Zero Product Property 2

This task is part of a series of tasks that lead students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression to help find its solutions (rather than memorizing steps without understanding). In this particular task, we are trying to get students to prove the zero product property, which is the lynchpin in understanding how to solve quadratic equations by factoring. In tasks that follow in this series, students will apply this property to solving quadratic equations and justifying their solutions.

Material Type: Activity/Lab

Zero Product Property 3

This task is part of a series of tasks that lead students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression to help find its solutions (rather than memorizing steps without understanding). Teachers should feel free to skip any tasks in the series that students have already mastered.

Material Type: Activity/Lab

Zero Product Property 4

This task is the fourth in a series of tasks that leads students to understand The Zero Product Property (ZPP) and apply it to solving quadratic equations. The emphasis is on using the structure of a factorable expression to justify the solution method (rather than memorizing steps without understanding). Teachers should feel free to skip any tasks in the series that students have already mastered.

Material Type: Activity/Lab

A Linear and Quadratic System

This task asks students to consider the linear and quadratic functions appearing on a coordinate plane.

Material Type: Activity/Lab

Author: Illustrative Mathematics

A-REI Collinear points

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...

Material Type: Activity/Lab

Author: Illustrative Mathematics

Taxi!

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

Material Type: Activity/Lab

Author: Illustrative Mathematics

A-REI Ideal Gas Law

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 aspects of the task and its potential use.

Material Type: Activity/Lab

Author: Illustrative Mathematics

Population and Food Supply

In this task students construct and compare linear and exponential functions and find where the two functions intersect. One purpose of this task is to demonstrate that exponential functions grow faster than linear functions even if the linear function has a higher initial value and even if we increase the slope of the line. This task could be used as an introduction to this idea.

Material Type: Activity/Lab

Author: Illustrative Mathematics

Two Squares are Equal

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4).

Material Type: Activity/Lab

Author: Illustrative Mathematics

Fishing Adventures 3

This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically. The progression of the content standards is 6.EE.8 to 7.EE.4 to A-REI.12.

Material Type: Activity/Lab

Author: Illustrative Mathematics

Solution Sets

The typical system of equations or inequalities problem gives the system and asks for the graph of the solution. This task turns the problem around. It gives a solution set and asks for the system that corresponds to it. The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing.

Material Type: Activity/Lab

Author: Illustrative Mathematics

A-REI Reasoning with linear inequalities

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: The following is a student solution to the inequality \frac{5}{18} - \frac{x-2}{9} \leq \frac{x-4}{6}. \begin{align} \frac{5}{18} - \frac{x-2}{9} & \le...

Material Type: Activity/Lab

Author: Illustrative Mathematics

A-REI Integer Solutions to Inequality

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: What is the sum of all integer solutions to $1\lt (x-2)^2\lt 25$?...

Material Type: Activity/Lab

Author: Illustrative Mathematics

Interpreting the Graph

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions, or as an assessment tool with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Material Type: Activity/Lab

Author: Illustrative Mathematics

Domains

The purpose of this task to help students think about an expression for a function as built up out of simple operations on the variable, and understand the domain in terms of values for which each operation is invalid (e.g., dividing by zero or taking the square root of a negative number).

Material Type: Activity/Lab

Author: Illustrative Mathematics

Do Two Points Always Determine a Linear Function II?

This task is designed as a follow-up to the task F-LE Do Two Points Always Determine a Linear Function? Linear equations and linear functions are closely related, and there advantages and disadvantages to viewing a given problem through each of these points of view. This task is intended to show the depth of the standard F-LE.2 and its relationship to other important concepts of the middle school and high school curriculum, including ratio, algebra, and geometry.

Material Type: Activity/Lab

Author: Illustrative Mathematics

Do Two Points Always Determine a Linear Function?

This problem allows the student to think geometrically about lines and then relate this geometry to linear functions. Or the student can work algebraically with equations in order to find the explicit equation of the line through two points (when that line is not vertical).

Material Type: Activity/Lab

Author: Illustrative Mathematics

Finding the domain

The purpose of this task is to introduce the idea of the domain of a function by linking it to the evaluation of an expression defining the function. By thinking through the evaluation step by step, students isolate the exact point where a given input results in an undefined output. In part (b), any domain that excludes x=3 is possible. It is conventional when given a function defined by an expression to take the domain to be the largest possible, but it is worth pointing that this is a convention, not a mathematical fact. As students gain a mature understanding of functions they learn that the domain is something that is specified when you define the function, it does not come already attached.

Material Type: Activity/Lab

Parabolas and Inverse Functions

This task assumes students have an understanding of the relationship between functions and equations. Using this knowledge, the students are prompted to try to solve equations in order to find the inverse of a function given in equation form: when no such solution is possible, this means that the function does not have an inverse.

Material Type: Activity/Lab

Author: Illustrative Mathematics

Points on a Graph

This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function.

Material Type: Activity/Lab

Author: Illustrative Mathematics