Section	Teacher-facing learning goals
Exponent Properties	Lesson 1 - Optional Comprehend why the properties of exponents hold for positive integers. Generalize from repeated reasoning to see how exponent properties help us extend the definition of exponents from positive integers to all integers. Lesson 2 - Optional Comprehend the meaning of square roots and cube roots. Determine side lengths of squares and cubes, and express their exact values using radicals. Lesson 3 Explain how to match expressions involving exponents with equivalent expressions. Justify the equivalence of \(\begin{array}{l}b^{1/n}\\\end{array}\) and \(\begin{array}{l}\sqrt[n]b\\\end{array}\) using the properties of exponents. Lesson 4 Justify the equivalence of \(b^{m/n}\) and \(\sqrt[n]{b^m}\) using the properties of exponents. Use graphs to estimate the value of expressions involving positive rational exponents. Lesson 5 Draw a graph representing negative rational exponents and use it to estimate values. Use both radicals and exponents to represent numbers.
Solving Equations with Square and Cube Roots	Lesson 6 Comprehend that the symbol \(\sqrt{x}\) denotes only the positive square root of \(x\). Explain why equations of the form \(x^2 = a\) can have two solutions, while equations of the form\(\sqrt{x} = a\) cannot have more than one solution. Lesson 7 Explain why squaring each side of an equation, or taking the square root of each side, may result in an inequivalent equation. Use graphs to represent solutions to equations involving squares and square roots. Lesson 8 Calculate solutions to equations involving cubes and cube roots and explain the solution method used. Compare and contrast the processes of solving equations with square roots and with cube roots. Justify using a graph that every number, positive or negative, has exactly one cube root. Lesson 9 - Optional Calculate solutions to radical equations. Create radical equations that have either zero, one, or two solutions. Explain the steps taken to solve radical equations.
A New Kind of Number	Lesson 10 Justify that \(x^2=\text-1\) has no real solutions by using a graph of \(y=x^2\). Represent imaginary numbers visually with the imaginary number line. Lesson 11 Comprehend that the symbol "\(i\)” means the positive square root of -1 and that an imaginary number is a real number times \(i\). Represent solutions to equations using \(i\) and the complex plane. Lesson 12 Determine powers of imaginary numbers and represent them on the complex plane. Determine the result of adding complex numbers, and represent it in the form \(a+bi\), where \(a\) and \(b\) are real numbers. Use the complex plane to visualize why the sum of two complex numbers is another complex number. Lesson 13 Calculate the result of multiplying two complex numbers. Justify that two equivalent expressions involving complex numbers are equivalent, using the fact that \(i^2=\text-1\). Lesson 14 - Optional Add, subtract, and multiply complex numbers, and represent the solutions in the form \(a+bi\). Explain reasoning and critique the reasoning of others when writing numbers in the form \(a+bi\). Generalize patterns in repeated reasoning to show what happens when ii is raised to different powers. Lesson 15 Determine what information is needed to find the complex numbers that were multiplied to get a certain result, and ask questions to elicit that information.
Solving Quadratics with Complex Numbers	Lesson 16 - Optional Explain (in writing) the reasoning used to solve a quadratic equation by completing the square and by using the quadratic formula. Justify whether two radical expressions are equivalent. Solve quadratic equations and explain the solution method. Lesson 17 Calculate complex solutions to quadratic equations by completing the square. Compare and contrast quadratic equations with real and with non-real complex solutions. Explain how to solve equations that have non-real complex solutions. Lesson 18 Calculate complex solutions using the quadratic formula and represent them in the form \(c \pm di\), where \(c\) and \(d\) are real numbers. Explain reasoning and critique the reasoning of others when solving quadratic equations. Explain when solutions are non-real when using the quadratic formula. Lesson 19 - Optional Create quadratic equations that will have either real or non-real solutions, then calculate the specific solutions. Justify that a quadratic equation will have either real or non-real solutions using graphical reasoning.

Alg2.3: Complex Numbers and Rational Exponents