. Predicting the Ones Digit Goals Eamine patterns in the eponential 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 eample, by eamining the ones digits of the powers of ( =, =, =, =, =, =, =, =, and so on) students see that the sequence,,, 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,, 9,, 9,, and so on. We say that the ones digits for the powers of 9 repeat in a cycle of. The ones digits for powers of are all that is, the ones digits repeat in a cycle of. The ones digits of the powers of any whole number repeat in a cycle of length,, or. Students create a powers table for a m for a = to a = 0 and m = to m =. They use the table to find patterns in order to predict the ones digits for powers such as 00, 0, and 0, and to estimate the standard form of these powers. Launch. Launch this problem by writing the values of y =, for whole-number -values from to. Write both the eponential and standard form for in the y-column. y or or or or or or or or 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 eponents 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:,,, and. If necessary, etend the table to convince students that this pattern continues. Can you predict the ones digit for? () What about 0? () Some students will continue the table or use their calculator to find. 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, and the fourth cycle starts with. The number is the third number in this fourth cycle, so its ones digit is. Students will not be able to find the ones digit of 0 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 epect students to answer this quickly. You might postpone 0 until later and give students more time to eplore. What other patterns do you see in the table? 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 is the cycle of repeating units digits and whether is the first, second, third, or fourth number in a cycle. If you divide the eponent by, the remainder tells you where the power is in a cycle. INVESTIGATION Investigation Patterns With Eponents 99
When you multiply two powers of, the eponent in the product is the sum of the eponents of the factors. For eample, =. Find an -value and values for the missing digits that will make this a true number sentence: = ( = 0, which gives 0 =,0,) Net, introduce Problem.. 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. Eplore. 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 net problem. Although the problem focuses on patterns in the ones digits, students may notice other interesting patterns as well. Here are some eamples: Some numbers occur more than once in the powers chart. For eample, appears three times: =, =, and =. When you multiply powers of, the eponent of the product is the sum of the eponents of the factors. For eample, = + =. The number of zeros in 0 n 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 eponents, continuing the table gets tedious, and the calculator rounds off after 0 digits. Students should begin to focus on the patterns in the ones digits. y or or or 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? (,, or, depending on the base. For eample, for the powers of, the ones digits, 9,, repeat, so the cycle is of length.) Which bases have cycles of length? (,,, and ) Which bases have a cycle of length? (,,, and 0) Which bases have a cycle of length? ( and 9) If you know the eponent, how can you use the pattern of the cycle to determine the ones digit of the power? For eample, the ones digits for n repeat in a cycle:,,,. How can we use this fact to find the ones digit of? (For powers of, the eponents,, 9,,,, and so on correspond to a ones digit of. Students may recognize that each of these numbers is one more than a multiple of. In other words, when you divide these numbers by you get a remainder of. The eponents,, 0,,, and so on, correspond to a ones digit of. Students may recognize that these numbers are more than a multiple of. The eponents,,,, 9, and so on correspond to a ones digit of. These numbers are more than a multiple of. Finally, the eponents,,, and so on correspond to a ones digit of.) 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 eample, the ones digits for powers of are the same as the ones digits for powers of,, and 9. The ones digits for powers of are the same as the ones digits for powers of,, and. If students struggle with this idea, ask: If you were to look at the ones digits for powers of, you would find that they follow the same pattern as the ones digits for the powers of. Why do you think this is true? What affects the ones digit? (To get successive powers of, you multiply by, so the ones digit will be the same as the ones digit of times the previous ones digit. For, the ones digit is ; for, the ones digit is because = ; for, the ones digit is because = ; and for 00 Growing, Growing, Growing
, the ones digit is because =. Then the ones digits start to repeat. A similar argument will work for all whole-number bases greater than 0.) In Question E, remind students to use their knowledge about the patterns of the ones digit to narrow the choices. For eample, for a =,, a must be,,, or 9. Obviously, is not a choice. And 9 would be close to 0, which has digits much too large a number. The number, has only digits. At this point, students should argue that is too large. They may use an argument similar to the following but with words, not symbols: 0, so =????? 0? 0? 0? 0? 0? 0, or?????? 0? 0? 0? 0? 0? 0, which equals? 0. We know that 0 has digits, which is too large, so the answer is =,. Be sure to make note of interesting patterns, reasoning, and questions that arise during the Eplore. Summarize. 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 net 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 eplain some of the patterns until the net problem, when the properties of eponents are developed. Go over some of the powers in Questions C and D. Be sure to have students eplain their strategies. After a student or group has eplained 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 eponent, and the ones digit, you can use the table in Figure. Save the completed powers table for the launch of Problem.. Be sure to assign ACE Eercise. It is needed for Problem.. Figure Pattern Ones Digit End of first cycle. The eponent is a multiple of. The eponent has a remainder of when divided by. The eponent has a remainder of when divided by. The eponent has a remainder of when divided by. End of second cycle. The eponent is a multiple of. 9 9 The eponent 9 has a remainder of when divided by. 0 0 The eponent 0 has a remainder of when divided by. The eponent has a remainder of when divided by. End of third cycle. The eponent is a multiple of. 0 0 The eponent has a remainder of when divided by. The eponent has a remainder of when divided by. The eponent has a remainder of when divided by. End of fourth cycle. The eponent is a multiple of. The eponent 0 has a remainder of when divided by. So the ones digit is. INVESTIGATION Investigation Patterns With Eponents 0
0 Growing, Growing, Growing
. Predicting the Ones Digit Mathematical Goals At a Glance PACING days Eamine patterns in the eponential 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 =, for = to. Write both the eponential and standard form for 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? Can you predict the ones digits for? What about 0? 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. Materials Transparencies.A and.b Labsheet. Vocabulary power Eplore 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? Which bases have a cycle of length? Which bases have a cycle of length? If you know the eponent, 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, you would find that they follow the same pattern as the ones digits for the powers of. 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 net 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 eplain their strategies. Materials Student notebooks large sheet of poster paper (optional) Investigation Patterns With Eponents 0
ACE Assignment Guide for Problem. Core, Other Applications, 9; Connections, ; Etensions,, ; unassigned choices from previous problems Adapted For suggestions about adapting ACE eercises, see the CMP Special Needs Handbook. Connecting to Prior Units, : Data Around Us Answers to Problem. A. Figure B. See the Eplore notes for patterns in the ones digits for each base. Here are some additional patterns students might notice. Square numbers a have ones digits,, 9,,,, 9,,, 0. There is symmetry around the. This will repeat with each 0 square numbers. Fourth powers (,,, etc.) have ones digits 0,,, and. The fifth powers (,,, etc.) have ones digits,,,,,,,, 9, and 0, in that order. C... The even powers of have as a ones digit... The even powers of 9 have as a ones digit... There is a cycle of length in the ones digits of the powers of. is the beginning of the fifth cycle... Any power of has a ones digit of.. 0. Any power of 0 has a ones digit of 0. D... The ones digit of is, so all powers of will have a ones digit of... The powers of have the same ones digits as the corresponding powers of... The powers of have the same ones digits as the corresponding powers of... The powers of 9 have the same ones digits as the corresponding powers of 9. E.. =,. 9 9 =,0,9. =,90,. The base must have a ones digit of. can be ruled out without directly computing; we know it is too small because it is less than 0 =,000,000. F.. =. 9 =,0, Figure Powers Table 9 9 0 0 9 9 00 9,000,9,0,09, 0,000,0,,,0, 9,09 00,000 9,09,,,9,,,000,000,,, 9,9,,09,,,99 0,000,000,, 90,,9,,,0,,,0, 00,000,000 Ones Digits of Powers,,,, 9,,,, 9,,,,, 9, 0 0 Growing, Growing, Growing