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Polynomials, Expressions, and Formulas

Polynomials, Expressions, and Formulas. Or, How I Learned to Stop Worrying and Love the Variable. Overview.

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Polynomials, Expressions, and Formulas

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  1. Polynomials, Expressions, and Formulas Or, How I Learned to Stop Worrying and Love the Variable

  2. Overview We begin this module with a study of numbers and the basic operations of addition, subtraction, multiplication, and division. The algebraic properties of these operations are introduced along with the conventions for the order in which the operations are performed in mathematical expressions. Lessons are included on numerical and variable expressions, as well as on writing, evaluating, and simplifying these expressions. These concepts are extended to polynomials, with lessons on simplifying, adding, subtracting, multiplying, dividing, and factoring polynomials. There are also lessons on more general formulas, including evaluating and rewriting formulas. On the following Topics slide there is a link to a table of contents for each topic. Clicking on a link on a Contents slide will take you to the first lesson slide for that topic. In many cases, there will be more than one slide on that topic. Work through as many as you need to master the topic, and then go on to the next topic. Clicking on the heading for a slide will take you back to the Contents slide for that topic; clicking on Return to Topics List on the Contents slide will take you back to the main Topics page.

  3. Topics Sets of Real Numbers Properties of Operations Order of Operations Numerical Expressions Variables and Expressions Writing Variable Expressions Evaluating Variable Expressions Simplifying Variable Expressions Polynomial Expressions Polynomial Multiplication Polynomial Division Factoring Polynomials Formulas & Polynomials in Two (or more) Variables Polynomial Functions

  4. CONTENTS: Sets of Real Numbers • Description and examples of subsets of the set of real numbers, including the rational numbers and irrational numbers • Subsets of the Real Numbers • Recognizing Irrational and Rational Numbers • Interactive Practice Problems: Recognizing Rational and Irrational Numbers Return to Topics List

  5. Sets of Real Numbers Link: Number Sets • Khan Academy video giving definitions of the various subsets of the real numbers • Subsets include natural numbers, whole numbers, integers, rational numbers, and irrational numbers, with examples of numbers in each subset • Time 5:17

  6. Sets of Real Numbers Link: Real Numbers • Mathispower4u video on sets of real numbers • Describes various sets of real numbers, size comparison of real numbers, and absolute values • Time 9:07

  7. Sets of Real Numbers Link: Recognizing Irrational Numbers • Khan Academy video explaining the difference between rational and irrational numbers • Time 2:18

  8. Sets of Real Numbers Link: Recognizing Rational and Irrational Numbers • Khan Academy interactive practice problems on recognizing rational and irrational numbers • A sequence of five practice problems is provided, along with • help with correct formats to enter answers • hints to help solve the problem, if needed • links to video lessons if you get stuck

  9. CONTENTS: Properties of Operations • Descriptions and examples of properties of operations on numbers, including commutative, associative, and distributive laws; identities and inverses for multiplication and addition • Terminology of Properties of Numbers • Commutative Property of Addition • Commutative Property of Multiplication • Associative Law of Addition • Associative Law of Multiplication • Identity Properties of 0 and 1 • Inverse Properties of Addition and Multiplication • The Distributive Property • Interactive Practice Problems: Distributive Property Return to Topics List

  10. Properties of Operations Link: Properties of Numbers 1 • Khan Academy video examples recognizing the associative and commutative properties of arithmetic • Time 2:03

  11. Properties of Operations Link: Number Properties Terminology 1 • Khan Academy video examples recognizing the distributive, associative, and commutative properties of arithmetic • Time 3:05

  12. Properties of Operations Link: Properties of Numbers 2 • Khan Academy video examples recognizing the properties of arithmetic • Time 1:52

  13. Properties of Operations Link: Commutative Law of Addition • Khan Academy video example applying the commutative and associative laws of addition to rewrite a number expression • Time 1:40

  14. Properties of Operations Link: Commutative Property for Addition • Khan Academy video example applying the commutative and associative laws of addition to rewrite a number expression • Time 3:23

  15. Properties of Operations Link: Commutative Property of Multiplication • Khan Academy video example applying the commutative law of multiplication to rewrite a number expression • Time 1:48

  16. Properties of Operations Link: Associative Law of Addition • Khan Academy video example applying the associative law of addition to rewrite a number expression • Time 2:10

  17. Properties of Operations Link: Associative Law of Multiplication • Khan Academy video example applying the associative law of multiplication to rewrite a number expression • Time 2:15

  18. Properties of Operations Link: Identity Property of 0 • Khan Academy video examples demonstrating the additive identity property of 0 • Uses the method of the number line and absolute values when adding a zero in a numerical expression with more than one operation • Time 4:46

  19. Properties of Operations Link: Identity Property of 1 • Khan Academy video examples demonstrating the multiplicative identity property of 1, explaining that you always get the original term number as the answer • Time 1:24

  20. Properties of Operations Link: Identity Property of 1 (Second Example) • Khan Academy video with more examples demonstrating the multiplicative identity property of 1 • Time 1:55

  21. Properties of Operations Link: Inverse Property of Addition • Khan Academy video examples of using the additive inverse to move from a given number back to zero • The number line method is used to visualize the impact of adding the opposite of the given number • Time 2:59

  22. Properties of Operations Link: Inverse Property of Multiplication • Khan Academy video examples demonstrating the multiplicative inverse property • Uses the operation of division and builds to multiplication by the reciprocal • Time 3:16

  23. Properties of Operations Link: The Distributive Property • Khan Academy video example demonstrating the distributive property of multiplication over addition, with an explanation of why the distributive property works • Time 4:56

  24. Properties of Operations Link: The Distributive Property 2 • Khan Academy video example demonstrating the distributive property of multiplication over subtraction • Time 2:11

  25. Properties of Operations Link: The Distributive Property Example 1 • Khan Academy video examples demonstrating the distributive property in several algebraic expressions • Examples require combining like terms and distributing fractions • Time 5:39

  26. Properties of Operations Link: Distributive Property Exercise • Khan Academy video examples demonstrating the distributive property of multiplication over addition in numerical expressions • One problem reviews factoring the greatest common divisor and writing a distributive expression • Time 3:03

  27. Properties of Operations Link: The Distributive Property • Math Planet text and video examples demonstrating the distributive property in several situations • Time 1:41

  28. Properties of Operations Link: Distributive Property • Khan Academy interactive practice problems on the distributive property • A sequence of five practice problems is provided, along with • help with correct formats to enter answers • hints to help solve the problem, if needed • links to video lessons if you get stuck

  29. CONTENTS: Order of Operations • Descriptions and examples of the conventions on the order of operations in numerical and variable expressions • Introduction to Order of Operations • Order of Operations Examples • Order of Operations With and Without Parentheses • Interactive Practice Problems: Order of Operations • Interactive Practice Problems: Expressions With Parentheses Return to Topics List

  30. Order of Operations Link: Introduction to Order of Operations • Khan Academy video introducing the order of operations on numbers • Examples show the impact of the agreed upon order for simplifying numerical expressions • Examples also explain operations at the same level and the “left to right” rule • Time 9:40

  31. Order of Operations Link: The Order of Operations • Mathispower4u video introducing the order of operations on numbers • Includes several examples applying the order of operations to simplify expressions • Time 6:31

  32. Order of Operations Link: Order of Operations • Khan Academy video example applying the order of operations on numbers to simplify a numerical expression • Time 4:27

  33. Order of Operations Link: Order of Operations with Exponents • Khan Academy video example applying the order of operations on numbers • Time 4:36

  34. Order of Operations Link: More Complicated Order of Operations Example • Khan Academy video advanced example applying the order of operations to a numerical expression involving parentheses • Time 5:20

  35. Order of Operations Link: Order of Operations Examples • Khan Academy video advanced examples applying the order of operations on numbers • Time 14:23

  36. Order of Operations Link: Evaluating an Expression With and Without Parentheses • Khan Academy video example demonstrating the effect of parentheses on the order of operations of numbers • Time 2:06

  37. Order of Operations Link: Order of Operations • Khan Academy interactive practice problems on order of operations of numbers • A sequence of five practice problems is provided, along with • help with correct formats to enter answers • hints to help solve the problem, if needed • links to video lessons if you get stuck

  38. Order of Operations Link: Order of Operations with Negative Numbers • Khan Academy interactive practice problems on order of operations of numbers • A sequence of five practice problems is provided, along with • help with correct formats to enter answers • hints to help solve the problem, if needed • links to video lessons if you get stuck

  39. Order of Operations Link: Expressions With Parentheses • Khan Academy interactive practice problems on evaluating numerical expressions containing parentheses • A sequence of five practice problems is provided, along with • help with correct formats to enter answers • hints to help solve the problem, if needed • links to video lessons if you get stuck

  40. CONTENTS: Numerical Expressions • Examples and interactive problems on writing numerical expressions from verbal phrases • Constructing Numerical Expressions Example • Interactive Practice Problems: Writing Numerical Expressions Return to Topics List

  41. Numerical Expressions Link: Constructing Numerical Expressions Example • Khan Academy Video example constructing a numerical expression from a verbal phrase • Time 1:22

  42. Numerical Expressions Link: Writing Numerical Expressions with Exponents • Khan Academy interactive practice problems on writing numerical expressions from verbal phrases • A sequence of five practice problems is provided, along with • help with correct formats to enter answers • hints to help solve the problem, if needed • links to video lessons if you get stuck

  43. Numerical Expressions Link: Writing Numerical Expressions with Exponents Word Problems • Khan Academy interactive practice problems on writing numerical expressions with exponents from word problems • A sequence of five practice problems is provided, along with • help with correct formats to enter answers • hints to help solve the problem, if needed • links to video lessons if you get stuck

  44. CONTENTS: Variables and Expressions • Description and examples of variables and expressions • What is a Variable? • Introduction to Variables and Expressions • Notations for Multiplication With Variables Return to Topics List

  45. Variables and Expressions Link: What is a Variable? • Khan Academy video example introducing the concept of a variable • Time 3:18

  46. Variables and Expressions Link: Introduction to Variables and Variable Expressions • Mathispower4u video introducing and explaining the concepts of variables and variable expressions • Also explains the difference between an expression and an equation • Includes examples of • writing variable expressions from verbal phrases • evaluating a variable expression at a number • Time 7:55

  47. Variables and Expressions Link: Why Aren't We Using the Multiplication Sign? • Khan Academy video examples demonstrating the use of variables and various notation for multiplication with variables • Time 4:58

  48. CONTENTS: Writing Variable Expressions • Translating verbal phrases into mathematical expressions • Constructing Algebraic Expressions from Verbal Phrases • Writing an Equation from a Word Problem • Interactive Practice Problems: Writing Simple Variable Expressions • Interactive Practice Problems: Writing Compound Variable Expressions • Interactive Practice Problems: Writing Expressions from Word Problems Return to Topics List

  49. Writing Variable Expressions Link: How Do You Turn a Simple Verbal Phrase Into an Algebraic Expression? • Virtual Nerd video introduction to translating verbal phrases into algebraic expressions • Examples are given of very basic phrases that require each of the operations of addition, subtraction, multiplication, and division • Translates words such as sum, total, difference, product, quotient, and ratio • Time 3:36

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