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 will exploring how chaning the equation of a parabola on Desmos.com will give you similar graphs. These equations are all in vertex form and will ask students to determine the vertex of each equation. Students will all be asked to graph each of the equations in the families as well. Original worksheet created by Curt Sauer.

Using video segments and web interactives from Get the Math, students engage in an exploration of mathematics, specifically reasoning and sense making, to solve real world problems. In this lesson, students focus on understanding the Big Ideas of Algebra: patterns, relationships, equivalence, and linearity; learn to use a variety of representations, including modeling with variables; build connections between numeric and algebraic expressions; and use what they have learned previously about number and operations, measurement, proportionality, and discrete mathematics as applications of algebra. Methodology includes guided instruction, student-partner investigations, and communication of problem-solving strategies and solutions. In the Introductory Activity, students view a video segment in which they learn how Elton Brand, an accomplished basketball player, uses math in his work and are presented with a mathematical basketball challenge. In Learning Activity 1, students solve the challenge that Elton posed in the video, which involves using algebraic concepts and reasoning to figure out the maximum height the basketball reaches on its way into the basket by using three key variables and Elton Brand's stats. As students solve the problem, they have an opportunity to use an online simulation to find a solution. Students summarize how they solved the problem, followed by a viewing of the strategies and solutions used by the Get the Math teams. In Learning Activity 2, students try to solve additional interactive basketball (projectile motion) challenges. In the Culminating Activity, students reflect upon and discuss their strategies and talk about the ways in which algebra can be applied in the world of sports and beyond.

This lesson is designed to help students develop strategies for solving optimization problems. Such problems typically involve using limited resources to greatest effect, as in, for example, the allocation of time and materials to maximize profit.
Before the lesson, students attempt the problem individually. You then review their work and formulate questions for students to answer in order to improve their solutions.At the start of the lesson, students work alone answering your questions.Students are then grouped and engage in a collaborative discussion of the same task. In the same small groups, students are given sample solutions to comment on and evaluate.In a whole-class discussion, students explain and compare solution strategies seen and used.Finally, students revise their individual solutions and comment on what they have learned.
Materials required
Each individual student will need a copy of the  task, some plain paper, a calculator, and a copy of the How Did You Work? questionnaire.Each small group of students will need copies of the Sample Responses to Discuss.Graph paper should be kept in reserve and used only when necessary or requested.
Time needed
Approximately 15 minutes before the lesson, a 1-hour lesson, and 10 minutes in a follow-up lesson.

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.

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.