of what it represents in the place to its left. itools: Base-Ten Blocks

Transcripción

1 Lesson 1.1 Investigate Place Value and Patterns CC.5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1_ of what it represents in the place to its left. Recognize the 10 to 1 relationship among place-value positions. Cómo puedes describir la relación entre dos valores posicionales? base ten blocks itools: Base-Ten Blocks 0 min. Investigate Activity to Build Conceptual Understanding activity is for students to recognize the respective values of digits in a number based on their place-value position. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress through instruction. the base-ten system to develop sound mathematical practices by asking Por qué los bloques de base diez son un buen modelo para comprender el valor posicional? Qué conclusiones puedes sacar de tu modelo? Cómo usarás la información de tu modelo? Qué sucede cuando te mueves a la izquierda del valor posicional? once they understand these place-value concepts. Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, Exercises Cómo puedes describir la relación entre dos valores posicionales? Cualquier valor posicional es diez veces mayor que el valor que tiene al lado derecho y 1 del valor a su lado izquierdo. 10 Chapter 1 7 New York City Implementation Guide NYC37

2 Lesson 1. Place Value of Whole Numbers CC.5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1_ of what it represents in the place to its left. Read and write whole numbers through hundred millions. Cómo puedes leer, escribir y representar números naturales hasta las centenas de millón? itools: Base-Ten Blocks Animated Math Models 0 min. activity is for students to understand the place value system for whole numbers through hundred millions. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress place value to develop sound mathematical practices by asking En qué situaciones sería mejor usar la forma en palabras o la forma desarrollada en lugar de la forma normal? Representan los números el mismo valor cuando se escriben en formas diferentes? Cómo sabes qué dígito tiene el mayor valor en un número? Qué palabra del vocabulario matemático describe cada conjunto de tres dígitos? once they understand these place value concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 8, 14, 17 or 18 0 Cómo puedes leer, escribir y representar números naturales hasta las centenas de millón? Identifico el valor posicional de cada uno de los dígitos en un periodo, nombro el número y el periodo y coloco una coma entre los periodos. Write Standard Form, Expanded Form, and Word Form at the top of the page. Write five numbers that are at least 8 digits long under Standard Form. Write the expanded form and the word form for each number under the appropriated heading. 8 Chapter 1 NYC38 New York City Implementation Guide

3 Lesson 1.3 Algebra Properties CC.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Use properties of operations to solve problems. Cómo puedes aplicar las propiedades de las operaciones para resolver problemas? Models Animated Math itools: Counters 0 min. activity is for students to use properties of multiplication and addition to solve problems. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress through instruction. operations and properties to develop sound mathematical practices by asking Qué hiciste primero? Por qué? Funcionará siempre ese método? Qué propiedades aplicaste para hallar la respuesta? Por qué puedes aplicar esa propiedad para este problema? once they understand these addition and multiplication properties. Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, making them especially rich. Exercises 10 1 or Cómo puedes aplicar las propiedades de las operaciones para resolver problemas? Puedo aplicar las propiedades asociativa, conmutativa, de identidad y distributiva para agrupar los números que sean más fáciles de sumar o multiplicar. Explain how you could mentally find by using the Distributive Property. Chapter 1 9 New York City Implementation Guide NYC39

4 Lesson 1.4 Algebra Powers of 10 and Exponents CC.5.NBT. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. Write and evaluate repeated factors in exponent form. Cómo puedes usar un exponente para mostrar las potencias de 10? base-ten blocks Animated Math Models itools: Base-Ten Blocks 0 min. activity is for students to use models and repeated multiplication to understand exponential form and powers of ten. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress through instruction. modeling and place value to develop sound mathematical practices by asking Por qué son los bloques de base diez un buen modelo para las potencias de diez? Qué conclusiones puedes sacar de tu modelo? Cómo cambiaría tu modelo si el número base no fuera diez? Qué otros patrones puedes hallar? once they understand these base-ten concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 11, 13, 14 Cómo puedes usar un exponente para mostrar las potencias de 10? Puedo contar el número de veces que se usa el 10 como factor que se repite y escribir ese número como exponente. Consider Write a pattern to find the value of the expression. 30 Chapter 1 NYC40 New York City Implementation Guide

5 Lesson 1.5 Algebra Multiplication Patterns CC.5.NBT. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. Use a basic fact and pattern to multiply mentally by multiples of 10, 100, and 1,000. Cómo puedes usar una operación básica y un patrón para multiplicar mentalmente por múltiplos de 10, 100 y 1,000? Animated Math Models 0 min. activity is for students to form two generalizations based on patterns. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress place value and patterns to develop sound mathematical practices by asking Qué operación usaste para representar la situación? Por qué? Cómo puedes usar un problema más simple como ayuda para hallar la respuesta? Qué propiedad puedes aplicar para resolver el problema? Por qué? Qué otros patrones hallaste? once they understand the application of these multiplication patterns. Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, making them especially rich. Exercises 0 3 or 4 6 Cómo puedes usar una operación básica y un patrón para multiplicar mentalmente por múltiplos de 10, 100 y 1,000? Comienzo con una operación básica y luego escribo más ecuaciones con ceros adicionales en los factores hasta que hallo el producto del problema original. Busco un patrón para escribir los productos. Do the products and have the same number of zeros? Explain. Chapter 1 31 New York City Implementation Guide NYC41

6 Lesson 1.6 Multiply by 1-Digit Numbers CC.5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm. Multiply by 1-digit numbers. Cómo multiplicas por números de 1 dígito? Real World Video, Ch. 1 Animated Math Models itools: Base-Ten Blocks 0 min. activity is for students to use the standard algorithm to multiply by 1-digit numbers. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress multiplication to develop sound mathematical practices by asking Qué operación usaste para representar la situación? Por qué representa la situación la multiplicación? Qué hiciste primero? Por qué? Cómo sabes que la respuesta es razonable? once they understand how to multiply using the standard algorithm Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, making them especially rich. Exercises 8 10 or 19 Cómo multiplicas por números de 1 dígito? Puedo usar el valor posicional y la reagrupación. Primero multiplico las unidades, luego las decenas y así sucesivamente, reagrupando a medida que sea necesario. Show how to solve the problem using place value with regrouping. Explain how you knew when to regroup. 3 Chapter 1 NYC4 New York City Implementation Guide

7 Lesson 1.7 Multiply by -Digit Numbers Note: The instructional time for this lesson can also be days. Use the extra time to complete some of the On Your Own exercises as a class. Discuss how place value supports the multiplication algorithm. CC.5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm. Multiply by -digit numbers. Cómo multiplicas por números de dígitos? Animated Math Models 0 min. activity is for students to use place value and regrouping to multiply by -digit numbers. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress multiplication and estimation to develop sound mathematical practices by asking Qué operación usaste para representar la situación? Cómo representa la situación la multiplicación? Has resuelto algún problema parecido al planteado? Cómo sabes que la respuesta es razonable? once they understand these multiplication concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 17 0 Cómo multiplicas por números de dígitos? Multiplico el primer factor por las unidades del segundo factor y reagrupo si es necesario. Multiplico el primer factor por las decenas del segundo factor y reagrupo si es necesario. Luego, sumo los dos productos parciales. Write a problem multiplying a 3-digit number by a -digit number. Show all the steps to solve it by using place value and regrouping and by using partial products. Chapter 1 33 New York City Implementation Guide NYC43

8 Lesson 1.8 Relate Multiplication to Division CC.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Use multiplication to solve division problems. Cómo se usa la multiplicación para resolver un problema de división? 0 min. activity is for students to use the Distributive Property to relate multiplication and division. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress multiplication and division to develop sound mathematical practices by asking Qué operación usaste para representar la situación? Por qué representa la situación la división? Por qué es una matriz rectangular un buen modelo para el problema planteado? Qué conclusiones puedes sacar de tu modelo? once they understand the relationship between multiplication and division. Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, making them especially rich. Exercises 8 10 or Cómo se usa la multiplicación para resolver un problema de división? Primero dibujo un modelo de área, lo divido en dos partes y escribo el divisor a la izquierda. Escribo dos múltiplos del divisor cuya suma sea igual al dividendo; divido esos números entre el divisor y escribo los cocientes. La respuesta es la suma de los cocientes. For the problem , draw two different ways to break apart the array. Use the Distributive Property to write products for each different way. 34 Chapter 1 NYC44 New York City Implementation Guide

9 Lesson 1.9 Problem Solving Multiplication and Division CC.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Use the strategy solve a simpler problem to solve problems. Cómo puedes usar la estrategia de resolver un problema más simple como ayuda para resolver un problema de división? 0 min. activity is for students to solve division problems by breaking difficult numbers into numbers that are easier to divide. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress multiples and properties to develop sound mathematical practices by asking Qué hiciste primero? Por qué? Qué estrategia podrías usar para facilitar los cálculos? Qué propiedades aplicaste para hallar la respuesta? Cómo sabes que la respuesta es razonable? Students can begin independent practice once they understand these multiplication and division concepts. Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, making them especially rich. Exercises 6, 8, 9 Cómo puedes usar la estrategia de resolver un problema más simple como ayuda para resolver un problema de división? Puedo usar esta estrategia para descomponer un dividendo en números más pequeños que sean más fáciles de dividir para resolver un problema. Rewrite Exercise 4 with different numbers. Solve the new problem and show your work. Chapter 1 35 New York City Implementation Guide NYC45

10 Lesson 1.10 Algebra Numerical Expressions CC.5.OA. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. Write numerical expressions. Cómo puedes describir una situación con una expresión numérica? 0 min. activity is for students to translate words into numerical expressions. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress operations and expressions to develop sound mathematical practices by asking Qué operación usaste para representar la situación? Cómo representa la operación la situación? Cómo lo sabes? Cómo decides qué operación realizar primero? Students can begin independent practice once they understand how to use numerical expressions. Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, making them especially rich. Exercises 9, 1, 13 or Cómo puedes describir una situación con una expresión numérica? Busco en el problema el lenguaje que representa las operaciones. Luego decido en qué orden se necesita completar las operaciones y escribo una expresión. Si hay más de una operación, coloco paréntesis alrededor de la primera operación a completarse. Write a numerical expression. Then write words to match the expression. 36 Chapter 1 NYC46 New York City Implementation Guide

11 Lesson 1.11 Algebra Evaluate Numerical Expressions CC.5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Use the order operations to evaluate numerical expressions. En qué orden deben evaluarse las operaciones para hallar la solución de un problema? Animated Math Models 0 min. activity is for students to learn the fundamentals of the order of operations. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress mathematical operations to develop sound mathematical practices by asking Qué hiciste primero? Por qué? Has resuelto un problema parecido a este? Descríbelo. Qué crees que sucederá si sumas antes de multiplicar? Cómo sabes que la respuesta es razonable? once they understand how to apply the order of operations to an expression. Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, making them especially rich. Exercises 1 14 En qué orden deben evaluarse las operaciones para hallar la solución de un problema? 1: realizar las operaciones entre paréntesis; : multiplicar y dividir de izquierda a derecha; 3: sumar y restar de izquierda a derecha. Give two examples that show how using parentheses can change the order in which operations are performed in an expression. Chapter 1 37 New York City Implementation Guide NYC47

12 Lesson 1.1 Algebra Grouping Symbols CC.5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Evaluate numerical expressions with parentheses, brackets, and braces. En qué orden deben evaluarse las operaciones para hallar una solución cuando hay paréntesis dentro de paréntesis? 0 min. activity is for students to extend their understanding of the order of operations to include grouping symbols. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress the order of operations to develop sound mathematical practices by asking these questions. Qué hiciste primero? Por qué? Qué operaciones usaste para representar la situación? Por qué? Qué sucedería si no usaras paréntesis? once they understand these algebra concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 8, 10, 11 En qué orden deben evaluarse las operaciones para hallar una solución cuando hay paréntesis dentro de paréntesis? Debo realizar las operaciones desde el conjunto más interno de símbolos de agrupación hasta el par más externo. El orden es paréntesis, corchetes y llaves. Explain how to use grouping symbols to organize information appropriately. 38 Chapter 1 NYC48 New York City Implementation Guide

13 Lesson.1 Place the First Digit CC.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Place the first digit in the quotient by estimating or using place value. Cómo puedes saber dónde colocar el primer dígito del cociente sin dividir? Real World Video, Ch. Animated Math Models itools: Base-Ten Blocks 0 min. evelopment is key. The goal of this activity is for students to use estimation to place the first digit in the quotient and to use the division algorithm. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress estimation and division to develop sound mathematical practices by asking Cómo te ayudaría la estimación a resolver este problema? Cuál es otra forma de resolver el problema? Cómo puedes usar vocabulario matemático en tu explicación? Cómo sabes que la respuesta es razonable? once they understand these division concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 19, 3, 5 Cómo puedes saber dónde colocar el primer dígito del cociente sin dividir? Mediante el uso de operaciones básicas y números compatibles. Puedo hacer una estimación del valor del cociente, lo cual debería permitirme colocar el primer dígito del cociente. Chapter 41 New York City Implementation Guide NYC49

14 Lesson. Divide by 1-Digit Divisors CC.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Divide 3- and 4-digit dividends by 1-digit divisors. Cómo resuelves y verificas problemas de división? Animated Math Models itools: Base-Ten Blocks 0 min. activity is for students to use the division algorithm and to use inverse operations to check their solutions. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress through instruction. estimation and division to develop sound mathematical practices by asking Qué operación usaste para representar la situación? Por qué? Cómo te ayudaría la estimación a resolver el problema? Cómo sabes que la respuesta es razonable? Students can begin independent practice once they understand these multiplication and division concepts. Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, making them especially rich. Exercises or 0 Cómo resuelves y verificas problemas de división? Multiplico el cociente por el divisor y luego sumo el residuo al producto. Si el resultado es idéntico al dividendo, se verifica el problema. Use a map to plan a trip in the United States. Find the number of miles between your current location and your destination, and divide the mileage by the number of days or hours that you wish to travel. 4 Chapter NYC50 New York City Implementation Guide

15 Lesson.3 Investigate Division with -Digit Divisors CC.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Model division with -digit divisors using base-ten blocks. Cómo puedes usar bloques de base diez para representar y comprender la división de números naturales? base-ten blocks itools: Base-Ten Blocks 0 min. Investigate Activity to Build Conceptual Understanding activity is for students to connect the area model for multiplication to division with -digit divisors. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress modeling and division to develop sound mathematical practices by asking Por qué los bloques de base diez son un buen modelo para la división? Qué conclusiones puedes sacar de tu modelo? Cómo cambiaría tu modelo si hubiera 10 estudiantes en cada hilera? Cómo apoya el dibujo rápido tu trabajo? Share and Show Approximately 0 min. Students can continue practice once they understand these division models and -digit divisors. Select exercises based on students depth of understanding. Use the exercises below if you prefer to assign fewer items. Exercises 6, 8, 10 Cómo puedes usar bloques de base diez para representar y comprender la división de números naturales? Primero, usaría bloques de unidades, decenas y centenas para representar el dividendo. Luego, usaría un bloque de centenas y bloques de decenas para mostrar 10 grupos que tengan el mismo número que el divisor. Usaría el resto de los bloques para formar más grupos del mismo tamaño. A medida que trabajo, quizás tenga que reagrupar centenas o decenas. Write a division problem that has 3-digit dividend and a divisor between 10 and 0. Show how to solve it by drawing a quick picture. Chapter 43 New York City Implementation Guide NYC51

16 Lesson.4 Partial Quotients CC.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Use partial quotients to divide by -digit divisors. Cómo puedes usar cocientes parciales para dividir entre divisores de dígitos? 0 min. activity is for students to learn a different method (partial quotients) for dividing with -digit divisors. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress division to develop sound mathematical practices by asking Qué operación representa la situación en el problema planteado? Por qué? Qué harás primero? Por qué? Funcionará siempre ese método? once they understand division with partial quotients. Select exercises based on students depth of understanding. The exercises below require higher order Exercises or 18 1 Cómo puedes usar cocientes parciales para dividir entre divisores de dígitos? Multiplico el divisor por diez y continúo restando ese producto del dividendo hasta que el resultado sea menor que ese producto. Luego, resto otros múltiplos del divisor hasta que el resultado sea menor que el divisor o cero. Explain how using a partial quotient to divide is similar to using the Distributive Property to multiply. 44 Chapter NYC5 New York City Implementation Guide

17 Lesson.5 Estimate with -Digit Divisors CC.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Estimate quotients using compatible numbers. Cómo puedes usar números compatibles para estimar cocientes? Animated Math Models 0 min. activity is for students to connect their understanding of basic facts, patterns, and compatible numbers to make an estimate that helps them place the first digit. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress number patterns and division to develop sound mathematical practices by asking Qué operación usaste para representar la situación? Qué estrategia podrías usar como ayuda para resolver el problema? Cómo descubriste el patrón? Cómo sabes que la respuesta es razonable? once they understand these estimation and division concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 1, 3, 4 or 5 7 Cómo puedes usar números compatibles para estimar cocientes? Redondeo el divisor a un múltiplo de 10 y el dividendo al múltiplo más cercano al nuevo divisor. Luego, uso un patrón de ceros de operaciones básicas para completar la estimación. Create a division problem with a -digit divisor. Using more than 1 set of compatible numbers, observe what happens when you estimate using a different divisor, a different dividend, and when both are different. Using a calculator, compare the estimates to the answer and describe the differences. Chapter 45 New York City Implementation Guide NYC53

18 Lesson.6 Divide by -Digit Divisors CC.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Divide by -digit divisors. Cómo puedes dividir entre divisores de dígitos? Animated Math Models 0 min. activity is for students to use the division algorithm to divide with -digit divisors. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress estimation and division to develop sound mathematical practices by asking Qué operación usaste para representar la situación? Por qué? Qué hiciste primero? Por qué? Cómo sabes que la respuesta es razonable? once they understand how to use the division algorithm. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 16 1 or 4 Cómo puedes dividir entre divisores de dígitos? Primero, uso números compatibles y una estimación para colocar el primer dígito. Luego, repito estos pasos: dividir, multiplicar, restar y comparar y, cuando sea necesario, reagrupar. Choose a problem that you solved in the lesson, and solve the same problem using the partial quotients method. Compare the methods to solve the problems. Name the method you like better, and explain why. 46 Chapter NYC54 New York City Implementation Guide

19 Lesson.7 Interpret the Remainder CC.5.NF.3 Interpret a fraction as division of the numerator by the denominator ( a_ 5 a 4 b). Solve word problems involving division of b whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. Also CC.5.NBT.6 Solve division problems and decide when to write a remainder as a fraction. Cuándo escribes el residuo en forma de fracción al resolver un problema de división? itools: Base-Ten Blocks 0 min. activity is for students to solve division problems and decide how to represent the remainder. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress division and fractions to develop sound mathematical practices by asking these questions. Qué hiciste primero? Por qué? Cómo sabes cuándo debes expresar el residuo como una fracción? Cómo sabes que las respuestas son equivalentes? Cómo puedes usar el vocabulario matemático en tu explicación? once they understand these division concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 4, 6, 7 Cuándo escribes el residuo en forma de fracción al resolver un problema de división? El residuo debe escribirse como fracción cuando debe ser parte del cociente. El contexto del problema determina si el cociente debe escribirse como una fracción. Chapter 47 New York City Implementation Guide NYC55

20 Lesson.8 Adjust Quotients CC.5.NBT.7 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Adjust the quotient if the estimate is too high or too low. Cómo puedes ajustar el cociente si tu estimación es muy alta o muy baja? 0 min. activity is to familiarize students with the conditions that are present when an estimate is too high or too low. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress division to develop sound mathematical practices by asking Qué hiciste primero? Por qué? Qué modelo podrías usar como ayuda para resolver el problema? Qué sucedería si no alinearas los valores posicionales? once they understand how to adjust quotients. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 4 Cómo puedes ajustar el cociente si tu estimación es muy alta o muy baja? Si el producto es mayor que el dividendo, disminuyo la estimación en 1. Si la diferencia es mayor que o igual al divisor, aumento la estimación en 1. Explain the different ways that you can use multiplication to estimate and solve division problems. 48 Chapter NYC56 New York City Implementation Guide

21 Lesson.9 Problem Solving Division CC.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Solve problems by using the strategy draw a diagram. Cómo puede el uso de la estrategia de dibujar un diagrama ayudarte a resolver un problema de división? 0 min. activity is for students to organize information and use a strategy to solve division problems. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress problem solving strategies and division to develop sound mathematical practices by asking Qué operación usaste para representar la situación? Por qué representa la situación la división? Qué estrategia podrías usar como ayuda para resolver el problema? Cómo apoya tu trabajo el dibujar un diagrama? once they understand these division concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 4 6 Cómo puede el uso de la estrategia de dibujar un diagrama ayudarte a resolver un problema de división? Dibujar un diagrama, como un modelo de barras, me ayuda a organizar la información de una manera que pueda entender. Create a word problem that uses division. Draw a bar model to help you write an equation to solve the problem. Chapter 49 New York City Implementation Guide NYC57

22 Lesson 3.1 Investigate Thousandths CC.5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Also CC.5.NBT.3a Model, read, and write decimals to thousandths. Cómo puedes describir la relación entre dos valores posicionales decimales? color pencils straightedge Animated Math Models 0 min. Investigate Activity to Build Conceptual Understanding activity is for students to extend their understanding of decimal place value to thousandths. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress decimal place value to develop sound mathematical practices by asking Por qué una cuadrícula de 10 X 10 es un buen modelo para los decimales? Qué conclusiones puedes sacar de tu modelo? Cómo cambiaría el modelo para mostrar milésimos? En qué se asemeja el modelo a un modelo de bloques de base diez? Problem Solving Approximately 10 min. once they understand these decimal place-value concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 17 0 Cómo puedes describir la relación entre dos valores posicionales decimales? El valor de un valor posicional es 10 veces el valor de la posición a su derecha y 1/10 el valor de la posición a su izquierda. Write four decimals with the digit 4 in a different place in each ones, tenths, hundredths, and thousandths. Then write a statement that compares the value of the digit 4 in the different decimals. 5 Chapter 3 NYC58 New York City Implementation Guide

23 Lesson 3. Place Value of Decimals CC.5.NBT.3a Read, write and compare decimals to thousandths. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., ( 1 10 ) ( ) 1 3 ( ). Also CC.5.NBT.1 Read and write decimals through thousandths. Cómo lees, escribes y representas decimales hasta los milésimos? Animated Math Models 0 min. activity is for students to read, write, and represent decimals through thousandths. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress decimal place value to develop sound mathematical practices by asking Por qué es una tabla de valor posicional un modelo útil para este problema? Qué conclusiones puedes sacar de la tabla de valor posicional? Cómo descubriste el patrón para el valor posicional? Qué otros patrones puedes hallar? Students can begin independent practice once they understand these place value and decimal concepts. Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, making them especially rich. Exercises 1 14 or SummarizeApproximately 5 min. Cómo lees, escribes y representas decimales hasta los milésimos? Puedo usar una tabla de valor posicional para ayudarme a ver el valor de cada dígito. Cada valor es 10 veces más el valor del lugar a su derecha. Un decimal puede escribirse en forma normal, en palabras o en forma desarrollada. El último dígito determina el nombre del decimal. Write Standard Form, Expanded Form, and Word Form at the top of the page. Under Standard Form, write five decimals that have at least 3 digits to the right of the decimal point. Write the expanded form and the word form for each number under the appropriate heading. Chapter 3 53 New York City Implementation Guide NYC59

24 Lesson 3.3 Compare and Order Decimals CC.5.NBT.3b Read, write and compare decimals to thousandths. Compare two decimals to thousandths based on meanings of the digits in each place, using., 5, and, symbols to record the results of comparisons. Compare and order decimals to thousandths using place value. Cómo puedes usar el valor posicional para comparar y ordenar decimales? Animated Math Models 0 min. activity is for students to compare and order decimals using place value or a place-value chart. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress decimals and place value to develop sound mathematical practices by asking Qué hiciste primero? Por qué? Has resuelto un problema parecido al problema dado? Qué sucedería si no alinearas los puntos decimales? Cómo sabes que la respuesta es razonable? once they understand these decimal and place value concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 0 or 3 6 Cómo puedes usar el valor posicional para comparar y ordenar decimales? Alineo los puntos decimales de los números a comparar u ordenar o uso una tabla de valor posicional. Comparo los dígitos en cada valor posicional, comenzando por el mayor valor. El dígito que es mayor es el número mayor. Si los dígitos son los mismos, me muevo al siguiente valor menor y comparo los dígitos. Write a word problem that can be solved by ordering three decimals to thousandths. Include a solution. 54 Chapter 3 NYC60 New York City Implementation Guide

25 Lesson 3.4 Round Decimals CC.5.NBT.4 Use place value understanding to round decimals to any place. Round decimals to any place. Cómo puedes usar el valor posicional para redondear decimales a un lugar dado? 0 min. activity is for students to round decimals to the nearest whole number, tenth or hundredth using a place-value chart and place value. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress decimals and place value to develop sound mathematical practices by asking Qué hiciste primero? Por qué? Has resuelto un problema parecido a este? Descríbelo. Cómo puedes usar lenguaje matemático en tu explicación? Qué crees que sucederá si el último dígito es un 5? once they understand these decimal and place value concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 11, 14, 17 or 0 3 Cómo puedes usar el valor posicional para redondear decimales a un lugar dado? Hallo el dígito en el lugar al que estoy redondeando. Luego, observo el dígito a su derecha. Si es mayor que o igual a 5, aumento en 1 el dígito en el lugar al que estoy redondeando y elimino los dígitos a su derecha. Si el dígito a su derecha es menor que 5, mantengo igual el dígito en el lugar al que estoy redondeando y elimino los dígitos a su derecha. Describe how to round to the nearest tenth. Chapter 3 55 New York City Implementation Guide NYC61

26 Lesson 3.5 Investigate Decimal Addition CC.5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Model decimal addition using base-ten blocks. Cómo puedes usar bloques de base diez para representar la suma de decimales? base-ten blocks 0 min. Investigate Activity to Build Conceptual Understanding activity is for students to use manipulatives to model decimal addition. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress decimals and addition to develop sound mathematical practices by asking Por qué son los bloques de base diez un buen modelo para la suma de decimales? Qué conclusiones puedes sacar de tu modelo? Funcionará siempre este método para sumar decimales? Cómo lo sabes? Problem Solving Approximately 10 min. once they understand these decimal and addition concepts. Select exercises based on students depth of understanding. The exercise below requires higher order making it especially rich. Exercise 8 Cómo puedes usar bloques de base diez para representar la suma de decimales? Represento cada sumando usando planos, largos y cubos pequeños. Sumo los centésimos y, de ser necesario, reagrupo cada 10 centésimos como 1 décimo. Sumo los décimos y, si es necesario, reagrupo cada 10 décimos como 1 unidad. Explain why drawing a quick picture is helpful when adding decimals. 56 Chapter 3 NYC6 New York City Implementation Guide

27 Lesson 3.6 Investigate Decimal Subtraction CC.5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Model decimal subtraction using base-ten blocks. Cómo puedes usar bloques de base diez para representar la resta de decimales? base-ten blocks 0 min. Investigate Activity to Build Conceptual Understanding activity is for students to use manipulatives to model decimal subtraction. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress decimals and subtraction to develop sound mathematical practices by asking Has resuelto un problema parecido al problema dado? Por qué son los bloques de base diez un buen modelo para la resta de decimales? Qué conclusiones puedes sacar de tu modelo? Funcionará siempre este modelo para restar decimales? Problem Solving Approximately 10 min. Students can begin independent practice once they understand these decimal and subtraction concepts. Select exercises based on students depth of understanding. The exercise below requires higher order thinking skills and critical reasoning, making it especially rich. Exercise 8 Cómo puedes usar bloques de base diez para representar la resta de decimales? Represento el primer número usando planos, largos y cubos pequeños. Luego represento el número que se está restando al retirar los cubos pequeños, los largos y luego los planos que representan el número. Reagrupo si es necesario. Describe a problem involving decimals that you would use a quick picture to solve. Then solve the problem. Chapter 3 57 New York City Implementation Guide NYC63

28 Lesson 3.7 Estimate Decimal Sums and Differences CC.5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Make reasonable estimates of decimal sums and differences. Cómo puedes hacer una estimación de sumas y diferencias de decimales? Animated Math Models 0 min. activity is for students to using rounding and benchmarks to estimate decimal sums and differences. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress decimals and estimation to develop sound mathematical practices by asking Qué operación elegiste para representar la situación? Por qué la suma representa la situación en el problema planteado? Qué estrategia podrías usar para facilitar el cálculo? Cómo sabes que una estimación es una respuesta razonable? once they understand these decimal and estimation concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 13, 15, 16 or Cómo puedes hacer una estimación de sumas y diferencias de decimales? Se puede usar el redondeo y las rectas numéricas con puntos de referencia para una estimación de sumas y diferencias de decimales. Explain why estimation is an important skill to know when adding and subtracting decimals. 58 Chapter 3 NYC64 New York City Implementation Guide

29 Lesson 3.8 Add Decimals CC.5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Add decimals using place value. Cómo puede el valor posicional ayudarte a sumar decimales? 0 min. activity is for students to use place value to add decimals. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress decimals and addition to develop sound mathematical practices by asking Qué hiciste primero? Por qué? Qué modelo podrías usar como ayuda para resolver el problema? Cómo apoya tu dibujo la respuesta? Qué sucedería si no alinearas los valores posicionales? once they understand these decimal addition concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises 1 15 SummarizeApproximately 5 min. Cómo puede el valor posicional ayudarte a sumar decimales? Alinear los valores posicionales de sumandos decimales me ayuda a sumar centésimos a centésimos, décimos a décimos, unidades a unidades y así sucesivamente. Describe an addition problem that you might need to regroup hundredths to solve. Chapter 3 59 New York City Implementation Guide NYC65

30 Lesson 3.9 Subtract Decimals CC.5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Subtract decimals using place value. Cómo puede el valor posicional ayudarte a restar decimales? Animated Math Models 0 min. activity is for students to use place value and regrouping to subtract decimals. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress decimals and subtraction to develop sound mathematical practices by asking Qué hiciste primero? Por qué? Qué modelo podrías usar como ayuda para resolver el problema? Cómo apoya tu dibujo la respuesta? Qué sucedería si no comenzaras con el menor valor posicional? Students can begin independent practice once they understand these decimal subtraction concepts. Select exercises based on students depth of understanding. The exercises below require higher order thinking skills and critical reasoning, making them especially rich. Exercises Cómo puede el valor posicional ayudarte a restar decimales? Alinear los valores posicionales me ayuda a restar centésimos de centésimos, décimos de décimos, unidades de unidades y así sucesivamente. Write a decimal subtraction problem that requires regrouping to solve. Then solve the problem. 60 Chapter 3 NYC66 New York City Implementation Guide

31 Lesson 3.10 Algebra Patterns with Decimals CC.5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Identify, describe, and create numeric patterns with decimals. Cómo puedes usar la suma o la resta para describir un patrón o crear una secuencia con decimales? Animated Math Models 0 min. activity is for students to identify patterns in decimal number sequences and write rules to define those patterns. As students work through Unlock the Problem, gauge their level of understanding to make better decisions about how to progress decimals and numerical patterns to develop sound mathematical practices by asking Qué hiciste primero? Por qué? Cómo descubriste el patrón para este problema? Qué regla estableciste para este patrón? Cómo sabes si la respuesta es razonable? once they understand these decimal pattern concepts. Select exercises based on students depth of understanding. The exercises below require higher order Exercises Cómo puedes usar la suma o la resta para describir un patrón o crear una secuencia con decimales? Para describir un patrón, necesito analizar las diferencias entre un término y el siguiente. Para crear una secuencia necesito usar la regla del patrón provista, que incluye el valor inicial. Give an example of a rule describing the pattern for a sequence. Then write the terms of the sequence for your rule. Chapter 3 61 New York City Implementation Guide NYC67