Section	Teacher-facing learning goals
Finding Unknown Inputs	Lesson 1 Explain (orally and in writing) the meaning of the solution to a quadratic equation in terms of a situation. Write a quadratic equation that represents geometric constraints. Lesson 2 Recognize the limitations of certain strategies used to solve a quadratic equation. Understand that the factored form of a quadratic expression can help us find the zeros of a quadratic function and solve a quadratic equation. Write quadratic equations and reason about their solutions in terms of a situation.
Solving Quadratic Equations	Lesson 3 Find the solutions to simple quadratic equations and justify (orally) the reasoning that leads to the solutions. Understand that a quadratic equation may have two solutions. Lesson 4 Given quadratic equations where one side is a product of factors and the other is zero, find the solution(s) and explain (orally and in writing) why the solutions make the equation true. Understand that the “zero product property” (in written and spoken language) means that if the product of two numbers is 0 then one of the factors must also be 0. Lesson 5 Coordinate (orally) graphs with no horizontal intercepts, quadratic functions with no (real) zeros, and quadratic equations with no (real) solutions. Describe (orally and in writing) the relationship between the solutions to quadratic equations of the form \(\text {expression}=0\) and the horizontal intercepts of the graph of the related function. Explain (orally and in writing) why dividing each side of a quadratic equation by a variable is not a reliable way to solve the equation. Lesson 6 Apply the distributive property to multiply two sums or two differences, using a rectangular diagram to illustrate the distribution as needed. Generalize the relationship between equivalent quadratic expressions in standard form and factored form, and use the generalization to transform expressions from one form to the other. Use a diagram to represent quadratic expressions in different forms and explain (orally and in writing) how the numbers in the factors relate to the numbers in the product. Lesson 7 Apply the distributive property to multiply a sum and a difference, using a diagram to illustrate the distribution as needed. Given a quadratic expression of the form \(x^2+bx+c\), where \(c\) is negative, write an equivalent expression in factored form. When multiplying a sum and a difference, explain (orally and in writing) how the numbers and signs of the factors relate to the numbers in the product. Lesson 8 Understand that multiplying a sum and a difference, \((x+m)(x-m)\), results in a quadratic with no linear term and explain (orally) why this is the case. When given quadratic expressions with no linear term, write equivalent expressions in factored form. Lesson 9 Recognize that the number of solutions to a quadratic equation can be revealed when the equation is written as \(\text {expression in factored form} = 0\). Use factored form and the zero product property to solve quadratic equation Lesson 10 Given a quadratic expression of the form \(ax^2+bx+c\), where aa is not 1, write an equivalent expression in factored form. Write a quadratic equation that represents a context, consider different methods for solving it, and describe (orally) the limitations of each method.
Completing the Square	Lesson 11 Comprehend that equations containing a perfect-square expression on both sides of the equal sign can be solved by finding square roots. Comprehend that perfect squares of the form \((x+n)^2\) are equivalent to \(x^2+2nx+n^2\). Use the structure of expressions to identify them as perfect squares. Lesson 12 Comprehend that to “complete the square” is to determine the value of \(c\) that will make the expression \(x^2+bx+c\) a perfect square. Describe (orally and in writing) how to complete the square. Solve quadratic equations of the form \(x^2+bx+c=0\) by rearranging terms and completing the square. Lesson 13 Express any quadratic equation in the form \((x+p)^2=q\) and solve the equation by finding square roots. Generalize (orally) a process for completing the square to express any quadratic equation in the form \((x+p)^2=q\). Solve quadratic equations in which the squared term has a coefficient of 1 by completing the square. Lesson 14 Generalize (orally) a process for completing the square to express any quadratic equation in the form \((kx+m)^2=q\). Solve quadratic equations in which the squared term has a coefficient other than 1 by completing the square. Lesson 15 Coordinate and compare (orally and in writing) solutions to quadratic equations obtained by completing the square and those obtained by graphing. Understand that the “plus-minus” symbol is used to represent both square roots of a number and that the square root notation expresses only the positive square root. Use radical and “plus-minus” symbols to express solutions to quadratic equations.
The Quadratic Formula	Lesson 16 Recognize that the solutions obtained using the quadratic formula are the same as those found by using factored form or by completing the square. Use the quadratic formula to solve quadratic equations of the form \(ax^2+bx+c=0\). Lesson 17 Interpret (orally and in writing) the solutions to quadratic equations in context. Practice using the quadratic formula to solve quadratic equations, rearranging the equations into \(ax^2+bx+c=0\) if not already given in this form. Lesson 18 Analyze and critique (orally and in writing) solutions to quadratic equations that are found using the quadratic formula. Determine whether a given value is a solution to a quadratic equation. Lesson 19 Explain (in writing) the steps used to derive the quadratic formula. Explain (orally and in writing) how the solutions obtained by completing the square are expressed by the quadratic formula. Understand that the quadratic formula can be derived by generalizing the process of completing the square. Lesson 20 Explain (orally and in writing) why the product of a non-zero rational and irrational number is irrational. Explain (orally and in writing) why the sum of a rational and irrational number is irrational. Explain (orally and in writing) why the sum or product of two rational numbers is rational. Lesson 21 Explain (orally and in writing) why the product of a non-zero rational and irrational number is irrational. Explain (orally and in writing) why the sum of a rational and irrational number is irrational. Explain (orally and in writing) why the sum or product of two rational numbers is rational.
Vertex Form Revisited	Lesson 22 Analyze and explain (orally and in writing) the steps for completing the square and understand how they transform a quadratic expression from standard to vertex form. Identify the vertex of a graph of a quadratic function when the expression that defines it is in vertex form. Write equivalent quadratic expressions in vertex form by completing the square. Lesson 23 Explain (orally and in writing) how an expression in vertex form can show whether the vertex of a graph represents the maximum or minimum of a quadratic function. Rewrite a quadratic expression in vertex form to identify the maximum or minimum value of the function the expression defines. Use the structure of a quadratic expression in vertex form to determine whether the vertex of its graph represents the maximum or minimum of the quadratic function.
Putting It All Together	Lesson 24 Choose and write the appropriate form for expressing a quadratic function to solve a problem. Interpret features of graphs and expressions that represent quadratic functions to gain information about the situations being modeled.

Alg1.7: Quadratic Equations