Unit 3: Unit Rates and Percentages
Lesson 8: More about Constant Speed

This lesson allows students to practice working with equivalent ratios, tables that represent them, and associated unit rates in the familiar context of speed, time, and distance. Students use unit rates (speed or pace) and ratios (of time and distance) to find unknown quantities (e.g., given distances and times, find a constant speed or pace; and given a speed or pace, solve problems about distance and time).

Unit 9: Putting It All Together
Lesson 5: More than Two Choices

This lesson is optional. It is the second of three lessons that explores the mathematics of voting. The activities in this lesson build on each other and on the previous lesson. As with all lessons in this unit, all related standards have been addressed in prior units; this lesson provides an optional opportunity to go more deeply and make connections between domains.

The five activities in this lesson deal with elections in which there are more than two choices. For example, if there are three choices, then the top vote getter might be approved by only 34% of the voters. Students explore several different rules for determining the winner: plurality, runoff, and instant runoff, and discover that the rules can give different results from the same set of voter preferences. They think about which voting rule more fairly represents the opinions of the voters. The mathematics in these activities emphasizes quantitative reasoning in a real-world situation (MP2 and MP4).

Most of the activities use students’ skills from earlier units to reason about ratios and proportional relationships (MP2) in the context of real-world problems (MP4). While some of the activities do not involve much computation, they all require serious thinking.

Most importantly, this lesson addresses topics that are important for citizens in a democracy to understand. Teachers may wish to collaborate with a civics/government teacher to learn how the fictional middle-school situations in this lesson relate to real-world elections.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

En el módulo 3, la comprensión de los estudiantes de la adición y la resta de las fracciones se extiende desde el trabajo anterior con equivalencia de fracción y decimales. Este módulo marca un cambio significativo lejos de la centralidad de los grados elementales de las diez unidades de base al estudio y el uso del conjunto completo de unidades fraccionarias desde el avance de grado 5, especialmente como se aplica al álgebra.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
In Module 3, students' understanding of addition and subtraction of fractions extends from earlier work with fraction equivalence and decimals. This module marks a significant shift away from the elementary grades' centrality of base ten units to the study and use of the full set of fractional units from Grade 5 forward, especially as applied to algebra.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

En el módulo 8, el módulo final del año, los estudiantes amplían su comprensión de las relaciones completas a través de la lente de la geometría. A medida que los estudiantes componen y descomponen formas, comienzan a desarrollar una comprensión de las fracciones unitarias como partes iguales de un todo.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
In Module 8, the final module of the year, students extend their understanding of part–whole relationships through the lens of geometry.  As students compose and decompose shapes, they begin to develop an understanding of unit fractions as equal parts of a whole.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

En este módulo de 40 días, los estudiantes se basan en su trabajo de grado 3 con fracciones unitarias mientras exploran la equivalencia de fracción y extienden esta comprensión a números mixtos. Esto lleva a la comparación de fracciones y números mixtos y a la representación de ambos en una variedad de modelos. Las fracciones de referencia juegan un papel importante en la capacidad de los estudiantes para generalizar y razonar sobre la fracción relativa y los tamaños de números mixtos. Luego, los estudiantes tienen la oportunidad de aplicar lo que saben para ser cierto para las operaciones de números enteros a los nuevos conceptos de fracción y operaciones de números mixtos.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
In this 40-day module, students build on their Grade 3 work with unit fractions as they explore fraction equivalence and extend this understanding to mixed numbers.  This leads to the comparison of fractions and mixed numbers and the representation of both in a variety of models.  Benchmark fractions play an important part in students’ ability to generalize and reason about relative fraction and mixed number sizes.  Students then have the opportunity to apply what they know to be true for whole number operations to the new concepts of fraction and mixed number operations.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

Este módulo de 20 días brinda a los estudiantes su primera oportunidad de explorar los números decimales a través de su relación con las fracciones decimales, expresando una cantidad dada tanto en la fracción como en las formas decimales. Utilizando la comprensión de las fracciones desarrolladas en todo el Módulo 5, los estudiantes aplican el mismo razonamiento a los números decimales, construyendo una base sólida para el trabajo de grado 5 con operaciones decimales.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
This 20-day module gives students their first opportunity to explore decimal numbers via their relationship to decimal fractions, expressing a given quantity in both fraction and decimal forms.  Utilizing the understanding of fractions developed throughout Module 5, students apply the same reasoning to decimal numbers, building a solid foundation for Grade 5 work with decimal operations.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

Módulo 4 de grado 5 extiende la comprensión del estudiante de las operaciones de fracción a la multiplicación y la división de fracciones y fracciones decimales. El trabajo procede de la interpretación de los gráficos de línea que incluyen mediciones fraccionales para interpretar las fracciones como división y razonamiento sobre la búsqueda de fracciones de conjuntos a través de la fracción por multiplicación de números enteros. El módulo procede a la fracción por multiplicación de fracción en formas de fracción y decimal. Una comprensión de la multiplicación como escala y multiplicación por N/N como multiplicación por 1 permite a los estudiantes razonar sobre productos y convertir fracciones en decimales y viceversa. Los estudiantes son presentados al trabajo de división con fracciones y fracciones decimales. Los casos de división se limitan a la división de números enteros por fracciones unitarias y fracciones unitarias por números enteros. Se introducen divisores de fracción decimal y la fracción equivalente y el pensamiento del valor del lugar permiten al alumno razonar sobre el tamaño de los cocientes, calcular los cocientes y colocar decimales con sensatez en los cocientes. A lo largo del módulo, se les pide a los estudiantes que razonen sobre estos conceptos importantes interpretando expresiones numéricas que incluyen operaciones de fracción y decimales y perseverar en la resolución de problemas de varios pasos en el mundo real que incluyen todas las operaciones de fracción compatibles con el uso de diagramas de cintas.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
Grade 5’s Module 4 extends student understanding of fraction operations to multiplication and division of both fractions and decimal fractions.  Work proceeds from interpretation of line plots which include fractional measurements to interpreting fractions as division and reasoning about finding fractions of sets through fraction by whole number multiplication.  The module proceeds to fraction by fraction multiplication in both fraction and decimal forms.  An understanding of multiplication as scaling and multiplication by n/n as multiplication by 1 allows students to reason about products and convert fractions to decimals and vice versa.  Students are introduced to the work of division with fractions and decimal fractions.  Division cases are limited to division of whole numbers by unit fractions and unit fractions by whole numbers.  Decimal fraction divisors are introduced and equivalent fraction and place value thinking allow student to reason about the size of quotients, calculate quotients and sensibly place decimals in quotients.  Throughout the module students are asked to reason about these important concepts by interpreting numerical expressions which include fraction and decimal operations and by persevering in solving real-world, multistep problems which include all fraction operations supported by the use of tape diagrams.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

En este módulo de 35 días de grado 3, los estudiantes extienden y profundizan la práctica de segundo grado con "acciones iguales" para comprender las fracciones como particiones iguales de un todo. Su conocimiento se vuelve más formal a medida que trabajan con los modelos de área y la línea numérica.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
In this 35-day Grade 3 module, students extend and deepen second grade practice with "equal shares" to understanding fractions as equal partitions of a whole. Their knowledge becomes more formal as they work with area models and the number line.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

Unit 2: Introducing Ratios
Lesson 12: Navigating a Table of Equivalent Ratios

The purpose of this lesson is to develop students’ ability to work with a table of equivalent ratios. It also provides opportunities to compare and contrast different ways of solving equivalent ratio problems.

Students see that a table accommodates different ways of reasoning about equivalent ratios, with some being more direct than others. They notice (MP8) that to find an unknown quantity, they can:

Find the multiplier that relates two corresponding values in different rows (e.g., “What times 5 equals 8?”) and use that multiplier to find unknown values. (This follows the multiplicative thinking developed in previous lessons.)
Find an equivalent ratio with one quantity having a value of 1 and use that ratio to find missing values.

All tasks in the lesson aim to strengthen students’ understanding of the multiplicative relationships between equivalent ratios—that given a ratio , an equivalent ratio may be found by multiplying both and by the same factor. They also aim to build students’ awareness of how a table can facilitate this reasoning to varying degrees of efficiency, depending on one’s approach.

Ultimately, the goal of this unit is to prepare students to make sense of situations involving equivalent ratios and solve problems flexibly and strategically, rather than to rely on a procedure (such as “set up a proportion and cross multiply”) without an understanding of the underlying mathematics.

To reason using ratios in which one of the quantities is 1, students are likely to use division. In the example above, they are likely to divide the 90 by 5 to obtain the amount earned per hour. Remind students that dividing by a whole number is the same as multiplying by its reciprocal (a unit fraction) and encourage the use of multiplication (as shown in the activity about hourly wages) whenever possible. Doing so will better prepare students to: 1) scale down, e.g., to find equivalent ratios involving values that are smaller than the given ones, 2) relate fractions to percentages later in the course, and 3) understand division of fractions (including the “invert and multiply” rule) in a later unit.

Unit 3: Unit Rates and Percentages
Lesson 17: Painting a Room

In this culminating lesson, students make material and cost estimates for a home improvement project, applying and integrating many concepts and skills from the past three units.

Students determine the area of the walls of a bedroom, estimate the amount of paint needed to paint them, and determine the cost associated with the project (MP4). Along the way, they reason about areas of two-dimensional figures, convert units of measurements, solve ratio and rate problems, and work with percentages. Though there is a single correct measure for the total area of the walls to be painted, the amount of paint needed will depend on some assumptions and decisions students make about the work involved. The problem requires students to make some decisions about how to approach the task and which tools to use (MP5).

Depending on instructional choices made, this lesson could take one or more class meetings. The time estimates for the two main activities are intentionally left blank because the time will vary based on instructional decisions made. Variables affecting the amount of time needed include how much guidance and autonomy students are given, how elaborate the presentation of their work is expected to be, and how much time is taken for sharing solutions at the end.

Unit 2: Introducing Ratios
Lesson 15: Part-Part-Whole Ratios

Up to this point, students have worked with ratios of quantities where the units are the same (e.g., cups to cups) and ratios of quantities where the units are different (e.g., miles to hours). Sometimes in the first case, the sum of the quantities makes sense in the context, and we can ask questions about the total amount as well as the component parts. For example, when mixing 3 cups of yellow paint with 2 cups of blue paint, we get a total of 5 cups of green paint. (Notice that this does not always work; 3 cups of water mixed with 2 cups of dry oatmeal will not make 5 cups of soggy oatmeal.) In the paint scenario, the ratio of yellow paint to blue paint to green paint is . Furthermore, if we double the amount of both yellow and blue paint, we will double the amount of green paint. In general, if the ratio of yellow to blue paint is equivalent, the ratio of yellow to blue to green paint will also be the equivalent. We can see this is always true because of the distributive property:

a : b : (a+b) is equivalent to 2a : 2b : (2a+2b) because 2a + 2b = 2(a+b).

These ratios are sometimes called “part-part-whole” ratios.

In this lesson, students learn about tape diagrams as a handy tool to represent ratios with the same units and as a way to reason about individual quantities (the parts) and the total quantity (the whole). Here students also see ratios expressed not in terms of specific units (milliliters, cups, square feet, etc.) but in terms of "parts" (e.g., the recipe calls for 2 parts of glue to 1 part of water).

Unit 3: Unit Rates and Percentages
Lesson 11: Percentages and Double Number Lines

In the previous lesson, students learned to find percentages of 100 and percentages of 1 in the context of money (100 cents and $1). In this lesson, they explore percentages of quantities other than 100 and 1 in a variety of contexts. All of the tasks use comparison contexts—describing one quantity relative to another quantity—rather than part-whole contexts.

Students continue to have double number lines as a reasoning tool to use if they want. In several cases the double number line is provided. There are two reasons for this. First, the equal intervals on the provided double number line are useful for reasoning about percentages. Second, using the same representation that was used earlier for other ratio and rate reasoning reinforces the idea of a percentage as a rate per 100 (MP7). It is perfectly acceptable, however, for students to use strategies other than double number lines for solving percentage problems.

Unit 3: Unit Rates and Percentages
Lesson 12: Percentages and Tape Diagrams

In previous lessons students used double number lines to reason about percentages. Double number lines show different percentages when a given amount is identified as 100%, and emphasize that percentages are a rate per 100. In this lesson they use tape diagrams. Tape diagrams are useful for seeing the connection between percentages and fractions.

Tape diagrams are useful in solving problems of the form A is B% of C when you are given two of the numbers and must find the third. When reasoning about percentages, it is important to indicate the whole as 100%, just as it is important to indicate the whole when working with fractions (MP6).

Unit 9: Putting It All Together
Lesson 6: Picking Representatives

This lesson is optional. The five activities in this third lesson on the mathematics on voting return to the situation of an election with two choices. However, rather than directly choosing the result, voters elect representatives, each of whom then casts a single vote for all the people they represent. The activities explore ways to “share” the representatives fairly between groups of people. In the first activity, numbers have been designed so that representatives (or computers) can be shared exactly proportionally between several groups. In later activities, it’s impossible to share representatives fairly; students may use division with decimal quotients or with remainders to try to find the least unfair way. The final activity asks students to gerrymander several districts: to divide it into sections in two ways to influence the final voting result in opposite ways. The mathematics here involves geometric properties of shapes on maps: area and connectedness, as well as some proportional reasoning.

Most of the activities use students’ skills from earlier units to reason about ratios and proportional relationships (MP2) in the context of real-world problems (MP4). While some of the activities do not involve much computation, they all require serious thinking and decision making (MP3).

Most importantly, this lesson addresses topics that are important for citizens in a democracy to understand. Teachers may wish to collaborate with a civics or government teacher to learn how the fictional middle-school situations in this lesson relate to real-world elections.

Unit 2: Introducing Ratios
Lesson 3: Recipes

This is the first of two lessons that develop the idea of equivalent ratios through physical experiences. A key understanding is that if we scale a recipe up (or down) to make multiple batches (or a fraction of a batch), the result will still be “the same” in some meaningful way. Students see this idea in two contexts, taste and color:

In this lesson, a mixture containing two batches of a recipe tastes the same as a mixture containing one batch. For example, 2 cups of water mixed thoroughly with 8 teaspoons of powdered drink mix tastes the same as 1 cup of water mixed with 4 teaspoons of powdered drink mix.
In the next lesson, a mixture containing two batches of a recipe for colored water will produce the same shade of the color as a mixture containing one batch. For example, 10 ml of blue mixed with 30 ml of yellow produces the same shade of green as 5 ml of blue mixed with 15 ml of yellow.
The fact that two equivalent ratios yield the same taste or produce the same color is a physical manifestation of the equivalence of the ratios. In this lesson, students start to use the term equivalent ratios.

Students see that scaling a recipe up (or down) requires multiplying the amount of each ingredient by the same factor, e.g., doubling a recipe means doubling the amount of each ingredient (MP7). They also gain more experience using a discrete diagram as a tool to represent a situation.

This task provides a context for indentifying different ways of representing half of a rectangle.

Unit 2: Introducing Ratios
Lesson 2: Representing Ratios with Diagrams

Students used physical objects to learn about ratios in the previous lesson. Here they use diagrams to represent situations involving ratios and continue to develop ratio language. The use of diagrams to represent ratios involves some care so that students can make strategic choices about the tools they use to solve problems. Both the visual and verbal descriptions of ratios demand careful interpretation and use of language (MP6).

Students should see diagrams as a useful and efficient ways to represent ratios. There is not really a right or wrong way to draw a diagram; what is important is that it represents the mathematics and makes sense to the student, and the student can explain how the diagram is being used. However, a goal of this lesson is to help students draw useful diagrams efficiently.

When students are asked to draw diagrams, they often include unnecessary details such as making each cup look like an actual cup, which makes the diagrams inefficient to use for solving problems. Examples of very simple diagrams help guide students toward more abstract representations while still relying on visual or spatial cues to support reasoning.

While students may say “for every 2 cups of juice there is 1 cup of soda,” note that for now, we will not suggest writing the association as 2:1. Equivalent ratios will be carefully developed in upcoming lessons. Diagrams are referred to as "discrete diagrams" in these materials, but students do not need to know this term. In student-facing materials they are simply called "diagrams."

The discrete diagrams in this lesson are meant to reflect the parallel structure of double number lines that students will learn later in the unit. But for now, students do not need to draw them this way as long as they can explain their diagrams and interpret discrete diagrams like the ones shown in the lesson.

Unit 2: Introducing Ratios
Lesson 11: Representing Ratios with Tables

Over the course of this unit, students learn to work with ratios using different representations. They began by using discrete diagrams to represent ratios and to identify equivalent ratios. Later, they reasoned more efficiently about ratios using double number lines. Here, they encounter situations in which using a double number line poses challenges and for which a different representation would be helpful. Students learn to organize a set of equivalent ratios in a table, which is a more abstract but also a more flexible tool for solving problems.

Although different representations are encouraged at different points in the unit, allowing students to use any representation that accurately represents a situation and encouraging them to compare the efficiency of different methods will develop their ability to make strategic choices about representations (MP5). Whatever choices they make, they should be encouraged to explain how their method works in solving a problem.

Unit 2: Introducing Ratios
Lesson 14: Solving Equivalent Ratio Problems

The purpose of this lesson is to give students further practice in solving equivalent ratio problems and introduce them to the info gap activity structure. The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Unit 2: Introducing Ratios
Lesson 16: Solving More Ratio Problems

In this lesson, students use all representations they have learned in this unit—double number lines, tables, and tape diagrams—to solve ratio problems that involve the sum of the quantities in the ratio. They consider when each tool might be useful and preferable in a given situation and why (MP5). In so doing, they make sense of situations and representations, and are strategic in their choice of solution method (MP1).