Unit 3 Number Systems and Operations

Lesson 1

Learning Focus

Add and subtract polynomials algebraically.

Add and subtract polynomials graphically.

Lesson Summary

In this lesson, we learned to add and subtract polynomials. We learned that the procedure used for adding and subtracting is analogous to adding whole numbers because polynomials have the same structure as whole numbers. Polynomials are added by adding like terms. When subtracting polynomials, we can avoid sign errors by adding the opposite of each term.

Lesson 2

Learning Focus

Multiply polynomials.

Raise binomials to powers.

Lesson Summary

In this lesson, we built on our understanding of area models from Algebra 1 to multiply polynomials. We learned to use either the box method or to distribute each term of the first factor to each term of the second factor. Both methods are based on the Distributive Property. We also learned an efficient method for raising binomials to powers using Pascal’s Triangle to help find the coefficient of each term in the expansion.

Lesson 3

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 4

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 5

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 6

Learning Focus

Support or challenge claims about different types of numbers and the result of adding, subtracting, multiplying, and dividing.

Support or challenge claims about the result of adding, subtracting, multiplying, and dividing polynomials.

Lesson Summary

In this lesson, we examined claims about the closure of sets of numbers and classes of functions under the operations of addition, subtraction, multiplication, and division. An example of such a claim is: The set of whole numbers is closed under division. A counterexample that shows this claim to be false is: $2 \div 7 = \frac{2}{7}$ . Since $\frac{2}{7}$ is a rational number, this example shows that dividing two whole numbers does not always result in a whole number.

Lesson 7

Learning Focus

Solve quadratic inequalities both graphically and algebraically.

Interpret solutions to quadratic inequalities that arise from context.

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 8

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.