Unit 2 Number and Operations in Base Ten: Place Value, Addition, and Subtraction


 Claude Parrish
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1 Unit 2 Number and Operations in Base Ten: Place Value, Addition, and Subtraction Introduction In this unit, students will review place value to 1,000,000 and understand the relative sizes of numbers in each place. Students will also practice representing numbers with base ten materials, and convert between numbers and expanded form. Students will order sets of two or more numbers, and will find the greatest and least number possible using the digits given. Students will add and subtract with and without regrouping, use pictures and charts to solve word problems, and recognize what is given and what they need to find in a word problem. Finally, students will round to the closest ten, hundred, or thousand, and estimate sums and differences by rounding each addend to the nearest ten, hundred, or thousand. Number and Operations in Base Ten C1
2 NBT51 Place Value Pages STANDARDS 5.NBT.A.3, 5.NBT.A.1 Vocabulary expanded form hundreds hundred thousands millions ones place value ten thousands tens thousands Goals Students will identify the place value and the actual value of digits in numbers up to 7 digits, and learn to write number words for numbers with up to 7 digits. PRIOR KNOWLEDGE REQUIRED Knows the number words one, ten, hundred, thousand, ten thousand, hundred thousand, million, and their corresponding numerals MATERIALS BLM Place Value Cards (p. C58) scissors tape or sticky tack Introduce place value. Photocopy BLM Place Value Cards and cut out the five cards. Write the number 65,321 on the board, leaving extra space between all the digits, and hold the ones card under the 3. ASK: Did I put the card in the right place? Is 3 the ones digit? Have a volunteer put the card below the correct digit. Invite volunteers to position the other cards correctly. Cards can be temporarily affixed to the board using tape or sticky tack. Exploring more about place value. Write 23,989 on the board and ask students to identify the place value of the underlined digit. (NOTE: If you give each student a copy of BLM Place Value Cards, individuals can hold up their answers. Have students cut out the cards before you begin.) Repeat with several numbers that have an underlined digit. Include whole numbers with anywhere from 1 to 5 digits. (MP.6) Vary the question slightly by asking students to find the place value of a particular digit without underlining it. (Example: Find the place value of the digit 4 in these numbers: 24,001, 42,305, 34,432, 32,847.) Continue until students can identify place value correctly and confidently. Include examples where you ask for the place value of the digit 0. Write 28,306 on the board. SAY: The number 28,306 is a 5digit number. What is the place value of the digit 2? (If necessary, point to each digit as you count aloud from the right: ones, tens, hundreds, thousands). SAY: The 2 is in the ten thousands place, so it stands for 20,000. What does the digit 8 stand for? (8,000) the 3? (300) the 0? (0 tens or just 0) the 6? (6) ASK: What is the value of the 6 in 2,608? (600) in 36,504? (6,000) in 63,504? (60,000) in 6,504? (6,000) ASK: In the number 6,831, what does the digit 3 stand for? (30) the 6? (6,000) the 1? (1) The 8? (800) C2 Teacher s Guide for AP Book 5.1
3 ASK: What is the value of the 0 in 340? (0) in 403? (0) in 8,097? (0) Write out the corresponding addition statements so students see that 0 always has a value of 0, no matter what position it is in. ASK: In the number 7,856, what is the tens digit? (5) the thousands? (7) ones? (6) hundreds? (8) Repeat for 3,050, 455, 77,077. Introduce hundred thousands and millions place value. Follow the same procedure as in the previous lesson to introduce the place values hundred thousands and millions. Write the correct spelling for all place values on the board. Exercises: Identify the place value of the digit 3 in each number. a) 312,607 b) 453,207 c) 3,762,906 d) 7,401,235 Answers: a) 300,000, b) 3,000, c) 3,000,000, d) 30 Determining how many times as much or as many. Review the concept how many times as many if necessary. Begin with twice as many by having students solve word problems such as: Carrie has three tickets to a concert. Rob has twice as many as Carrie. How many tickets does Rob have? Draw three tickets on the board and write Carrie s name beside them. Then write Rob s name to the right of Carrie s and SAY: I know that Rob has twice as many tickets as Carrie, so I just have to draw Carrie s number of tickets twice. Carrie Rob Have students solve word problems with three times as many and emphasize that you have to draw the lesser number of objects three times to make the greater number. ASK: How many times as much money is a dollar worth than a dime? (10) than a nickel? (20) than a penny? (100) How many times as much money is a dime worth than a nickel? (2) than a penny? (10) How many times as much money is a quarter worth than a nickel? (5) a dollar worth than a quarter? (4) How many times as many blocks are in a hundreds block than in a tens block? (10) than a ones block? (100) How many times as many blocks are in a thousands block than in a hundreds block? (10) than a tens block? (100) than a ones block? (1,000) Remind students that we can speak of numbers as being twice as many or three times as many or four times as many as another number. Exercises: Fill in the blanks. a) 12 = 3, so 12 is times as many as 3. b) 20 = 4, so 20 is times as many as 4. Number and Operations in Base Ten 51 C3
4 c) 100 = 10, so 100 is times as many as 10. d) 1,000 = 10, so 1,000 is times as many as 10. e) 30,000 = 300, so 30,000 is times as many as 300. f) 4,000 is times as many as 40. g) 400,000 is times as many as 400. Answers: a) 4, b) 5, c) 10, d) 100, e) 100, f) 100, g) 1,000 ASK: What is the value of the first 1 in the number 1,716? (1,000) What is the value of the second 1? (10) How many times as many is the first 1 worth than the second 1? (100) How many tens blocks would you need to make a thousands block? (100) Repeat with more numbers in which the digit 1 is repeated (examples: 1,341, 11, 101, 21,314). ASK: What is the value of the first 3 in 3,231? (3,000) the second 3? (30) How many times as many is the first 3 worth than the second 3 in 3,231? (100) Exercises: Copy and complete the chart. Add rows for b) through d). First Number Second Number How many times as many? a) 3, b) 5,000 and 50 c) 500,000 and 5,000 d) 800,000 and 8 Answers: b) 100, c) 100, d) 100,000 Bonus: Give students who finish quickly greater numbers with the two threes moving farther and farther apart, ending with a number such as: 823,450,917,863,401 Have students fill in a chart like the following: (MP.7, MP.8) Number How many places to the right of the first 3 is the second 3? How many times as many is the first 3 worth than the second 3? How many 0s are in the number from the previous column? 3, , ,383 1,233 Make up your own number What pattern do you notice? Does the pattern help you solve the Bonus problem? Answer: The second 3 is 9 places over from the first 3 in the Bonus, so to write how many times as many the first 3 is worth than the second 3, write 1 with 9 zeroes: 1,000,000,000. C4 Teacher s Guide for AP Book 5.1
5 Reading and writing multidigit numbers. Write on the board: 32,000 ASK: How many thousands are there? Show students how to write thirtytwo thousand. Exercises: Write number words for multiples of 1,000. a) 17,000 b) 20,000 c) 489,000 d) 638,000 e) 704,000 Answers: a) seventeen thousand, b) twenty thousand, c) four hundred eightynine thousand, d) six hundred thirtyeight thousand, e) seven hundred four thousand Exercises: Write the number words for 5digit and 6digit numbers. a) 23,802 b) 254,006 c) 30,109 d) 140,019 e) 632,540 Answers: a) twentythree thousand eight hundred two, b) two hundred fiftyfour thousand six, c) thirty thousand one hundred nine, d) one hundred forty thousand nineteen, e) six hundred thirtytwo thousand five hundred forty Write out the place value words for numbers with up to 12 digits: ones tens hundreds thousands ten thousands hundred thousands millions ten millions hundred millions billions ten billions hundred billions Point out that after the thousands, there is a new word at every three place values. This is why we put commas between every three digits in numbers so that we can see when a new word will be used. This helps us to quickly identify and read multidigit numbers. Demonstrate this using the number 3,456,720,603, which is read as three billion four hundred fiftysix million seven hundred twenty thousand six hundred three. Then write another multidigit number on the board 42,783,089,320 and ASK: How many billions are in this number? (42) How many millions? (783) How many thousands? (89) Then read the whole number together. Extensions 1. Identify and write the numbers given specific criteria and constraints. a) Write a number between 12,730 and 12,370. b) Write a 5digit even number with the tens digit 6. c) Write an odd number greater than 9,860. d) Write a number with the tens digit one more than the ones digit. Number and Operations in Base Ten 51 C5
6 Harder (MP.1) e) Write a 5digit number with the tens digit eight less than the ones digit. f) Write a 6digit number in which the first three digits are odd. g) Write a number between 50 and 60 with both digits the same. h) Write a 3digit even number with all its digits the same. i) Find the sum of the digits in each of these numbers: 37, 48, 531, 225, 444, 372. j) Write a 3digit number in which all digits are the same and the sum of the digits is 15. k) Which number has the tens digit one less than the ones digit: 324,540, 880, 11,034, 909? l) Write a 4digit number in which the digits are all equal and the sum of the digits is 12. Make up more such questions, or have students make up their own. Sample answers: a) 12,573, b) 10,468, c) 9,957, d) 4,187, e) 81,619, f) 131,589 Answers: g) 55; h) 222, 444, 666, or 888; i) 10, 12, 9, 9, 12, 12; j) 555, k) 11,034, l) 3,333 (MP.7) 2. Which has a greater place value in the following numbers, the 3 or the 6? How many times as many is the digit worth? a) 63 b) 623 c) 6,342 d) 36 e) 376 f) 3,006 g) 6,731 h) 7,362 i) 9,603 j) 3,568 k) 3,756 l) 3,765 If students need help, ask them how they could turn each question into a problem they already know how to solve. For example, if it was 323 instead of 623, you would know how to solve it. (the first 3 is worth 100 times as many as than the second 3) How is 623 different from 323? (6 is two times as many as 3, so the 6 is worth 200 times as many as the 3) Answers: a) 6, 20 times; b) 6, 200 times; c) 6, 20 times; d) 3, 5 times; e) 3, 50 times; f) 3, 500 times; g) 6, 200 times; h) 3, 5 times; i) 6, 200 times; j) 3, 50 times; k) 3, 500 times; l) 3, 50 times; m) 6, 200 times C6 Teacher s Guide for AP Book 5.1
7 NBT52 Representation in Expanded Form Pages STANDARDS 5.NBT.A.3 Goals Students will practice representing numbers with base ten materials, and convert between numbers and expanded form. Vocabulary digit hundreds block numeral ones block place value tens block PRIOR KNOWLEDGE REQUIRED Knows place values for ones, tens, hundreds, thousands, and ten thousands Is familiar with base ten materials Knows the number words for place values of one, ten, hundred, thousand, ten thousand, hundred thousand, million, and their corresponding numerals MATERIALS 5 hundreds blocks per student 25 tens blocks per student 15 ones blocks per student Thousands block Hundreds block Tens block Ones block Introduce the standard way to represent numbers using blocks. As students group and manipulate base ten materials throughout the lesson, monitor the models they create on their desks and/or have students sketch their answers on paper so you can verify their understanding. Students can work individually or in small groups. Give each student three tens blocks and fifteen ones blocks. Ask them to make the number 17 and to explain how their model represents that number. PROMPTS: How many tens blocks do you have? (1) How many ones are in a tens block? (10) How many ones are there altogether? (17) Repeat with more 2digit numbers (examples: 14, 19, 28, 34, 32, 25). Then give each student 3 hundreds blocks and have them make 3digit numbers (examples: 235, 129, 316). Use prompts, if needed, to help students break down their models. For the number 235, ASK: How many ones are in the hundreds blocks? (200) tens blocks? (30) ones blocks? (5) How many ones are there altogether? (235) Have students work in pairs to make the following: 12 using exactly 12 blocks, 22 using 13 blocks, 25 using 16 blocks, and 31 using 13 blocks. Some of these models will be nonstandard (example: 22 using 13 blocks = 1 tens block and 12 ones blocks, instead of the standard 2 tens blocks and 2 ones blocks). Encourage students to make a standard model first, then ask themselves whether they need more blocks or fewer blocks. (they will always need more blocks) Which blocks can they trade to keep the value the same but increase the number of blocks? (answers will vary) Number and Operations in Base Ten 52 C7
8 Review how to draw the base ten blocks, especially the thousands cube, as shown on p. 13 in the AP Book. Have students draw models for various 4digit numbers (examples: 2,541, 1,082, 3,204). Review expanded form. ASK: How much is the 4 worth in 459? (400) the 5? (50) the 9? (9) Have a volunteer write the expanded form of 459 on the board. (459 = ) Exercises: Write the numbers in expanded form. a) 352 b) 896 c) 784 Answers: a) 352 = ; b) 896 = ; c) 784 = Ensure that all students have written at least the first one correctly. Review expanded form of numbers that contain 0. Ask a volunteer to write 350 in expanded form on the board. (350 = ) Have another volunteer write the expanded form of 305. (305 = ) Exercises: Write the numbers in expanded form. a) 207 b) 270 c) 702 d) 720 Answers: a) 207 = ; b) 270 = ; c) 702 = ; d) 720 = Proceed to 4 and 5digit numbers, then to 6 and 7digit numbers Exercises: Write the numbers in expanded form. a) 6,103 b) 7,064 c) 29,031 d) 30,402 e) 234,401 f) 7,032,530 g) 8,411,206 Answers: a) 6,103 = 6, ; b) 7,064 = 7, ; c) 29,031 = 29, ; d) 30,402 = 30, ; e) 234,401 = 200, , , ; f) 7,032,530 = 7,000, , ; g) 8,411,206 = 8,000, , , Review expanded form using number words. Write on the board: 32,427 = ten thousands + thousands + hundreds + tens + ones Have a volunteer fill in the blanks. Repeat with the numbers 4,589; 38,061; 5,770, 804; 40,300. Proceed to 6 and 7digit numbers, adding the words millions and hundred thousands. Then model writing the words millions, hundred thousands, ten thousands, thousands, hundreds, tens, and ones, as you write the expanded form of the number 8,642,503: 8,642,503 = 8 millions + 6 hundred thousands + 4 ten thousands + 2 thousands + 5 hundreds + 0 tens + 3 ones C8 Teacher s Guide for AP Book 5.1
9 (MP.7) Exercise: Fill in the blanks, then rewrite the number words in the correct order. 5,423,891 = ones + hundred thousands + tens + ten thousands + thousands + millions + hundreds Answer: 5 millions + 4 hundred thousands + 2 ten thousands + 3 thousands + 8 hundreds + 9 tens + 1 one Writing numbers from the expanded form. ASK: What is ? Is there a 0 in the number? How do you know? What would happen if we didn t write the 0 digit because we thought the 0 didn t matter, and we just wrote the 5 and the 7 as 57? Does = 57? (no) Emphasize that in expanded form we do not need to write the 0 because expanded form is an addition statement (or sum) and the 0 does not add anything. But the 0 means something in multidigit numbers. It makes sure that each digit s place value is recorded properly. Mathematicians call 0 a place holder because of this. Exercises: Add. a) b) c) d) e) f) 4, g) 5, h) 7, i) 80, j) 60, Bonus: 30, ,000, Answers: a) 302, b) 230, c) 405, d) 504, e) 740, f) 4,378, g) 5,061, h) 7,003, i) 80,300, j) 60,049, Bonus: 8,030,410 Give students examples of 3, 4, 5, 6, and 7digit sums for which they need to fill in the blanks (examples: = 253; 50, = 54,020; 7,000, = 7,040,300) Bonus: Provide examples in which the order of the addends differs from the order of the digits. (example: 4,327 = ,000) ASK: What is the sum of the digits in 234? (9) in 5,121? (9) Ask students to find a number that has digits with a sum of 11. Encourage students to find more than one answer. (137; 5,132; 902; 830; 416; 623; 3,044) Have students make (or sketch) models of their numbers using base ten materials. ASK: How many blocks did you need? How can you find the number of blocks from the digits? Exercises: What is the sum of the tens digit and the thousands digit? a) 2,735 b) 5,745 c) 39,602 d) 412,777? Answers: a) 5, b) 9, c) 9, d) 9 Challenge students to find a number with sum of the tens digit and the thousands digit equal to 11. (7,148; 9,020; 8,932; 6,456) Number and Operations in Base Ten 52 C9
10 Review the concept of twice as many and three times as many. Exercises 1. Which digit is twice the ones digit? a) 432 b) 6,723 c) 45,621 d) 308,754 e) 67, Which digit is three times the tens digit? a) 513 b) 89,736 c) 907,508 d) 36,422 Answers: 1. a) 4, b) 6, c) 2, d) 8, e) 0; 2. a) 3, b) 9, c) 0, d) 6 Exercises: Do these problems individually. a) Write a number whose hundreds digit is twice its ones digit. b) In the number 4,923, find a digit that is twice another digit. c) In the number 265,321, which digit is exactly three times the tens digit? d) Write an odd number whose ten thousands digit is twice its hundreds digit. Answers: a) 613 (answers may vary), b) 4, c) 6, d) 80,413 (answers may vary) Extensions (MP.1) 1. Write an odd number whose ten thousands digit is twice its hundreds digit. Sample answer: 61, Solve these puzzles using base ten blocks: a) I am greater than 20 and less than 30. My ones digit is one more than my tens digit. b) I am a 2digit number: Use 6 blocks to make me. Use twice as many tens blocks as ones blocks. c) I am a 3digit number. My digits are all the same. Use 9 blocks to make me. d) I am a 2digit number. My tens digit is 5 more than my ones digit. Use 7 blocks to make me. e) I am a 3digit number. My tens digit is one more than my hundreds digit and my ones digit is one more than my tens digit. Use 6 blocks to make me. f) Show 1,123 using exactly 16 blocks. (There are 2 answers.) g) I am a 4digit number. My digits are all the same. Use 12 blocks to make me. C10 Teacher s Guide for AP Book 5.1
11 Answers: a) 23, 24, 25, 26, 27, 28, or 29; b) 42, c) 333, d) 61, e) 123, f) 11 hundreds blocks + 2 tens blocks + 3 ones blocks or 1 thousands block + 12 tens blocks + 3 ones blocks, g) 3, Solve these puzzles by only imagining the base ten blocks. Use base ten blocks for harder questions if necessary. Questions (a) through (e) have more than one answer share your answers with a partner. a) I have more tens than ones. What number could I be? b) I have the same number of ones and tens blocks. What number could I be? c) I have twice as many tens blocks as ones blocks. What 2digit number could I be? d) I have six more ones than tens. What number could I be? e) I have an equal number of ones, tens, and hundreds, and twice as many thousands as hundreds. What number could I be? f) I have one set of blocks that make the number 13 and one set of blocks that make the number 22. Can I have the same number of blocks in both sets? g) I have one set of blocks that make the number 23 and one set of blocks that make the number 16. Can I have the same number of blocks in both sets? Answers: a) Answers can be any number in which the tens digit is higher than the ones digit, such as 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, etc. As the tens digit increases by one, the number of ones digits that can be used also increases by one; b) 11, 22, 33, 44, 55, 66, 77, 88, 99; c) 21, 42, 63, 84; d) 17, 28, 39; e) 2,111, 4,222, 8,444; f) yes, they can both have 4 blocks: 13 = 1 tens block + 3 ones blocks, 22 = 2 tens blocks + 2 ones blocks; g) no (MP.3) 4. Make a chart to investigate how many base ten blocks are needed to make different numbers. Use these example numbers in the chart: 84, 23, 916, 196, 1,003, 200, 4,321, 9,022. When students are done the chart, ASK: a) Do all 3digit numbers need more blocks than all 2digit numbers? (no) Are all 3digit numbers greater than all 2digit numbers? (yes) Why is this the case? (3digit numbers are hundreds and 2digit numbers are tens) b) How can you find the number of base ten blocks needed just by looking at the digits? (add the digits) c) What is the greatest number of base ten blocks needed for a 1digit number? (9) a 2digit number? (18) a 3digit number? (27) a 4digit number? (36) Describe the pattern. (all are multiples of 9) Number and Operations in Base Ten 52 C11
12 d) Make an organized list to find all 2digit numbers that need the exact number of blocks. i) 1 block ii) 2 blocks iii) 3 blocks iv) 15 blocks Answers: i) 10; ii) 11, 20; ii) 12, 21, 30; iv) 69, 78, 87, 96 e) Make an organized list to find all 3digit numbers that need the exact number of blocks. i) 1 block ii) 2 blocks iii) 3 blocks iv) 4 blocks v) 5 blocks Answers: i) 100; ii) 101, 110, 200; iii) 102, 111, 120, 201, 210, 300; iv) 103, 112, 121, 130, 202, 211, 220, 301, 310, 400; v) 104, 113, 122, 131, 140, 203, 212, 221, 230, 302, 320, 401, 410, Which of the following are correct representations of 352? a) b) 1 hundred + 20 tens + 2 ones c) 2 hundreds + 15 tens + 2 ones d) 34 tens + 12 ones Answers: a) and c) represent 352 Make up more problems like this and have students make up problems of their own. C12 Teacher s Guide for AP Book 5.1
13 NBT53 Comparing and Ordering Pages MultiDigit Numbers STANDARDS 5.NBT.A.3 Goals Students will order sets of two or more numbers, and will find the greatest and least number possible using the digits given. Vocabulary expanded form greater than > less than < PRIOR KNOWLEDGE REQUIRED Can compare pairs of numbers Knows place values for ones, tens, hundreds, thousands, and ten thousands Knows the number words for place values of one, ten, hundred, thousand, ten thousand, hundred thousand, million, and their corresponding numerals MATERIALS grid paper (optional) 3 dice per pair of students 3 small boxes, bags, or other containers per pair of students 1digit to 5digit number cards, 1 per student (MP.7) Comparing numbers that differ by only one digit. Write the numbers 435 and 425 in expanded form on the board. ASK: Which number is greater? (435) How can you easily tell from the expanded form that 435 is greater than 425? (30 is more than 20, and everything else is the same) Circle the 30 and 20 to emphasize this. 435 = = Exercises: Use expanded form to identify the greater number. a) 352 or 452 b) 405 or 401 c) 398 or 358 d) 541 or 241 Answers: a) 452, b) 405, c) 398, d) 541 Point out that you can compare the numbers 435 and 425 just by looking at the digits that are different. SAY: Since the tens digits are the only ones that are different, the number with the greater tens digit is greater Exercises: Circle the digits that are different, then write the greater number. a) 542 b) 706 c) 3,240 d) 432, , ,706 Answers: a) hundreds digit, 642; b) ones digit, 706; c) hundreds digit, 3,540; d) ten thousands digit, 432,706 Number and Operations in Base Ten 53 C13
14 Comparing numbers that differ by more than one digit. Tell students that you want to compare two 3digit numbers. Write on the board: ASK: Which number has more hundreds? (342) more tens? (257) more ones? (257) Which number do you think is greater the one with the most hundreds, the most tens, or the most ones? (most hundreds) Why? (hundreds are greater than tens and ones) Encourage students who have trouble seeing that the number with the most hundreds is greater to think in terms of money in which a hundred cents is a dollar, ten cents is a dime, and 1 cent is a penny. The question then becomes: Which is more money: 3 dollars, 4 dimes, and 2 pennies; or 2 dollars, 5 dimes, and 7 pennies? The money amount that has more dollars is greater. Exercises: Circle the greater number. a) 731 or 550 b) 642 or 713 c) 519 or 382 Answers: a) 731, b) 713, c) 519 Encourage students who are struggling to compare the number of hundreds more easily by writing the second number below the first number on grid paper, if necessary. 582 = = Now tell students that you want to compare 582 and 574. Demonstrate how to split the numbers, as shown in the margin. SAY: The hundreds are the same, so you just have to compare 82 and 74. ASK: Which is greater, 82 or 74? (82) How do you know? (it has more tens) Point out that when comparing any two numbers, start at the left and circle the first digits that are different those will tell you which number is greater. Have volunteers circle the first, leftmost digits that are different in these pairs of numbers: a) 632 b) 873 c) 2,741 d) 352, , ,631 Exercises: Write the greater number in each pair above. Answers: a) 641, b) 873, c) 2,750, d) 352,418 Exercises: Counting from the left, underline the first digit that is different. Then circle the greater number. a) 718 and 523 b) 5,234 and 6,234 c) 9,843,275 and 9,846,192 d) 874,321 and 874,132 Answers: a) 718, b) 6,234, c) 9,846,192, d) 874,321 NOTE: Have students who are struggling use grid paper to line up the place values ones with ones, tens with tens, and so on. Example: C14 Teacher s Guide for AP Book 5.1
15 Remind students that < means is less than because the wider end is on the right, so the lesser number is first. Similarly, > means is greater than because the wider end is on the left so the greater number is first. Exercises: Write < or > for each pair of numbers. a) 32, ,329 b) 912,401 91,401 c) 8,322,407 8,322,415 d) 62,713 68,290 Answers: a) 32,417 < 276,329; b) 912,401 > 91,401; c) 8,322,407 < 8,322,415; d) 62,713 < 68,290 (MP.1) Write on the board: (MP.3) Ask students to make all the possible 3digit numbers from these digits. (358, 385, 538, 583, 835, 853) Tell them to try to do it in an organized way. Take suggestions for how to do that. (start with the hundreds digit; write all the numbers that have the hundreds digit 3 first, then work on the numbers with the hundreds digit 5, then 8) How many numbers are there altogether? (6, 2 starting with each digit) Which of these numbers is greatest? (835) Which is least? (358) Have students reflect: Could they have found the greatest number using these digits without listing all the possible numbers? SAY: I want to find the greatest number possible using each of the digits 1, 3, 5, and 8 once. Do I need to find all the numbers I can make from these digits to find the greatest number? (no, you can simply write the numbers in order from greatest to least starting from the left) Exercises: Create the greatest possible number using each digit once. a) 2, 9, 7, 8, 4 b) 4, 4, 8, 3, 2 c) 3, 7, 0, 6, 1 Answers: a) 98,742, b) 84,432, c) 76,310 Then ask students to find the least possible number using those same digits. (Note that a number cannot begin with the digit 0, so the least number in c) is 10,367.) Finally, ask students to use the same digits to find a number in between the greatest and least possible numbers. Exercises: Order the groups of three 4digit numbers from least to greatest. a) 3,458, 3,576, 3,479 b) 4,987, 6,104, 6,087 c) 4,387, 2,912, 3,006 Answers: a) 3,458, 3,479, 3,576; b) 4,987, 6,087, 6,104; c) 2,912, 3,006, 4,387 Combine 3digit and 4digit numbers and have students order them from least to greatest (example: 3,407, 410, 740). Have students order similar groups of 5digit numbers. Bonus: Have students order longer lists of numbers. Number and Operations in Base Ten 53 C15
16 Finally, combine numbers with different numbers of digits. For example, have students order from least to greatest 2,354, 798, 54,211. (798, 2,354, 54,211) Have students use the digits 0 to 9 each once so that, >,. ASK: Can you place the digits so that, >,? (no, a 6digit number is always greater than a 4digit number) Exercises 1. Use the digits 0, 1, and 2 to create a number that is: a) more than 200 b) between 100 and 200 c) a multiple of Use the digits 4, 6, 7, and 9 to create a number that is: a) between 6200 and 6500 b) larger than 8000 c) an even number between 9500 and Use the digits 0, 3, 4, 7, and 8 to create a number that is: a) a multiple of 10 b) between 73,900 and 75,000 c) more than 40,000 and less than 70, Find the correct digit. a) < < b) < < ACTIVITIES 1 2 (MP.3) 1. Students work with a partner. Partner 1 rolls 3 dice. Partner 2 makes the greatest possible number with the digits Partner 1 rolled. Partners then switch roles. HARDER: Students work in pairs and compete with each other to make the greatest (or least, or closest to 500) 3digit number. Use three small boxes, bags, or other containers for the hundreds, tens, and ones digit, but choose where to put each roll before doing the next roll. For example, if Partner 1 is trying for the least number and rolls a 2, Partner 2 might choose to place it in the hundreds box. If the next roll is a 1, Partner 2 will want to move the 2 but will not be allowed to. 2. Students use number cards to line up for lunch in order from greatest to least. The number cards should range from 1digit to 5digit numbers. Discuss with students how they can organize themselves into groups. Are there any groups in which all the numbers in one group will be greater than all the numbers in other groups? If necessary, prompt students with questions such as: Are all the 2digit numbers greater than all the 1digit numbers? How do you know? Should the 2digit numbers go before the 1digit C16 Teacher s Guide for AP Book 5.1
17 numbers, or after? Who should go right before the 2digit numbers? Who should go first? Before having any group go, ensure that all the groups know which order they go in. ASK: Who belongs in the first group? Have them show their cards. Repeat with the second, third, fourth, and fifth groups. Extensions (MP.1) 1. How many whole numbers are greater than 4,000 but less than 4,350? Explain how you know. Answer: 349 numbers; any number from 4,001 to 4, What is the greatest 6digit number you can create so that: a) the number is a multiple of 5 b) the ten thousands digit is twice the tens digit Answers: a) 999,995; b) 989, Write the number 98,950 on the board and challenge students to find all the greater numbers that use the same digits. Answers: 99,058; 99,085; 99,508; 99,580; 99,805; 99, Write the number 75,095 on the board. Have each student find a number that differs from it in only one digit. Have them ask a partner whether their number is greater or less than 75,095. Whose number is greatest? Have them work in groups of 4, 5, or 6 to order all the numbers they made. Sample answers: The greatest number is 95,095, the least number is 15,095, and the closest number is 75,094 or 75,096. Number and Operations in Base Ten 53 C17
18 NBT54 Regrouping Pages STANDARDS preparation for 5.NBT.B.7 Goals Students will write numbers as sums of ones, tens, hundreds, and thousands in different ways. Vocabulary digits number sentence place value regrouping PRIOR KNOWLEDGE REQUIRED Is familiar with base ten materials Knows place values for ones, tens, hundreds, thousands, and ten thousands Knows the number words for place values of one, ten, hundred, thousand, ten thousand, hundred thousand, million, and their corresponding numerals Review base ten blocks. Draw on the board: ASK: How many ones blocks are there altogether? (34) Ask for strategies for counting the blocks. (count by fives, then by ones, or count by sevens, then by ones; using multiplication: multiply 7 by 5, then subtract 1, or multiply 5 by 6, then add 4, or multiply 7 by 4, then add 6) ASK: Do we have enough to trade for a tens block? (yes) How do you know? (because there are more than 10 ones blocks) How many tens blocks can we trade for? (3) How do you know? (there are more than 30 blocks) Have a volunteer group sets of 10. Where do you see that number in 34? (the 3 represents the number of tens) What does the 4 tell you? (the number of ones left over) Draw base ten models with more than ten ones and have students practice trading ten ones blocks for a tens block. They should draw models to record their trades in their notebooks. Example: 4 tens + 19 ones = 5 tens + 9 ones Students should also be comfortable translating these pictures into number sentences: = = = 59 C18 Teacher s Guide for AP Book 5.1
19 Using regrouping. ASK: What number is 6 tens + 25 ones? How can we regroup the 25 ones to solve this question? 25 = 2 tens + 5 ones = , so 6 tens + 25 ones = tens 25 ones This means 6 tens + 25 ones = 8 tens + 5 ones, as summarized by this chart: Tens Ones = = 5 Have students practice using a chart like this to regroup numbers. Then have them regroup numbers without using the chart: 3 tens + 42 ones = tens + ones Continue with problems that require: 1) regrouping tens to hundreds (example: 4 hundreds + 22 tens). 2) regrouping ones to tens and tens to hundreds. Include numbers with 9 tens and more than 10 ones, so they will not have to regroup tens to hundreds until after they regroup the ones (example: 2 hundreds + 9 tens + 14 ones). Use this to explain the importance of grouping the ones first you can only tell if there are enough tens to make a hundred after you have grouped all the ones into tens. 3) regrouping hundreds to thousands. 4) regrouping thousands to ten thousands, hundreds to thousands, tens to hundreds, and ones to tens. Again, include examples with 9 hundreds and more than 10 tens, or 9 thousands, 9 hundreds, 9 tens, and more than 10 ones. Examples: a) 3 ten thousands + 4 thousands + 15 hundreds + 5 tens + 3 ones b) 4 ten thousands + 17 thousands + 9 hundreds + 18 tens + 2 ones c) 5 ten thousands + 12 thousands + 9 hundreds + 3 tens + 38 ones d) 4 ten thousands + 9 thousands + 13 hundreds + 14 tens + 7 ones e) 5 ten thousands + 12 thousands + 19 hundreds + 5 tens + 12 ones f) 9 ten thousands + 8 thousands + 16 hundreds + 34 tens + 11 ones g) 7 ten thousands + 5 thousands + 9 hundreds + 12 tens + 8 ones h) 6 ten thousands + 9 thousands + 9 hundreds + 24 tens + 7 ones i) 8 ten thousands + 9 thousands + 9 hundreds + 9 tens + 26 ones j) 8 ten thousands + 9 thousands + 13 hundreds + 8 tens + 25 ones Number and Operations in Base Ten 54 C19
20 Extensions 1. Show the regrouping in Questions 2 and 3 on AP Book p. 17 with base ten blocks. 2. Teach students the Egyptian system for writing numerals to help them appreciate the utility of place value. Write the following numbers using both the Egyptian and our Arabic systems: Invite students to study the numbers for a moment, then ASK: What is different about the Egyptian system for writing numbers? (It uses symbols instead of digits. You have to show the number of ones, tens, and so on individually if you have 7 ones, you have to draw 7 strokes. In our system, a single digit (7) tells you how many ones there are.) Review the ancient Egyptian symbols for 1, 10, and 100, and introduce the symbols for 1,000 and 10,000: 1 = (stroke) 10 = (arch) 100 = (coiled rope) 1,000 = (lotus leaf) 10,000 = (finger) Ask students to write a few numbers the Egyptian way and to translate those Egyptian numbers into regular numerals. Emphasize that the order in which you write the symbols does not matter: 234 = Have students write a number that is very long to write the Egyptian way. (example: 798) ASK: How is our system more convenient? Why is it helpful to have a place value system in which the ones, tens, and so on are always in the same place? Students might want to invent their own number system using the Egyptian system as a model. Answer: To show 999 we need just 3 symbols, but the Egyptian way needs 27 symbols. 3. If you teach students Egyptian writing, you can then ask them to show regrouping using Egyptian writing. C20 Teacher s Guide for AP Book 5.1
21 NBT55 MultiDigit Addition Pages STANDARDS 5.NBT.B.7 Goals Students will add up to 6digit numbers with and without regrouping. Vocabulary algorithm expanded form place value chart regrouping standard notation for addition PRIOR KNOWLEDGE REQUIRED Is familiar with base ten materials Knows place values for ones, tens, hundreds, thousands, and ten thousands Knows place values for ones, tens, hundreds, thousands, and ten thousands MATERIALS 1 die per pair of students Review regrouping when using base ten materials for addition. Tell students you want to add 27 and 15. Begin by drawing base ten models of 27 and 15 on the board: 27 = 15 = Then write the addition statement and combine the two models to represent the sum: = ASK: How many ones do we have in total? (12) how many tens? (3) Replace 10 ones with 1 tens block. ASK: Now how many ones do we have? (2) How many tens? (4) How many do we have altogether? (42) = Exploring addition with regrouping using place value charts. Use a tens and ones chart to summarize how you regrouped the ones: Tens Ones After combining the base ten materials 4 2 After regrouping 10 ones blocks as 1 tens block Number and Operations in Base Ten 55 C21
22 Exercises: Draw the base ten materials and the tens and ones charts for each addition statement. a) 36 b) 28 c) 46 d) Bonus e) 32 f) Selected solution: a) 7 tens blocks + 11 ones blocks = 8 tens blocks + 1 ones block = 81 Answers: b) 65, c) 82, d) 47, Bonus: e) 91, f) 74 Using standard notation for addition with regrouping 2 digits. Ask students if they really need the base ten materials or if they can add without them. Draw the chart on the board: Tens Ones Add each digit separately 8 3 Regroup 10 ones as 1 ten: = = SAY: When you use a chart, you can add the tens and ones first, then regroup. When you do the sum directly, you regroup right away: = 13, which is 1 ten + 3 ones, so you put the 3 in the ones column and add the 1 to the tens column. Demonstrate the first step of writing the sum of the ones digits: Ask students how many ones there are when you add the ones digits, and how you are regrouping them. Tell them that when we regroup ten ones for a ten, we put the 1 on top of the tens column. Mathematicians call this process carrying the 1. Ask students for reasons why this name is appropriate for the notation. Have students do the first step for several problems, then move on to problems in which students need to do both steps. Some students may need to begin by having the first step done for them so they can focus only on completing the second step. C22 Teacher s Guide for AP Book 5.1
23 Include examples in which the numbers add up to more than 100 (examples: , ). Exercises: Add the numbers. a) b) c) d) e) f) g) Bonus: Answers: a) 85, b) 84, c) 90, d) 85, e) 113, f) 114, g) 87, Bonus: 166 Adding 3digit numbers using expanded form. Have volunteers draw on the board base ten models for 152 and 273. Tell students that you want to add these numbers. ASK: How many hundreds, tens, and ones are there altogether? (3 hundreds + 12 tens + 5 ones) Do we need to regroup? (yes) How do you know? (the tens digit is more than 9) How can we regroup? (since there are 12 tens, we can trade 10 of them for 1 hundred) After regrouping, how many hundreds, tens, and ones are there? (4 hundreds + 2 tens + 5 ones) What number is that? (425) Write out and complete the following statements as you work through the example: 153 hundred + tens + ones hundreds + tens + ones hundreds + tens + ones After regrouping: hundreds + tens + ones Have students add more pairs of 3digit numbers. Provide examples in the following sequence: either the ones or the tens need to be regrouped (examples: , ). both the ones and the tens need to be regrouped (examples: , ). the tens need to be regrouped, but you don t realize it until you regroup the ones (examples: , ) Using standard notation for addition with regrouping 3 digits. Now show students the standard algorithm alongside a hundreds, tens, and ones chart for the first example you did together ( ): Hundreds Tens Ones = = Number and Operations in Base Ten 55 C23
24 Point out that after regrouping the tens, you add the 1 hundred that you carried over from the tens at the same time as the hundreds from the two numbers, so you get hundreds. Have students add more 3digit numbers using the chart and the algorithm at first, then using only the algorithm. Do examples in the same sequence as before: either the ones or tens need to be regrouped; both the ones and tens need to be regrouped; the tens need to be regrouped after the ones have been regrouped. Exercises: Add using the standard algorithm. a) b) c) d) e) f) g) h) Bonus: Answers: a) 575, b) 815, c) 1,490, d) 1,769, e) 953, f) 1,486, g) 905, h) 1,000, Bonus: 1,995 Adding 4, 5, 6digit numbers. When students have mastered 3digit numbers, repeat the lesson with 4digit numbers, then with 5digit numbers. Do not assume that students will be so familiar with the method that you can skip steps in the process. Use a chart alongside the algorithm to start, and provide examples in which different digits need to be regrouped. Finally, include both 4 and 5digit numbers as addends in the same sum (example: 32, ,736). Exercises 1. Add the 4digit numbers. a) 1, ,217 b) 3, ,590 c) 6, ,258 d) 1, ,958 e) 4, ,347 f) 4, ,325 Bonus: 3, , ,132 Answers: a) 8,575, b) 6,563, c) 14,953, d) 9,193, e) 13,005, f) 10,000, Bonus: 11, Add the 5digit and 6digit numbers. a) 81, ,584 b) 13, ,756 c) 94, ,647 d) 862, ,857 e) 394, ,656 f) 146, ,925 Bonus: g) 15, , ,209 h) 3,789, ,654,987 Answers: a) 94,557, b) 93,241, c) 113,005, d)1,061,452, e) 810,004, f) 1,000,000, Bonus: g) 75,214, h) 9,444,443 C24 Teacher s Guide for AP Book 5.1
25 ACTIVITIES 1 2 Game for two players: 1. Each player makes a copy of this grid: + 2. Players take turns rolling a die and writing the number rolled in one of their own grid boxes. The winner is the player who creates two 3digit numbers with the greatest sum. Variations: the addends have different numbers of digits (example: 3 digits and 4 digits, 3 digits and 5 digits, both 5 digits) add 3 numbers instead of 2 the winner is the player whose sum is closest to a given number, such as 400 the winner is the player who has the least sum Extensions 1. Add the numbers. a) b) 3, , ,132 c) 15, , ,209 d) 17, , ,117 Answers: a) 1,995, b) 11,834, c) 75,214, d) 78,309 (MP.3) (MP.1) 2. When students are comfortable adding 3, 4, and 5digit numbers, review with the class what a palindrome is. Ask a volunteer to articulate what the rule is for determining whether a number is a palindrome. (a palindrome is a number whose digits are in the same order when written from right to left, as when written from left to right) Write on the board: ,532 2,332 4,332 12,334,321 Point to each number, one at a time, and ask students whether the number is a palindrome. (yes, no, yes, yes, no, no) Students can signal their answers by giving a thumbs up or thumbs down. Repeat with several more numbers. When students are comfortable with the new terminology, ask them to list the 2digit palindromes. (11, 22, 33, and so on to 99) Tell them that you are going to show them how to turn any number into a palindrome by using addition. Write the number 13 on the board and ask students Number and Operations in Base Ten 55 C25
26 to find numbers you can add to 13 to make a palindrome. (9, 20, 31, 42, 53, 64, 75, 86) Tell students that one of these numbers can be obtained from the original 13 in a very easy way. Which number is that? (31) How can we obtain 31 from 13? (use the same digits, but in reverse order) Have students add these numbers to their reverse: 35, 21, 52 (example: = 88). Do they always get a palindrome? (yes) Challenge students to find a 2digit number for which a palindrome does not result by adding it to its reverse. Tell them to follow the same process with their resulting number. For example, if they started with 69, they will see that = 165 is not a palindrome, so they could repeat the process with 165 ( = 726). Ask how many have palindromes now. Have students continue repeating the process until they end up with a palindrome. Starting with 69 (or 96), the sequence of numbers they get will be: 69 (or 96), 165, 726, 1,353, 4,884. Have students repeat the process starting with various 2digit numbers (examples: 54, 74, 37, 38, 56, 28), then with multipledigit numbers (examples: 341, 576, 195, 197, 8,903, 9,658, 18,271). Tell students that most numbers will eventually become palindromes, but that mathematicians have not yet proven whether all numbers will. Over 2,000,000 steps have been tried (using a computer, of course) on the number 196, and mathematicians have still not found a palindrome. (MP.1) Ask students to find a number that will not become a palindrome, even after five or ten steps. 3. Systematically list all palindromes from 100 to 200 (it s easy to give a rule for these) or from 100 to 1,000. Answers: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191; a general rule for these 3digit numbers might be 1d1, where d can be any digit. (MP.1, MP.3) 4. ASK: When you add two palindromes, do you always get a palindrome? When do you get a palindrome and when do you not? Can you change the question slightly to make it have a yes answer? (When you add two palindromes whose digits are all less than 5, do you always get another palindrome?) 5. Find a number that, when added to its reverse, results in the following number. a) 584 b) 766 c) 1,251 d) 193 Which one is not possible? Explain how you know. Answers: a) 292, b) 383, c) 625, d) not possible because the difference between ones and hundreds is more than 1 C26 Teacher s Guide for AP Book 5.1
27 NBT56 MultiDigit Subtraction Pages STANDARDS 5.NBT.B.7 Goals Students will subtract with and without regrouping. Vocabulary borrowing minuend regrouping standard algorithm PRIOR KNOWLEDGE REQUIRED Understands subtraction as taking away Is familiar with base ten materials Knows place values for ones, tens, hundreds, thousands, and ten thousands Knows the number words for place values of one, ten, hundred, thousand, ten thousand, hundred thousand, million, and their corresponding numerals Can separate the tens and the ones MATERIALS base ten blocks cards numbered 1 to 8, one set per student Review subtraction without regrouping. Tell students that you want to use base ten materials to subtract Have a volunteer draw a base ten model of 48 on the board. SAY: I want to take away 32. How many tens blocks should I remove? (3) Demonstrate crossing them out. ASK: How many ones blocks should I remove? (2) Cross those out, too. ASK: What do I have left? How many tens? (1) How many ones? (6) What is 48 32? (16) Have student volunteers do other problems on the board with no regrouping (examples: 97 46, 83 21, 75 34). (51, 62, 41) Have classmates explain the steps the volunteers are following. Then have students create and solve similar problems in their notebooks. Check students work to make sure the first number is not less than the second number to avoid negative results. Give students examples of base ten models with the subtraction shown and have them complete tens and ones charts: Tens Ones Have students create and subtract more 2digit numbers (no regrouping) using the chart and base ten models together. Number and Operations in Base Ten 56 C27
28 When students have mastered this, have them subtract by writing out the tens and ones (as in Question 2 on p. 22 in the AP Book): 46 = 4 tens + 6 ones  13 = 1 ten + 3 ones = 3 tens + 3 ones = 33 Then have students separate the tens and ones using only numerals (as in Question 3 on p. 23 in the AP Book): 36 = = = = 12 Now ask students to subtract: (MP.3) ASK: Which strategy did you use? Is there a quick way to subtract without using base ten materials, or tens and ones charts, or separating the tens and ones? (yes subtract each digit from the one above) What are you really doing in each case? (subtracting the ones from the ones, and the tens from the tens, and putting the resulting digits in the right places) Have students draw a base ten model of 624 and show how to subtract 310. Have them subtract using the standard algorithm (example: by lining up the digits) and check to see whether they got the same answer both ways. Repeat with 4 and 5digit numbers that do not require regrouping (example: 34,586 12,433). Ask students how they learned to subtract Have a volunteer demonstrate on the board. ASK: Can you use the same method to subtract 46 28? (no) Why not? (there aren t enough ones) Should you be able to subtract 28 from 46? (yes) If you have 46 things, does it make sense to take away 28 of them? (yes) Challenge students to think of a way to change the problem to one that looks like a problem they did last time. Tell them that they are allowed to change one of the numbers, then adjust their answers. Have them work in pairs. Possible answers may include: find 46 26, then subtract 2 more because you didn t subtract enough find 48 28, then subtract 2 because you added 2 to get 48 find 46 20, then subtract 8 because you didn t subtract enough (you can find 26 8 by counting down) find 38 28, then find by counting down Have volunteers come to the board to show their strategies and how they needed to adjust their answer. Restate, add to, or correct students explanations when necessary. Then have all students solve similar C28 Teacher s Guide for AP Book 5.1
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