# NCTM Content Standard/National Science Education Standard

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1 Title: BASE-ic Space Travel Brief Overview: This unit introduces the concepts of bases and exponents (or powers) in order to gain a deeper understanding of place value. Students will assume the role of intergalactic space travelers. Traveling in crews of four, they will journey to galaxies that use number systems other than base 10, including base 5 and base 2. To facilitate this travel, they must purchase fuel by learning about these bases, compare them to one another, and convert numbers between bases. NCTM Content Standard/National Science Education Standard Understand numbers, ways of representing numbers, relationships among numbers, and number systems: Understand the place value structure of the base-ten number system and be able to represent and compare whole numbers and decimals Recognize equivalent representations for the same number and generate them by decomposing and composing numbers Compute fluently and make reasonable estimates: Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools Grade/Level: Grade 5 (Accelerated Program) Duration/Length: 3 days (60 minutes per day) Student Outcomes: Students will: Distinguish between base and exponent (or power) Understand that in the base 10 system, each place value increases by the power of 10 Represent numbers using base subscript and exponent superscript Convert base 10 numbers to base 5 and numbers in base 5 to base 10 Explore conversions between bases other than base 5 and base 10

2 Materials and Resources: Day 1 Spacecraft Design (Student Resource 1) Base 10 materials (cubes, flats, units) for each group to have at least one of each Board/Overhead Example (Teacher Resource 1) Place Value Pocket Chart Directions (Teacher Resource 2A-B) You might want to make a pocket chart to use as a visual reference for students. Colored construction paper Glue sticks Scissors Markers 3 x 5 Index cards 10 per student (You may want to make a set of digit cards and trim to 2 x 5.) I Have/Who Has with Exponents (Teacher Resource 3) Cut apart each card to make 1 set per group of 12 students. Boarding Pass Day 1 exit ticket (Student Resource 2) Boarding Pass Day 1 Key (Teacher Resource 4) Day 2 Quintopian Creature (Teacher Resource 5) 40 colored chips per group (10 each of 4 different colors: white, red, blue, yellow) Additional Place Value Pocket Charts (Teacher Resource 7) Board/Overhead Example (Teacher Resource 6A-C) Other base 5 information can be found at: o htm o o Galaxy Quintopia (Student Resource 3A-B) Galaxy Quintopia Key (Teacher Resource 8A-B) Boarding Pass Day 2 exit ticket (Student Resource 4) Boarding Pass Day 2 Key (Teacher Resource 9) Day 3 Biaculean Creature (Teacher Resource 10) Venn Diagram (Student Resource 5) Venn Diagram Key (Teacher Resource 11) Additional Place Value Pocket Charts (Teacher Resource 6 from Day 2) Make an eight-column chart by taping two charts together and labeling with base 2 notation. Board/Overhead Example (Teacher Resource 12A-D) Galaxy Biaculus (Student Resource 6A-C) Galaxy Biaculus Key (Teacher Resource 13)

10 Summative Assessment: Authors: Distribute A Ticket Home Summative Assessment (Student Resource 8A-B) to board the spacecraft and travel back home to the Milky Way. Say: It s time to travel! You need to purchase fuel to get to your home base! In order to do that, you must make conversions from base 10 to base 5 to base 2 and back to base 10. Good luck on your mission. Graded assessments (see Teacher Resource 15A-B) will determine the level of performance. Marian Dingle Guilford Elementary School Howard County Public Schools Cheryl D. Madden Eastwood Center Baltimore County Public Schools

11 Student Resource 1 Name of Spacecraft Crewmembers

12 Student Resource 2 Boarding Pass #1 Day 1 Complete this boarding pass in order to board your spacecraft and travel to our first destination: Use the number 6,475 to answer the following questions: 1. What is the value of the digit 4? 2. In what place is the 4? 3. Write that place in exponential form. 4. What is the value of the digit 7? 5. In what place is the 7? 6. Write that place in exponential form. 7. In what place is the digit 6? 8. Write that place in exponential form. Use the expression 6 3 to answer the following questions: 9. Which digit is the base? 10. Which digit is the exponent? It s time to travel! Let s blast off! Tomorrow we will land at our first destination, Galaxy Quintopia!

13 Student Resource 3A Galaxy Quintopia Today our spacecraft arrived in the Galaxy of Quintopia. The inhabitants of Quintopia use the Base 5 place value system. You and your spacecrew need to translate Quintopian numbers into Planet Earth s Base 10 system! Look at the following example: 5 3 = = = = x 25 = 75 4 x 5 = 20 3 x 1 = 3 So, = = Now you try! Use the Base 5 Place Value Chart shown above to help you = x 25 + x 5 + x = = x x 25 + x 5 + x = = x 25 + x 5 + x = = = = 10

14 Now try converting from Base 10 into Base 5! Look at this example: = 5 Student Resource 3B Page Two Think There is one group of 125 in = 47. There is one group of 25 in = 22. There are four groups of 5 in = 2. There are two groups of 1 in = 1 x x x x 1 = = x x 25 + x 5 + x 1 = = x 25 + x 5 + x 1 = = x x 25 + x 5 + x 1 = 5

15 Student Resource 4 Boarding Pass #2 Day 2 Complete this boarding pass in order to board your spacecraft and travel to our next destination. Use the place value chart to help you. Be sure to show your work! 5 3 = = = = x 25 = 75 4 x 5 = 20 3 x 1 = 3 Convert this Base 5 number to Base = x x 25 + x 5 + x = 10 Convert this Base 10 number to Base = x x 25 + x 5 + x 1 = 5 It s time to travel! Let s blast off! Tomorrow we will land at our next destination, Galaxy Biaculus!

16 Base 5 Base 10 Student Resource 5

17 Student Resource 6A Galaxy Biaculus Today our spacecraft arrived in the Galaxy of Biaculus. The inhabitants of Biaculus use the Base 2 place value system. Since we just came from Galaxy Quintopia, you and your space crew need to translate Quintopian base 5 numbers into Biaculean base 2 numbers! Example: Convert to base x 125 = x 25 = 0 2 x 5 = 10 3 x 1 = First convert to base 10: 1 x x x x 1 = 3 = = Now convert to base x128=128 0x64=0 0x32=0 0 x 16 = 0 1 x 8 = 8 0 x 4 =0 1 x 2 = 2 0x 1 = Think There is one group of 128 in = 10. There are zero groups of 64, zero groups of 32, and zero groups of 16 in 10. There is one group of 8 in = 2. There are zero groups of 4 in 2, one group of 2 in 2, and zero groups of 1. Answer: Now you try, using the above charts to help you.

18 Student Resource 6B Galaxy Biaculus (page 2) Convert these numbers from base 5 to base = x 25 + x 5 + x = = x x 64 + x 32 + x 16 + x 8 + x 4 + x = = x x 25 + x 5 + x = = x x 64 + x 32 + x 16 + x 8 + x 4 + x = 2

19 Student Resource 6C Galaxy Biaculus (page 3) Almost home! In order to get there, you and your space crew should practice converting Biaculean numbers back into Earth s base 10 numbers! Look at the following example: x 16 = 16 1 x 8 = 8 0 x 4 =0 1 x 2 = 2 1 x 1 = So, = = Now you try! Use the Base 2 Place Value Chart shown above to help you = x x 64 + x 32 + x 16 + x 8 + x 4 + x 2 x = = x x 64 + x 32 + x 16 + x 8 + x 4 + x 2 x = 10 CHALLENGE: Can you convert to a base 5 number?

20 Student Resource 7 Quintopian Practice Use the chart to convert 4321 in base 5 to base 10. Base 5 Places Step Step 2 4 x 125 = x 25 =75 2 x 5 = 10 1 x 1 = 1 Step Step Write the digits in the chart. 2. Multiply the digit by the place. 3. Add the values of each place. 4. The sum is the number in base 10. Now, use the chart to help convert from base 5 to base Step 1 Step 2 Step 3 Step 4 Base 5 Places Step 1 Step 2 Step 3 Step 4 Base 5 Places

21 Student Resource 8A A Ticket Home Day 3 In order to train future space cadets in BASE-ic space travel, please complete the following to show all that you have learned. Good luck on your mission. 1. The following number is written in base 10. What does the digit 4 represent? A 4 B 40 C 400 D 4, In the base 10 number below, the 3 is in what place? 3, A ones B tens C hundreds D thousands 3. In the following number, the digit 3 is called the: 2 3 A base B exponent C square D cube 4. The number 100 can be represented by which expression? A 10 0 B 10 1 C 10 2 D 10 3

22 Student Resource 8B A Ticket Home Day 3 Brief Constructed Response Complete this boarding pass in order to board your spacecraft and travel back to our home base, Planet Earth: Part A Convert the following base 5 number to base 10: = 10 Part B Use what you know about place value to explain why your answer is correct. Use numbers and/or words in your explanation. Let s go home!!

23 Teacher Resource 1A Board/Overhead Example Day 1 Write a 4-digit number on the board, as below: 3562 Ask students to identify places and record underneath each digit thousand hundred ten one Then write the numeral of each place thousand hundred ten one Ask students to help you represent each number using the number 10. You will eventually have this: thousand hundred ten one x 10 x x (leave blank) Explain that there is another way to represent numbers that are multiplied by themselves. Define base as a number that is multiplied by itself and exponent or power as the number of times it is multiplied. It is written like this: base exponent or base power Ask the class these questions: How many times is the 10 multiplied in the thousands place? (3) How many times is the 10 multiplied in the hundreds place? (2) How many times is the 10 multiplied in the tens place? (1) How many times is the 10 multiplied in the ones place? (0) Since 10 multiplied by itself 0 times in the ones place, it is written as Add that any base raised to the zero power is one.

24 Teacher Resource 1B Board/Overhead Example Day 1 Add each exponential expression underneath each place, so it looks like this: thousand hundred ten one x 10 x x (leave blank) Ask: How do we write the ones place? (10 to the zero power.) How do we write the tens place? (10 to the first power.) How do we write the hundreds place? (10 to the second power.) How do we write the thousands place? (10 to the third power.)

25 Teacher Resource 2A Place Value Pocket Chart Directions Student Materials: one piece of 9 x 12 construction paper glue stick marker index cards scissors Directions: 1. Holding the construction paper horizontally, fold approximately one-half of the bottom portion upward. 2. Crease the fold so that it lies flat and forms a pocket at the bottom. pocket 3. Fold the paper horizontally in half, then in half again, so that three creases are formed, dividing the sheet into fourths. Unfold. Glue pockets at outer edges. glue glue

26 Teacher Resource 2B 4. Use a marker to draw lines on the place value pocket chart over each vertical crease. 5. Label the bottom of each pocket with the appropriate base and power and corresponding word name. (for Base 10 charts: 10 0 ones; 10 1 tens; 10 2 hundreds; 10 3 thousands; for Base 5 charts: 5 0 ones; 5 1 fives; 5 2 twentyfives; 5 3 one hundred twenty-fives etc.) 10 3 thousands 10 2 hundreds 10 1 tens 10 0 ones 6. Have students trim 10 index cards to approximately 2 x 5 (trim off about one inch so that the cards fit easily into each pocket). Have students label each card with a digit from 0 through 9. The cards will be inserted into the pockets to demonstrate place value of digits in various bases.

27 I Have Who Has with Exponential Numbers Teacher Resource 3 I have 100. Who has 2 2? I have 4. Who has 3 2? I have 9. Who has 4 2? I have 16. Who has 5 2? I have 25. Who has 6 2? I have 36. Who has 9 2? I have 81. Who has 7 2? I have 49. Who has 8 2? I have 64. Who has 2 3? I have 8. Who has 3 3? I have 27. Who has 5 3? I have 125. Who has 10 2?

28 Teacher Resource 4 Boarding Pass #1 Day 1 Answer Key Complete this boarding pass in order to board your spacecraft and travel to our first destination: Use the number 6,475 to answer the following questions: 1. What is the value of the digit 4? 2. In what place is the 4? 3. Write that place in exponential form. 400 hundreds What is the value of the digit 7? 5. In what place is the 7? 70 tens 6. Write that place in exponential form In what place is the digit 6? 8. Write that place in exponential form. Use the expression 6 3 to answer the following questions: 9. Which digit is the base? 10. Which digit is the exponent? thousands It s time to travel! Let s blast off! Tomorrow we will land at our first destination, Galaxy Quintopia!

29 Quintopian Creature Teacher Resource 5

30 Teacher Resource 6A Board/Overhead Example Day 2 Write a 2-digit number on the board, as below: Explain: The number reads, seventy-two, base 10. The base is written as a subscript. Since we have always dealt with base 10 numbers, we have dropped the base notation until now. Now, write the first four places for the base 5 system. Remember, each digit increases by a power of 5, and that the digit to the right is always the unit or one, which is five to the zero power. Now, explain that the next place has a value of 5, which is 5 to the first power. one 1 five one 5 1 The next place has a value of 25, which is five to the second power. Similarly, the next place has a value of 125, which is five to the third power. one hundred twenty-five twenty-five five one x 5 x 5 5 x Base System Conversions: Our next challenge is to convert our base 10 number to base 5. Step 1: What is the highest place in base 5 that is less than or equal to 72? (25) How many groups of 25 are in 72? (2)

31 Teacher Resource 6B Board/Overhead Example Day 2 So, we write a 2 in the 25 place. 2 one hundred twenty-five twenty-five five one Since we have 2 groups of 25 which makes 50, subtract 50 from 72 to get 22. Step 2: Now go to the next place to the right, which is the 5 place. How many groups of 5 are in 22? (4) So, we write a 4 in the 5 place. 2 4 one hundred twenty-five twenty-five five one Since we have 2 groups of 5 which makes 20, subtract 20 from 22 to get 2. Step 3: Now go to the next place to the right, which is the 1 place. How many groups of 1 are in 2? (2) So, we write a 2 in the 1 place one hundred twenty-five twenty-five five one So, = 242 5

32 Teacher Resource 7 Additional Place Value Pocket Charts Base Five 5 3 one hundred 5 2 twenty- 5 1 fives 5 0 ones twenty-fives fives Base Two 2 7 one hundred 2 5 thirtyseconds twentyeighths 2 6 sixtyfourths 2 4 sixteens 2 3 eights 2 2 fours 2 1 twos 2 0 ones

33 Teacher Resource 8A Galaxy Quintopia Answer Key Today our spacecraft arrived in the Galaxy of Quintopia. The inhabitants of Quintopia use the Base 5 place value system. You and your spacecrew need to translate Quintopian numbers into Planet Earth s Base 10 system! Look at the following example: 5 3 = = = = x 25 = 75 4 x 5 = 20 3 x 1 = 3 So, = = Now you try! Use the Base 5 Place Value Chart shown above to help you = 1 x x 5 + 4x = = 4 x x x 5 + 0x = = 3 x x x = = = = 66 10

34 Teacher Resource 8B Now try converting from Base 10 into Base 5! Look at this example: = 5 Think There is one group of 125 in = 47. There is one group of 25 in = 22. There are four groups of 5 in = 2. There are two groups of 1 in = 1 x x x x 1 = = 1 x x x x = = 2 x x x = = 2 x x x x =

35 Teacher Resource 9 Boarding Pass #2 Day 2 Complete this boarding pass in order to board your spacecraft and travel to our next destination. Use the place value chart to help you. Be sure to show your work! 5 3 = = = = x 25 = 75 4 x 5 = 20 3 x 1 = 3 Convert this Base 5 number to Base = 3 x x x x = Convert this Base 10 number to Base = 1 x x x x = It s time to travel! Let s blast off! Tomorrow we will land at our next destination, Galaxy Biaculus!

36 Biaculean Creature Teacher Resource 10

37 Venn Diagram Answer Key Base 5 Base 10 Sample Answers: five digits, 0 5 places are ones, fives, twentyfives, one hundred twenty-fives, etc. Sample Answers: digits 0 5 places for different values values grow exponentially Sample Answers: ten digits, 0 9 places are ones, tens, hundreds, thousands, etc. Teacher Resource 11

38 Teacher Resource 12A Board/Overhead Example Day 3 Write a 3-digit base 5 number on the board, as below: Convert base 5 to base 2 in two parts. First, convert base 5 to base 10, then base 10 to base 2. Write the digits in their base 5 places. Remember, each digit increases by a power of 5, and that the digit to the right is always the unit or one, which is five to the zero power twenty-five five one Convert to base 10: Add: = 93 3 x 25 = 75 Answer: x 5 = 15 3 x 1 = 3 Now convert this number to base 2 by writing the place values for base 2. You ll need more places because the denominations increase at a slower rate. Start with the one place and add each place to the left by multiplying by 2, one at a time (as in Day 1 and Day 2). sixty-four thirty-two sixteen eight four two one x2x2x2x2x2 2x2x2x2x2 2x2x2x2 2x2x2 2x Step 1: What is the highest place in base 2 that is less than or equal to 93? (64) How many groups of 64 are in 93? (1)

39 Teacher Resource 12B Board/Overhead Example Day 3 So, we write a 1 in the 64 place. 1 sixty-four thirty-two sixteen eight four two one Since we have 1 group of 64 which makes 64, subtract 64 from 93 to get 29. Step 2: Now go the next place to the right which is the 32 place. How many groups of 32 are in 29? (0) So, we write a 0 in the 32 place. 1 0 sixty-four thirty-two sixteen eight four two one Since we have 0 groups of 32 which makes 0, subtract 0 from 29 to get 29. Step 3: Now go the next place to the right which is the 16 place. How many groups of 16 are in 29? (1) So, we write a 1 in the 16 place sixty-four thirty-two sixteen eight four two one Since we have 1 group of 16 which makes 16, subtract 16 from 29 to get 13. Step 4: Now go the next place to the right which is the 8 place.

40 Teacher Resource 12C Board/Overhead Example Day 3 How many groups of 8 are in 13? (1) So, we write a 1 in the 8 place sixty-four thirty-two sixteen eight four two one Since we have 1 group of 8 which makes 8, subtract 8 from 13 to get 5. Step 5: Now go the next place to the right which is the 4 place. How many groups of 4 are in 5? (1) So, we write a 1 in the 4 place sixty-four thirty-two sixteen eight four two one Since we have 1 group of 4 which makes 4, subtract 4 from 5 to get 1. Step 6: Now go the next place to the right which is the 2 place. How many groups of 2 are in 1? (0) So, we write a 0 in the 2 place sixty-four thirty-two sixteen eight four two one Since we have 0 groups of 4 which makes 0, subtract 0 from 1 to get 1.

41 Teacher Resource 12D Board/Overhead Example Day 3 Step 7: Now go the next place to the right which is the 1 place. How many groups of 1 are in 1? (1) So, we write a 1 in the 1 place sixty-four thirty-two sixteen eight four two one So, Observation: = = The process of conversion to base 2 may seem tedious, but when deciding how many groups are in a given place, the choices are either 0 or 1. Thus, the mental math is simpler than in other base conversions.

42 Teacher Resource 13A Galaxy Biaculus Answer Key Today our spacecraft arrived in the Galaxy of Biaculus. The inhabitants of Biaculus use the Base 2 place value system. Since we just came from Galaxy Quintopia, you and your space crew need to translate Quintopian base 5 numbers into Biaculean base 2 numbers! Example: Convert to base x 125 = x 25 = 0 2 x 5 = 10 3 x 1 = First convert to base 10: 1 x x x x 1 = 3 = = Now convert to base x128=128 0x64=0 0x32=0 0 x 16 = 0 1 x 8 = 8 0 x 4 =0 1 x 2 = 2 0x 1 = Think There is one group of 128 in = 10. There are zero groups of 64, zero groups of 32, and zero groups of 16 in 10. There is one group of 8 in = 2. There are zero groups of 4 in 2, one group of 2 in 2, and zero groups of 1. Answer: Now you try, using the above charts to help you.

43 Teacher Resource 13B Galaxy Biaculus Answer Key Convert these numbers from base 5 to base = 4 x x x = = 1 x x x x x x x = = 2 x x x x = = 1 x x x x x x x x =

44 Teacher Resource 13C Galaxy Biaculus Answer Key Almost home! In order to get there, you and your space crew should practice converting Biaculean numbers back into Earth s base 10 numbers! Look at the following example: x 16 = 16 1 x 8 = 8 0 x 4 =0 1 x 2 = 2 1 x 1 = So, = = Now you try! Use the Base 2 Place Value Chart shown above to help you = (1 x 32) + (0 x 16) + (1 x 8) + (1 x 4) + (0 x 2)+ (1 x 1) = = 1 x x x x x x x 2+ 0 x = CHALLENGE: Can you convert to a base 5 number?

45 Teacher Resource 14 Quintopian Practice Answer Key Use the chart to convert 4321 in base 5 to base 10. Base 5 Places Step Step 2 4 x 125 = x 25 =75 2 x 5 = 10 1 x 1 = 1 Step Step Write the digits in the chart. 2. Multiply the digit by the place. 3. Add the values of each place. 4. The sum is the number in base 10. Now, use the chart to help convert from base 5 to base Base 5 Places Step Step 2 3 x x 25 4 x 5 4 x 1 Step Step Base 5 Places Step Step 2 1 x x 25 4 x 5 1 x 1 Step Step

46 Teacher Resource 15A A Ticket Home Day 3 Answer Key In order to train future space cadets in BASE-ic space travel, please complete the following to show all that you have learned. Good luck on your mission. 3. The following number is written in base 10. What does the digit 4 represent? A 4 B 40 C 400 D 4, In the base 10 number below, the 3 is in what place? 3, A ones B tens C hundreds D thousands 3. In the following number, the digit 3 is called the: 2 3 A base B exponent C square D cube 4. The number 100 can be represented by which expression? A 10 0 B 10 1 C 10 2

47 D 10 3

48 Teacher Resource 15B A Ticket Home Day 3 Answer Key Brief Constructed Response Complete this boarding pass in order to board your spacecraft and travel back to our home base, Planet Earth: Part A Convert the following base 5 number to base 10: = Part B Use what you know about place value to explain why your answer is correct. Use numbers and/or words in your explanation. To convert from Base 5 to Base 10, it is important to understand the place value of each digit. In Base 5, the digit farthest to the right has a value of one, so you multiply that digit by one. The next digit represents five, so you multiply that digit by five. The next place has a value of 25, so you multiply that digit by twenty-five. You continue in this manner, and each place in Base 5 is worth five times more than the previous place. After the value of each digit is determined, the values are added to calculate the Base 10 equivalent. See the chart below: Base 5 Places Step Step 2 0 x 125 = 0 1 x 25 =25 4 x 5 = 20 3 x 1 = 3 Step Step Let s go home!!

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