Using this tool, students build these classic fractals: the Koch snowflake, a fractal tree, a reduced square, and the Sierpinksi triangle. As these shapes grow and change using an iterative process, students can observe patterns in the images created and in the table of values as the fractals progress through several stages.

This module brings together the ideas of similarity and congruence and the properties of length, area, and geometric constructions studied throughout the year.  It also includes the specific properties of triangles, special quadrilaterals, parallel lines and transversals, and rigid motions established and built upon throughout this mathematical story.  This module's focus is on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page.

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

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.

In this 20-day module students explore area as an attribute of two-dimensional figures and relate it to their prior understandings of multiplication. Students conceptualize area as the amount of two-dimensional surface that is contained within a plane figure.  They come to understand that the space can be tiled with unit squares without gaps or overlaps.  They make predictions and explore which rectangles cover the most area when the side lengths differ.  Students progress from using square tile manipulatives to drawing their own area models and manipulate rectangular arrays to concretely demonstrate the arithmetic properties. The module culminates with students designing a simple floor plan that conforms to given area specifications.

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

This 40-day final module of the year offers students intensive practice with word problems, as well as hands-on investigation experiences with geometry and perimeter.  The module begins with solving one- and two-step word problems based on a variety of topics studied throughout the year, using all four operations.  Next students explore geometry.  Students tessellate to bridge geometry experience with the study of perimeter.  Line plots, familiar from Module 6, help students draw conclusions about perimeter and area measurements.  Students solve word problems involving area and perimeter using all four operations.  The module concludes with a set of engaging lessons that briefly review the fundamental Grade 3 concepts of fractions, multiplication, and division.

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

In this 25-day module, students work with two- and three-dimensional figures.  Volume is introduced to students through concrete exploration of cubic units and culminates with the development of the volume formula for right rectangular prisms.  The second half of the module turns to extending students’ understanding of two-dimensional figures.  Students combine prior knowledge of area with newly acquired knowledge of fraction multiplication to determine the area of rectangular figures with fractional side lengths.  They then engage in hands-on construction of two-dimensional shapes, developing a foundation for classifying the shapes by reasoning about their attributes.  This module fills a gap between Grade 4’s work with two-dimensional figures and Grade 6’s work with volume and area.

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

The goal of this task is to use geometry study the structure of beehives. Beehives have a tremendous simplicity as they are constructed entirely of small, equally sized walls. In order to as useful as possible for the hive, the goal should be to create the largest possible volume using the least amount of materials. In other words, the ratio of the volume of each cell to its surface area needs to be maximized. This then reduces to maximizing the ratio of the surface area of the cell shape to its perimeter.

A main concern of shoe engineers is creating shoes that provide the right amount of arch support to prevent (or fix) common gait misalignments that lead to injury. During this activity, students look at their own footprints and determine whether they have either of the two most prominent gait misalignments: overpronation (collapsing arches) or supination (high arches). Knowing the shape of a person's foot, and their natural arch movement is necessary to design shoes to fix these gain alignments.

Students teams determine the size of the caverns necessary to house the population of the state of Alabraska from the impending asteroid impact. They measure their classroom to determine area and volume, determine how many people the space could sleep, and scale this number up to accommodate all Alabraskans. They work through problems on a worksheet and perform math conversions between feet/meters and miles/kilometers.

The applets in this Interactive Geometry Dictionary (IGD) will allow students an opportunity to explore finding the area of some common shapes. The applets demonstrate how to find the area of a triangle using the area of a parallelogram, which in turn can be found using the area of a rectangle. This tool also supports the lesson "What's My Area" cataloged separately.

This Java interactive tool can be used to create dynamic drawings on an isometric dot grid, and to explore volume, surface area, and congruence concepts. Users can draw figures using edges, faces, or cubes and can shift, rotate, color, decompose, and view figures in 2‑D or 3‑D with this applet. Instructions on using and exploring with the tool are included on the page. A related multi-lesson unit from Illuminations for middle school students is linked to the side.

In this 5-lesson unit, students engage in measurement activities involving length, area, volume, time, and weight, using objects, pictures and symbols. Students practice measuring using standard and nonstandard units. Some lessons are introduced using children's literature.

In this activity, learners slide shapes to create unusual tiled patterns. Learners transform a rectangle into a more interesting shape and then make a tessellation by repeating that shape over and over again. Learners will also calculate the area of a rectangle. This activity works best as a "centers" activity.

This lesson unit is intended to help you assess how well students are able to: Interpret a situation and represent the variables mathematically; select appropriate mathematical methods to use; explore the effects on the area of a rectangle of systematically varying the dimensions whilst keeping the perimeter constant; interpret and evaluate the data generated and identify the optimum case; and communicate their reasoning clearly.

This goal of this task is to give students familiarity using the formula for the area of a circle while also addressing measurement error.

In this series of three hands-on activities, students develop an understanding of the relationship between fractions, decimals, and percents based upon models of hundredths. Each activity requires students to display their one hundred pieces of candy in a different way: linear model, grid model (rectangular area), and region model (circle graph/pie chart). This lesson includes student worksheets, assessment questions, extension suggestions, and a link to a circle graph tool.

(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 25 días, los estudiantes trabajan con figuras dos y tridimensionales. El volumen se introduce a los estudiantes a través de la exploración concreta de unidades cúbicas y culmina con el desarrollo de la fórmula de volumen para los prismas rectangulares correctos. La segunda mitad del módulo se convierte en extender a los estudiantes la comprensión de las figuras bidimensionales. Los estudiantes combinan el conocimiento previo del área con el conocimiento recién adquirido de la multiplicación por fracción para determinar el área de las figuras rectangulares con longitudes laterales fraccionadas. Luego participan en la construcción práctica de formas bidimensionales, desarrollando una base para clasificar las formas razonando sobre sus atributos. Este módulo llena un vacío entre el trabajo de Grado 4 S con figuras bidimensionales y el trabajo de grado 6 con volumen y área.

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

English Description:
In this 25-day module, students work with two- and three-dimensional figures.  Volume is introduced to students through concrete exploration of cubic units and culminates with the development of the volume formula for right rectangular prisms.  The second half of the module turns to extending students’ understanding of two-dimensional figures.  Students combine prior knowledge of area with newly acquired knowledge of fraction multiplication to determine the area of rectangular figures with fractional side lengths.  They then engage in hands-on construction of two-dimensional shapes, developing a foundation for classifying the shapes by reasoning about their attributes.  This module fills a gap between Grade 4’s work with two-dimensional figures and Grade 6’s work with volume and area.

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 reúne las ideas de similitud y congruencia y las propiedades de la longitud, el área y las construcciones geométricas estudiadas durante todo el año. También incluye las propiedades específicas de los triángulos, cuadriláteros especiales, líneas paralelas y transversales, y movimientos rígidos establecidos y construidos sobre esta historia matemática. El enfoque de este módulo está en las posibles relaciones geométricas entre un par de líneas de intersección y un círculo dibujado en la página.

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

English Description:
This module brings together the ideas of similarity and congruence and the properties of length, area, and geometric constructions studied throughout the year.  It also includes the specific properties of triangles, special quadrilaterals, parallel lines and transversals, and rigid motions established and built upon throughout this mathematical story.  This module's focus is on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page.

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 20 días, los estudiantes exploran el área como un atributo de figuras bidimensionales y lo relacionan con su comprensión previa de multiplicación. Los estudiantes conceptualizan el área como la cantidad de superficie bidimensional que está contenida dentro de una figura plana. Llegan a comprender que el espacio puede estar mortal con cuadrados unitarios sin huecos o superposiciones. Hacen predicciones y exploran qué rectángulos cubren la mayor cantidad de área cuando las longitudes laterales difieren. Los estudiantes progresan del uso de manipulaciones de baldosas cuadradas hasta dibujar sus propios modelos de área y manipular matrices rectangulares para demostrar concretamente las propiedades aritméticas. El módulo culmina con estudiantes que diseñan un plano de planta simple que se ajusta a las especificaciones de área dadas.

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

English Description:
In this 20-day module students explore area as an attribute of two-dimensional figures and relate it to their prior understandings of multiplication. Students conceptualize area as the amount of two-dimensional surface that is contained within a plane figure.  They come to understand that the space can be tiled with unit squares without gaps or overlaps.  They make predictions and explore which rectangles cover the most area when the side lengths differ.  Students progress from using square tile manipulatives to drawing their own area models and manipulate rectangular arrays to concretely demonstrate the arithmetic properties. The module culminates with students designing a simple floor plan that conforms to given area specifications.

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 final del año de 40 días ofrece a los estudiantes una práctica intensiva con problemas de palabras, así como experiencias prácticas de investigación con geometría y perímetro. El módulo comienza con la resolución de problemas de palabras de uno y dos pasos basados ​​en una variedad de temas estudiados durante todo el año, utilizando las cuatro operaciones. A continuación, los estudiantes exploran la geometría. Estudiantes Tessellate para la experiencia de la geometría de puente con el estudio del perímetro. Las parcelas de línea, familiares del Módulo 6, ayudan a los estudiantes a sacar conclusiones sobre las mediciones de perímetro y área. Los estudiantes resuelven problemas de palabras que involucran área y perímetro utilizando las cuatro operaciones. El módulo concluye con un conjunto de lecciones atractivas que revisan brevemente los conceptos fundamentales de grado 3 de fracciones, multiplicación y división.

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

English Description:
This 40-day final module of the year offers students intensive practice with word problems, as well as hands-on investigation experiences with geometry and perimeter.  The module begins with solving one- and two-step word problems based on a variety of topics studied throughout the year, using all four operations.  Next students explore geometry.  Students tessellate to bridge geometry experience with the study of perimeter.  Line plots, familiar from Module 6, help students draw conclusions about perimeter and area measurements.  Students solve word problems involving area and perimeter using all four operations.  The module concludes with a set of engaging lessons that briefly review the fundamental Grade 3 concepts of fractions, multiplication, and division.

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