1
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential; Task 3.3.2
TASK 3.3.2: ONE GRAIN OF RICE
Solutions
Read a portion of One Grain of Rice by Demi with the students. Stop after Rani has been
given 4 grains of rice.
If the book is not available, you can tell the story yourself, of a girl that is given one grain
of rice and then each day the number of grains of rice she is given is doubled.
Complete the following table showing the correct amount of rice for each day.
Table 1
Day
Process
Grains of Rice
0
1
2
3
4
5
6
7
8
9
10
…
1=20
2=21
2⋅2=22
2⋅2⋅2=23
2⋅2⋅2⋅2=24
2⋅2⋅2⋅2⋅2=25
2·2·2·2·2·2
= 266.
2⋅2⋅2⋅2⋅2⋅2=2
2⋅2⋅2⋅2⋅2⋅2⋅2=27
2⋅2⋅2⋅2⋅2⋅2⋅2⋅2=28
2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2=29
2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2=210
…
2⋅2⋅2⋅…⋅2=230
1
2
4
8
16
32
64
128
256
512
1024
…
30
30 factors
1,073,741,824
…
…
…
n
2⋅2⋅2⋅…⋅2=2n
2n
n factors
2. Write a function for the number of grains of rice Rani will receive on the nth day.
Write the function as follows.
After n days, how many 2’s will be multiplied? In other words, how many
factors of 2 are there in the expression? n
How can you write 2⋅2⋅2⋅2⋅2⋅2 ? 26
So the function would be f(n) = 2n, where n is the day in question.
3. Find a viewing window for the problem situation.
Use the following questions to lead participants to finding a suitable viewing window.
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 3.3.2
o What does x stand for in this problem? number of days
o What values make sense for x in this problem? (answers will vary)
Sample answer. 1 to 10 days.
o What does y stand for in this problem? number of grains of
rice received each day
o What values make sense for y in this problem. (answers will vary)
Sample answer: number of grains of rice 210, which is 1024.
Sketch your graph.
Identify your window:
Xmin:
-1
Xmax:
12
Xscl:
1
Ymin:
-1
Ymax:
1200
Yscl:
50
4. Justify your window choice.
Sample answer: The variable x stands for the number of days, so one to ten days
appears reasonable. The variable y stands for the number of grains of rice, so 0
through 1024 will show the grains of rice for days 1 through 10.
5. What operation is being repeated in this problem? multiplication
6. What types of functions results from this operation? exponential
Use the function rule to find the following:
7. On the 18th day, how many grains of rice will Rani receive?
218 = 262,144 grains of rice Rani will receive on the 18th day.
6. Rani received 8,388,608 grains of rice. What day is it? Show how you found your
solution.
2n = 8,388,608
Calculator sequence:
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 3.3.2
{1, 8388608}
Enter
{1, 8388608}
Enter
nter
The calculator thinks the first number is answer #1,
and the second number is answer # 2.
{ANS(1)+1,ANS(2)÷2}
Enter
{2,4194304}
Enter
Enter
nter
{4,1048576}
Enter
Enter
nter
{5,524288}
Enter
Continue to press enter until the second answer of 2 is displayed. The count is the
number of days. 23 answer {23, 2}nter
Alternative: While not an Algebra I solution, point out to participants that this equation,
2n = 8388608 can be solved using natural log, as follows:
2 n = 8388608
ln 2 n = ln8388608
( )
n ln 2 = ln8388608
ln8388608
ln 2
n = 23
n=
Questions for discussion:
•
•
•
•
•
•
What operation is being repeated in this problem? division
What function results from repeated division? exponential function
In Task 1, what other repeated operation did we use? repeated addition and
multiplication
What function results from repeated addition? linear functions
What kind of graphs result form repeated addition? linear
What kind of graphs result from repeated multiplication? exponential
9. It is difficult to visualize this amount of rice (8,388,608 grains of rice). What if each
grain of rice was a penny? What is the value in dollars of 8,388,608 pennies?
$83,886.08
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 3.3.2
TASK 3.3.2 ONE GRAIN OF RICE
After reading, in One Grain of Rice by Demi, that Rani has been given four grains of rice,
complete the following table showing the correct amount of rice for each day.
Table I
Day
Process
Grains of Rice
0
1
1
2
2
4
3
4
5
6
7
10
…
…
…
…
…
30
…
n
2. Write a function for the number of grains of rice Rani will receive on the nth day.
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 3.3.2
3. Find a viewing window for the problem situation.
Sketch your graph.
Identify your window:
Xmin:
Xmax:
Xscl:
Ymin:
Ymax:
Yscl:
4. Justify your window choice.
5. What operation is being repeated in this problem?
6. What function results from this operation?
Use the function rule to find the following:
7. On the 18th day, how many grains of rice will Rani receive?
8. Rani received 8,388,608 grains of rice. What day is it? Show how you found your
solution.
9. It is difficult to visualize this amount of rice (8,388,608 grains of rice). What if each
grain of rice was a penny? What is the value in dollars of 8,388,608 pennies?
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.