In this module, students learn about translations, reflections, and rotations in the …
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.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
In Module 3, students learn about dilation and similarity and apply that …
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.
In the first topic of this 15 day module, students learn the …
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. 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. Students apply their knowledge of linear equations and their graphs from Module 4 to graphs of linear functions. 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. They learn to interpret the equation y=mx+b as defining a linear function whose graph is a line. Students compare linear functions and their graphs and gain experience with non-linear functions as well. 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.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Module 7 begins with work related to the Pythagorean Theorem and right …
Module 7 begins with work related to the Pythagorean Theorem and right triangles. 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: Lessons 15 and 16, M3: Lessons 13 and 14). In Modules 2 and 3, students used the Pythagorean Theorem to determine the unknown length of a right triangle. In cases where the side length was an integer, students computed the length. When the side length was not an integer, students left the answer in the form of x2=c, where c was not a perfect square number. Those solutions are revisited and are the motivation for learning about square roots and irrational numbers in general.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Module 2 explores two-dimensional and three-dimensional shapes. Students learn about flat and …
Module 2 explores two-dimensional and three-dimensional shapes. Students learn about flat and solid shapes independently as well as how they are related to each other and to shapes in their environment. Students begin to use position words when referring to and moving shapes. Students learn to use their words to distinguish between examples and non-examples of flat and solid shapes.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Kindergarten comes to a close with another opportunity for students to explore …
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.
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 …
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.
Students see that geometric shapes can be found in all sorts of …
Students see that geometric shapes can be found in all sorts of structures as they explore the history of the Roman Empire with a focus on how engineers 2000 years ago laid the groundwork for many structures seen today. Through a short online video, brief lecture material and their own online research directed by worksheet questions, students discover how the Romans invented a structure known today as the Roman arch that enabled them to build architecture never before seen by humankind, including the amazing aqueducts. Students calculate the slope and its total drop and angle over its entire distance for an example aqueduct. Completing this lesson prepares students for the associated activity in which teams build and test model aqueducts that meet specific constraints. This lesson serves as an introduction to many other geometry—and engineering-related lessons—including statics and trusses, scale modeling, and trigonometry.
An interactive applet and associated web page that show the definition of …
An interactive applet and associated web page that show the definition of a horizontal line in coordinate geometry. The applet has two points that the user can drag which define a line. The line flagged when it is horizontal (slope=0) and the equation of the line is shown. The grid, details and coordinates can be turned on and off. The applet can be printed exactly as it appears on the screen to make handouts. The web page has a discussion on how to test for horizontal, the line equation and has links to other pages relating to coordinate geometry. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.
The purpose of this task is for students to apply the concepts …
The purpose of this task is for students to apply the concepts of mass, volume, and density in a real-world context. There are several ways one might approach the problem, e.g., by estimating the volume of a person and dividing by the volume of a cell.
This is a mathematical modeling task aimed at making a reasonable estimate …
This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.
In this problem, the variables a,b,c, and d are introduced to represent …
In this problem, the variables a,b,c, and d are introduced to represent important quantities for this esimate: students should all understand where the formula in the solution for the number of leaves comes from. Estimating the values of these variables is much trickier and the teacher should expect and allow a wide range of variation here.
his is a version of ''How thick is a soda can I'' …
his is a version of ''How thick is a soda can I'' which allows students to work independently and think about how they can determine how thick a soda can is. The teacher should explain clearly that the goal of this task is to come up with an ''indirect'' means of assessing how thick the can is, that is directly measuring its thickness is not allowed.
Engineers create and use new materials, as well as new combinations of …
Engineers create and use new materials, as well as new combinations of existing materials to design innovative new products and technologies—all based upon the chemical and physical properties of given substances. In this activity, students act as materials engineers as they learn about and use chemical and physical properties including tessellated geometric designs and shape to build better smartphone cases. Guided by the steps of the engineering design process, they analyze various materials and substances for their properties, design/test/improve a prototype model, and create a dot plot of their prototype testing results.
An interactive applet and associated web page that provide step-by-step animated instructions …
An interactive applet and associated web page that provide step-by-step animated instructions on how to measure angles using a protractor. Specifically, it uses a protractor to measure two angles that form a vertical pair, verifying they have the same measure. The animation can be run either continuously like a video, or single stepped to allow classroom discussion and thought between steps. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.
Create a human sundial to measure time using shadows. Activity from Weekly …
Create a human sundial to measure time using shadows. Activity from Weekly STEM in a Bag. Colorado Americorp agents in Araphahoe, Denver, Garfield, Larimer, and Weld Counties. Work supported by the Corporation for National and Community Service under Americorps grant number 18AFHCO0010008. Opinions or points of view expressed in this lesson are those of the authors and do not necessarily represent the official position of or a position that is endorsed by the Corporation or the Americorps program. This resource is also available in Spanish in the linked file.
This rich task is an excellent example of geometric concepts in a …
This rich task is an excellent example of geometric concepts in a modeling situation and is accessible to all students. In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.
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