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 3-digit 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 PRACTICES 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 2-digit 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 NYC10 New York City Implementation Guide

2 Nombre Redondear a la decena o la centena más próxima Pregunta esencial Cómo puedes redondear números? SOLUCIONA el problema Lección 1.2 ESTÁNDAR COMÚN CC.3.NBT.1 Use place value understanding and properties of operations to perform multi-digit arithmetic. Al redondear un número, hallas un número que te indica alrededor de cuánto o de cuántos. El bate de béisbol de Mara mide 32 pulgadas de longitud. Cuál es su longitud redondeada a la decena de pulgadas más próxima? De una manera Usa una recta numérica para redondear. A Redondea 32 a la decena más próxima Halla entre qué decenas está el número. 32 está entre _ 30 y _ está más cerca de _ 30 que de _ redondeado a la decena más próxima es _. 30 MÉTODOS MATEMÁTICOS Menciona otros tres números que se redondeen en 30 al redondearlos a la decena más próxima. Explícalo. Entonces, la longitud del bate de Mara redondeada a la decena de pulgadas más próxima es _ 30 pulgadas. B Redondea 174 a la centena más próxima Halla entre qué centenas está el número. 174 está entre _ 100 y _ está más cerca de _ 200 que de _ Entonces, 174 redondeado a la centena más próxima es _. Respuesta posible: 28, 31, 34; Explicación posible: 28 está entre 20 y 30 pero está más cerca de 30; 31 y 34 están entre 30 y 40 pero están más cerca de 30. Capítulo 1 9 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 3-digit 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 3-digit 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 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 NYC12 New York City Implementation Guide

4 Inténtalo! Redondea 718 a la decena y a la centena más próxima. Ubica y rotula 718 en las rectas numéricas. Revise el trabajo de los estudiantes. A Decena más próxima B Centena más próxima está más cerca de _ está más cerca de _ 700 que de _. 710 que de _ Entonces, 718 se redondea en _. 700 Entonces, 718 se redondea en _. De otra manera Usa el valor posicional. A Redondea 63 a la decena más próxima. Piensa: El dígito en el lugar de las unidades indica si el número está más cerca de 60 o de 70. 3, 5 Entonces, el dígito de las decenas queda igual. Escribe 6 en el lugar de las decenas. 63 Halla el lugar al que quieres redondear. Observa el dígito de la derecha. Si el dígito es menor que 5, el dígito en el lugar de redondeo queda igual. Si el dígito es igual a o mayor que 5, el dígito en el lugar de redondeo aumenta en uno. Escribe ceros en los dígitos a la derecha del lugar de redondeo. Escribe cero en el lugar de las unidades. Entonces, 63 redondeado a la decena más próxima es _. 60 B Redondea 457 a la centena más próxima. Piensa: El dígito en el lugar de las decenas indica si el número está más cerca de 400 o de Entonces, el dígito de las centenas aumenta en uno. Escribe 5 en el lugar de las centenas. Escribe cero en el lugar de las decenas y de las unidades. Entonces, 457 redondeado a la centena más próxima es _ Charla matemática: Explicación posible: Cuando uso el valor posicional, observo el dígito a la derecha del lugar de redondeo para ver de qué decena o centena está más cerca. Si el dígito que sigue es igual a o mayor que 5, estará más cerca de la decena o la centena que sigue. Cuando uso una recta numérica, puedo ver de qué decena o centena está más cerca el número. MÉTODOS MATEMÁTICOS Explica en qué se parecen usar el valor posicional y usar una recta numérica. 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 PRACTICES 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 NYC14 New York City Implementation Guide

6 Nombre Comunicar y mostrarn Ubica y rotula 46 en la recta numérica. Redondea a la decena más próxima está entre _ 40 y _ está más cerca de _ 50 que de _ redondeado a la decena más próxima es _. 50 Redondea a la decena más próxima. Revise el trabajo de los estudiantes _ _ _ 50 MÉTODOS MATEMÁTICOS Cuál es el mayor número que se redondea en 50 al redondearlo a la decena más próxima? Cuál es el menor número? Explícalo. mayor número: 54; menor número: 45; Explicación posible: Como este grupo de números se redondea en 50 (45, 46, 47, 48, 49, 50, 51, 52, 53, 54), el mayor número es 54 y el menor número es 45. Redondea a la centena más próxima _ _ _ 700 Por tu cuenta Ubica y rotula 548 en la recta numérica. Redondea a la centena más próxima. Revise el trabajo de los estudiantes está entre _ 500 y _ está más cerca de _ 500 que de _ redondeado a la centena más próxima es _. 500 Redondea a la decena y a la centena más próxima _ 580 _ Capítulo 1 Lección 2 11 New York City Implementation Guide NYC15

7 Problem Solving MATHEMATICAL PRACTICES 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 PRACTICES 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 NYC16 New York City Implementation Guide

8 Resolución de problemas Usa la tabla para resolver los problemas 16 a 18. MÉTODOS MATEMÁTICOS Representar Razonar Interpretar 16. Qué día fueron alrededor de 900 visitantes a la exposición de las jirafas? Visitantes de la exposición de las jirafas domingo 17. Qué cantidad de visitantes redondeada a la decena más próxima fue el domingo a la exposición de las jirafas? 890 visitantes 18. Qué dos días fueron alrededor de 800 visitantes por día a la exposición de las jirafas? lunes y jueves Día Domingo Lunes Martes Miércoles Jueveses Viernes Sábado Cantidad de visitantes Escribe cinco números que se redondeen en 360 al redondearlos a la decena más próxima. Resultado posible: 356, 357, 359, 361, Cuál es el error? Camilo dijo que 555 redondeado a la decena más próxima es 600. Cuál es el error de Camilo? Explícalo. El resultado debe ser 560. Explicación posible: Camilo redondeó a la centena más próxima en lugar de a la decena más próxima. 21. Preparación para la prueba Cuánto es 438 redondeado a la decena más próxima? A 450 B 440 C 430 D PARA PRACTICAR MÁS: Cuaderno de práctica de los estándares, págs. P5 y P6 PRÁCTICA ADICIONAL: Cuaderno de práctica de los estándares, pág. P27 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 real-world 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 PRACTICES 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 NYC18 New York City Implementation Guide

10 Nombre Representar con matrices Pregunta esencial Cómo puedes usar matrices para resolver problemas de división? Lección 6.6 ESTÁNDAR COMÚN CC.3.OA.3 Represent and solve problems involving multiplication and division. Investigar Materiales fichas cuadradas Puedes usar matrices para representar la división y hallar grupos iguales. A. Cuenta 30 fichas cuadradas. Forma una matriz para hallar cuántas hileras de 5 hay en 30. B. Forma una hilera de 5 fichas cuadradas. C. Sigue hasta que hayas formado todas las hileras de 5 fichas cuadradas posibles. 6 hileras Cuántas hileras de 5 formaste? Sacar conclusionesn 1. Explica cómo usaste las fichas cuadradas para hallar el número de hileras de 5 fichas que hay en 30. Explicación posible: Dispuse las 30 fichas cuadradas en hileras de 5 fichas cuadradas. Luego conté el número de hileras que formé. Había 6 hileras. Entonces, hay 6 hileras de 5 en Qué ecuación de multiplicación podrías escribir para la matriz? Explícalo ; Explicación posible: Hay 6 hileras de 5 fichas cuadradas cada una. 6 grupos de 5 son Aplica Indica cómo usar una matriz para hallar cuántas hileras de 6 hay en 30. Respuesta posible: Formo hileras de 6 hasta usar las 30 fichas cuadradas. Hay 5 hileras de 6 en 30. Capítulo 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? 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 NYC20 New York City Implementation Guide

12 Hacer conexionesn Puedes escribir una ecuación de división para mostrar cuántas hileras de 5 hay en 30. Completa el siguiente dibujo para mostrar la matriz que formaste en la sección Investigar. Puedes dividir para hallar el número de hileras iguales o para hallar el número de objetos en cada hilera j 6 En 30, hay _ hileras de 5 fichas cuadradas. Entonces, _. Inténtalo! Cuenta 24 fichas cuadradas. Forma una matriz que tenga el mismo número de fichas cuadradas en 4 hileras. Coloca 1 ficha cuadrada en cada una de las 4 hileras. Luego sigue colocando 1 ficha cuadrada en cada hilera hasta que hayas usado todas las fichas cuadradas. Dibuja tu matriz abajo. Cuántas fichas cuadradas hay en cada hilera? 6 fichas cuadradas Explicación posible: En la matriz se muestra cuántas fichas cuadradas hay en cada hilera, y eso me ayuda a resolver el problema de división. MÉTODOS MATEMÁTICOS Explica cómo formar una matriz te ayuda a dividir. Qué ecuación de división puedes escribir para tu matriz? ó 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 NYC22 New York City Implementation Guide

14 Nombre Comunicar y mostrar Usa fichas cuadradas para formar una matriz. Resuelve. 1. Cuántas hileras de 3 hay en 18? 2. Cuántas hileras de 6 hay en 12? 6 hileras 2 hileras Revise las matrices de los estudiantes. 3. Cuántas hileras de 7 hay en 21? 3 hileras 4. Cuántas hileras de 8 hay en 32? 4 hileras Revise las matrices Forma una matriz. Luego escribe una ecuación de división. de los estudiantes fichas cuadradas en 5 hileras fichas cuadradas en 2 hileras fichas cuadradas en 4 hileras fichas cuadradas en 9 hileras Cuántas hileras de 3 hay en 15? 10. Cuántas hileras de 8 hay en 24? Explicación posible: Cuento el número de hileras cuando conozco el número que hay en cada hilera. Cuento el número de fichas cuadradas que hay en cada hilera cuando conozco el número de hileras. MÉTODOS MATEMÁTICOS Explica cuándo debes contar el número de hileras para hallar el resultado y cuándo debes contar el número de fichas cuadradas que hay en cada hilera para hallar el resultado. 11. Muestra dos maneras en que podrías formar una matriz con fichas cuadradas para Sombrea cuadrados en la cuadrícula para mostrar las matrices. Revise los dibujos de los estudiantes. Capítulo 6 Lección New York City Implementation Guide NYC23

15 Unlock the Problem MATHEMATICAL PRACTICES 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 PRACTICES 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 NYC24 New York City Implementation Guide

16 MÉTODOS MATEMÁTICOS Representar Razonar Interpretar SOLUCIONA el problema PREPARACIÓN PARA LA PRUEBA 12. Thomas tiene 28 plántulas de tomate para plantar en su jardín. Quiere plantar 4 plántulas en cada hilera. Cuántas hileras de plántulas de tomate plantará Thomas? A 5 B 6 C 7 D 8 a. Qué debes hallar? cuántas hileras de plántulas de tomate plantará Thomas b. Qué operación podrías usar para resolver el problema? Respuesta posible: la división c. Dibuja una matriz para hallar el número de hileras de plántulas de tomate. Revise los dibujos de los estudiantes. Debe haber una matriz con 7 hileras de 4 fichas cuadradas. d. De qué otra manera podrías haber resuelto el problema? Respuesta posible: Podría haber usado la resta repetida. e. Completa las oraciones. Thomas tiene 28 plántulas de tomate. Quiere plantar 4 plántulas en cada. hilera Entonces, Thomas plantará 13. Faith planta 36 flores en 6 hileras iguales. Cuántas flores hay en cada hilera? A 42 B 30 C 7 D 6 7 hileras de plántulas de tomate. f. Rellena el círculo del resultado correcto arriba. 14. El sábado se vendieron 20 plantas en una tienda. Cada cliente compró 5 plantas. Cuántos clientes compraron plantas? A 3 B 4 C 5 D PARA PRACTICAR MÁS: Cuaderno de práctica de los estándares, págs. P115 y P116 PRÁCTICA ADICIONAL: Cuaderno de práctica de los estándares, pág. P124 New York City Implementation Guide NYC25

17 Lesson 1.12 Problem Solving Model Addition and Subtraction Common Core Standard CC.3.OA.8 Solve two-step 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 two-step 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 PRACTICES 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 two-step 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 NYC26 New York City Implementation Guide

18 Nombre Resolución de problemas Representar la suma y la resta Pregunta esencial Cómo puedes usar la estrategia hacer un diagrama para resolver problemas de suma y de resta de un paso y de dos pasos? RESOLUCIÓN DE PROBLEMAS Lección 1.12 ESTÁNDAR COMÚN CC.3.OA.8 Solve problems involving the four operations, and identify and explain patterns in arithmetic. SOLUCIONA el problema Sami anotó 84 puntos en la primera ronda de un videojuego nuevo. En la segunda ronda, anotó 21 puntos más que en la primera. Cuál fue el puntaje total de Sami? Puedes usar un modelo de barras para resolver el problema. Qué debo hallar? Debo hallar el. puntaje total de Sami Lee el problema Qué información debo usar? Sami anotó _ 84 puntos en la primera ronda. Completa el modelo de barras para mostrar la cantidad de puntos que anotó Sami en la segunda ronda. Anotó _ 21 puntos más en la segunda ronda. Resuelve el problema Cómo usaré la información? Dibujaré un modelo de barras para mostrar la cantidad de puntos que anotó Sami en cada ronda. Luego usaré el modelo para decidir qué operación usar. Completa otro modelo de barras para mostrar el puntaje total de Sami. 84 puntos 21 puntos 84 puntos 105 puntos puntos puntos Cuántos puntos anotó Sami en la segunda ronda? 105 puntos 2. Cuál fue el puntaje total de Sami? 105 puntos 5 Capítulo 1 51 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 NYC28 New York City Implementation Guide

20 Haz otro problema Anna anotó 265 puntos en un videojuego. Greg anotó 142 puntos. Cuántos puntos más anotó Anna que Greg? Puedes usar un modelo de barras para resolver el problema. Qué debo hallar? Debo hallar cuántos puntos más anotó Anna que Greg. Lee el problema Qué información debo usar? Resuelve el problema Cómo usaré la información? Anota los pasos que seguiste para resolver el problema. Debo completar el modelo de barras para mostrar el puntaje de cada uno. Anna anotó 265 puntos. Greg anotó 142 puntos. Usaré un modelo de barras para mostrar los puntos que anotó cada uno. Luego usaré ese modelo para decidir qué operación usar. Debo restar para hallar la parte desconocida. Anna 265 puntos Greg 142 puntos puntos 3. Cuántos puntos más anotó Anna que Greg? 123 puntos 4. Cómo sabes que tu resultado es razonable? Respuesta posible: Puedo usar números amigos para restar mentalmente. Resto 2 de 142 y obtengo 140. Luego resto 2 de 265 y obtengo 263; Cómo te ayudó el dibujo a resolver el problema? Respuesta posible: El modelo de barras me ayudó a ver que debía restar y qué números debía restar. 52 MÉTODOS MATEMÁTICOS Explica cómo cambiaría la longitud de cada barra del modelo si Greg anotara más puntos que Anna, pero el puntaje total quedara igual. La barra de Anna sería la más corta y la barra de Greg sería la más larga. 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 two-step 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 PRACTICES 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 NYC30 New York City Implementation Guide

22 Nombre Comunicar y mostrarn 1. En la elección de la escuela, Sara obtuvo 73 votos. Benji obtuvo 25 votos menos que Sara. Cuántos estudiantes votaron en total? Primero, halla cuántos estudiantes votaron por Benji. Piensa: Escribe los números en las barras. Entonces, Benji obtuvo _ 48 votos. A continuación, halla la cantidad total de votos. Piensa: Sara Benji SOLUCIONA el problema 73 votos 5 votos votos 25 votos 48 Pistas Usa la pizarra de Resolución de problemas. Elige una estrategia que conozcas. votos votos Escribe los números en las barras. Entonces, 121 estudiantes votaron en total. 2. Qué pasaría si en otra elección hubiera que votar por 3 estudiantes y la cantidad total de votos fuera la misma? Cómo sería el modelo de barras de la cantidad total de votos? Cuántos votos podría obtener cada estudiante? Respuesta posible: Habría 3 barras. Cantidad de votos posibles: 55, 30 y 36 5 votos 121 votos 3. Plantea un problema Usa el modelo de barras que está a la derecha. Escribe un problema para él. Problema posible: Russ y Juan coleccionan estampillas. Entre ambos, coleccionaron 157 estampillas. Russ coleccionó 89 estampillas. Cuántas estampillas coleccionó Juan? 4. Resuelve tu problema. Sumarás o restarás? Respuesta posible: Restaré; Capítulo 1 Lección 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 PRACTICES Essential Question How can you use the strategy draw a diagram to solve one- and two-step 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 NYC32 New York City Implementation Guide

24 Por tu cuenta 5. En la Tienda Tecnológica de Tony hay una liquidación. Había 142 computadoras. En la liquidación se vendieron 91 computadoras. Cuántas computadoras no se vendieron? 51 computadoras MÉTODOS MATEMÁTICOS Representar Razonar Interpretar 6. En la liquidación se vendieron 257 videojuegos. Esta cantidad es 162 videojuegos más que los que se vendieron la semana anterior a la liquidación. Cuántos videojuegos se vendieron la semana anterior a la liquidación? 95 videojuegos 7. En una semana se vendieron 128 celulares. La semana siguiente se vendieron 37 celulares más que la semana anterior. Cuántos celulares se vendieron en esas dos semanas? 293 celulares 8. El lunes la cantidad de clientes que fueron a la tienda redondeada a la centena más próxima fue 400. Cuál es la mayor cantidad de clientes que pueden haber ido a la tienda? Explícalo. 449 clientes; Explicación posible: Para redondear en 400, el número mayor debe tener un 4 en el lugar de las decenas para que el lugar de las centenas quede igual. El número mayor en el lugar de las unidades es Preparación para la prueba La cantidad de computadoras portátiles vendidas en un día fue 42. Esa cantidad es 18 menos que la cantidad de computadoras de escritorio vendidas. Cuántas computadoras de escritorio se vendieron? A 24 C 60 B 50 D PARA PRACTICAR MÁS: Cuaderno de práctica de los estándares, págs. P25 y P26 PRÁCTICA ADICIONAL: Cuaderno de práctica de los estándares, pág. P28 New York City Implementation Guide NYC33