Grains of Rice
Grains of Rice
Overview
Concepts
•
Students will analyze and find rules for the growth pattern of powers of two.
Algebra
Materials
Other Versions of the Story
•
•
•
Other versions of this story include The King's Chessboard by David Birch,
The Token Gift by Hugh William McKibbon, and A Grain Of Rice by Helena
Clare Pittman. However, these other versions start with one grain of rice on
the first square, which makes the mathematics less accessible for some
middle school students. That is, the function rules are more difficult to derive
and more complicated to express on the calculator.
TI-73 Explorer™
The Rajah's Rice by David Barry
Student activity sheets
Introduction
Chandra’s Request
“All I ask for is rice. If Your Majesty pleases, place two
grains of rice on the first square of this chessboard.
Place four grains on the second square, eight on the
next, and so on, doubling each pile of rice till the last
square.”
From The Rajah’s Rice
How many grains of rice would Chandra receive in all?
Procedure
1. Distribute the activity sheets to students.
2. Read the first part of the book or tell the first part of
the story through Chandra’s making the request for
her reward.
3. After a discussion about the number of squares on a
chessboard, have each student make an estimate for
the total number of grains of rice that Chandra would
receive.
•
Students rarely have a feel for the huge number of
grains of rice the request would yield. Do not tell
them the answer at this point. Explain that they will
do some exploration and then have a chance to revise
their estimates.
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Grains of Rice
•
Students may not know that there are 64 squares on a
chessboard. It is a nice addition to have such a board
in the class.
•
You may want to let your students calculate the first
few values for the number of grains of rice by hand,
but they will soon need to use a calculator as the
values quickly become unwieldy.
4. Have students work alone or in pairs to find the
number of grains of rice for each square on the first
row of the chessboard. Then let students share their
results.
5. As the numbers get larger, students can use repeated
multiplication or the constant function to generate
the powers of two on the Home Screen, using the
following keystrokes:
Repeated Multiplication
•
To use repeated multiplication, simply press 2 (for
the number of grains of rice on the first square) and
then press b.
•
Press ¯ Á and b again.
•
Continue to press b to find the number of grains
of rice on each successive square (Figure 1).
Figure 1
Using the Constant Function
•
To set up the constant function, press †, press
¯ Á b. See Figure 2.
•
Return to the Home screen by pressing y l.
Type 1 to represent the first square, and press @.
Continue to press @ to see the number of grains
for each successive square as many times as desired.
•
Figure 2
The constant function has the advantage of displaying
the square number as well as the number of squares
needed for successive stages of the pattern (Figure 3).
Figure 3
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Grains of Rice
6. It is helpful to start a table like the one shown below
on the board and to encourage students to look for
patterns in their results.
Square #
# of Grains
Total # of
Grains so Far
1
2
2
2
4
6
3
8
14
4
16
30
5
32
62
6
64
126
7
128
254
8
256
510
•
Students are usually able to recognize the doubling
pattern quickly in the form of an iterative rule that
says “Each square has twice as many grains of rice as
the previous square.”
•
However, they may have trouble expressing this
pattern in the form of an explicit rule using a
variable. It is helpful to have students include the
numbers used to generate each value in the pattern.
•
For example, the 8 grains of rice on the third square
of the chessboard can be expressed not as 4 (from
square 2) x 2, but rather as (2)(2)(2) (from square 1
and square 2), etc., as shown below.
Square #
•
# of Grains
1
2
2
(2)(2) = 4
3
(2)(2)(2) = 8
4
(2)(2)(2)(2) = 16
5
(2)(2)(2)(2)(2) = 32
6
(2)(2)(2)(2)(2)(2) = 64
7
(2)(2)(2)(2)(2)(2)(2) =128
8
(2)(2)(2)(2)(2)(2)(2)(2)=256
n
2n
Thus, students can more easily conclude that the rule
for the number of grains of rice on any square n is,
“Number of grains = 2n.”
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Grains of Rice
7. Once students find the explicit rule for the number of
grains of rice on any square, they also have the basis
for writing the rule for the total number of grains of
rice so far.
•
Again, students will usually first discover the
iterative rule, “The total so far is always 2 less than
the number of grains on the next square.” For
example, they notice that the total of 14 grains of rice
so far for the third square is 2 less than the 16 grains
of rice on the fourth square.
•
Again, it is helpful for students to write the values in
their table using a different notation to help them
discover a way to express the explicit form of the rule
for this pattern.
•
Thus the table might look like the one shown below:
Square
#
# of
Grains
Total # of Grains so Far
1
21
22 – 2 = 4 – 2 = 2
2
22
23 – 2 = 8 – 2 = 6
3
23
24 – 2 = 16 – 2 = 14
4
24
25 – 2 = 32 – 2 = 30
5
25
26 – 2 = 64 – 2 = 62
6
26
27 – 2 = 128 – 2 = 126
7
27
28 – 2 = 256 – 2 = 254
8
28
29 – 2 = 512 – 2 = 510
n
2n
2(n + 1) – 2
8. It may be appropriate to wait until students generate
the values for several rows of squares on the
chessboard before asking them to derive the explicit
rules. At some point, however, the students must
write the rules using y and x to generate a table of
values for this situation.
•
For the number of grains on any square, y = 2x.
•
For the total number of grains so far, y = 2(x + 1) – 2.
•
Then students should enter the rules in the & menu
of the calculator. To enter the rules into the & menu,
press & and type the rules to the right of Y1 and Y2.
See Figure 4.
Figure 4
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Grains of Rice
•
Before they create and view the table, ask students to
make a new estimate for the total number of grains of
rice that will be found on the last square of the
chessboard.
•
This time, their estimates will be more realistic,
although there are only a few students who are able
to come close to the actual number even at this point
in the investigation.
Using a Table
1. Next, students create a table to show the values
generated by the explicit rules.
•
To set up the table, press f and enter the values
shown in Figure 5.
•
Have students interpret particular values in the table
to assess their understanding. For example, in the
table for y = 2x, scroll down the table to the value in
the Y1 column when x = 32, and ask the meaning of
that value. Students should interpret this as, “A total
of 4,294,967,296 grains of rice would be on the
thirty-second square of the chessboard.”
•
Notice that the exact value (versus the value written
in scientific notation) can be viewed at the bottom of
the screen when the cursor is placed on the entry in
the table. Press i to see the information shown
in Figures 6 to 8.
Figure 5
Figure 6
2. Once students are comfortable interpreting values in
the table, they should go to row 64 and answer the
question asked at the beginning of the activity.
Note: The exact value for 2(64 + 1) – 2 is too large to be
displayed. Instead, the value is displayed as
3.68934881 E19.
•
Some classes will need a separate investigation about
scientific notation while others will enjoy finding the
exact total number of grains of rice on the
chessboard.
Figure 7
Figure 8
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Grains of Rice
Using a Graph
According to the level of experience students have with
coordinate graphing, it may be appropriate for them to
graph some values of the function in a scatterplot.
1. First, have students enter the data for squares 1-10 of
the pattern into lists.
2. Enter the number of each square 1-10 into L1, the
number of grains for each square into L2, and the
corresponding total number of grains so far into L3.
3. To enter the data into lists, first clear any data in the
lists.
•
Press 3.
•
Use the arrow keys to place the cursor in the header
for L1.
•
Press ‘ and Í. Items in the list should
clear.
•
Repeat these steps for L2 and L3.
•
Next, enter the data in the lists. For each item, type in
the digit(s), and press b. See Figure 9.
Figure 9
Note: You may use a formula to generate the values in
L2 and L3 by substituting L1 for x in the rules
used in the & menu.
4. Thus, you may use the following keystrokes to enter
a formula for L2.
•
Place the cursor in the header of L2.
•
Press 2 7 v and select L1. Press Í. See
Figure 10.
Figure 10
5. Similarly, you may use the following keystrokes to
enter a formula for L3.
•
Place the cursor in the header of L3.
•
Press 2 7 (then v and select L1 then + 1) and – 2
and press b. See Figure 11.
Note: If you place quotation marks around a formula,
that formula will appear in the header of the list
when you place the cursor there. This is a
valuable tool for students who may need
reminding of the formula they used to generate
the list.
Figure 11
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Grains of Rice
6. Set up the first scatterplot.
•
Clear any equations in the & menu.
•
Press y o for e. If any plot besides Plot1 is
on, select 4:PlotsOff to quickly turn all plots off.
•
Then re-enter the STAT PLOTS window by pressing
y o.
•
Press 1 to select Plot1, and use the settings shown in
Figure 12.
Figure 12
7. Have students choose appropriate values so the
viewing window fits the data. See Figure 13 for an
example Window setting.
Figure 13
8. Press * to view the scatterplot. See a sample
scatterplot in Figure 14.
•
Have students use the ) feature to identify
points that correspond with data in the lists and to
interpret the meanings of each of those points.
•
For example, they should interpret the point (5, 32)
as, “On the fifth square of the chessboard, there
would be 32 grains of rice.” For the scatterplot record
of this, see Figure 15.
Figure 14
9. Use the following questions to guide discussion
about the scatterplot:
•
What is an appropriate label for the x-axis in this
plot?
•
What is an appropriate label for the y-axis in this
plot?
•
When you trace from one point for any stage number
to the point for the next stage number in the plot, how
much does the x-value increase? Explain why.
•
When you trace from one point for any stage number
to the point for the next stage number in the plot, how
much does the y-value increase? Explain why.
•
Do the coordinates for every point in the scatterplot
follow the rule you found for the number of grains of
rice on any square? Explain how.
Figure 15
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Grains of Rice
•
It is important that students discuss the differences
between this graph and the graph of a linear function.
They should notice and verbalize that the rate of
change in this function is not constant, but increasing,
and that the rate of change increases very rapidly.
•
For example, the difference between the number of
grains of rice on squares 1 and 2 is 2 whereas the
difference between the number of grains of rice on
squares 2 and 3 is 4 and the difference between the
values for squares 3 and 4 is 8.
•
Hopefully, students will relate this rate of change to
the fact that the variable is the exponent in the
function rule. Otherwise, you should point this out.
You should tell them that such a function is called an
exponential function.
10. Similarly, students should set up, display, and
discuss Plot2 to show the total grains of rice on the
chessboard. Ask questions similar to those listed
above for Plot1 to assess student understanding.
•
Define Plot2 as shown in Figure 16.
•
Use the Window settings shown in Figure 17.
Figure 16
Figure 17
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The sample scatterplot is shown in Figure 18.
11. Both plots may be viewed at the same time for
comparisons.
•
Figure 18
Simply turn on both Plot1 and Plot2 in the e
menu (Figure 19).
Figure 19
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Grains of Rice
•
Compare Plot1 and Plot2. See Figures 20 and 21.
12. Students can benefit from using another way to solve
this problem using the TI-73 Explorer™.
•
Although the TI-73 Explorer™ does not have a
sequence mode, there is an alternate method for
generating and displaying the values for the number
of grains of rice and the total number of grains of rice
so far for each square of the chessboard.
Figure 20
13. On the Home Screen, use the following sequence of
keystrokes to set initial values for this problem,
make rules, and generate values using those rules:
•
Press t. Move the cursor onto {. Press b.
•
Move the cursor onto Done. Press b. This will
paste { on the Home Screen.
•
Type 1,2,2.
•
Press t, Move the cursor onto }. Press b.
•
Move the cursor onto Done. Press b. This will
paste } on the Home Screen to end the entry.
•
Press b. You should see the screens shown in
Figures 22 and 23.
•
So far, the keystrokes have set up the starting values.
The first number (1) is the square number, the second
number (2) is the number of grains of rice on that
square, and the third number (2) is the total number
of grains so far (Figure 24).
Figure 21
Figure 22
Figure 23
14. To enter the rules for each of these 3 beginning
values, use the following sequences of keystrokes:
•
Press t. Move the cursor onto {. Press b.
•
Move the cursor onto Done. Press b. This will
paste { on the Home Screen.
•
Type the following:
¢(1) + 1, ¢(2)*2, ¢(2)*2 + ¢ (3)
•
Press t. Move the cursor onto }. Press b.
Figure 24
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Grains of Rice
•
Move the cursor onto Done. Press b. This will
paste } on the Home Screen to end the entry. See
Figures 25 to 27.
15. Continue to press to see as many sets of values as
you wish.
•
Note that there are other ways to enter the rules for
the starting values {1 2 2}, for example,
{¢(1)+1, ¢(2)*2, ¢(3)*2+2}.
Figure 25
16. Follow-up investigations could include validating or
disproving Chandra’s statement in the book that,
“Add all 64 squares together and you get India,
covered knee deep in rice.”
Figure 26
Figure 27
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Grains of Rice
Grains of Rice
For use with The Rajah’s Rice by David Barry
“All I ask for is rice. If your majesty pleases, place two grains of rice
on the first square of this chessboard. Place four grains on the second square,
eight on the next, and so on, doubling each pile of rice to the last square.”
How many grains of rice would Chandra receive in all?
1. Explore this problem by answering the following questions.
•
How many squares are there on each row of a chessboard? __________
•
Estimate how many grains of rice will be given by the Rajah for the last square of the 8 X 8
chessboard to grant Chandra’s request.
•
If two grains of rice are placed on the first square of the first row of the chessboard, how
many grains will be placed on each of the remaining squares of the first row?
2nd square
•
3rd square
4th square
5th square
6th square
7th square
8th square
Continue the doubling pattern for the next few rows of the chessboard. How many grains of
rice will be needed for the last square of each of the following?
The last square of the second row
The last square of the third row
The last square of the fourth row
•
Now estimate again how many total grains of rice will be needed for the last square of the
chessboard.
•
Read the last page of The Rajah’s Rice to see how close your final estimate is to the actual
result. Did the results surprise you? Explain.
Extension
1. Calculate the number of cubic yards of rice the Rajah would need to grant Chandra’s request.
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