Word (.docx)
Acrobat (.pdf)
- 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.
- 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.
Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.
- Assessment Limits :
In items that require the student to transform a quadratic equation to
vertex form, b/a must be an even integer.In items that require the student to solve a simple quadratic equation
by inspection or by taking square roots, equations should be in the
form ax² = c or ax² + d = c, where a, c, and d are rational numbers and
where c is not an integer that is a perfect square and c – d is not an
integer that is a perfect square.In items that allow the student to choose the method for solving a
quadratic equation, equations should be in the form ax² + bx + c = d,
where a, b, c, and d are integers.Items may require the student to recognize that a solution is nonreal
but should not require the student to find a nonreal solution. - Calculator :
Neutral
- Clarification :
Students will rewrite a quadratic equation in vertex form by completing the square.Students will use the vertex form of a quadratic equation to complete steps in the derivation of the quadratic formula.
Students will solve a simple quadratic equation by inspection or by taking square roots.
Students will solve a quadratic equation by choosing an appropriate method (i.e., completing the square, the quadratic formula, or factoring).
Students will validate why taking the square root of both sides when solving a quadratic equation will yield two solutions.
Students will recognize that the quadratic formula can be used to find complex solutions.
- Stimulus Attributes :
The formula must be given in the item for items that can only be solved using the quadratic formula.Items should be set in a mathematical context.
Items may use function notation.
- Response Attributes :
Items may require the student to complete a missing step in the
derivation of the quadratic formula.Items may require the student to provide an answer in the form
(x – p)² = q.Items may require the student to recognize equivalent solutions to
the quadratic equation.Responses with square roots should require the student to rewrite
the square root so that the radicand has no square factors.
- Test Item #: Sample Item 1
- Question:
Matthew solved the quadratic equation shown.
4x²-24x+7=3
One of the steps that Matthew used to solve the equation is shown.
Drag the values into the boxes to complete the step and the solution.
- Difficulty: N/A
- Type: GRID: Graphic Response Item Display
- Test Item #: Sample Item 2
- Question:
An equation is shown.
3x²+7x=1
The formula
can be used to solve the equation.Click on the blank to enter a numeric expression that is one solution to the given equation.
- Difficulty: N/A
- Type: EE: Equation Editor
Related Courses
Related Access Points
Related Resources
Educational Games
Formative Assessments
Lesson Plans
Original Student Tutorials
Problem-Solving Tasks
Tutorials
Unit/Lesson Sequence
Virtual Manipulative
Worksheet
STEM Lessons - Model Eliciting Activity
Students will rank sports players designing methods, using different indicators, and working with quadratic equations.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
MFAS Formative Assessments
Students are asked to explain how to recognize when the quadratic formula results in complex solutions.
Students are shown four quadratic equations and asked to choose the best method for solving each equation.
Original Student Tutorials Mathematics - Grades 9-12
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.
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.
Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.
Learn how to solve rational functions by getting common denominators in this interactive tutorial.
Student Resources
Original Student Tutorials
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.
Type: Original Student Tutorial
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.
Type: Original Student Tutorial
Learn how to solve rational functions by getting common denominators in this interactive tutorial.
Type: Original Student Tutorial
Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.
Type: Original Student Tutorial
Educational Games
In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Educational Game
In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.
Type: Educational Game
Problem-Solving Tasks
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.
Type: Problem-Solving Task
This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). 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, while 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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
Tutorials
You will learn in this video how to solve Quadratic Equations using the Quadratic Formula.
Type: Tutorial
You will learn int his video how to solve the Quadratic Equation by Completing the Square.
Type: Tutorial
In this video tutorial students will learn how to solve quadratic equations by square roots.
Type: Tutorial
Virtual Manipulative
This resource can be used to assess students' understanding of solving quadratic equation by taking the square root. A great resource to view prior to this is "Solving quadratic equations by square root' by Khan Academy.
Type: Virtual Manipulative
Parent Resources
Problem-Solving Tasks
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.
Type: Problem-Solving Task
This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). 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, while 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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
Virtual Manipulative
This resource can be used to assess students' understanding of solving quadratic equation by taking the square root. A great resource to view prior to this is "Solving quadratic equations by square root' by Khan Academy.
Type: Virtual Manipulative
