Goals
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Examine patterns in the exponential and standard forms of powers of whole numbers•
Use patterns in powers to estimate the onesdigits 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 amfor
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, 650, and 1720, and to estimate the standard form of these powers.
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 2xin the y-column.
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:
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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 215is the third number in this fourth cycle, so its ones digit is 8.
Students will not be able to find the ones digit of 250by 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 250until later and give students more time to explore.
•
What other patterns do you see in the table?Here are some patterns students may notice. (Some of these have already been mentioned.)
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The ones digits are all even.•
To predict the ones digit, all you need to know is the cycle of repeating units digits and whether 2xis 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.x 1 2 3 4 5 6 7 8 y 21 or 2 22 or 4 23 or 8 24 or 16 25 or 32 26 or 64 27 or 128 28 or 256
Launch
5.1
5.1
Predicting the Ones Digit
5.1
I N V E S T I G A T I O N 5•
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.•
Find an x-value and values for the missingdigits 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.
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 10nis 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.
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 2nrepeat 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 forExplore
5.1
5.1
x 3 4 7 y 23 or 8 24 or 16 27 or 128124, 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 912would 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 712is too large. They may use an argument similar to the
following but with words, not symbols: 72ø 50, so 712 = 72 ? 72 ? 72 ? 72 ? 72 ? 72 ø 50 ? 50 ? 50 ? 50 ? 50 ? 50, or
5 ? 5 ? 5 ? 5 ? 5 ? 5 ? 10 ? 10 ? 10 ? 10 ? 10 ? 10, which equals 56? 106.
We know that 106has 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.
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.
Summarize
5.1
5.1
I N V E S T I G A T I O N 5 Figure 1 x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 50 Ones Digit 2 4 8 6 2 4 8 6 2 4 8 6 2 4 8 6 4 2x 21 ! 2 22 ! 4 23 ! 8 24 ! 16 25 ! 32 26 ! 64 27 ! 128 28 ! 256 29 ! ■ 210 ! ■ 211 ! ■ 212 ! ■ 213 ! ■ 214 ! ■ 215 ! ■ 216 ! ■ 250 ! ■ PatternEnd of first cycle. The exponent 4 is a multiple of 4.
The exponent 5 has a remainder of 1 when divided by 4. The exponent 6 has a remainder of 2 when divided by 4. The exponent 7 has a remainder of 3 when divided by 4.
End of second cycle. The exponent 8 is a multiple of 4.
The exponent 9 has a remainder of 1 when divided by 4. The exponent 10 has a remainder of 2 when divided by 4. The exponent 11 has a remainder of 3 when divided by 4.
End of third cycle. The exponent 12 is a multiple of 4.
The exponent 13 has a remainder of 1 when divided by 4. The exponent 14 has a remainder of 2 when divided by 4. The exponent 15 has a remainder of 3 when divided by 4.
End of fourth cycle. The exponent 16 is a multiple of 4.
The exponent 50 has a remainder of 2 when divided by 4. So the ones digit is 4.
Predicting the Ones Digit
5.1
Launch
Explor
e
Summarize
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 powersMaterials
•
Transparencies 5.1A and 5.1B•
Labsheet 5.1 Vocabulary•
power Materials•
Student notebooks•
large sheet of poster paper (optional)At a Glance
Launch this problem by writing the values of y = 2x, for x = 1 to 8. Write both the exponential and standard form for 2xin 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?•
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.
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.
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. Go over some of the powers in Questions C and D. Be sure to have students explain their strategies.
ACE Assignment Guide
for Problem 5.1
Core 1–7, 54
Other Applications8, 9; Connections 44, 45;
Extensions 52, 53, 55; unassigned choices from previous problems
Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs
Handbook.
Connecting to Prior Units 44, 45: Data Around Us
Answers to Problem 5.1
A. Figure 2
B. See the Explore notes for patterns in the ones digits for each base. Here are some additional patterns students might notice.
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Square numbers a2have ones digits 1, 4, 9, 6, 5, 6, 9, 4, 1, 0. There is symmetry around the 5. This will repeat with each 10 square numbers.•
Fourth powers (14, 24, 34, etc.) have ones digits 0, 1, 5, and 6.•
The fifth powers (15, 25, 35, etc.) have ones digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, in that order.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.
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.
4.5. Any power of 5 has a ones digit of 5.
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.
3.7. The powers of 17 have the same ones digits as the corresponding powers of 7.
4.1. The powers of 29 have the same ones digits as the corresponding powers of 9.
E. 1. 312 = 531, 441
2.99= 387,420,489
3.156= 11,390,625. The base must have a ones digit of 5. 56can be ruled out without directly computing; we know it is too small because it is less than 106 = 1,000,000.
F. 1. 77 = 823543 2.98 = 43,046,721 Figure 2 x 1 2 3 4 5 6 7 8 Ones Digits of Powers 1x 1 1 1 1 1 1 1 1 1 2x 2 4 8 16 32 64 128 256 2, 4, 8, 6 3x 3 9 27 81 243 729 2,187 6,561 3, 9, 7, 1 4x 4 16 64 256 1,024 4,096 16,384 65,536 4, 6 5x 5 25 125 625 3,125 15,625 78,125 390,625 5 6x 6 36 216 1,296 7,776 46,656 279,936 1,679,616 6 7x 7 49 343 2,401 16,807 117,649 823,543 5,764,801 7, 9, 3, 1 8x 8 64 512 4,096 32,768 262,144 2,097,152 16,777,216 8, 4, 2, 6 9x 9 81 729 6,561 59,049 531,441 4,782,969 43,046,721 9, 1 10x 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 0 Powers Table
