of R1 in R3.
15. What generalization can you make about the areas of rectangles similar to the 3x4
rectangle R1?
Rectangles that are similar to our 3 x 4 rectangle will have areas that satisfy the equation
A=12n2 where n is the multiplier of the length and width.
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 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.3
TASK 3.1.3: DIAGONAL PATTERNS
1. A 3x4 rectangle is sketched on the grid below. Calculate the exact length of the
diagonal. Record in Table I along with the process that you used. What is the
name of the process?
Table I
Length
3
Width
Process
Length of Diagonal
4
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 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.3
2. Sketch on the grid the rectangle that has its lower left hand corner at the point
(0, 0) and whose length and width is twice that of the original rectangle R1. Label
this rectangle R2. Calculate the length of the diagonal of R2. Record the
information in the table. How is the length of the diagonal of R2 related to the
length of the diagonal of R1?
3. Sketch on the grid the rectangle that has its lower left hand corner at the point
(0, 0) and whose length and width is three times that of the original rectangle R1.
Label this rectangle R3. Calculate the length of the diagonal of R3. Record the
information in the table. How is the length of the diagonal of R3 related to the
length of the diagonal of R1?
4. Repeat (3) for R4 and R5, rectangles that have their lower left hand corner at the
point (0,0) and whose length and width is 4 and 5 times respectively of the
original rectangle R1. How are the lengths of the diagonals of R4 and R5 related
to the length of the diagonal of R1?
5. Describe the function rule that fits this data.
6. Are the lengths of the diagonals of the rectangles proportional to the length of
diagonal of R1? Do they form a direct variation?
7. Plot on your grid the line that passes through the point (0, 0) and the upper right
hand corner of the original rectangle R1. What do you notice? What is the
equation of this line? What does the slope represent in this situation? Explain the
relation of the pattern that you found in calculating the slopes of the diagonal with
the slope of this 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.
8
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.3
8. How does the line that you drew in (7) help you to explain the results that you
found in Table 1?
9. In rectangles defined like those discussed here, if the diagonal of such a rectangle
with a width of 12 is 15, what is the length of the diagonal of a rectangle with
width of 60? Explain how you determine your answer.
10. Suppose that the original rectangle
R1 had length 4 and width 3. As
in the previous exercises and on
the grid, construct rectangles that
are have 2, 3, and 4 times the
length and width of this rectangle.
What is the length of the diagonals
of each of the rectangles? How
does this compare with the
previous results? What
mathematical property explains
this?
11. How is the line that passes through the upper right hand corner of the new set of
rectangles related to the line you found for the first set?
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.
9
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.3
12. Go back now to the original set of rectangles that you created. Calculate the areas
of rectangles R1 – R5 to complete the table below.
Table II
Length
Width
3
4
Process
Area of Rectangle
13. How is the area of R2 related to the area of R1? Use the drawing of R1 and R2 on
your grid to justify your relationship.
14. Calculate the area of R3. How is the area of R3 related to the area of R1? Use
the drawings on your grid to justify your relationship.
15. What generalization can you make about the areas of rectangles similar to the 3x4
rectangle R1?
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.