Students work as engineers and learn to conduct controlled experiments by changing one experimental variable at a time to study its effect on the experiment outcome. Specifically, they conduct experiments to determine the angular velocity for a gear train with varying gear ratios and lengths. Student groups assemble LEGO MINDSTORMS(TM) NXT robots with variously sized gears in a gear train and then design programs using the NXT software to cause the motor to rotate all the gears in the gear train. They use the LEGO data logging program and light sensors to set up experiments. They run the program with the motor and the light sensor at the same time and analyze the resulting plot in order to determine the angular velocity using the provided physics-based equations. Finally, students manipulate the gear train with different gears and different lengths in order to analyze all these factors and figure out which manipulation has a higher angular velocity. They use the equations for circumference of a circle and angular velocity; and convert units between radians and degrees.

A gear is a simple machine that is very useful to increase the speed or torque of a wheel. In this activity, students learn about the trade-off between speed and torque when designing gear ratios. The activity setup includes a LEGO(TM) MINDSTORMS(TM) NXT pulley system with two independent gear sets and motors that spin two pulleys. Each pulley has weights attached by string. In a teacher demonstration, the effect of adding increasing amounts of weight to the pulley systems with different gear ratios is observed as the system's ability to lift the weights is tested. Then student teams are challenged to design a gear set that will lift a given load as quickly as possible. They test and refine their designs to find the ideal gear ratio, one that provides enough torque to lift the weight while still achieving the fastest speed possible.

This lesson unit is intended to help you assess how well students working with square numbers are able to: choose an appropriate, systematic way to collect and organize data, examining the data for patterns; describe and explain findings clearly and effectively; generalize using numerical, geometrical, graphical and/or algebraic structure; and explain why certain results are possible/impossible, moving towards a proof.

This hands-on activity shows you how to build basic architectural shapes out of toothpicks and gumdrops. Learners explore how different shapes are more stable than others, and are introduced to ideas about "stretching and squashing"--that is, about tension and compression.

This task presents students with some creative geometric ways to represent the fraction one half. The goal is both to appeal to students' visual intuition while also providing a hands on activity to decide whether or not two areas are equal.

In this lesson, students will how to name angles, measure and classify angles, identify congruent angles, and how to find angle measures using Angle Addition Postulate.

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.

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

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 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 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 plane—the 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 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 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.

These games support student development of spatial sense and foster familiarity with the mathematical vocabulary of geometry.

The purpose of this lesson is to teach the students about how a spacecraft gets from the surface of the Earth to Mars. The lesson first investigates rockets and how they are able to get us into space. Finally, the nature of an orbit is discussed as well as how orbits enable us to get from planet to planet specifically from Earth to Mars.

This activity is an easy way to demonstrate the fundamental properties of polar and non-polar molecules (such as water and oil), how they interact, and the affect surfactants (such as soap) have on their interactions. Students see the behavior of oil and water when placed together, and the importance soap (a surfactant) plays in the mixing of oil and water which is why soap is used every day to clean greasy objects, such as hands and dishes. This activity is recommended for all levels of student, grades 3-12, as it can easily be scaled to meet any desired level of difficulty.

Students are introduced to gear transmissions and gear ratios using LEGO MINDSTORMS(TM) NXT robots, gears and software. They discover how gears work and how they can be used to adjust a vehicle's power. Specifically, they learn how to build the transmission part of a vehicle by designing gear trains with different gear ratios. Students quickly recognize that some tasks require vehicle speed while others are more suited for vehicle power. They are introduced to torque, which is a twisting force, and to speed the two traits of all rotating engines, including mobile robots using gears, bicycles and automobiles. Once students learn the principles behind gear ratios, they are put to the test in two simple design activities that illustrate the mechanical advantages of gear ratios. The "robot race" is better suited for a quicker robot while the "robot push" calls for a more powerful robot. A worksheet and post-activity quiz verify that students understand the concepts, including the tradeoff between torque and speed.

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 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 distances around the classroom. They practice using different scales of measurement in metric units and estimation.

This hands-on activity explores five different forms of erosion (chemical, water, wind, glacier and temperature). Students rotate through stations and model each type of erosion on rocks, soils and minerals. The students record their observations and discuss the effects of erosion on the Earth's landscape. Students learn about how engineers are involved in the protection of landscapes and structures from erosion. Math problems are included to help students think about the effects of erosion in real-world scenarios.