1 ENGAGE. 2 TEACH and TALK GO. Round to the Nearest Ten or Hundred


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1 Lesson 1.2 c Round to the Nearest Ten or Hundred Common Core Standard CC.3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100. Lesson Objective Round 2 and 3digit numbers to the nearest ten or hundred. Essential Question How can you round numbers? Vocabulary round 1 ENGAGE Access Prior Knowledge Discuss with students situations where you do not need to know an exact number, but knowing about how much or about how many is sufficient. For example, the length of a car is about 10 feet. There are about 100 seats in the auditorium. The height of a house is about 20 feet. How are these numbers alike? They all have zeros. They all tell about how many. 2 TEACH and TALK GO Online Unlock the Problem When would you round a number? Discuss the problem. Be sure that students understand that 32 is an exact number and they need to round 32 to the nearest ten. What is an example of a rounded number? Numbers with a zero at the end can be examples of rounded numbers, such as 10, 20, 30, and so on. What are the tens that are closest to the number 32? 30 and 40 One Way MATHEMATICAL Animated Math Models Why is a number line a good way to think about which numbers should be rounded? A number line can show how far apart numbers are from each other so they can be compared easily. Why does the first number line include tens and not hundreds? Possible answer: because 32 is a 2digit number and I am rounding 32 to the nearest 10 Between which two tens is 32? Which ten is it closer to? Explain. 30 and is closer to 30. It is only 2 numbers away from 30 but 8 numbers away from 40. Use Math Talk to focus students thinking on the fact that more than one number would round to 30. What makes a number able to be rounded to 30? It must be 25 or greater, or less than 35. In which direction would you round 28 to get to 30? up In which direction would you round 34 to get to 30? down You might draw a number line on the board from 20 to 40 so students can see that the numbers and would round to 30. Look at the second number line. How is rounding to the nearest hundred similar to rounding to the nearest ten? Possible answer: when I round to the nearest ten, I look at the number line to see which ten the number is closer to. When I round to the nearest hundred, I look at the number line to see which hundred the number is closer to. What is 144 rounded to the nearest hundred? 100 Lesson _MNYCETS225572_C01L02TE.indd 1 4/5/13 8:50 AM NYC10 New York City Implementation Guide
2 Name Round to the Nearest Ten or Hundred Essential Question How can you round numbers? Lesson 1.2 COMMON CORE STANDARD CC.3.NBT.1 Use place value understanding and properties of operations to perform multidigit arithmetic. UNLOCK the Problem When you round a number, you find a number that tells you about how much or about how many. Mia s baseball bat is 32 inches long. What is its length rounded to the nearest ten inches? One Way Use a number line to round. A Round 32 to the nearest ten Find which tens the number is between. 32 is between _ 30 and _ is closer to _ 30 than it is to _ rounded to the nearest ten is _. 30 So, the length of Mia s bat rounded to the nearest ten inches is _ 30 inches. B Round 174 to the nearest hundred Find which hundreds the number is between. 174 is between _ 100 and _ is closer to _ 200 than it is to _. 100 So, 174 rounded to the nearest hundred is _. 200 MATHEMATICAL Name three other numbers that round to 30 when rounded to the nearest ten. Explain. Possible answer: 28, 31, 34; possible explanation: 28 is between 20 and 30 but it is closer to 30; 31 and 34 are between 30 and 40 but they are closer to 30. Chapter 1 9 Untitled214 3_MNYCETS225572_C01L02TE.indd 9 2 4/11/2011 4/5/13 10:42:15 8:51 AM AM New York City Implementation Guide NYC11
3 c Try This! Discuss with students that there are different ways to round a number. Then, have students complete Parts A and B. How is rounding a 3digit number, like 718, to the nearest ten similar to rounding the number to the nearest hundred? Possible answer: in each case, I find the two tens or hundreds the number is between and then I decide which is closer. How is rounding a 3digit number to the nearest ten different than rounding to the nearest hundred? Possible answers: the number lines I use are different. To round to the nearest ten, I need to find the two tens that the number is between. To round to the nearest hundred, I need to find the two hundreds that the number is between. I write a zero for the digit in the ones place when rounding to ten. I write a zero for the digits in the tens and ones places when rounding to hundred. Another Way Students should recognize that the result of rounding using place value is the same as rounding on a number line. How might rounding using place value be quicker? Possible answer: I don t have to draw a number line to see the numbers. How is rounding to the nearest ten and rounding to the nearest hundred using place value the same? In each case, I look at the digit to the right of the place I am rounding to. If the digit is less than 5, the digit in the rounding place stays the same. If the digit is 5 or greater, the digit in the rounding place increases by 1. I write zeros for the digits to the right of the rounding place. Use Math Talk to show students how using place value is similar to using a number line. How can you look at the ones place to tell if 54 should be rounded up or down? If the number in the ones place is 1, 2, 3, or 4, it should be rounded down. If the number in the ones place is 5, 6, 7, 8, or 9, it should be rounded up. Which place value digit should you look at to decide if 168 should be rounded to 200? Look at the tens place value. In which place value digit should you look at to decide if 81 should be rounded to 100? Look at the tens place value. In which place value digit should you look at to decide if 81 should be rounded to 80? Look at the ones place value. COMMON ERRORS COMMON ERRORS Error Students may round numbers incorrectly because they do not look at the place to the immediate right. Example To round 718 to the nearest hundred, students may look at the 8 and round to 800. Springboard to Learning Have students circle the place to which they are rounding and underline the number to the immediate right before rounding. Lesson _MNYCETS225572_C01L02TE.indd 3 4/5/13 8:50 AM NYC12 New York City Implementation Guide
4 Try This! Round 718 to the nearest ten and hundred. Locate and label 718 on the number lines. Check students work. A Nearest Ten B Nearest Hundred is closer to _ 720 than it is 718 is closer to _ 700 than it is to _. 710 to _ So, 718 rounds to _. So, 718 rounds to _. 700 Another Way Use place value. A Round 63 to the nearest ten. Think: The digit in the ones place tells if the number is closer to 60 or l, 5 So, the tens digit stays the same. Write 6 as the tens digit. Write zero as the ones digit. So, 63 rounded to the nearest ten 63 Find the place to which you want to round. Look at the digit to the right. If the digit is less than 5, the digit in the rounding place stays the same. If the digit is 5 or greater, the digit in the rounding place increases by one. Write zeros for the digits to the right of the rounding place. is _. 60 B Round 457 to the nearest hundred. Think: The digit in the tens place tells if the number is closer to 400 or l 5 5 So, the hundreds digit increases by one. Write 5 as the hundreds digit. Write zeros as the tens and ones digits. So, 457 rounded to the nearest hundred is _ Math Talk: Possible explanation: when you use place value, you look at the digit to the right of the rounding place to see to which ten or hundred the digit is closer. If the next digit is 5 or greater, it will be closer to the next ten or hundred. When you use a number line, you can see to which ten or hundred the number is closer. MATHEMATICAL Explain how using place value is similar to using a number line. 10 Untitled _MNYCETS225572_C01L02TE.indd 4 3/15/2011 3:51:50 AM 4/5/13 8:49 AM New York City Implementation Guide NYC13
5 3 PRACTICE Share and Show Guided Practice The first problem connects to the learning model. Have students use the MathBoard to explain their thinking. Use Math Talk to focus on students understanding of rounding. Encourage students to explain their thinking. Use Exercises 6 and 7 for Quick Check. Students should show their answers for the Quick Check on the Math Board. If If Then Quick Check Quick Check Rt I Rt a student misses Exercises 6 and 7 Differentiate Instruction with RtI Tier 1 Activity, p. 9B Reteach 1.2 Soar to Success Math 15.15, Why are you being asked to find which hundred 548 is closer to? That is how you figure out how to round the number. In Exercises 13 15, is it possible to have the same answer when rounding to the nearest ten and hundred? Explain. Yes, when a number is rounded to the closest ten, it is possible that the number rounds to a hundred, such as Exercise 14. The closest ten to 298 is 300. The closest hundred to 298 is also 300. Go Deeper MATHEMATICAL To extend their thinking, ask students to find the greatest number that rounds to 500. Remind them that they are rounding to the hundreds place, not the tens place. Would you be rounding up or down to find the greatest number that rounds to 500? down What is that number? 549 c On Your Own Independent Practice If students complete Exercises 6 and 7 correctly, they may continue with Independent Practice. Encourage students to complete the On Your Own section independently, but provide guidance as necessary. Ask questions to make sure students know what they need to find out. In Exercises 10 12, why is 550 not a possible answer? I am asked to round to the nearest hundred, and 550 is not a hundred. Lesson _MNYCETS225572_C01L02TE.indd 5 4/5/13 8:49 AM NYC14 New York City Implementation Guide
6 Name Share and Show N Locate and label 46 on the number line. Round to the nearest ten is between _ 40 and _ is closer to _ 50 than it is to _ rounded to the nearest ten is _. 50 Round to the nearest ten. Check students work _ _ _ 50 MATHEMATICAL What is the greatest number that rounds to 50 when rounded to the nearest ten? What is the least number? Explain. greatest 54; least 45; Possible explanation: since this group of numbers rounds to 50 (45, 46, 47, 48, 49, 50, 51, 52, 53, 54), the greatest number is 54 and the least number is 45. Round to the nearest hundred _ _ _ 700 On Your OwnN Locate and label 548 on the number line. Round to the nearest hundred. Check students work is between _ 500 and _ is closer to _ 500 than it is to _ rounded to the nearest hundred is _. 500 Round to the nearest ten and hundred _ 580 _ Chapter 1 Lesson 2 11 Untitled _MNYCETS225572_C01L02TE.indd 6 3/15/2011 3:51:51 AM 4/5/13 8:49 AM New York City Implementation Guide NYC15
7 Problem Solving MATHEMATICAL For Exercises 16 18, students use information from a table and rounding to the nearest ten or hundred to solve problems. In Exercise 18, encourage students to discuss strategies they can use to determine which numbers round to 800. Problem To solve Exercise 19, students will have to use higher order thinking skills. Remind them of when a number is rounded up to the nearest ten and when a number is rounded down to the nearest ten. Does 351 round to 360 when rounding to the nearest ten? Explain. No, 351 is closer to 350 than it is to 360, so it rounds to 350 instead of 360. Does 357 round to 360 when rounding to the nearest ten? Explain. Yes, the closest ten to 357 is 360. How can you use this thinking to find other numbers that round to 360? I can think of which numbers are closest to 360 without rounding to another ten. For example, 356, 359, 361, and 364 all round to SUMMARIZE MATHEMATICAL Essential Question How can you round numbers? I can use a number line or place value. Math Journal Describe how to round 678 to the nearest hundred. Test Prep Coach Test Prep Coach helps teachers to identify common errors that students can make. In Exercise 21, if students selected: A or C, they rounded to the wrong ten. D, they rounded to the nearest hundred. Lesson _MNYCETS225572_C01L02TE.indd 7 4/5/13 8:49 AM NYC16 New York City Implementation Guide
8 Problem Solving Use the table for MATHEMATICAL Model Reason Make Sense 16. On which day did about 900 visitors come to the giraffe exhibit? Sunday 17. To the nearest ten, how many visitors came to the giraffe exhibit on Sunday? 890 visitors 18. On which two days did about 800 visitors come to the giraffe exhibit each day? Visitors to the Giraffe Exhibit Day Number of Visitors Sunday 894 Monday 793 Tuesday 438 Wednesday 362 Thursday 839 Friday 725 Saturday 598 Monday and Thursday 19. Write five numbers that round to 360 when rounded to the nearest ten. Possible answer: 356, 357, 359, 361, What s the Error? Cole said that 555 rounded to the nearest ten is 600. What is Cole s error? Explain. The answer should be 560. Possible explanation: Cole rounded to the nearest hundred instead of the nearest ten. 21. Test Prep What is 438 rounded to the nearest ten? A 450 B 440 C 430 D FOR MORE PRACTICE: Standards Practice Book, pp. P5 P6 FOR EXTRA PRACTICE: Standards Practice Book, p. P27 Untitled _MNYCETS225572_C01L02TE.indd 8 4/11/ :42:23 A 4/5/13 8:48 A New York City Implementation Guide NYC17
9 Lesson 6.6 Investigate Model with Arrays Common Core Standard CC.3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Also CC.3.OA.2 Lesson Objective Model division by using arrays. Essential Question How can you use arrays to solve division problems? 1 ENGAGE Materials square tiles Access Prior Knowledge Have students use square tiles to review making arrays for multiplication. Remind students that an array is a set of objects arranged in rows. There is the same number of tiles in each row. What realworld examples can arrays represent? Possible answers: marching bands or chairs set up in equal rows Have students model Why did you make 4 rows of 5? Possible answer: 4 x 5 means 4 groups, or rows, of 5 How did you find the product? Possible answer: I skip counted by 5 four times to get 20. c What should you do if you put the array together and do not have an equal number of tiles in each row? I must count again to see if I made an error, because an array must have an equal number of tiles in each row. Draw Conclusions What number did you divide? Explain. 30; possible answer: I started with 30 tiles. What number did you divide by? Explain. 5; possible answer: I made rows of 5 tiles each. Problem Exercise 3 requires students to generalize from the 5 by 6 array to finding a new array with 6 tiles in each row. Use the same array to describe a strategy for finding the number of rows of 6 tiles that are in 30. Possible answer: I knew that , so Count the number of tiles in a row in the first array; Think: What number times 6 equals 30? How is this array different from the one you put together in the Investigate section that shows how many rows of 5 there are in 30? The final number, 30, is the same for both arrays. This one has 6 tiles in one row, while the first array had 5 tiles in one row. c 2 TEACH and TALK GO Online Investigate MATHEMATICAL itools Work together with students to complete the steps of the activity. Be sure students make equal rows of 5. Explain that making equal rows is necessary so that the array is easy to follow and analyze. Remind students of the difference between a row and the number of tiles in the row. In which direction must you work to make a row of tiles? across from left to right In which direction must you count to find how many rows are in your completed array? from top to bottom Lesson _MNYCETS225572_C06L06TE.indd 9 4/5/13 8:57 AM NYC18 New York City Implementation Guide
10 Name Model with Arrays Essential Question How can you use arrays to solve division problems? Lesson 6.6 COMMON CORE STANDARD CC.3.OA.3 Represent and solve problems involving multiplication and division. Investigate Materials square tiles You can use arrays to model division and find equal groups. A. Count out 30 tiles. Make an array to find how many rows of 5 are in 30. B. Make a row of 5 tiles. C. Continue to make as many rows of 5 tiles as you can. How many rows of 5 did you make? 6 rows Draw ConclusionsN 1. Explain how you used the tiles to find the number of rows of 5 in 30. Possible explanation: I placed the 30 tiles in rows of 5 tiles. Then I counted the number of rows I made. There were 6 rows. So, there are 6 rows of 5 in What multiplication equation could you write for the array? Explain ; Possible explanation: there are 6 rows of 5 tiles each. 6 groups of 5 are Apply Tell how to use an array to find how many rows of 6 are in 30. Possible answer: make rows of 6 until all 30 tiles are used. There are 5 rows of 6 in 30. Chapter Untitled _MNYCETS225572_C06L06TE.indd 10 4/4/ :31:55 AM 4/5/13 8:57 AM New York City Implementation Guide NYC19
11 c Make Connections Help students connect the array to a division equation. Why is 30 the dividend? because that is the number of tiles being divided, or separated into equal groups What represents the quotient in this array? The number of rows is the quotient. Try This! After students find the quotient, have them share how they wrote the division equation that the array represents. Then check their quotients. To divide, you have used drawing equal groups or circling equal groups, repeated subtraction, counting back on a number line, and arrays. Which method do you think is the easiest? Explain. Have several students explain their choices. Students explanations may include the following: Drawing or circling the groups it s easier when you can see the problem in a picture. Repeated subtraction its faster to keep subtracting than to make a drawing. Number line it s easy to count the jumps to get the quotient. Array it s easy to put the tiles in equal rows and then count the number of rows. Use Math Talk to focus on students understanding of using an array to divide. Explain that students can count a whole row of an array as one number in a division problem. So, if an array has 4 rows, students can think: The answer to that problem should be found by looking at the number of tiles in a row in the array, which is 6. Similarly, if there are 6 tiles in a row in the array, students can think: They can find the answer by looking at the number of rows, which is 4. Why is it important to make neat rows when making an array? If neat rows are not made, it is easy to make mistakes counting the rows and number of titles in a row, and therefore get the wrong answer. COMMON ERRORS Error Students may make an incorrect array for a division problem. Example How many rows of 3 are in 18? COMMON ERRORS Springboard to Learning Review with students that if the problem asks for rows of 3, you put that number in each row and count the number of rows to get the answer. If the problem gives the number of rows, you start by making that many rows with one tile in each row. You keep adding one tile to each row until all the tiles are used. Then you count the numbers of tiles in each row. Lesson _MNYCETS225572_C06L06TE.indd 11 4/5/13 8:57 AM NYC20 New York City Implementation Guide
12 Make ConnectionsN You can write a division equation to show how many rows of 5 are in 30. Show the array you made in Investigate by completing the drawing below. You can divide to find the number of equal rows or to find the number in each row j There are _ 6 rows of 5 tiles in 30. So, _. 6 Try This! Count out 24 tiles. Make an array with the same number of tiles in 4 rows. Place 1 tile in each of the 4 rows. Then continue placing 1 tile in each row until you use all the tiles. Draw your array below. Possible explanation: it helps me solve the division problem by showing how many tiles are in each row. MATHEMATICAL Explain how making an array helps you divide. How many tiles are in each row? 6 tiles What division equation can you write for your array? or Untitled _MNYCETS225572_C06L06TE.indd 12 4/4/ :31:57 AM 4/5/13 8:57 AM New York City Implementation Guide NYC21
13 3 PRACTICE Quick Check Rt I Share and Show Guided Practice Exercises 5 8 are examples of partitive division. Before students complete the page, ask a volunteer to explain how he or she will find the answer to Exercise 5. Separate 25 tiles into 5 groups by placing one tile in each of 5 rows. Place one tile at a time in each row until all tiles are used. Count the number of tiles in each row to find the quotient. Remind students to write the division equations their arrays represent. Use Exercises 2 and 6 for Quick Check. Students should show their answers for the Quick Check on the MathBoard. Use Math Talk to focus on students understanding of using arrays to model division. How are the number of rows in an array and the number of tiles in a row related? You must find both to know how to divide using an array. If you know the total and one of the numbers, you can find the other number. Give students an extra example to make sure they understand the concept. Look at Exercise 9 again. What does the number 3 in the problem tell you? how many are in each row What does the number 5 in the problem tell you? the number of rows in the array What division problem can you write if you know that there is a total of 15 tiles and there are 5 tiles in a row? What does the 3 tell you about the array? It tells the number of rows. If Then a student misses Exercises 2 and 6 Differentiate Instruction with RtI Tier 1 Activity, p. 231B Reteach 6.6 Soar to Success Math Lesson _MNYCETS225572_C06L06TE.indd 13 4/5/13 8:57 AM NYC22 New York City Implementation Guide
14 Name Share and Show Use square tiles to make an array. Solve. 1. How many rows of 3 are in 18? Check students arrays. 2. How many rows of 6 are in 12? 6 rows 2 rows 3. How many rows of 7 are in 21? 3 rows 4. How many rows of 8 are in 32? 4 rows Make an array. Then write a division equation. Check students arrays tiles in 5 rows tiles in 2 rows tiles in 4 rows tiles in 9 rows How many rows of 3 are in 15? 10. How many rows of 8 are in 24? Possible explanation: you count the number of rows when you know the number in each row. You count the number of tiles in each row when you know the number of rows. MATHEMATICAL Explain when you count the number of rows to find the answer and when you count the number of tiles in each row to find the answer. 11. Show two ways you could make an array with tiles for Shade squares on the grid to record the arrays. Check students drawings. Chapter 6 Lesson Untitled _MNYCETS225572_C06L06TE.indd 14 4/4/ :31:58 A 4/5/13 8:57 AM New York City Implementation Guide NYC23
15 Unlock the Problem MATHEMATICAL In Exercise 12, students draw an array to solve a problem in context. In Step d, have students share other strategies they could use to solve the problem. Do you find repeated subtraction to be easier or harder than making arrays? Explain. Answers will vary. Encourage students to express their thoughts about making arrays and why they find them easier or harder than repeated subtraction. As students express their thoughts about arrays and what they find difficult or easy about them, address their apprehensions. You may find that students are unsure about whether they have made the correct number of rows or put the correct number of objects in each row. 4 SUMMARIZE MATHEMATICAL Essential Question How can you use arrays to solve division problems? Possible answer: I can find how many equal groups by placing that number of tiles in each row of an array until all tiles are used. The number of rows is the answer. I can divide the tiles into a number of rows, placing 1 tile at a time in each row, until all the tiles are used. The number of tiles in each row is the answer. Math Journal Draw an array to show how to arrange 20 chairs into 5 equal rows. Explain what each part of the array represents. Remind them that a row is the distance across from left to right. If they are showing that Thomas planted 4 seedlings in a row, they must draw 4 objects in one single row. Then they can find the number of rows it takes to reach 28. Test Prep Coach Test Prep Coach helps teachers to identify common errors that students can make. For Exercise 13, if students selected: A They added 36 and 6. B They subtracted 6 from 36. C They incorrectly divided by 6. Lesson _MNYCETS225572_C06L06TE.indd 15 4/5/13 8:57 AM NYC24 New York City Implementation Guide
16 UNLOCK the Problem MATHEMATICAL 12. Thomas has 28 tomato seedlings to plant in his garden. He wants to plant 4 seedlings in each row. How many rows of tomato seedlings will Thomas plant? A 5 B 6 C 7 D 8 Model Reason Make Sense TEST PREP a. What do you need to find? how many rows of tomato seedlings Thomas will plant b. What operation could you use to solve the problem? Possible answer: division c. Draw an array to find the number of rows of tomato seedlings. Check students drawings. There should be an array with 7 rows of 4 tiles. d. What is another way you could have solved the problem? Possible answer: I could have used repeated subtraction. e. Complete the sentences. Thomas has 28 tomato seedlings. He wants to plant Faith plants 36 flowers in 6 equal rows. How many flowers are in each row? A 42 B 30 C 7 D 6 seedlings in each. row So, Thomas will plant 7 rows of tomato seedlings. f. Fill in the bubble for the correct answer choice above. 14. There were 20 plants sold at a store on Saturday. Customers bought 5 plants each. How many customers bought the plants? A 3 B 4 C 5 D FOR MORE PRACTICE: Standards Practice Book, pp. P115 P116 FOR EXTRA PRACTICE: Standards Practice Book, p. P124 Untitled _MNYCETS225572_C06L06TE.indd /4/2011 4/5/1310:32:03 8:57 AM New York City Implementation Guide NYC25
17 Lesson 1.12 Problem Solving Model Addition and Subtraction Common Core Standard CC.3.OA.8 Solve twostep word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Lesson Objective Solve addition and subtraction problems by using the strategy draw a diagram. Essential Question How can you use the strategy draw a diagram to solve one and twostep addition and subtraction problems? 1 ENGAGE Access Prior Knowledge Introduce the lesson by asking students: Have you ever played a computer game with another person? What kind of game was it? Did you keep score? How did you know who won? 2 TEACH and TALK c Unlock the Problem MATHEMATICAL Have students read the problem. Point out that a bar model is a diagram that can help them decide what operation to use to solve a problem. Guide students to read each question in the graphic organizer and answer it before solving the problem. What question are you trying to answer? What was Sami s total score? What information do you know? He scored 84 points in the first round and 21 more points in the second round. Did Sami score more points in the first round of the game or the second round? He scored more points in the second round. How did you use the first bar model? I used the first bar model to figure out how many points Sami scored in the second round. How did you use the information from the problem to label the first bar model? I labeled the longer bar 84 to show the points Sami scored in the first round. I labeled the shorter bar 21 to show how many more points he scored in the second round. The number under the bars shows the total number of points Sami scored in the second round. What does the second bar model show? The second bar model shows Sami s total score for the two rounds. How are the bar models related to the addition sentences that represent them? Possible answer: the addends are in the bars, and the sum is represented by the number under the bars. Students may have difficulty with the fact that this is a twostep problem. They may neglect to complete the second step of the problem and think that the answer is 105 points. Tell students that they should reread the problem carefully after they find their answer to be sure that they have answered the question that is asked. Why is it important to check the original problem when you are finished to make sure you answered the right question? It is possible to get an answer that is mathematically correct, but answers the wrong question. Lesson _MNYCETS225572_C01L12TE.indd 17 4/5/13 9:02 AM NYC26 New York City Implementation Guide
18 Name Problem Solving Model Addition and Subtraction Essential Question How can you use the strategy draw a diagram to solve one and twostep addition and subtraction problems? PROBLEM SOLVING Lesson 1.12 COMMON CORE STANDARD CC.3.OA.8 Solve problems involving the four operations, and identify and explain patterns in arithmetic. UNLOCK the Problem Sami scored 84 points in the first round of a new computer game. He scored 21 more points in the second round than in the first round. What was Sami s total score? You can use a bar model to solve the problem. What do I need to find? I need to find Sami s. total score Read the Problem What information do I need to use? Sami scored _ 84 points in the first round. He scored _ 21 more points than that in the second round. Complete the bar model to show the number of points Sami scored in the second round. Solve the Problem How will I use the information? I will draw a bar model to show the number of points Sami scored in each round. Then I will use the bar model to decide which operation to use. Complete another bar model to show Sami s total score. 84 points 21 points 84 points 105 points points points 1 5 n 5 n 1 5 s s 1. How many points did Sami score in the second round? 105 points 2. What was Sami s total score? 189 points Chapter 1 51 Untitled75 3_MNYCETS225572_C01L12TE.indd /7/2011 4/5/13 9:34:47 9:02 A New York City Implementation Guide NYC27
19 c Try Another Problem Have students read the problem and then answer the questions in the graphic organizer and solve the problem. Invite students to share their diagrams and explanations. Who scored more points? How does the bar model show this? Anna; the bar for Anna s points is longer. How does the bar model help you solve the problem? Possible answers: it shows that the unknown part is the difference between Anna s and Greg s scores. What numbers do you need to subtract to solve the problem? I need to subtract Greg s score from Anna s score. How is the bar model related to the subtraction sentence that represents it? Possible answer: the number in the shorter bar is subtracted from the number in the longer bar to find the difference, which is represented by the unknown quantity to the right of the shorter bar. In problem 4, there are several ways to check for reasonableness, including estimation. Invite students to share their answers and discuss the different ways to estimate, such as rounding or using compatible numbers, to check for reasonableness. You may suggest that students place completed Try Another Problem graphic organizers in their portfolios. COMMON ERRORS Error Students may have difficulty determining how to label the bars in the bar model. Example In Try Another Problem, students may draw a shorter bar for Anna s points than for Greg s points and, therefore, label it incorrectly. Springboard to Learning Remind students that although the bars in a bar model do not have to be in exact proportion, longer bars should represent greater numbers. Have students first determine whose score is greater before drawing their bar models. Use Math Talk to focus on students understanding of how to use bar models to solve a problem. Ask students to look at the bar models again and compare the length of each bar model compared to Anna and Greg s scores. What would the bar model look like if Greg scored more points than Anna? Greg s bar model would be longer than Anna s. Whose bar model would show the unknown part with the gray box? Greg s bar model Lesson _MNYCETS225572_C01L12TE.indd 19 4/5/13 9:01 AM NYC28 New York City Implementation Guide
20 Try Another Problem Anna scored 265 points in a computer game. Greg scored 142 points. How many more points did Anna score than Greg? You can use a bar model to solve the problem. What do I need to find? I need to find how many more points Anna scored than Greg. Anna 265 points Read the Problem What information do I need to use? Solve the Problem Record the steps you used to solve the problem. I need to complete the bar model to show each person s score. Anna scored 265 points. Greg scored 142 points. How will I use the information? I need to subtract to find the unknown part n n I will use a bar model to show the points each person scored. Then I will use the bar model to decide which operation to use. Greg 142 points points 3. How many more points did Anna score than Greg? 123 points 4. How do you know your answer is reasonable? Possible answer: I can use friendly numbers to subtract mentally. I subtract 2 from 142 to get 140. Then I subtract 2 from 265 to get 263; How did your drawing help you solve the problem? Possible answer: the bar model helped me see that I had to subtract and what numbers I needed to subtract. MATHEMATICAL Explain how the length of each bar in the model would change if Greg scored more points than Anna but the totals remained the same. Anna s bar would be the shorter bar and Greg s bar would be the longer bar. 52 Untitled _MNYCETS225572_C01L12TE.indd 20 4/7/2011 9:34:49 AM 4/5/13 9:01 AM New York City Implementation Guide NYC29
21 3 PRACTICE Share and Show Guided Practice Be sure to point out to students that Exercise 1 is a twostep problem. Ask students to determine what they need to find, what information they need to use, and how they can use the information. Problem Exercise 2 requires that students use higher order thinking skills. The problem varies the scenario presented in Exercise 1. Students should connect the number and length of the bars to the numeric label of each bar. What would be used to represent the third student? A third bar model Is there just one answer for the number of votes that each student gets? Explain. No; There are many possibilities for the number of votes each student gets. The numbers can be any three numbers that add up to 121 votes. Use Exercises 3 and 4 for Quick Check. Students should show their answers for the Quick Check on the MathBoard. Go Deeper MATHEMATICAL To extend their thinking, have students write a problem and draw their own bar model to solve it. Explain that the problem can be about anything they wish, and can have two or more bar models. Remind students that they would need at least two bar models so that two quantities can be compared. What kind of problem can you write that would use a bar model to help solve it? The problem could be about comparing scores, lengths or weights of items, or any comparison in which one quantity is known, part of another quantity is known, and the total is unknown. Quick Check Rt I If a student misses Exercises 3 and 4 Then Differentiate Instruction with RtI Tier 1 Activity, p. 51B Reteach 1.2 Soar to Success Math Lesson _MNYCETS225572_C01L12TE.indd 21 4/5/13 9:01 AM NYC30 New York City Implementation Guide
22 Name Share and ShowN 1. Sara received 73 votes in the school election. Ben received 25 fewer votes than Sara. How many students voted in all? First, find how many students voted for Ben. Sara UNLOCK the Problem 73 votes Tips Use the problem solving MathBoard. Choose a strategy you know. Think: n Write the numbers in the bars. So, Ben received _ 48 votes. Next, find the total number of votes. Ben votes votes votes Think: s Write the numbers in the bars. So, 121 students voted in all. 2. What if there were 3 students in another election and the total number of votes was the same? What would the bar model for the total number of votes look like? How many votes might each student get? 73 votes 48 votes votes votes Possible answer: there would be 3 bars. Possible number of votes: 55, 30, and Pose a Problem Use the bar model at the right. Write a problem to match it. 89 Possible problem: Russ and Juan collect stamps. Together they collected 157 stamps. Russ collected 89 stamps. How many stamps did Juan collect? 4. Solve your problem. Will you add or subtract? Possible answer: subtract; Chapter 1 Lesson Untitled _MNYCETS225572_C01L12TE.indd 22 4/7/2011 9:34:51 A 4/5/13 9:01 AM New York City Implementation Guide NYC31
23 c On Your Own Independent Practice If students complete Exercises 3 and 4 correctly, they may continue with Independent Practice. Encourage students to solve the problems independently, but provide assistance as needed. If students struggle with the On Your Own problems, ask them what they are trying to find out, and what they already know about the problem. Encourage them to use the Show Your Work area of the page and then show it to you to see if their thinking and calculations are correct. Circle parts of their work that contains errors, and guide them through the process of making corrections. Problem Exercise 8 requires that students use higher order thinking skills to solve the problem. What can you do to find out the greatest number that could be rounded to 400? I must think of the greatest number that can be rounded down to 400. What number would be in the hundreds place? Explain. 4; I know I will be rounding down because I am looking for the greatest number that rounds to 400. What number would be in the tens place? Explain. 4; I know that any number that rounds to 400 must be less than 450. What number would be in the ones place? Explain. 9; If I have a 4 in the tens place, the greatest number that can be in the ones place and still round down to 400 would be 9. The answer is 449. Test Prep Coach Test Prep Coach helps teachers to identify common errors that students can make. In Exercise 9, if students selected: A They subtracted instead of added. B They did not add the regrouped ten. D They added incorrectly. 4 SUMMARIZE MATHEMATICAL Essential Question How can you use the strategy draw a diagram to solve one and twostep addition and subtraction problems? Possible answer: I can draw a bar model to see if I need to add or subtract. Math Journal Write an addition or subtraction problem and draw a diagram to solve it. Lesson _MNYCETS225572_C01L12TE.indd 23 4/5/13 9:01 AM NYC32 New York City Implementation Guide
24 On Your Own MATHEMATICAL Model Reason Make Sense 5. Tony s Tech Store is having a sale. The store had 142 computers in stock. During the sale, 91 computers were sold. How many computers were not sold? 51 computers 6. The number of computer games sold during the sale was 257. This is 162 more than the number sold the week before the sale. How many computer games were sold the week before the sale? 95 computer games 7. In one week, 128 cell phones were sold. The following week, 37 more cell phones were sold than the week before. How many cell phones were sold in those two weeks? 293 cell phones 8. On Monday, the number of customers in the store, rounded to the nearest hundred, was 400. What is the greatest number of customers that could have been in the store? Explain. 449 customers; Possible explanation: to round to 400, the greatest number must have a 4 in the tens place so that the hundreds place stays the same. The greatest number in the ones place is Test Prep The number of laptop computers sold in one day was 42. That is 18 fewer than the total number of desktop computers sold. How many desktop computers were sold? A 24 C 60 B 50 D FOR MORE PRACTICE: Standards Practice Book, pp. P25 P26 FOR EXTRA PRACTICE: Standards Practice Book, p. P28 Untitled75 3_MNYCETS225572_C01L12TE.indd /7/2011 4/5/13 9:34:52 9:01 AM New York City Implementation Guide NYC33
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