M$5 Course Outline



The Bronx High School of Science Mathematics Department

Valerie Reidy, Principal Rosemarie Jahoda, Assistant Principal

M$5 Course Outline

Unit 1

1. Properties of Real Numbers

a. Be able to define and identify

-the commutative property

-the associative property

-the identity element under addition (the additive identity)

-the identity element under multiplication (the multiplicative identity)

-an element’s inverse under addition (the additive inverse of an element)

-an element’s inverse under multiplication (the multiplicative inverse of an element)

2. Linear Equations

a. Know how to identify a linear equation

b. Know how to solve a linear equation

3. Linear Inequalities

a. Know how to solve a linear inequality

b. Know how to graph your solution to a linear inequality on a number line

c. Multiplying or dividing by a negative number will flip the direction of the inequality sign.

4. Compound Inequalities

a. Know how to solve a compound inequality (both conjunctive and disjunctive)

← Conjunctive:

← Disjunctive:

b. Be able to identify an inequality as conjunctive or disjunctive.

c. Know how to graph your solution to a compound inequality on a number line

Practice:

5. Absolute Value Equations and Inequalities

a. Know how to solve an absolute value equation:

b. Know how to solve an absolute value inequality:

c. Know how to graph your solution to an absolute value inequality:

d. Pages 393-396 in the red book

Unit 2

6. Laws of Exponents

a. Know all the properties of exponents (See “Properties of Exponents” and “Properties of Exponents 2” on the website. Powerpoint is required.

← Website:

b. Be able to solve any problem involving exponents

c. Pages 752-759 in the red book

Practice:

7. Operations with Polynomials

a. Know how to add and subtract polynomials (combine like terms)

b. Know how to multiply polynomials (distribution)

c. Know how to divide polynomials (factor, then divide)

8. Factoring Polynomials

a. Know how to factor polynomials with a = 1

b. Know how to factor polynomials with a =/= 1 (see “Factoring Polynomials” on the website)

← Website:

c. Pages 333-338 in the red book

9. Factor by Grouping

a. Know how to factor by grouping (involved in 8b above):

10. Quadratic Equations

a. Know how to solve by factoring.

b. Know how to solve by using the quadratic formula.

c. Pages 422-423

11. The Quadratic Formula

a. Know it. End of story.

Unit 3

12. Quadratic Inequalities

a. Be able to solve a quadratic inequality:

←

←

b. Pages 966-970 in the red book

13. Multiplying and Dividing Rational Expressions

a. Know how to simplify the product of two rational expressions (Factor, cancel)

b. Know how to simplify the quotient of two rational expressions (Multiply by the inverse of the denominator, factor, cancel)

← See “Dividing Rational Expressions” on the website

c. Pages 349-353 in the red book

14. Sums and Differences of Rational Expressions

a. Be able to add/substract two rational expressions (get common denominators, add/subtract your numerators while remembering to distribute your subtraction sign, then factor and cancel)

b. Pages 355-359 in the red book

15. Complex Fractions

a. be able to simplify a complex fraction (get the expression to be a single fraction divided by a single fraction, then multiply by the inverse of the denominator).

b. Pages 364-365 in the red book

16. Solving Rational Equations

a. Be able to solve rational equations (get common denominators for EVERY term in the problem, then eliminate the denominators. Solve the remaining equation.)

b. Be able to solve work problems

← Practice: See “Work Problems” on the website

← See “Work Problems Presentation” on the website

← Website:

c. Pages 368-375 in the red book

Unit 4

17. Simplifying Radical Expressions

a. Define and identify a “principle square root”

b.

c.

d. Pages 402-406 in the red book

18. Sums of Radicals:

a. Know how to combine like terms involving radicals

b. Pages 407-409 in the red book

19. Multiplying and Dividing Radicals

a. Remember that we can’t have radicals in the denominator, so we must multiply both the numerator and the denominator by the conjugate of the denominator (Rationalizing the denominator)

b. Pages 411-421 in the red book

20. Products of Binomials Containing Radicals

a. Pages 412-413 in the red book

21. Equations Containing Radicals:

a. Know how to solve an equation involving a radical

b. You must check your answer. It is an essential part of how to solve these problems.

c. Pages 427-429 in the red book

Unit 5

22. The Imaginary Unit:

a. Be able to define the imaginary unit.

b. Simplify the imaginary unit raised to a power. (ex. i879= ?)

←

c. Page 921-925 in the red book

23. The Complex Numbers

a. Know how to write any number as a complex number (a + bi form)

b. Pages 925-931 in the red book

24. Multiplying and Dividing Complex Numbers:

a. Be able to multiply any two complex numbers (Remembering that i2 = -1)

b. Be able to divide any two complex numbers (Multiply both the numerator and denominator by the conjugate of the denominator, we can’t have i in the denominator)

c. Be able to find the multiplicative inverse of a complex number.

d. Pages 931-941 in the red book

25. Solving Quadratic Equations & Absolute Value of Complex Numbers

a. Be able to represent a complex number as a vector

b. Be able to find the absolute value of a complex number (find the magnitude, M, of the vector representing the complex number: [pic])

c. Be able to solve a quadratic equation and express the solutions as a complex number, in a + bi form.

d. Pages 944-947 in the red book

26. Nature of the Roots:

a. Be able to describe the roots of a quadratic equation

b. Be able to find the discriminant of a quadratic equation: b2-4ac

c. PLEASE NOTE: THIS WEBSITE SAYS THERE ARE 3 CASES, BUT THERE ARE 4. READ THE FIRST CASE CAREFULLY TO SEE WHY THERE ARE ACTUALLY 4 (they combine two).

d. Pages 947-954 in the red book

27. Writing Quadratic Equations Given the Roots

a. Be able to write a quadratic equation by using the sum and product of the roots.

b. Remember that traditionally, polynomials are written with integer (not fractional) coefficients.

c. If a radical or complex number is a root of an equation, its conjugate is also going to be a root.

d. Pages 954-961 in the red book

28. Solving for a Coefficient, Constant, or Root of a Quadratic Equation

a. Given a root (or the roots) of an equation, be able to identify each of the coefficients.

b. Pages 954-961 in the red book

Unit 6

29. Relations

a. Remember that a relation is ANY set of ordered points.

b. Pages 482-487 in the red book

30. Functions

a. Be able to identify if a set of ordered points/graph/equation is a function.

←

←

b. Be able to define a function.

c. Pages 487-495 in the red book

31. Domain of a Function:

a. Be able to state the domain of a function (or more accurately, the domain that makes a relation a function)

b. Pages 490-491 in the red book

32. Function Notation

33. Applications of Linear Functions

34. Operations with Functions

a. Know how to add, subtract, multiply, and divide functions

b. Know the notation used for each of the above operations

35. Composing Functions:

a. Given two equations f(x) and g(x), be able to write their compositions f(g(x)) and g(f(x)).

b. The composition of a function and its inverse function is the identity function, f(x) = x

c. Pages 542-554 in the red book

36. Determining the Inverse of a Function

a. Pages 548-554 in the red book

37. Quadratic Functions

a. Given a question about projectile motion (e.g. “a ball is throw into the air…” etc), be able to find the maximum/minimum algebraically and on your calculator, as well as the roots of the equation

b. Be able to interpret what these numbers mean (e.g. if a maximum is at (2, 5), the maximum height attained by the object is 5 units).

c. Practice:

Unit 7

38. Review of Transformations

39. Rotations and Rotational Symmetry

40. Compositions of Transformations

41. Glide Reflections and Isometry

For 38-41 above, be able to apply each transformation to a point or set of points (like a triangle). FOCUS ON KNOWING THE NOTATION!

Remember: A rotation around the origin can be written as [pic] or [pic] (if we were rotating 180 degrees, the number would change based on the problem)

Pages 559-602 in the red book

42. Transforming Quadratic and Absolute Value Functions

a. Pages 514-522 in the red book

43. Transforming Circles

a. Pages 523-528 in the red book

Unit 8

44. Ellipse

a. Pages 523-528 in the red book

45. Hyperbola

a. Pages 528-533 in the red book

For 42-45, review how to write the equations for those shapes, and how changing the equation changes where the shape could be found on the xy coordinate plane.

If we are asked to analyze a conic section based on the graph, know how to identify and state the critical points, such as the vertices and the center.

46. Inverse Variation:

a. Be able to define find the constant of variation, and apply it to various situations

b. Use the equation xy=k

c. On the website given here, note that they have always solved for y. We wrote the equation differently in class, but it is the same basic equation.

d. Pages 539-542 in the red book

47. Solving Systems of Conics

a. Use the method of substitution to eliminate one variable, then solve for the other.

b. Use your solution for the first variable to find the second variable. You may find up to 4 answers for a single system of conic equations problem.

c. Pages 962-964 in the red book

Good luck! (Don’t be Paige, as seen below):

[pic]

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