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5.1
Predicting the Ones Digit
Goals
• Examine patterns in the exponential and
standard forms of powers of whole numbers
• Use patterns in powers to estimate the ones
digits for unknown powers
In this problem, students start by looking at
patterns in the ones digits of powers. For example,
by examining the ones digits of the powers of 2
(21 = 2, 22 = 4, 23 = 8, 24 = 16, 25 = 32,
26 = 64, 27 = 128, 28 = 256, and so on)
students see that the sequence 2, 4, 8, 6 repeats
over and over. We say that the ones digits repeat
in a cycle of four. The ones digits for powers of 9
are 9, 1, 9, 1, 9, 1, and so on. We say that the ones
digits for the powers of 9 repeat in a cycle of 2.
The ones digits for powers of 5 are all 5—that is,
the ones digits repeat in a cycle of 1. The ones
digits of the powers of any whole number repeat
in a cycle of length 1, 2, or 4.
Students create a powers table for a m for
a = 1 to a = 10 and m = 1 to m = 8. They use
the table to find patterns in order to predict the
ones digits for powers such as 2100, 6 50, and 17 20,
and to estimate the standard form of these powers.
Launch 5.1
Launch this problem by writing the values of
y = 2x, for whole-number x-values from 1 to 8.
Write both the exponential and standard form for
2x in the y-column.
x
Let students look for patterns in the table.
Collect their discoveries on a large sheet of paper.
If students can and are ready, let them give
reasons for why the patterns occur. You may want
to come back to these patterns later.
Use the Getting Ready to focus students on the
patterns among the exponents and ones digits in
the standard form of powers.
Suggested Questions Ask:
• Look at the column of y-values in the table.
What pattern do you see in how the ones digits
of the standard form change?
Students should notice that the ones digits
repeat in cycles of four: 2, 4, 8, and 6. If necessary,
extend the table to convince students that this
pattern continues.
• Can you predict the ones digit for 215? (8)
What about 250? (4)
Some students will continue the table or use their
calculator to find 215. Hopefully, at least some
students will use the pattern to reason as follows:
The ones digits occur in cycles of four. The third
complete cycle ends with 212, and the fourth cycle
starts with 213. The number 215 is the third number
in this fourth cycle, so its ones digit is 8.
Students will not be able to find the ones digit
of 250 by using their calculators; the result will be
displayed in scientific notation, and the ones digit
will not be shown. Students will need to use the
pattern previously described. Don’t expect
students to answer this quickly. You might
postpone 250 until later and give students more
time to explore.
• What other patterns do you see in the table?
y
or
2
2
22
or
4
3
23 or
8
4
24
or 16
5
25
or 32
6
26 or 64
7
27 or 128
8
28 or 256
Here are some patterns students may notice.
(Some of these have already been mentioned.)
• The ones digits are all even.
• To predict the ones digit, all you need to know
I N V E S T I G AT I O N
1
21
is the cycle of repeating units digits and whether
2x is the first, second, third, or fourth number in
a cycle.
• If you divide the exponent by 4, the remainder
tells you where the power is in a cycle.
5
Investigation 5
Patterns With Exponents
99
• When you multiply two powers of 2, the
exponent in the product is the sum of the
exponents of the factors. For example,
23 3 24 = 27.
x
y
3
23 or
4
24
7
27 or 128
8
or 16
• Find an x-value and values for the missing
digits that will make this a true number
sentence: 2x = _ _ _ _ _ _ 6 (x = 20, which
gives 220 = 1,048,576)
Next, introduce Problem 5.1. Make sure
students understand how the table is organized.
Students should fill out the powers table in
Question A on their own and then work in groups
of two or three for the rest of the problem.
Explore 5.1
Students should have no trouble filling in the
table. Encourage them to take some care in
completing the table because they will use it as a
reference in the next problem.
Although the problem focuses on patterns in
the ones digits, students may notice other
interesting patterns as well. Here are some
examples:
• Some numbers occur more than once in the
powers chart. For example, 64 appears three
times: 26 = 64, 43 = 64, and 82 = 64.
• When you multiply powers of 2, the exponent of
the product is the sum of the exponents of the
factors. For example, 23 3 24 = 23 + 4 = 27.
• The number of zeros in 10n is n.
For Questions C and D, some students may try
to continue the table or use their calculator to
find the ones digits. But for very large exponents,
continuing the table gets tedious, and the
calculator rounds off after 10 digits. Students
should begin to focus on the patterns in the
ones digits.
100 Growing, Growing, Growing
Suggested Questions If students are having
trouble with using the patterns of the ones digits,
ask:
• What are the lengths of the cycles of repeating
ones digits? (1, 2, or 4, depending on the
base. For example, for the powers of 3, the
ones digits 3, 9, 7, 1 repeat, so the cycle is of
length 4.)
• Which bases have cycles of length 4? (2, 3, 7,
and 8)
• Which bases have a cycle of length 1? (1, 5, 6,
and 10)
• Which bases have a cycle of length 2?
(4 and 9)
• If you know the exponent, how can you use
the pattern of the cycle to determine the ones
digit of the power? For example, the ones
digits for 2n repeat in a cycle: 2, 4, 8, 6. How
can we use this fact to find the ones digit of
221? (For powers of 2, the exponents 1, 5, 9,
13, 17, 21, and so on correspond to a ones
digit of 2. Students may recognize that each
of these numbers is one more than a multiple
of 4. In other words, when you divide these
numbers by 4 you get a remainder of 1. The
exponents 2, 6, 10, 14, 18, and so on,
correspond to a ones digit of 4. Students may
recognize that these numbers are 2 more than
a multiple of 4. The exponents 3, 7, 11, 15, 19,
and so on correspond to a ones digit of 8.
These numbers are 3 more than a multiple of
4. Finally, the exponents 4, 8, 12, 16 and so on
correspond to a ones digit of 6.)
Question D asks about powers of bases greater
than those in the table. Students should reason
that the ones digit of a power is determined by
the ones digits of the base. For example, the ones
digits for powers of 12 are the same as the ones
digits for powers of 2, 22, and 92. The ones digits
for powers of 17 are the same as the ones digits
for powers of 7, 57, and 87. If students struggle
with this idea, ask:
• If you were to look at the ones digits for powers
of 12, you would find that they follow the same
pattern as the ones digits for the powers of 2.
Why do you think this is true? What affects the
ones digit? (To get successive powers of 12, you
multiply by 12, so the ones digit will be the
same as the ones digit of 2 times the previous
ones digit. For 121, the ones digit is 2; for 122,
the ones digit is 4 because 2 3 2 = 4; for 123,
the ones digit is 8 because 4 3 2 = 8; and for
124, the ones digit is 6 because 8 3 2 = 16.
Then the ones digits start to repeat. A similar
argument will work for all whole-number bases
greater than 10.)
In Question E, remind students to use their
knowledge about the patterns of the ones digit to
narrow the choices. For example, for
a12 = 531,441, a must be 1, 3, 7, or 9. Obviously, 1
is not a choice. And 912 would be close to 1012,
which has 13 digits—much too large a number.
The number 531,441 has only 6 digits. At this
point, students should argue that 712 is too large.
They may use an argument similar to the
following but with words, not symbols:
7 2 ø 50, so 712 = 7 2 ? 7 2 ? 7 2 ? 7 2 ? 7 2 ? 7 2 ø
50 ? 50 ? 50 ? 50 ? 50 ? 50, or
5 ? 5 ? 5 ? 5 ? 5 ? 5 ? 10 ? 10 ? 10 ? 10 ? 10 ? 10, which
equals 5 6 ? 10 6.
We know that 106 has 7 digits, which is too large,
so the answer is 312 = 531, 441.
Be sure to make note of interesting patterns,
reasoning, and questions that arise during the
Explore.
Figure 1
1
2
21 !
2
2
4
4
3
8
8
2 !
2 !
4
24 ! 16
6
Ones Digit
2
3
5
It might be helpful to have a large poster of the
completed powers table that students can refer to,
both during this summary and for the next
problem. You might also use the transparency
provided for this problem.
Ask for general patterns students found. Post
these on a sheet of chart paper. Ask students to give
reasons for the patterns. Students may not be able to
explain some of the patterns until the next problem,
when the properties of exponents are developed.
Go over some of the powers in Questions C
and D. Be sure to have students explain their
strategies. After a student or group has explained
a strategy, ask the rest of the class to verify the
reasoning or to pose questions to the presenter so
everyone is convinced of the validity of the
reasoning. For further help with the patterns in
the length of the cycle, the exponent, and the ones
digit, you can use the table in Figure 1.
Save the completed powers table for the launch
of Problem 5.2.
Be sure to assign ACE Exercise 54. It is needed
for Problem 5.2.
Pattern
2x
x
Summarize 5.1
End of first cycle. The exponent 4 is a multiple of 4.
6
5
The exponent 5 has a remainder of 1 when divided by 4.
2
6
The exponent 6 has a remainder of 2 when divided by 4.
4
7
2 ! 32
2 ! 64
7
2 ! 128
The exponent 7 has a remainder of 3 when divided by 4.
8
8
28 ! 256
End of second cycle. The exponent 8 is a multiple of 4.
6
2 ! ■
The exponent 9 has a remainder of 1 when divided by 4.
2
2
10
! ■
The exponent 10 has a remainder of 2 when divided by 4.
4
11
2
11
! ■
The exponent 11 has a remainder of 3 when divided by 4.
8
12
212 ! ■
End of third cycle. The exponent 12 is a multiple of 4.
6
9
10
9
The exponent 13 has a remainder of 1 when divided by 4.
2
2
! ■
The exponent 14 has a remainder of 2 when divided by 4.
4
15
2
15
! ■
The exponent 15 has a remainder of 3 when divided by 4.
8
16
216 ! ■
End of fourth cycle. The exponent 16 is a multiple of 4.
6
50
250 ! ■
The exponent 50 has a remainder of 2 when divided by 4.
So the ones digit is 4.
4
14
Investigation 5
Patterns With Exponents 101
5
! ■
14
13
I N V E S T I G AT I O N
2
13
102 Growing, Growing, Growing
At a Glance
5.1
Predicting the Ones Digit
1
PACING 12 days
Mathematical Goals
• Examine patterns in the exponential and standard forms of powers of
whole numbers
• Use patterns in powers to estimate the ones digits for unknown powers
Launch
Launch this problem by writing the values of y = 2x, for x = 1 to 8. Write
both the exponential and standard form for 2x in the y-column.
Let students look for patterns. Use the Getting Ready to focus students
on the patterns.
• Look at the column of y-values in the table. What pattern do you see in
how the ones digits of the standard form change?
Materials
•
Transparencies 5.1A
and 5.1B
•
Labsheet 5.1
Vocabulary
•
power
• Can you predict the ones digits for 215? What about 250?
• What other patterns do you see in the table?
Students should fill out the powers table in Question A on their own and
then work in groups of two or three for the rest of the problem.
Explore
If students are having trouble using the patterns of the ones digits, ask:
• What are the lengths of the cycles of repeating ones digits?
• Which bases have a cycle of length 4?
• Which bases have a cycle of length 1?
• Which bases have a cycle of length 2?
• If you know the exponent, how can you use the pattern of the cycle to
determine the ones digit of the power?
• If you were to look at the ones digits for powers of 12, you would find
that they follow the same pattern as the ones digits for the powers of 2.
Why do you think this is true?
• What affects the ones digit?
In Question E, remind students to use their knowledge about the
patterns of the ones digit to narrow the choices down.
Make note of interesting patterns, reasoning, and questions that arise.
Summarize
Display a completed powers table on chart paper or a transparency for
students to refer to, both during this summary and for the next problem.
Ask for general patterns. Ask students to give reasons for the patterns.
Materials
•
•
Student notebooks
large sheet of poster
paper (optional)
Go over some of the powers in Questions C and D. Be sure to have
students explain their strategies.
Investigation 5
Patterns With Exponents 103
ACE Assignment Guide
for Problem 5.1
C. 1. 6. The even powers of 4 have 6 as a ones
digit.
2. 1. The even powers of 9 have 1 as a ones
digit.
Core 1–7, 54
Other Applications 8, 9; Connections 44, 45;
3. 3. There is a cycle of length 4 in the
ones digits of the powers of 3. 17 is the
beginning of the fifth cycle.
Extensions 52, 53, 55; unassigned choices from
previous problems
Adapted For suggestions about adapting
4. 5. Any power of 5 has a ones digit of 5.
ACE exercises, see the CMP Special Needs
Handbook.
Connecting to Prior Units 44, 45: Data Around Us
5. 0. Any power of 10 has a ones digit of 0.
D. 1. 1. The ones digit of 31 is 1, so all powers of 31
will have a ones digit of 1.
2. 4. The powers of 12 have the same ones
digits as the corresponding powers of 2.
Answers to Problem 5.1
A. Figure 2
3. 7. The powers of 17 have the same ones
digits as the corresponding powers of 7.
B. See the Explore notes for patterns in the ones
digits for each base. Here are some additional
patterns students might notice.
4. 1. The powers of 29 have the same ones
digits as the corresponding powers of 9.
• Square numbers a 2 have ones digits 1, 4, 9,
E. 1. 312 = 531, 441
6, 5, 6, 9, 4, 1, 0. There is symmetry around
the 5. This will repeat with each 10 square
numbers.
2. 99 = 387,420,489
3. 156 = 11,390,625. The base must have a
ones digit of 5. 56 can be ruled out without
directly computing; we know it is too small
because it is less than 10 6 = 1,000,000.
• Fourth powers (14, 24, 34, etc.) have ones
digits 0, 1, 5, and 6.
• The fifth powers (15, 25, 35, etc.) have ones
F. 1. 77 = 823543
digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, in that
order.
2. 98 = 43,046,721
Figure 2
Powers Table
x
1x
1
1
2
3
4
5
6
7
8
9
10
2
1
4
9
16
25
36
49
64
81
100
3
1
8
27
64
125
216
343
512
729
1,000
4
1
16
81
256
625
1,296
2,401
4,096
6,561
10,000
5
1
32
243
1,024
3,125
7,776
16,807
32,768
59,049
100,000
6
1
64
729
4,096
15,625
46,656
117,649
262,144
531,441
1,000,000
7
1
128 2,187 16,384
78,125
279,936
823,543
2,097,152
4,782,969
10,000,000
8
1
256 6,561 65,536 390,625 1,679,616 5,764,801 16,777,216 43,046,721 100,000,000
2x
3x
4x
Ones
1 2, 4, 3, 9,
Digits of
8, 6 7, 1
Powers
104 Growing, Growing, Growing
4, 6
5x
5
6x
6
7x
8x
7, 9, 3, 1
9x
8, 4, 2, 6
10x
9, 1
0