OA4-13 Rounding on a Number Line Pages 80 81

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1 OA4-13 Rounding on a Number Line Pages STANDARDS 3.NBT.A.1, 4.NBT.A.3 Goals Students will round to the closest ten, except when the number is exactly halfway between a multiple of ten. PRIOR KNOWLEDGE REQUIRED Number lines Concept of closer MATERIALS number cards, string or tape, rope, a ring or hoop (see Activity) The closest ten. Show a number line from 0 to 10 on the board: Circle the 2 and ask if the 2 is closer to the 0 or to the 10. When students answer 0, draw an arrow from the 2 to the 0 to show the distance. Repeat with several examples and then ASK: Which numbers are closer to 0? Which numbers are closer to 10? Which number is a special case? Why is it a special case? Tell students you want to round numbers to the nearest ten. ASK: Would you round 6 to 0 or to 10? (10) Why? (because it is closer to 10 than to 0) Exercises: a) 3 rounded to the nearest ten is. b) 2 rounded to the nearest ten is. c) 8 rounded to the nearest ten is. d) 9 rounded to the nearest ten is. Give students a minute to answer these questions, then take up the answers by having students signal rounding up to 10 with a thumbs up and signal rounding down to 0 with a thumbs down. Then draw an incorrect number line with numbers not equally spaced, so that 4 appears closer to 10 than to 0. ASK: Is 4 closer to 0 or to 10 on this number line? (closer to 10) Why? (because you drew the number line incorrectly) Explain that when mathematicians say that 4 is closer to 0 than to 10, they mean 4 is closer to 0 than to 10 on any number line where the numbers are equally spaced. The number line has to be drawn properly. For the same reason, graph paper is printed with equally spaced lines, and there is equal spacing between measurements on thermometers, measuring cups, rulers, and many other instruments. Operations and Algebraic Thinking 4-13 D-1

2 Connection Real World Ask students whether they ve seen other number lines used to measure different things (e.g., on thermometers, measuring cups, radio stations). Ask students to identify cases where it s acceptable to round a measurement (e.g., indoor or outdoor temperature, percent of precipitation). Ask them to identify cases where it is important not to round numbers from a number line or measurement (e.g., body temperature, radio station, amount of medication). Conclude by saying that rounding on a number line or a measurement can be useful, but we should not do it in all situations. Then draw a number line from 10 to 30, with 10, 20, and 30 a different color than the other numbers Circle various numbers (not 15 or 25) and ask volunteers to draw an arrow showing which number they would round to if they had to round to the nearest ten. Repeat with a number line from 50 to 70, again writing the multiples of 10 in a different color. Then repeat with number lines from 230 to 250 or 370 to 390, etc. Ask students for a general rule to tell which ten a number is closest to. What digit should they look at? (the ones digit) How can they tell from this digit which multiple of ten a number is closest to? Then ask students to determine the closest multiple of ten given two choices instead of a number line. Example: Is 24 closer to 20 or 30? Is 276 closer to 270 or 280? ACTIVITY Write the numbers from 30 to 40 on cards. Use different colors to make the numbers 30 and 40 stand out from the rest. Attach the 11 cards to a rope using tape or by perforating the cards and attaching them with string. Make sure that the cards are equally spaced by 10 cm. Also, make sure that the entire line is centered on the rope. This is achieved most easily if the card labeled 35 is attached to the center of the rope and used as a reference for the placement of the rest of the cards. The ends of the rope will be longer than is indicated in the diagram below Take a ring or hoop and pull the rope through it. A hoop used for cross-stitching would be a good size D-2 Teacher s Guide for AP Book

3 Ask two volunteers to hold the number line taut. Ask a volunteer to find the middle number between 30 and 40. ASK: How do you know that this number is in the middle? What do you have to check? (the distance to the ends 30 of 31 the rope make a 35 volunteer do that) 38 Let 39 a 40 volunteer stand behind the line holding the middle number (35). Explain to students that the volunteers and the number line make a rounding machine. The machine will automatically round the number to the nearest ten. Put the ring on 32. Ask the volunteer holding the middle of the line to pull up from 35, so that the ring slides to Try more numbers. Ask students to explain why the machine works. You can repeat this activity with the multiples of 10 going up to 100 or with any other numbers. Do not include numbers that require rounding the digit 5. Operations and Algebraic Thinking 4-13 D-3

4 OA4-14 Rounding on a Number Line Pages (Hundreds and Thousands) STANDARDS 4.NBT.A.4 Goals Students will round to the closest hundred or thousand, except when the number is halfway between a multiple of a hundred or a thousand. PRIOR KNOWLEDGE REQUIRED Number lines Concept of closer The closest hundred. Repeat the previous lesson with a number line from 0 to 100 that shows only the multiples of At first, ask students whether numbers that are multiples of 10 (30, 70, 60, and so on) are closer to 0 or 100. Example: Is 40 closer to 0 or 100? Draw an arrow to show this. Repeat with several examples, then ASK: Which multiples of 10 are closer to 0 and which multiples of 10 are closer to 100? Which number is a special case? Why is it a special case? Then ask students about numbers that are not multiples of 10. First ask them where they would place the number 33 on the number line. Have a volunteer show this. Then ask the rest of the class if 33 is closer to 0 or to 100. Repeat with several numbers. Repeat with a number line from 100 to 200 and another number line from 700 to 800. Ask students for a general rule to tell which multiple of a hundred a number is closest to. What digit should they look at? (the tens digit) How can they tell from this digit which multiple of a hundred a number is closest to? When is there a special case? Emphasize that the number is closer to the higher multiple of 100 if its tens digit is 6, 7, 8, or 9, and it s closer to the lower multiple of 100 if its tens digit is 1, 2, 3, or 4. If the tens digit is 5, then any ones digit except 0 will make it closer to the higher multiple. Only when the tens digit is 5 and the ones digit is 0 do we have a special case where it is not closer to either. Repeat the lesson for thousands, emphasizing the importance of considering the value of the hundreds digit. Extensions 1. Rounding to the nearest ten, how many numbers round down to... a) 20? (four: 21, 22, 23, 24) b) 200? (four: 201, 202, 203, 204) c) 2,000? (four: 2,001, 2,001, 2,003, 2,004) D-4 Teacher s Guide for AP Book 4.1

5 2. a) Rounding to the nearest ten, how many numbers round down to 30? (four: 31, 32, 33, 34) b) Rounding to the nearest hundred, how many numbers round down to 300? (forty-nine: 301, 302, 303,, 349) c) Rounding to the nearest thousand, how many numbers round down to 3,000? (four hundred ninety-nine: 3,001, 3,002, 3,003,, 3,499) Operations and Algebraic Thinking 4-14 D-5

6 OA4-15 Rounding Pages STANDARDS 3.NBT.A.1, 4.NBT.A.3 Goals Students will round whole numbers to the nearest ten, hundred, or thousand. PRIOR KNOWLEDGE REQUIRED Knowing which multiples of ten, a hundred, or a thousand a number is between Finding which multiple of ten, a hundred, or a thousand a given number is closest to Review rounding 2-digit numbers. Have students round 2-digit numbers to the nearest ten. Do not at first include numbers that have ones digit 5. Ask students how they know which multiples of ten the number is between. ASK: How many tens are in 37? How many would be one more ten? So 37 is between 3 tens and 4 tens; that means it s between 30 and 40. Which multiple of ten is it closer to? ASK: How many tens are in 94? How many would be one more ten? What number is ten tens? So 94 is between what two multiples of 10? Which multiple of ten is it closer to? Repeat this line of questioning for 97. Connection Real World Rounding 2-digit numbers when the ones digit is 5. Tell students that when the ones digit is 5, a number is not closer to either the smaller or the larger ten, but we always round up. Give students many examples to practice with: 25, 45, 95, 35, 15, 5, 85, 75, 55, 65. If some students find this hard to remember, you could share the following analogy: I am trying to cross the street, but there is a big truck coming, so when I am partway across I have to decide whether to keep going or turn back. If I am less than halfway across, it makes sense to turn back because I am less likely to get hit. If I am more than halfway across, it makes sense to keep going because I am again less likely to get hit. But if I am exactly halfway across, what should I do? Each choice gives me the same chance of getting hit. Have students discuss what they would do and why. Remind them that they are, after all, trying to cross the street. So actually, it makes sense to keep going rather than to turn back. That will get them where they want to be. Here is another way to help students remember which way to round: Write out all the 2-digit numbers that have the same tens digits, for example, 3 (30, 31,..., 39). ASK: Which numbers should we round to 30 because they re closer to 30 than 40? Which numbers should we round to 40 because they re closer to 40 than to 30? How many are in each list? Where should we put 35 so that the lists are equal? Rounding 3-digit numbers to the nearest ten. SAY: To round 3-digit numbers to the nearest ten we look at the ones digit, the same way we do when we round 2-digit numbers to the nearest ten. The difference is that there is now a digit we need to ignore (the hundreds digit). D-6 Teacher s Guide for AP Book 4.1

7 Write on the board: SAY: The ones digit is less than 5, so we round down. Round 372 to 370. Exercises: Round each number to the nearest ten. a) 174 b) 885 c) 341 d) 936 Bonus e) 3,456 f) 28,712 Answers: a) 170, b) 890, c) 340, d) 940, e) 3,460, f) 28,710 Rounding 3-digit numbers to the nearest hundred. Write on the board: 240, 241, 242, 243, 244, 245, 246, 247, 248, 249 ASK: What do these numbers all have in common? (3 digits, hundreds digit 2, tens digit 4) Are they closer to 200 or 300? (200) How do you know? PROMPT: On a number line, what number is halfway from 200 to 300? (250) Point out that all these numbers are less than halfway from 200 to 300, so they are closer to 200 than to 300. ASK: Did you need to look at the ones digit to check that these numbers are closer to 200 than to 300? (no) What digit did you look at? (the tens digit) Summarize by saying that, to round to the nearest hundred, we need to look at the tens digit. Then tell students to look at these numbers: 250, 251, 252, 253, 254, 255, 256, 257, 258, 259. ASK: Which hundred are these numbers closest to? Are they all closest to 300 or is there one that s different? Why is that one a special case? If you saw that the tens digit was 5 but you didn t know the ones digit, and you had to guess if the number was closer to 200 or 300, what would your guess be? Would the number ever be closer to 200? Tell students that when rounding a number to the nearest hundred, mathematicians decided to make it easier and say that if the tens digit is a 5, you always round up. It doesn t make any more sense to round 250 to 200 than to 300, so you might as well round it up to 300 like you do all the other numbers that have hundreds digit 2 and tens digit 5. Then ASK: When rounding a number to the nearest hundred, what digit do we need to look at? (the tens digit) Write on the board: Round to the nearest hundred: Have a volunteer underline the hundreds digit because that is what they are rounding to. Then ask another volunteer to draw an arrow to the digit that determines whether to round up or round down, so the board now looks like: ASK: Where is that digit compared to the underlined digit? (It is the next one to the right.) How do you know when to round down and when to round up? Have another volunteer decide in each case whether to round up or down and write the correct rounded number. Operations and Algebraic Thinking 4-15 D-7

8 Repeat with 4-digit numbers, having students again round to the nearest hundred. Examples: 5,439, 2,964, 8,249. Then round 4-digit and 5-digit numbers to the nearest thousand. Examples: 4,520, 73,618, 2,388, 28,103, 87,093, 9,843. To help ensure that students round correctly, suggest that they always underline the digit they are rounding to, then point their pencil tip at the digit to the right of the one they underlined. This digit will tell them whether to round up or down. If any students are having trouble with rounding, teach them to round on a grid as shown on AP Book 4.1 p (Lesson OA4-6). Extension Another way to round a number to the nearest ten. First, add 5. Then, replace the ones digit in the answer with 0. Example: = = To round to the nearest hundred, add 50 instead of 5. The rounded number will be the answer with the ones and tens digits replaced with 0. Example: = = You can explain why this works as follows: The number 842 is between 800 and 900. Any number between 800 and 900 will round up to 900 if it is at least halfway to 900. When you add half of 100 to a number that is less than halfway to 900, you get a number still in the 800s; when you add half of 100 to a number that is at least halfway to 900, you get a number that is in the 900s. You can liken this to pouring liquid into a container that is half full. If the amount you are pouring is at least half a container, you will reach the top, and maybe overflow. If the amount you are pouring is less than half a container, you will not reach the top. Challenge students to make up a rule for using this method to round to the nearest thousand. D-8 Teacher s Guide for AP Book 4.1

9 OA4-16 Rounding on a Grid Pages STANDARDS 4.NBT.A.3 Goals Students will round whole numbers to the nearest ten, hundred, thousand, ten thousand, or hundred thousand. PRIOR KNOWLEDGE REQUIRED Knowing which multiples of ten, a hundred, or a thousand a number is between Finding which multiple of ten, a hundred, or a thousand a given number is closest to. Rounding on a grid (without regrouping). Show students how numbers can be rounded on a grid. Follow the steps shown below to round 12,473 to the nearest thousand. Step 1: As before, underline the digit you are rounding to and put your pencil on the digit to the right. If the digit under your pencil is 5, 6, 7, 8, or 9, you will round up; write round up beside (or above) the grid. If the digit under your pencil is 0, 1, 2, 3, or 4, you will round down; write round down. The hundreds digit here is 4, so we write round down. Round Down Step 2: To round up, add 1 to the underlined digit; to round down, keep the digit the same. In this case, we are rounding down, so we copy the Isolate Step 2. If any students are struggling with Step 2, make up several examples where the first step is done for them so that they can focus only on rounding the underlined digit up or down. Sample questions: Exercises: Round the underlined digit up or down as indicated: Round Down Round Up Round Up Once all students have mastered Step 2, move on Operations and Algebraic Thinking 4-16 D-9

10 Step 3: Change all numbers to the right of the rounded digit to zeros. Leave all numbers to the left of the rounded digit as they were. The number 12,473 rounded to the nearest thousand is 12,000. Round Down Rounding with regrouping. When students have mastered rounding without regrouping, give them several examples that demand regrouping as well. Warn them that the digits to the left of the rounded digit might change now. Round the 9 up to 10 and then regroup the 10 hundreds as 1 thousand and add it to the 7 thousand. Round Up Regroup Another way to do this rounding. Do an example together: Underline the digit you want to round to and decide whether to round up or down, as before, then change all numbers to the right of the rounded digit to zeros. We obtain (for the same example as above): Round Up Then ASK: Which two hundreds is the number between? PROMPT: How many hundreds are in 17,978? (179 hundreds are in 17,978, so the number is between 179 hundreds and 180 hundreds) Remind students that we are rounding up (point to the picture) because the tens digit is 7 the number is closer to 180 hundreds than to 179 hundreds. Complete the rounding by writing 180, not 179, as the number of hundreds. Round Up Point out that both ways of rounding get the same answer. Rounding larger numbers. Show grids with larger numbers, where regrouping does not occur. Ask students to round 538,226 to the nearest ten thousand. Here is the result they should obtain: Round Up D-10 Teacher s Guide for AP Book 4.1

11 Repeat with numbers where there is regrouping. Ask students to round 745,391 to the nearest ten. Here is the result they should obtain: Round Up Round three more large numbers that require regrouping as a class: a) round 439,889 to the nearest thousand b) round 953,219 to the nearest hundred thousand c) round 595,233 to the nearest ten thousand Students can round more such numbers individually. Exercises: Round each number to the given digit. a) thousands b) hundreds c) ten thousands d) hundreds Extensions 1. Regrouping twice when rounding. If you have an advanced class, you can try teaching this to the whole class. SAY: Sometimes rounding forces you to regroup two or more numbers. Let s see what to do when this happens. We want to round 3,999 to the nearest ten. First, we round 90 up to Then we regroup the 10 tens as 1 hundred and add it to the 9 hundreds. This gives 10 hundreds (1,000) Operations and Algebraic Thinking 4-16 D-11

12 Now we regroup the 10 hundreds as 1 thousand and add it to the 3 thousands To finish, we complete the rounded number by adding any missing zeros SAY: Let s do another example with a bigger number, such as 799,994. We will round this number to the nearest hundred. First, we round 900 up to 1, Then we regroup the 10 hundreds as 1 thousand and add them to the 9 thousands in the original number. Now we have 10 thousands ASK: What s the next step? (We regroup the 10 thousands as 1 ten thousand and add it to the 9 ten thousands in the original number. Now we have 10 ten thousands, or 1 hundred thousand (100,000).) ASK: What s the next step? (We add 1 to the hundred thousands.) ASK: How do we finish the rounding? (We complete the rounded number by adding any missing zeros.) D-12 Teacher s Guide for AP Book 4.1

13 Have students round 3,997 to the nearest hundred and to the nearest ten; 73,992 to the nearest hundred and to the nearest ten; and 39,997 to the nearest hundred and to the nearest thousand. 2. If 48,329 is rounded to 48,300, what digit has it been rounded to? (MP.1) 3. a) Write down all numbers that round to 40 when rounded to the nearest ten. How many such numbers are there? b) Write down all numbers that round to 800 when rounded to the nearest ten. How many such numbers are there? c) What is the smallest number that rounds to 800 when rounded to the nearest hundred? What is the largest number that rounds to 800 when rounded to the nearest hundred? How many numbers round to 800 when rounded to the nearest hundred? Hint: If you wrote down all the numbers from 1 to the largest number you found, and took away all the numbers that don t round to 800, how many numbers would still be in the list? Answers a) 35, 36, 37, 38, 39, 40, 41, 42, 43, 44; there are 10 such numbers b) 795, 796, 797, 798, 799, 800, 801, 802, 803, 804; again, there are 10 such numbers c) 750 is the smallest such number, and 849 is the largest. There are 849 numbers from 1 to 849. We don t want to include all the numbers from 1 to 749. So there are = 100 numbers in the list. Operations and Algebraic Thinking 4-16 D-13

14 OA4-17 Estimating Sums and Differences Pages STANDARDS 4.OA.A.3 Goals Students will estimate sums and differences by rounding each addend to the nearest ten, hundred, thousand, ten thousand, or hundred thousand. Vocabulary estimating the approximately equal to sign ( ) PRIOR KNOWLEDGE REQUIRED Rounding to the nearest ten, hundred, thousand, ten thousand, or hundred thousand Estimations in calculations. Show students how to estimate by rounding each number to the nearest ten: = 80. SAY: Since 52 is close to 50 and 34 is close to 30, will be close to, or approximately, Mathematicians have invented a sign to mean approximately equal to. It s a squiggly equal sign:. So we can write It would not be right to put = 80 because they are not actually equal; they are just close to, or approximately equal. Connection Real World Tell students that when they round up or down before adding, they aren t finding the exact answer, they are just estimating. They are finding an answer that is close to the exact answer. ASK: When do you think it might be useful to estimate answers? Sample answer: in a grocery store, estimating total price or change expected. Have students estimate the sums of 2-digit numbers by rounding each to the nearest ten. Remind them to use the approximately equal to sign. Exercises: a) b) c) d) e) f) = 70 Then ASK: How would you estimate 93 21? Write the estimated difference on the board (see margin). Have students estimate the differences of 2-digit numbers by again rounding each to the nearest ten. Exercises: a) b) c) d) e) f) Then have students practice estimating the sums and differences of 3-digit numbers by rounding to the nearest ten. (Examples: , ) 3- and 4-digit numbers by rounding to the nearest hundred. (Examples: , 4,501 1,511) 4- and 5-digit numbers by rounding to the nearest thousand. (Examples: 7, ,278, 13,891 11,990, 3, ,984) 5- and 6-digit numbers by rounding to the nearest ten thousand. (Examples: 54, ,447, 679, ,928) D-14 Teacher s Guide for AP Book 4.1

15 6- digit numbers by rounding to the nearest hundred thousand. (Examples: 928, ,219, 467, ,234) Is the estimate too high or too low? Write on the board: SAY: I estimated to be 70. Do you think this is higher than the actual answer or lower? (lower) Why? PROMPT: Is 30 more or less than 33? (less) Is 40 more or less than 41? (less) SAY: I rounded both numbers down, so the sum I get will be less than the actual sum. Have students verify this by calculating the actual sum. (74; indeed, 70 is less than 74) (MP.8) Exercises: Calculate both the actual sums and the rounded sums. Circle the larger sum. a) b) c) Answers: a) 73, 70, b) 87, 80, c) 115, 110. The actual sum should be circled in all cases. ASK: Which sum is larger, the actual sum or the rounded sum? (always the actual sum) Why was the actual sum always larger? (because the rounded numbers were smaller than the actual numbers; we always rounded down) Exercises: Calculate both the actual sums and the rounded sums. Circle the larger sum. a) b) c) Answers: a) 84, 90, b) 115, 120, c) 93, 100. The rounded sum should be circled in all cases. ASK: Which sum is larger, the actual sum or the rounded sum? (always the rounded sum) Why was the rounded sum always larger? (because the rounded numbers were larger than the actual numbers) Point out that when both numbers are rounded up, the rounded sum is larger, and when both numbers are rounded down, the rounded sum is smaller. Exercises: Predict whether Ahmed s estimate is too high or too low, then check your prediction by calculating the actual sum. a) Ahmed estimates as = 130. b) Ahmed estimates as = 1,500. c) Ahmed estimates as = 940. d) Ahmed estimates 23, ,706 as 20, ,000 = 30,000. e) Ahmed estimates 65, ,964 as 66, ,000 = 90,000. Answers a) Too low because 60 is less than 63 and 70 is less than 71, so will be less than Indeed, = 134 is more than 130. Operations and Algebraic Thinking 4-17 D-15

16 b) Too high because 800 is more than 752 and 700 is more than 689, so will be more than Indeed, = 1,441. c) Too low. Indeed, = 946. d) Too low. Indeed, 23, ,706 = 38,618. e) Too high. Indeed, 65, ,964 = 89,496. (MP.2) Recognizing when an answer is reasonable or not. For example, Daniel added 273 and 385, and got the answer 958. Does this answer seem reasonable? Students should see that even rounding both numbers up gives a sum less than 900, so this answer can t be correct. Do the following answers seem reasonable? Invite students to explain using estimates and perform the actual calculation to check their answers. a) Xian added 444 and 222 and got 888. b) Melissa added 196 and 493 and got 709. c) Enrico added 417 and 634 and got 951. (MP.8) Rounding to smaller place values is more accurate. SAY: Let s try estimating the sum by rounding to the tens and then to the hundreds. ASK: Which way do you think will give an answer closer to the actual sum? Write the sum on the board and get students to help you round the numbers to the given place value and then do the calculation. nearest ten: nearest hundred: ,180 1,200 Then calculate the sum of the two numbers and compare it with the two values we just obtained by estimating. ( = 1,181) ASK: The sum is closest to which answer, the one obtained by rounding the tens or the hundreds? (the tens) Explain that the lower the place value we round to in our estimation, the closer we get to the actual sum. Discuss how this is similar to measuring. Measuring to the nearest millimeter is more accurate and gives more information than measuring to the nearest centimeter because millimeters are smaller than centimeters. Do the same type of exercise with two 4-digit numbers: 5, ,213. Round to the tens: hundreds: thousands: 5,938 5,940 5,938 5,900 5,938 6, ,213 8, ,213 8, ,213 8,000 14,150 14,100 14,000 The actual sum is 14,151, so again rounding to the closest ten is the most accurate. Now estimate the sum 2, ,432 by rounding each number to the nearest: a) ten b) hundred c) thousand d) ten thousand D-16 Teacher s Guide for AP Book 4.1

17 Have students put their answers in order from closest to the actual answer to furthest from the actual answer. What do students notice? (rounding to smaller place values is more accurate) Point out that the answer to part d) above is = 0. Emphasize that rounding to too big a place value can become absurd. SAY: It would be like rounding the distance from my desk to your desk to the nearest mile. (MP.5) Choosing between speed and accuracy. ASK: Was adding more accurate when we rounded to the nearest tens, hundreds, or thousands? (tens) ASK: Was adding faster when we rounded to the nearest tens, hundreds, or thousands? (thousands) Why? (Adding 6,000 and 8,000 is as easy as adding 6 and 8, two 1-digit numbers, but adding 5,900 and 8,200 is like adding 59 and 82, two 2-digit numbers; 1-digit numbers are easier to add than 2-digit numbers.) Point out that we often need to choose between being fast and being more accurate. Sometimes we need more accuracy, and sometimes we need to be faster. Extensions 1. a) Estimate by rounding both numbers to the nearest hundred. Is your estimate higher or lower than the actual answer? b) Estimate by rounding both numbers to the nearest ten. Is your estimate higher or lower than the actual answer? Bonus: Make up another question where rounding to the nearest hundred is lower than the actual answer, but rounding to the nearest ten is higher than the actual answer. (MP.3) 2. Have students investigate when rounding one number up and one number down is better than rounding each to the nearest hundred by completing the following chart and circling the estimate that is closest to the actual answer: Actual Answer Round to the Nearest Hundred Round One Up and Round One Down 1, = 1, = 1,500 Operations and Algebraic Thinking 4-17 D-17

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