# All Lessons

## Solve Equations with Unknown Coefficients

(A) Solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides

A1.12.E(E) Solve mathematic and scientific formulas, and other literal equations, for a specified variable.

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## Add, Subtract, and Multiply Polynomials

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## Simplify Expressions with Roots

(10) Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions.

A1.10.A(A) Add and subtract polynomials of degree one and degree two

A1.10.B(B) Multiply polynomials of degree one and degree two

A1.10.C(C) Determine the quotient of a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend

A1.10.D(D) Rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property

A1.10.E(E) Factor, if possible, trinomials with real factors in the form ax2 + bx + c, including perfect square trinomials of degree two

A1.10.F(F) Decide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial.

A1.11(11) Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms.

A1.11.A(A) Simplify numerical radical expressions involving square roots

A1.11.B(B) Simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents.

8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

HSN.RN.1HSN.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^3 = (5^(1/3))^3 to hold, so (5^(1/3))^3 must equal 5.

HSN.RN.2HSN.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

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## Solve Multi-Step Linear Inequalities

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## Factor Quadratic Expressions - Part 1

(8) Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data.

A1.8.A(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

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## Factor Quadratic Expressions - Part 2

(8) Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data.

A1.8.A(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

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## Solve Exponential Equations

(C) Write exponential functions in the form f(x) = abx (where b is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay

A2.5.D(D) Solve exponential equations of the form y = abx where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions

PCAL.5.I(I) Generate and solve exponential equations in mathematical and real-world problems

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## Solve Quadratic Equations by Completing the Square

HSA.REI.4.A 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.

HSA.REI.4.BHSA.REI.4.B Solve quadratic equations by inspection (e.g., for x^2 = 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.

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## Graph Linear Functions by Plotting Ordered Pairs

(C) Contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation

A1.3.C(C) Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

HSA.REI.10HSA.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

HSF.IF.7.AHSF.IF.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima.

This expressions and equations lesson teaches students how to graph linear functions by plotting ordered pairs. The lesson includes research-based strategies and strategic questions that prepare students for assessments. In this lesson, students will plot ordered pairs from a linear function on the coordinate plane.

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## Identify Quadratic Function Graphs

(7) Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations.

A1.7.A(A) Graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry

A1.7.B(B) Describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions

A1.7.C(C) Determine the effects on the graph of the parent function f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d.

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## Identify and Graph Transformed Functions

(E) Determine the effects on the graph of the parent function f(x) = x when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d

A1.7.C(C) Determine the effects on the graph of the parent function f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d.

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## Find Zeros of Quadratic Functions by Factoring

(7) Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations.

A1.7.B(B) Describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions

HSF.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

HSF.IF.8.AHSF.IF.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

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## Identify Key Points from Quadratic Function Graphs

(7) Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations.

A1.7.A(A) Graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry

A1.7.B(B) Describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions

A1.7.C(C) Determine the effects on the graph of the parent function f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d.

A1.8(8) Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data.

A1.8.A(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

HSF.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

HSF.IF.8.AHSF.IF.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

HSF.IF.8.BHSF.IF.8.B Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)ᵗ, y = (0.97)ᵗ, y = (1.01)12ᵗ, y = (1.2)ᵗ/10, and classify them as representing exponential growth or decay.