Module 2 builds on students’ previous work with units and with functions from Algebra I, and with trigonometric ratios and circles from high school Geometry. The heart of the module is the study of precise definitions of sine and cosine (as well as tangent and the co-functions) using transformational geometry from high school Geometry. This precision leads to a discussion of a mathematically natural unit of rotational measure, a radian, and students begin to build fluency with the values of the trigonometric functions in terms of radians. Students graph sinusoidal and other trigonometric functions, and use the graphs to help in modeling and discovering properties of trigonometric functions. The study of the properties culminates in the proof of the Pythagorean identity and other trigonometric identities.

**NOTE: The New York State Education Department shut down the EngageNY website in 2022. In order to maintain educators' access, nearly all resources have been uploaded to archive.org and the resource links above have been updated to reflect their new locations.**

Play with objects on a teeter totter to learn about balance. Test what you've learned by trying the Balance Challenge game.

Students begin their sixth grade year investigating the concepts of ratio and rate. They use multiple forms of ratio language and ratio notation, and formalize understanding of equivalent ratios. Students apply reasoning when solving collections of ratio problems in real world contexts using various tools (e.g., tape diagrams, double number line diagrams, tables, equations and graphs). Students bridge their understanding of ratios to the value of a ratio, and then to rate and unit rate, discovering that a percent of a quantity is a rate per 100. The 35 day module concludes with students expressing a fraction as a percent and finding a percent of a quantity in real world concepts, supporting their reasoning with familiar representations they used previously in the module.

**NOTE: The New York State Education Department shut down the EngageNY website in 2022. In order to maintain educators' access, nearly all resources have been uploaded to archive.org and the resource links above have been updated to reflect their new locations.**

This lab exercise exposes students to a potentially new alternative energy source hydrogen gas. Student teams are given a hydrogen generator and an oxygen generator. They balance the chemical equation for the combustion of hydrogen gas in the presence of oxygen. Then they analyze what the equation really means. Two hypotheses are given, based on what one might predict upon analyzing the chemical equation. Once students have thought about the process, they are walked through the experiment and shown how to collect the gas in different ratios. By trial and error, students determine the ideal combustion ratio. For both volume of explosion and kick generated by explosion, they qualitatively record results on a 0-4 scale. Then, students evaluate their collected results to see if the hypotheses were correct and how their results match the theoretical equation. Students learn that while hydrogen will most commonly be used for fuel cells (no combustion situation), it has been used in rocket engines (for which a tremendous combustion occurs).

This resource may be used as a primary source for a general education/quantitative reasoning (QR) math course delivered in the inquiry based learning style. It may also be used to supplement a QR course with occasional in-class active learning activities.

Play with the left and right hands in different ways, and explore ratio and proportion. Start on the Discover screen to find each challenge ratio by moving the hands. Then, on the Create screen, set your own challenge ratios. Once you've found a challenge ratio, try to move both hands while maintaining the challenge ratio through proportional reasoning.

Students learn how different characteristics of shapes—side lengths, perimeter and area—change when the shapes are scaled, either enlarged or reduced. Student pairs conduct a “scaling investigation” to measure and calculate shape dimensions (rectangle, quarter circle, triangle; lengths, perimeters, areas) from a bedroom floorplan provided at three scales. They analyze their data to notice the mathematical relationships that hold true during the scaling process. They see how this can be useful in real-world situations like when engineers design wearable or implantable biosensors. This prepares students for the associated activity in which they use this knowledge to help them reduce or enlarge their drawings as part of the process of designing their own wearables products. Pre/post-activity quizzes, a worksheet and wrap-up concepts handout are provided.

Students apply their knowledge of scale and geometry to design wearables that would help people in their daily lives, perhaps for medical reasons or convenience. Like engineers, student teams follow the steps of the design process, to research the wearable technology field (watching online videos and conducting online research), brainstorm a need that supports some aspect of human life, imagine their own unique designs, and then sketch prototypes (using Paint®). They compare the drawn prototype size to its intended real-life, manufactured size, determining estimated length and width dimensions, determining the scale factor, and the resulting difference in areas. After considering real-world safety concerns relevant to wearables (news article) and getting preliminary user feedback (peer critique), they adjust their drawn designs for improvement. To conclude, they recap their work in short class presentations.