This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: In the triangle pictured above show that \left(\frac{|AB|}{|AC|}\right)^2 + \left(\frac{|BC|}{|AC|}\right)^2 = 1 Deduce that $\sin^2{\theta} + \cos^2{\...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Below is a picture of a right triangle with $a$ the measure of angle $A$: Joyce knows that the sine of $a$ is the length of the side opposite $A$ divid...

This problem uses the same numbers and asks essentially the same mathematical questions as "6.NS Bake Sale," but that task requires students to apply the concepts of factors and common factors in a context.

Students use the engineering design process to solve a real-world problem shoe engineering! Working in small teams, they design, build and test a pair of wearable platform or high-heeled shoes, taking into consideration the stress and strain forces that it will encounter from the shoe wearer. They conclude the activity with a "walk-off" to test the shoe designs and discuss the design process.

Students design, build and test shoes to develop new styles and improve existing designs.

Students use wood, wax paper and oil to investigate the importance of lubrication between materials and to understand the concept of friction. Using wax paper and oil placed between pieces of wood, the function of lubricants between materials is illustrated. Students extend their understanding of friction to bones and joints in the skeletal system and become aware of what engineers can do to help reduce friction in the human body as well as in machines.

Students conduct experiments to determine the flow rate of faucets by timing how long it takes to fill gallon jugs. They do this for three different faucet flow levels (quarter blast, half blast, full blast), averaging three trials for each level. They convert their results from gallons per second (gps) to cubic feet per second (cfs).

In this activity, students are introduced to faults. They will learn about different kinds of faults and understand their relationship to earthquakes. The students will build cardboard models of the three different types of faults as they learn about how earthquakes are formed.

Students have the opportunity to create and interpret data from simple daily questions.

This task provides students the opportunity to make use of units to find the gas need (N-Q.1). The key point is for them to explain their choices. This task provides an opportunity for students to practice MP2, Reason abstractly and quantitatively, and MP3, Construct viable arguments and critique the reasoning of others.

This task allows students to relate addition and subtraction problems to money in a context that introduces the concept of scarcity.

Filtering is the process of removing or separating the unwanted part of a mixture. In signal processing, filtering is specifically used to remove or extract part of a signal, and this can be accomplished using an analog circuit or a digital device (such as a computer). In this lesson, students learn the impact filtering can have on different types of signals, the concepts of frequency and spectrum, and the connections these topics have to real-world signals such as musical signals. Students also learn the roles that these concepts play in designing different types of filters. The lesson content prepares students for the associated activity in which they use an online demo and a variety of filters to identify the message in a distress signal heavily corrupted by noise.

Students learn the basic principles of filtering as well as how to apply digital filters to extract part of an audio signal by using an interactive online demo website. They apply this knowledge in order to isolate a voice recording from a heavily noise-contaminated sound wave. After completing the associated lesson, expect students to be able to attempt (and many successfully finish) this activity with minimal help from the instructor.

This simple-looking problem reveals much about how well students understand unit fractions as well as representing fractions on a number line.

In this activity students will learn the basic concept of Global Positioning Systems (GPS) using triangulation and measurement on a small scale in the classroom. Students discover how GPS and navigation integrate mathematic and scientific concepts to create a standard for locating people and objects. This activity helps students understand both the need for and methods of navigation.

Students create their own simple compasses using thread, needle and water in a bowl and learn how it works.

The purpose of this task is to have students work on a sequence of area problems that shows the advantage of increasingly abstract strategies in preparation for developing general area formulas for parallelograms and triangles.

This task asks students to find and use two different common denominators to add the given fractions. The purpose of this question is to help students realize that they can use any common denominator to find a solution, not just the least common denominator.