The focus of the lesson is to review and strengthen understanding of the Pythagorean Thm and the Converse of the Pythagorean Thm. In addition review and strengthen the students algebra skills regarding square roots.This cover Ohio Standards in geometry:Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
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The attached PDF provides learners with a unique way to find geometric shapes through pictures. In the case, the search is for Trapezoids and Parallelograms. Use, revise, or be inspired by the ideas and create your own to support your learners.
Module 1 embodies critical changes in Geometry as outlined by the Common Core. The heart of the module is the study of transformations and the role transformations play in defining congruence. The topic of transformations is introduced in a primarily experiential manner in Grade 8 and is formalized in Grade 10 with the use of precise language. The need for clear use of language is emphasized through vocabulary, the process of writing steps to perform constructions, and ultimately as part of the proof-writing process.
Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2. åÊTo be able to discuss similarity, students must first have a clear understanding of how dilations behave. åÊThis 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. åÊAn in-depth look at similarity within right triangles follows, and finally the module ends with a study of right triangle trigonometry.
Module 3, Extending to Three Dimensions, builds on studentsåÕ 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.
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 planeåÑthe 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 inequalitiesåÊand work to constrain the robot motion along line segments within the region.
This module brings together the ideas of similarity and congruence and the properties of length, area, and geometric constructions studied throughout the year. åÊIt also includes the specific properties of triangles, special quadrilaterals, parallel lines and transversals, and rigid motions established and built upon throughout this mathematical story. åÊThis module's focus is on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page.
In Module 5, students consider partåÐwhole relationships through a geometric lens. The module opens with students identifying the defining parts, or attributes, of two- and three-dimensional shapes, building on their kindergarten experiences of sorting, analyzing, comparing, and creating various two- and three-dimensional shapes and objects. Students combine shapes to create a new whole: a composite shape. They also relate geometric figures to equal parts and name the parts as halves and fourths. The module closes with students applying their understanding of halves to tell time to the hour and half hour.
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.åÊ 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.åÊ Next students explore geometry.åÊ Students tessellate to bridge geometry experience with the study of perimeter.åÊ Line plots, familiar from Module 6, help students draw conclusions about perimeter and area measurements.åÊ Students solve word problems involving area and perimeter using all four operations.åÊ The module concludes with a set of engaging lessons that briefly review the fundamental Grade 3 concepts of fractions, multiplication, and division.
This 20-day module introduces points, lines, line segments, rays, and angles, as well as the relationships between them. Students construct, recognize, and define these geometric objects before using their new knowledge and understanding to classify figures and solve problems. With angle measure playing a key role in their work throughout the module, students learn how to create and measure angles, as well as create and solve equations to find unknown angle measures. In these problems, where the unknown angle is represented by a letter, students explore both measuring the unknown angle with a protractor and reasoning through the solving of an equation. Through decomposition and composition activities as well as an exploration of symmetry, students recognize specific attributes present in two-dimensional figures. They further develop their understanding of these attributes as they classify two-dimensional figures based on them.
In this 25-day module, students work with two- and three-dimensional figures.åÊ Volume is introduced to students through concrete exploration of cubic units and culminates with the development of the volume formula for right rectangular prisms.åÊ The second half of the module turns to extending studentsåÕ understanding of two-dimensional figures.åÊ Students combine prior knowledge of area with newly acquired knowledge of fraction multiplication to determine the area of rectangular figures with fractional side lengths.åÊ They then engage in hands-on construction of two-dimensional shapes, developing a foundation for classifying the shapes by reasoning about their attributes.åÊ This module fills a gap between Grade 4åÕs work with two-dimensional figures and Grade 6åÕs work with volume and area.
In this 40-day module, students develop a coordinate system for the first quadrant of the coordinate plane and use it to solve problems.åÊ Students use the familiar number line as an introduction to the idea of a coordinate, and they construct two perpendicular number lines to create a coordinate system on the plane.åÊ Students see that just as points on the line can be located by their distance from 0, the planeåÕs coordinate system can be used to locate and plot points using two coordinates.åÊ They then use the coordinate system to explore relationships between points, ordered pairs, patterns, lines and, more abstractly, the rules that generate them.åÊ This study culminates in an exploration of the coordinate plane in real world applications.
In this module, students utilize their previous experiences in order to understand and develop formulas for area, volume, and surface area.åÊ Students use composition and decomposition to determine the area of triangles, quadrilaterals, and other polygons.åÊ Extending skills from Module 3 where they used coordinates and absolute value to find distances between points on a coordinate plane, students determine distance, perimeter, and area on the coordinate plane in real-world contexts.åÊ Next in the module comes real-life application of the volume formula where students extend the notion that volume is additive and find the volume of composite solid figures.åÊ They apply volume formulas and use their previous experience with solving equations to find missing volumes and missing dimensions.åÊ The final topic includes deconstructing the faces of solid figures to determine surface area.åÊ To wrap up the module, students apply the surface area formula to real-life contexts and distinguish between the need to find surface area or volume within contextual situations.
In Module 6, students delve further into several geometry topics they have been developing over the years.åÊ Grade 7 presents some of these topics, (e.g., angles, area, surface area, and volume) in the most challenging form students have experienced yet.åÊ Module 6 assumes students understand the basics.åÊ The goal is to build a fluency in these difficult problems.åÊ The remaining topics, (i.e., working on constructing triangles and taking slices (or cross-sections) of three-dimensional figures) are new to students.
In this module, students learn about translations, reflections, and rotations in the plane and, more importantly, how to use them to precisely define the concept of congruence. Throughout Topic A, on the definitions and properties of the basic rigid motions, students verify experimentally their basic properties and, when feasible, deepen their understanding of these properties using reasoning. All the lessons of Topic B demonstrate to students the ability to sequence various combinations of rigid motions while maintaining the basic properties of individual rigid motions. Students learn that congruence is just a sequence of basic rigid motions in Topic C, and Topic D begins the learning of Pythagorean Theorem.
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.åÊ The module begins with the definition of dilation, properties of dilations, and compositions of dilations.åÊ One overarching goal of this module is to replace the common idea of åÒsame shape, different sizesåÓ with a definition of similarity that can be applied to geometric shapes that are not polygons, such as ellipses and circles.
In the first topic of this 15 day module, students learn the concept of a function and why functions are necessary for describing geometric concepts and occurrences in everyday life.åÊ Once a formal definition of a function is provided, students then consider functions of discrete and continuous rates and understand the difference between the two.åÊ Students apply their knowledge of linear equations and their graphs from Module 4 to graphs of linear functions.åÊ Students inspect the rate of change of linear functions and conclude that the rate of change is the slope of the graph of a line.åÊ They learn to interpret the equation y=mx+b as defining a linear function whose graph is a line.åÊ Students compare linear functions and their graphs and gain experience with non-linear functions as well.åÊ In the second and final topic of this module, students extend what they learned in Grade 7 about how to solve real-world and mathematical problems related to volume from simple solids to include problems that require the formulas for cones, cylinders, and spheres.
Module 7 begins with work related to the Pythagorean Theorem and right triangles.åÊ Before the lessons of this module are presented to students, it is important that the lessons in Modules 2 and 3 related to the Pythagorean Theorem are taught (M2:åÊ Lessons 15 and 16, M3:åÊ Lessons 13 and 14).åÊ In Modules 2 and 3, students used the Pythagorean Theorem to determine the unknown length of a right triangle.åÊ In cases where the side length was an integer, students computed the length.åÊ When the side length was not an integer, students left the answer in the form ofåÊx2=c, whereåÊcåÊwas not a perfect square number.åÊ Those solutions are revisited and are the motivation for learning about square roots and irrational numbers in general.
Module 2 explores two-dimensional and three-dimensional shapes. åÊStudents learn about flat and solid shapes independently as well as how they are related to each other and to shapes in their environment. åÊStudents begin to use position words when referring to and moving shapes.åÊ Students learn to use their words to distinguish between examples and non-examples of flat and solid shapes.
Kindergarten comes to a close with another opportunity for students to explore geometry in Module 6. Throughout the year, students have built an intuitive understanding of two- and three-dimensional figures by examining exemplars, variants, and non-examples. They have used geometry as a context for exploring numerals as well as comparing attributes and quantities. To wrap up the year, students further develop their spatial reasoning skills and begin laying the groundwork for an understanding of area through composition of geometric figures.