Just as rigid motions are used to define congruence in Module 1, …
Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2. To be able to discuss similarity, students must first have a clear understanding of how dilations behave. This is done in two parts, by studying how dilations yield scale drawings and reasoning why the properties of dilations must be true. Once dilations are clearly established, similarity transformations are defined and length and angle relationships are examined, yielding triangle similarity criteria. An in-depth look at similarity within right triangles follows, and finally the module ends with a study of right triangle trigonometry.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Module 3, Extending to Three Dimensions, builds on students understanding of congruence …
Module 3, Extending to Three Dimensions, builds on students understanding of congruence in Module 1 and similarity in Module 2 to prove volume formulas for solids. The student materials consist of the student pages for each lesson in Module 3. The copy ready materials are a collection of the module assessments, lesson exit tickets and fluency exercises from the teacher materials.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
In this module, students explore and experience the utility of analyzing algebra …
In this module, students explore and experience the utility of analyzing algebra and geometry challenges through the framework of coordinates. The module opens with a modeling challenge, one that reoccurs throughout the lessons, to use coordinate geometry to program the motion of a robot that is bound within a certain polygonal region of the planethe room in which it sits. To set the stage for complex work in analytic geometry (computing coordinates of points of intersection of lines and line segments or the coordinates of points that divide given segments in specific length ratios, and so on), students will describe the region via systems of algebraic inequalities and work to constrain the robot motion along line segments within the region.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
This module brings together the ideas of similarity and congruence and the …
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.
This lesson unit is intended to help teachers assess how well students …
This lesson unit is intended to help teachers assess how well students are able to use geometric properties to solve problems. In particular, the lesson will help you identify and help students who have the following difficulties: solving problems by determining the lengths of the sides in right triangles; and finding the measurements of shapes by decomposing complex shapes into simpler ones. The lesson unit will also help students to recognize that there may be different approaches to geometrical problems, and to understand the relative strengths and weaknesses of those approaches.
This is a way to include Project Based Learning (PBL) with 4th-grade …
This is a way to include Project Based Learning (PBL) with 4th-grade geometry standards. This is a project students have a chance to work on individually as well as collaboratively to show real-world application of their skills and knowledge learned through the unit.
Understanding vocabulary is critical in order to fully understand triangle relationships. In …
Understanding vocabulary is critical in order to fully understand triangle relationships. In this activity, students will review the vocabulary associated with triangles: circumcenter, incenter, centroid, and orthocenter.
I had my students cut out the flashcards during class. I was very surprised some students took a lot of time cutting out the cards since they cut around the boxes instead of making straight cuts.
I would suggest before students are given the scissors, to model the quickest way to cut out the cards.
Students learn about common geometry tools and then learn to use protractors …
Students learn about common geometry tools and then learn to use protractors (and Miras, if available) to create and measure angles and reflections. The lesson begins with a recap of the history and modern-day use of protractors, compasses and mirrors. After seeing some class practice problems and completing a set of worksheet-prompted problems, students share their methods and work. Through the lesson, students gain an awareness of the pervasive use of angles, and these tools, for design purposes related to engineering and everyday uses. This lesson prepares students to conduct the associated activity in which they “solve the holes” for hole-in-one multiple-banked angle solutions, make their own one-hole mini-golf courses with their own geometry-based problems and solutions, and then compare their “on paper” solutions to real-world results.
A rigorous introduction designed for mathematicians into perturbative quantum field theory, using …
A rigorous introduction designed for mathematicians into perturbative quantum field theory, using the language of functional integrals. Basics of classical field theory. Free quantum theories. Feynman diagrams. Renormalization theory. Local operators. Operator product expansion. Renormalization group equation. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and string theory.
This is a second-semester graduate course on the geometry of manifolds. The …
This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.
Students are presented with a real-world application of where mathematics is used …
Students are presented with a real-world application of where mathematics is used in video game design. In the task, students are asked to solve a challenge using the corrdinate plane, slope, equations, and x- and y- axes.
In this Illumination activity, students act as reporters at the Super Bowl. …
In this Illumination activity, students act as reporters at the Super Bowl. Students study four pictures of things that they would typically find at a football game: players, a scoreboard, a crowd, and a concession stand. Students are asked to create problem situations that correspond to their interpretation of each of the pictures. The lesson includes a student worksheet and extension questions.
This interactive lesson encourages young students to solve problems by estimating angles …
This interactive lesson encourages young students to solve problems by estimating angles and distances. They use an applet to give LOGO-like commands, e.g. forward (length), turn (right or left) to make a path that moves a turtle to a pond. Students can create a Path 1 and Path 2 and try to minimize the total path length. There is a newer applet (Turtle Pond, cataloged separately) that allows for adding or editing the commands and a choice of right angles only, or angles in multiples of 15 degrees. The lesson provides suggestions for implementation and discussion questions.
In this lesson, students will investigate error. As shown in earlier activities …
In this lesson, students will investigate error. As shown in earlier activities from navigation lessons 1 through 3, without an understanding of how errors can affect your position, you cannot navigate well. Introducing accuracy and precision will develop these concepts further. Also, students will learn how computers can help in navigation. Often, the calculations needed to navigate accurately are time consuming and complex. By using the power of computers to do calculations and repetitive tasks, one can quickly see how changing parameters likes angles and distances and introducing errors will affect their overall result.
The first of these word problems is a multiplication problem involving equal-sized …
The first of these word problems is a multiplication problem involving equal-sized groups. The next two reflect the two related division problems, namely, "How many groups?" and "How many in each group?"
The purpose of this task is to show three problems that are …
The purpose of this task is to show three problems that are set in the same kind of context, but the first is a straightforward multiplication problem while the other two are the corresponding "How many groups?" and "How many in each group?" division problems.
In this activity students practice measuring techniques by measuring different objects and …
In this activity students practice measuring techniques by measuring different objects and distances around the classroom. They practice using different scales of measurement in metric units and estimation.
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