1
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential; Task 3.3.3
TASK 3.3.3: STARS, STARS, STARS
Solutions
Procedure Take a sheet of paper and fold it in half. Now
fold the new section in half again. Repeat this
process twice more. Now, unfold the piece of
paper. You should have a sheet of paper with
16 sections marked by the fold lines similar to
the one shown here. Number the sections as
shown. Take one star and place in the center
of section 1. Take two stars and place in
section 2. Take four stars and place in section
3. You will continue to use this pattern and
stick stars to each section of the paper until
each section is filled.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1. Estimate the number of stars you will need to complete the page using
this pattern. Request from the leader your estimate of the stars needed to
do this. You may request stars from your leader only one time.
(Answers will vary)
2. Using the pattern established above, continue to add stars to the sections
until you have the correct number of stars in each section or until you
run out of stars. Each time you complete a section, record the
information in Table 1.
Table I
Section
Process
Number of Stars
1
20=1
1
2
3
4
5
6
7
8
9
10
…
n
1
1*2=2
2
2
1*2*2=2
1*2*2*2=23
1*2*2*2*2=24
1*2*2*2*2*2=25
1*2*2*2*2*2*2=26
1*2*2*2*2*2*2*2=27
1*2*2*2*2*2*2*2*2=28
1*2*2*2*2*2*2*2*2*2=29
…
1*2*2*2*…*2=2n-1
(n -1)factors of 2
4
8
16
32
64
128
256
512
…
2n-1
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 3. Exponential; Task 3.3.3
3. Write a function for how many stars will be on the section number n.
r = 2n-1
4. How many sections were you able to complete with the stars you requested from
your leader? Why? (answers will vary)
5. Find a viewing window for the problem situation.
Sketch your graph:
Identify your window:
Xmin:
-1
Xmax: 16
Xscl:
1
Ymin:
-1
Ymax:
1024
Yscl:
1
5. Justify your window choice.
Sample answer: The variable x stands for the number of sections, so one to sixteen
sections appear reasonable. The variable y stands for the number of stars, so 0
through 1024 will show the number of stars for sections 1 through 10.
6. What operation is being repeated in this problem? multiplication
7.
If you continue with a second sheet with 16 sections, how many stars will you
have in the sixteenth section of the second sheet of paper (the 32nd section
overall)? 2,147,483,648
Explain how you determine your answer.
Method 1:
Enter a sequence in the graphing calculator to show the number of sections and
the number of stars.
{1,1}
Enter
{1,1}
Enter
nter
The calculator thinks the first number is answer #1,
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 3. Exponential; Task 3.3.3
and the second number is answer # 2.
{ANS(1)+1,ANS(2)*2}
Enter
{2,2}
Enter
Enter
nter
{3,4}
Enter
Enter
nter
{4,8}
Enter
Continue until you have {32, 2,147,483,648}
nter
Method 2
Use the formula r = 2n-1
There will be 32 sections in all, so n = 32.
r = 232!1
r = 231
r = 2,147, 483, 648
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 3. Exponential; Task 3.3.3
TASK 3.3.3: STARS, STARS, STARS
Procedure: Take a sheet of paper and fold it in half. Now
fold the new section in half again. Repeat this
process twice more. Now, unfold the piece of
paper. You should have a sheet of paper with
16 sections marked by the fold lines similar to
the one shown here. Number the sections as
shown. Take one star and place in the center
of section 1. Take two stars and place in
section 2. Take four stars and place in section
3. You will continue to use this pattern and
stick stars to each section of the paper until
each section is filled.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1. Estimate the number of stars you will need to complete the page using
this pattern. Request from the leader your estimate of the stars needed to
do this. You may request stars from your leader only one time.
2. Using the pattern established above, continue to add stars to the sections
until you have the correct number of stars in each section or until you
run out of stars. Each time you complete a section, record the
information in Table I.
Table 1
Section
1
2
3
4
5
6
7
8
9
10
…
n
Total
Process
Number of Stars
1
2
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.
5
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential; Task 3.3.3
3. Write a function for how many stars will be on the section number n.
4. How many sections were you able to complete with the stars you requested from
your leader? Why?
4. Find a viewing window for the problem situation.
Sketch your graph:
Identify your window:
Xmin:
Xmax:
Xscl:
Ymin:
Ymax:
Yscl:
5. Justify your window choice.
6. What operation is being repeated in this problem?
7. If you continue with a second sheet with 16 sections, how many stars will you have in
the sixteenth section of the second sheet of paper (the 32nd section overall)?
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.