Cluster Standards

This cluster includes the following benchmarks.

Visit the specific benchmark webpage to find related instructional resources.

MAFS.912.A-REI.2.3 :
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
MAFS.912.A-REI.2.4 : Solve quadratic equations in one variable.
1. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.
2. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Cluster Information

Number:

MAFS.912.A-REI.2

Title:

Solve equations and inequalities in one variable. (Algebra 1 - Major Cluster) (Algebra 2 - Supporting Cluster)

Type:

Cluster

Subject:

Mathematics - Archived

Grade:

912

Domain-Subdomain

Algebra: Reasoning with Equations & Inequalities

Cluster Access Points

This cluster includes the following Access Points.

MAFS.912.A-REI.2.AP.3a : Solve linear equations in one variable, including coefficients represented by letters.
MAFS.912.A-REI.2.AP.3b : Solve linear inequalities in one variable, including coefficients represented by letters.
MAFS.912.A-REI.2.AP.4a : Solve quadratic equations by completing the square.
MAFS.912.A-REI.2.AP.4b : Solve quadratic equations by using the quadratic formula.
MAFS.912.A-REI.2.AP.4c : Solve quadratic equations by factoring.

Cluster Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

Original Student Tutorials

Highs and Lows Part 2: Completing the Square: Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 2 of a 2 part series. Click HERE to open part 1.
Highs and Lows Part 1: Completing the Square: Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 1 of a 2 part series. Click HERE to open Part 2.
Solving Rational Equations: Using Common Denominators: Learn how to solve rational functions by getting common denominators in this interactive tutorial.
Solving Rational Equations: Cross Multiplying: Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.
Solving Inequalities and Graphing Solutions Part 2: Learn how to solve and graph compound inequalities and determine if solutions are viable in part 2 of this interactive tutorial series.

Click HERE to open Part 1.
Solving Inequalities and Graphing Solutions: Part 1: Learn how to solve and graph one variable inequalities, including compound inequalities, in part 1 of this interactive tutorial series.

Click HERE to open Part 2.

Formative Assessments

Quadratic Formula - 1: Students are asked to derive the quadratic formula by completing the square.
Quadratic Formula - 2: Students are asked to complete the derivation of the quadratic formula.
Solving a Literal Linear Equation: Students are given a literal linear equation and asked to solve for a specific variable.
Solving a Multistep Inequality: Students are asked to solve a multistep inequality.
Solve for Y: Students are asked to solve a linear inequality in one variable.
Solve for X: Students are asked to solve a linear equation in one variable.
Solve for N: Students are asked to solve a linear equation in one variable with fractional coefficients.
Solve for M: Students are asked to solve a linear equation in one variable.
Complete the Square - 1: Students are asked to solve a quadratic equation by completing the square.
Complete the Square - 2: Students are asked to solve a quadratic equation by completing the square.
Complete the Square - 3: Students are asked to solve a quadratic equation by completing the square.
Complex Solutions?: Students are asked to explain how to recognize when the quadratic formula results in complex solutions.
Which Strategy?: Students are shown four quadratic equations and asked to choose the best method for solving each equation.

Lesson Plans

Solving Quadratic Equations by Completing the square: Students will model the process of completing the square (leading coefficient of 1) with algebra tiles, and then practice solving equations using the completing the square method. This lesson provides a discovery opportunity to conceptually see why the process of squaring half of the b value is considered completing the square.
Solving Quadratics - Exploring Different Methods: Students will explore how different methods find the solutions (roots) to quadratic equations including, factoring, graphing, and the quadratic formula.
Ranking Sports Players (Quadratic Equations Practice): In this Model Eliciting Activity, MEA, students will rank sports players by designing methods, using different indicators, and working with quadratic equations.

Model-Eliciting-Activities, MEAs, allow students to critically analyze data sets, compare information, and require students to explain their thinking and reasoning. While there is no one correct answer in an MEA, students should work to explain their thinking clearly and rationally. Therefore, teachers should ask probing questions and provide feedback to help students develop a coherent, data-as-evidence-based approach within this learning experience.
Wine Glass Lab: Resonance and the Wave Equation: This activity is designed to help students understand the concept of resonance through the application of the wave equation to sounds produced by a singing wine glass.
Sorting Equations and Identities: This lesson is intended to help you assess how well students are able to:
- Recognize the differences between equations and identities.
- Substitute numbers into algebraic statements in order to test their validity in special cases.
- Resist common errors when manipulating expressions such as 2(x – 3) = 2x – 3; (x + 3)² = x² + 3².
- Carry out correct algebraic manipulations.
It also aims to encourage discussion on some common misconceptions about algebra.
Solving Quadratic Equations: Cutting Corners: Assess how well students can apply and solve quadratics in one variable. In particular, identify and help students who have difficulties with making sense of a real-life situation and deciding on the math to apply to the problem. Students will be solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and/or factoring. They will also analyze sample responses and interpret results in the context of the situation.
Method to My Mathness: In this lesson, students will complete proof tables to justify the steps taken to solve multi-step equations. Justifications include mathematical properties and definitions..
The Quadratic Quandary: Students will sort various quadratic equations by the method they would use for solving (ie. factoring, quadratic formula). Then as a class they justify their placements and eventually discover that there are many ways to solve and that some make sense in different situations, however there is no real "correct" method for each equation type.

Problem-Solving Tasks

Braking Distance: This task provides an exploration of a quadratic equation by descriptive, numerical, graphical, and algebraic techniques. Based on its real-world applicability, teachers could use the task as a way to introduce and motivate algebraic techniques like completing the square, en route to a derivation of the quadratic formula.
Two Squares are Equal: This classroom task is designed to elicit a variety of different methods of solving a quadratic equation. Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation. In contrast, others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.
Springboard Dive: The problem presents a context where a quadratic function arises. Careful analysis, including graphing of the function, is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.

Tutorials

Solving Quadratic Equations Using the Quadratic Formula: You will learn in this video how to solve Quadratic Equations using the Quadratic Formula.
Learning How to Complete the Square: You will learn int his video how to solve the Quadratic Equation by Completing the Square.
Solving Quadratic Equations by Square Roots: In this video tutorial students will learn how to solve quadratic equations by square roots.
Multiplying And Dividing With Inequalities: This video discusses multiplication and division of inequalities with negative numbers to solve the inequality.