The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even.

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 each of the equations below, rewrite the equation, solving for the indicated variable If $F$ denotes a temperature in degrees Fahrenheit and $C$ is ...

At this point in the unit, students have learned about Pascal's law, Archimedes' principle, Bernoulli's principle, and why above-ground storage tanks are of major concern in the Houston Ship Channel and other coastal areas. In this culminating activity, student groups act as engineering design teams to derive equations to determine the stability of specific above-ground storage tank scenarios with given tank specifications and liquid contents. With their floatation analyses completed and the stability determined, students analyze the tank stability in specific storm conditions. Then, teams are challenged to come up with improved storage tank designs to make them less vulnerable to uplift, displacement and buckling in storm conditions. Teams present their analyses and design ideas in short class presentations.

A collection of relevant lessons to supplement your units in Algebra I/II. Mix these lessons into your course to show students the algebraic reasoning behind social issues, public health, the environment, business, sports, and more.

Students design, build and test model roller coasters using foam tubing. The design process integrates energy concepts as they test and evaluate designs that address the task as an engineer would. The goal is for students to understand the basics of engineering design associated with kinetic and potential energy to build an optimal roller coaster. The marble starts with potential energy that is converted to kinetic energy as it moves along the track. The diameter of the loops that the marble traverses without falling out depends on the kinetic energy obtained by the marble.

Bernoulli's principle relates the pressure of a fluid to its elevation and its speed. Bernoulli's equation can be used to approximate these parameters in water, air or any fluid that has very low viscosity. Students learn about the relationships between the components of the Bernoulli equation through real-life engineering examples and practice problems.

This lesson begins with a demonstration prompting students to consider how current generates a magnetic field and the direction of the field that is generated. Through formal lecture, students learn Biot-Savart's law in order to calculate, most simply, the magnetic field produced in the center of a circular current carrying loop. For applications, students find it is necessary to integrate the field produced over all small segments in an actual current carrying wire.

This lab demonstrates Ohm's law as students set up simple circuits each composed of a battery, lamp and resistor. Students calculate the current flowing through the circuits they create by solving linear equations. After solving for the current, I, for each set resistance value, students plot the three points on a Cartesian plane and note the line that is formed. They also see the direct correlation between the amount of current flowing through the lamp and its brightness.

As a continuation of the theme of potential and kinetic energy, this lesson introduces the concepts of momentum, elastic and inelastic collisions. Many sports and games, such as baseball and ping-pong, illustrate the ideas of momentum and collisions. Students explore these concepts by bouncing assorted balls on different surfaces and calculating the momentum for each ball.

Students investigate the property dependence between concentrations and boiling point. In section 1, students first investigate the boiling point of various liquid solutions. In section 2, they analyze data collected by the entire class to generate two boiling point curves, one for salt solutions and one for sugar solutions. Finally, in section 3, students use the data they have analyzed to determine how to create a solution that has a particular boiling point and is a cost-effective design.

Use a series of interactive models and games to explore electrostatics. Learn about the effects positive and negative charges have on one another, and investigate these effects further through games. Learn about Coulomb's law and the concept that both the distance between the charges and the difference in the charges affect the strength of the force. Explore polarization at an atomic level, and learn how a material that does not hold any net charge can be attracted to a charged object. Students will be able to:

This activity utilizes hands on learning with the conservation of energy with the inclusion of elastic potential energy. Students use pogo sticks to experience the elastic potential energy and its conversion to gravitational potential energy.

This task asks students to use inverse operations to solve the equations for the unknown variable, or for the designated variable if there is more than one. Two of the equations are of physical significance and are examples of Ohm's Law and Newton's Law of Universal Gravitation.

This lesson unit is intended to help teachers assess how well students are able to: use the Pythagorean theorem to derive the equation of a circle; and translate between the geometric features of circles and their equations.

Students explore material properties by applying some basic principles of heat transfer. They use calorimeters to determine the specific heat of three substances: aluminum, copper and another of their choice. Each substance is cooled in a freezer and then placed in the calorimeter. The temperature change of the water and the substance are used in heat transfer equations to determine the specific heat of each substance. The students compare their calculated values with tabulated data.

Students examine how the power output of a photovoltaic (PV) solar panel is affected by temperature changes. Using a 100-watt lamp and a small PV panel connected to a digital multimeter, teams vary the temperature of the panel and record the resulting voltage output. They plot the panel's power output and calculate the panel's temperature coefficient.

This lesson unit is intended to help teachers assess how well students are able to use geometric properties to solve problems. In particular, it will help you identify and help students who have difficulty: decomposing complex shapes into simpler ones in order to solve a problem; bringing together several geometric concepts to solve a problem; and finding the relationship between radii of inscribed and circumscribed circles of right triangles.

Students explore the basics of DC circuits, analyzing the light from light bulbs when connected in series and parallel circuits. Ohm's law and the equation for power dissipated by a circuit are the two primary equations used to explore circuits connected in series and parallel. Students measure and see the effect of power dissipation from the light bulbs. Kirchhoff's voltage law is used to show how two resistor elements add in series, while Kirchhoff's current law is used to explain how two resistor elements add when in parallel. Students also learn how electrical engineers apply this knowledge to solve problems. Power dissipation is particularly important with the introduction of LED bulbs and claims of energy efficiency, and understanding how power dissipation is calculated helps when evaluating these types of claims. This activity is designed to introduce students to the concepts needed to understand how circuits can be reduced algebraically.

In this lesson, students are introduced to both potential energy and kinetic energy as forms of mechanical energy. A hands-on activity demonstrates how potential energy can change into kinetic energy by swinging a pendulum, illustrating the concept of conservation of energy. Students calculate the potential energy of the pendulum and predict how fast it will travel knowing that the potential energy will convert into kinetic energy. They verify their predictions by measuring the speed of the pendulum.

Students are given an engineering challenge: A nearby hospital has just installed a new magnetic resonance imaging facility that has the capacity to make 3D images of the brain and other body parts by exposing patients to a strong magnetic field. The hospital wishes for its entire staff to have a clear understanding of the risks involved in working near a strong magnetic field and a basic understanding of why those risks occur. Your task is to develop a presentation or pamphlet explaining the risks, the physics behind those risks, and the safety precautions to be taken by all staff members. This 10-lesson/4-activity unit was designed to provide hands-on activities to teach end-of-year electricity and magnetism topics to a first-year accelerated or AP physics class. Students learn about and then apply the following science concepts to solve the challenge: magnetic force, magnetic moments and torque, the Biot-Savart law, Ampere's law and Faraday's law. This module is built around the Legacy Cycle, a format that incorporates findings from educational research on how people best learn.