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: Consider three points in the plane, $P=(-4, 0), Q=(-1, 12)$ and $R=(4, 32)$. Find the equation of the line through $P$ and $Q$. Use your equation in (a...
Students learn about linear programming (also called linear optimization) to solve engineering design problems. As they work through a word problem as a class, they learn about the ideas of constraints, feasibility and optimization related to graphing linear equalities. Then they apply this information to solve two practice engineering design problems related to optimizing materials and cost by graphing inequalities, determining coordinates and equations from their graphs, and solving their equations. It is suggested that students conduct the associated activity, Optimizing Pencils in a Tray, before this lesson, although either order is acceptable.
This lesson unit is intended to help teachers assess how well students are able to use linear inequalities to create a set of solutions. In particular, the lesson will help teachers identify and assist students who have difficulties in: representing a constraint by shading the correct side of the inequality line; and understanding how combining inequalities affects a solution space.
Students create model elevator carriages and calibrate them, similar to the work of design and quality control engineers. Students use measurements from rotary encoders to recreate the task of calibrating elevators for a high-rise building. They translate the rotations from an encoder to correspond to the heights of different floors in a hypothetical multi-story building. Students also determine the accuracy of their model elevators in getting passengers to their correct destinations.
In this activity, students study gas laws at a molecular level. They vary the volume of a container at constant temperature to see how pressure changes (Boyle's Law), change the temperature of a container at constant pressure to see how the volume changes with temperature (Charles’s Law), and experiment with heating a gas in a closed container to discover how pressure changes with temperature (Gay Lussac's Law). They also discover the relationship between the number of gas molecules and gas volume (Avogadro's Law). Finally, students use their knowledge of gas laws to model a heated soda can collapsing as it is plunged into ice water.
This lesson unit is intended to help teachers assess how well students are able to: interpret a situation and represent the constraints and variables mathematically; select appropriate mathematical methods to use; explore the effects of systematically varying the constraints; interpret and evaluate the data generated and identify the optimum case, checking it for confirmation; and communicate their reasoning clearly.
Students use balloons (a polymer) to explore preconditioning a viscoelastic material behavior that is important to understand when designing biomedical devices. They improve their understanding of preconditioning by measuring the force needed to stretch a balloon to the same displacement multiple times. Students gain experience in data collection and graph interpretation.
Adult education classrooms are commonly comprised of learners who have widely disparate levels of mathematical problem-solving skills. This is true regardless of what level a student may be assessed at when entering an adult education program or what level class they are placed in. Providing students with differentiated instruction in the form of Push and Support cards is one way to level this imbalance, keeping all students engaged in one high-cognitive task that supports and encourages learners who are stuck, while at the same time, providing extensions for students who move through the initial phase of the task quickly. Thus, all
students are continually moving forward during the activity, and when the task ends, all students have made progress in their journey towards developing conceptual understanding of mathematical ideas along with a productive disposition, belief in one’s own ability to successfully engage with mathematics.
Students use their understanding of projectile physics and fluid dynamics to find the water pressure in water guns. By measuring the range of the water jets, they are able to calculate the theoretical pressure. Students create graphs to analyze how the predicted pressure relates to the number of times they pump the water gun before shooting.
This web-based graphing activity explores the similarities and differences between Velocity vs. Time and Position vs. Time graphs. It interactively accepts user inputs in creating "prediction graphs", then provides real-time animations of the process being analyzed. Learners will annotate graphs to explain changes in motion, respond to question sets, and analyze why the two types of graphs appear as they do. It is appropriate for secondary physical science courses, and may also be used for remediation in preparatory high school physics courses. This item is part of the Concord Consortium, a nonprofit research and development organization dedicated to transforming education through technology. Users must register to access full functionality of all the tools available with SmartGraphs.
This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.
Students explore how the efficiency of a solar photovoltaic (PV) panel is affected by the ambient temperature. They learn how engineers predict the power output of a PV panel at different temperatures and examine some real-world engineering applications used to control the temperature of PV panels.