In Module 3, students learn about dilation and similarity and apply that knowledge to a proof of the Pythagorean Theorem based on the Angle-Angle criterion for similar triangles.  The module begins with the definition of dilation, properties of dilations, and compositions of dilations.  One overarching goal of this module is to replace the common idea of “same shape, different sizes” with a definition of similarity that can be applied to geometric shapes that are not polygons, such as ellipses and circles.

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.

This review lesson about basic shapes (polygons)contains varied hands-on exercises and tilings about triangles, quadrilaterals, pentagons, and hexagons.

Brush up on your math skills relating to the area of parallelograms then try some practice problems to test your understanding.

Brush up on your math skills relating to the area of rhombuses then try some practice problems to test your understanding.

Brush up on your math skills relating to the area of trapezoids then try some practice problems to test your understanding.

Brush up on your math skills relating to the area of triangles then try some practice problems to test your understanding.

Learn about the properties of rectangles, rhombuses, and squares, and practice identifying them. Includes hints for questions. CCSS.Math.Content.3.G.A.1 Understand that shapes in different categories

Khan Academy learning modules include a Community space where users can ask questions and seek help from community members. Educators should consult with their Technology administrators to determine the use of Khan Academy learning modules in their classroom. Please review materials from external sites before sharing with students.

This tool allows you to create any geometric shape imaginable. Squares, triangles, rhombi, trapezoids and hexagons can be created, colored, enlarged, shrunk, rotated, reflected, sliced, and glued together.

This task asks students to identify which of the six polygons have the same area. Students may complete the task using a variety of techniques including decomposing shapes, using transformations (rotations, reflections, translations) to move one or more parts of the figure to another part to more easily calculate the area, enclosing the polygon inside a larger rectangle and then subtract the areas of the "extra" pieces, etc.

A detailed introduction to quadrilaterals, the different types, and the vocabulary used to discuss their properties and to classify them. [8:24]

Khan Academy learning modules include a Community space where users can ask questions and seek help from community members. Educators should consult with their Technology administrators to determine the use of Khan Academy learning modules in their classroom. Please review materials from external sites before sharing with students.

Gain a basic understanding of polygons by watching this easy to understand video tutorial. Additional resources are available as part of a paid subscription service. [7:26]

Gain a basic understanding of quadrilaterals by watching this easy to understand video tutorial. Additional resources are available as part of a paid subscription service. [9:49]

This video [4:39] provides an opportunity to learn about the characteristics of polygons. As you watch and listen to the teacher and student interact it helps clarify the thinking behind applying this concept.

Constructions and Angles

Unit Overview

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Use a protractor and ruler.
Identify different types of triangles and quadrilaterals and their characteristics.

Lesson Flow

After an initial exploratory lesson involving a paper folding activity that gets students thinking in general about angles and figures in a context, the unit is divided into two concept development sections. The first section focuses on types of angles—adjacent, supplementary, complementary, and vertical—and how they are manifested in quadrilaterals. The second section looks at triangles and their properties, including the angle sum, and how this affects other figures.

In the first set of conceptual lessons, students explore different types of angles and where the types of angles appear in quadrilaterals. Students fold paper and observe the angles formed, draw given angles, and explore interactive sketches that test many cases. Students use a protractor and ruler to draw parallelograms with given properties. They explore sketches of parallelograms with specific properties, such as perpendicular diagonals. After concluding the investigation of the angle types, students move on to the next set.

In the second set of conceptual development lessons, students focus on triangles. Students again fold paper to create figures and certain angles, such as complementary angles.

Students draw, using a protractor and ruler, other triangles with given properties. Students then explore triangles with certain known and unknown elements, such as the number of given sides and angles. This process starts with paper folding and drawing and continues with exploration of interactive sketches. Students draw conclusions about which cases allow 0, 1, 2, or an infinite number of triangles. In the course of the exploration, students discover that the sum of the measure of the interior angles of a triangle is 180°. They also learn that the sum of the measures of the interior angles of a quadrilateral is 360°. They explore other polygons to find their angle sum and determine if there is a relationship to angle sum of triangles. The exploration concludes with finding the measure of the interior angles of regular polygons and speculating about how this relates to a circle.

Lastly, students solve equations to find unknown angle measures. Using their previous experience, students find the remaining angle measures in a parallelogram when only one angle measure is given. Students also play a game similar to 20 Questions to identify types of triangles and quadrilaterals. Having completed the remaining lessons, students have a four-day Gallery to explore a variety of problems.

The unit ends with a unit assessment.

Students learn to classify triangles by the size of the angles and the measures of the sides.Key ConceptsTriangles are polygons with three sides.Scalene triangles have all sides with a different length and all angles with a different measure.Isosceles triangles have two sides with the same length and two angles with the same measure.Equilateral triangles have all sides with the same length and all angles with the same measure.Acute triangles have all angles with a measure less than 90°.Obtuse triangles have one angle with a measure greater than 90°.Right triangles have one angle with a measure of 90°.ELL: Keep in mind that consistency at the beginning is very important as students begin to learn and apply math vocabulary.Goals and Learning ObjectivesExplore conditions that result in triangles.Identify types of triangles based on the measure of the angles or the measures of the sides.

Lesson OverviewStudents explore different polygons by drawing them, measuring angles, and manipulating interactive sketches to find the angle sum for any polygon. Students also explore the angle measures in regular polygons.Key ConceptsThe angle sum in a triangle is 180°. A quadrilateral can be composed of two triangles, so the angle sum of a quadrilateral is 360°.The number of triangles that compose a polygon is two less than the number of sides (angles). The sum of the interior angles in a polygon with n sides is 180° (n – 2).Goals and Learning ObjectivesFind angle sums in polygons.Generalize to find the angle sum for any polygon.Find interior angle measures for regular polygons. 

Zooming In On Figures

Unit Overview

Type of Unit: Concept; Project

Length of Unit: 18 days and 5 days for project

Prior Knowledge

Students should be able to:

Find the area of triangles and special quadrilaterals.
Use nets composed of triangles and rectangles in order to find the surface area of solids.
Find the volume of right rectangular prisms.
Solve proportions.

Lesson Flow

After an initial exploratory lesson that gets students thinking in general about geometry and its application in real-world contexts, the unit is divided into two concept development sections: the first focuses on two-dimensional (2-D) figures and measures, and the second looks at three-dimensional (3-D) figures and measures.
The first set of conceptual lessons looks at 2-D figures and area and length calculations. Students explore finding the area of polygons by deconstructing them into known figures. This exploration will lead to looking at regular polygons and deriving a general formula. The general formula for polygons leads to the formula for the area of a circle. Students will also investigate the ratio of circumference to diameter ( pi ). All of this will be applied toward looking at scale and the way that length and area are affected. All the lessons noted above will feature examples of real-world contexts.
The second set of conceptual development lessons focuses on 3-D figures and surface area and volume calculations. Students will revisit nets to arrive at a general formula for finding the surface area of any right prism. Students will extend their knowledge of area of polygons to surface area calculations as well as a general formula for the volume of any right prism. Students will explore the 3-D surface that results from a plane slicing through a rectangular prism or pyramid. Students will also explore 3-D figures composed of cubes, finding the surface area and volume by looking at 3-D views.
The unit ends with a unit examination and project presentations.

Lesson OverviewStudents will compare the formula for the area of a regular polygon to discover the formula for the area of a circle.Key ConceptsThe area of a regular polygon can be found by multiplying the apothem by half of the perimeter. If a circle is thought of as a regular polygon with many sides, the formula can be applied.For a circle, the apothem is the radius, and p is C.A=a(p2)→A=rC2→A=rπd2→A=rπ2r2→A=rπr=πr2 GoalsDerive the formula for the area of a circle.Apply the formula to find the area of circles.SWD: Consider the prerequisite skills for this lesson: understanding and applying the formula for the area of a regular polygon. Students with disabilities may need direct instruction and guided practice with this skill.Students should understand these domain-specific terms:apothemparallelogramderivationheightapproximate (estimate)scatter plotpiperimetercircumferenceIt may be helpful to preteach these terms to students with disabilities.