Unit 3 Number Systems and Operations

Lesson 1

Learning Focus

Explore the meaning of a fraction as an exponent.

Lesson Summary

In this lesson, we developed meaning for using a fraction as an exponent. We began with a geometric sequence and found a way to fit data at the points halfway between whole number inputs in a way that would maintain the multiplicative behavior of the sequence. We continued to work with different fractional inputs and gave meaning to using a variety of fractions as exponents.

Lesson 2

Learning Focus

Relate the key features of exponential functions to properties of negative exponents.

Rewrite exponential expressions that involve negative exponents.

Lesson Summary

In this lesson, we noticed several characteristics of the graphs and tables of exponential functions that can be explained using our understanding of negative exponents. We also used the rules of exponents to change the form of numeric expressions that contain negative exponents.

Lesson 3

Learning Focus

Examine how the properties of exponents work with rational exponents.

Write equivalent exponential functions using different growth factors.

Lesson Summary

In this lesson, we continued to explore the meaning of rational exponents, including negative integer exponents and fractional exponents. We learned that the properties of exponents can be applied to all rational exponents, not just integer exponents.

Lesson 4

Learning Focus

Change the form of radical expressions using properties of exponents.

Lesson Summary

In this lesson, we learned how to change the form of complicated radical and exponential expressions using the properties of radicals and exponents. Strategies for changing the form of radical expressions can be explained by converting the radical expressions to exponential form.

Lesson 5

Learning Focus

Divide polynomials.

Write equivalent multiplication statements after dividing.

Know when one polynomial is a factor of another polynomial.

Lesson Summary

In this lesson, we learned that polynomials can be divided using long division like whole numbers. We learned to use technology to check our work and to avoid errors in subtraction by adding the opposite of the terms to be subtracted. We found that, like numbers, a polynomial is a factor of another polynomial if it divides with no remainder. We learned two ways to write equivalent multiplication statements when there was a remainder after dividing.

Lesson 6

Learning Focus

Write quadratic functions in vertex, factored, and standard form.

Find roots of a quadratic function.

Use the roots of a quadratic function to write the function in factored form.

Lesson Summary

In this lesson, we examined solutions to quadratic equations and connected them with the graph of the function. Solutions for a quadratic equation can be used to write the function in factored form in a process that is the reverse of solving an equation by factoring. We found that when the graph of the quadratic function did not cross the $x$ -axis, the quadratic formula gave solutions that included the square root of a negative number. Although there is not a real number that can be squared to get a negative number, these expressions seemed to work like square roots. Since the Fundamental Theorem of Algebra predicts that all quadratic functions will have two roots, the nature of solutions that involve the square root of a negative number still needs to be resolved.

Lesson 7

Learning Focus

Relate irrational numbers to physical quantities such as the hypotenuse of a right triangle.

Understand expressions that contain negative numbers inside a square root, like $\sqrt{- 1}$ .

Add, subtract, and multiply complex numbers.

Lesson Summary

In this lesson, we connected irrational numbers to the measure of geometric figures and showed where a given irrational number is located on the number line. We found irrational solutions of quadratic equations and used the solutions to write the equation in factored form. We also learned of a new set of numbers defined in terms of $\sqrt{- 1} = i$ and $i^{2} = - 1$ . We examined how these numbers fit into the number system and performed arithmetic operations on them.

Lesson 8

Learning Focus

Solve quadratic inequalities both graphically and algebraically.

Interpret solutions to quadratic inequalities that arise from context.

Solve a system of equations that contains both a quadratic and linear equation.

Lesson Summary

In this lesson, we developed a strategy for solving quadratic inequalities. The procedure involves solving the related quadratic equation and then using the graph or testing values to find the intervals that are solutions to the inequality. If the inequality represents a real context, the solutions must be interpreted so that they fit the situation.

Lesson 9

Learning Focus

Graph complex numbers in the complex plane.

Use vectors to add, subtract, and multiply complex numbers.

Divide complex numbers.

Find the distance and the midpoint between two complex numbers.

Lesson Summary

In this lesson we wrote formulas and used vectors to justify the basic operations on complex numbers. We represented complex numbers as vectors and points on the complex plane. Vector representation provided a way to examine the size of a complex number, called the modulus. We learned to divide complex numbers, to find the distance between two complex numbers, and the average of two complex numbers.