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Understanding Expressions, Equations, and Functions: An Introductory Guide

This chapter provides an overview of the foundational concepts of algebra, including terms, variables, and types of expressions. It emphasizes the distinction between expressions and equations, highlighting that while expressions can be simplified, only equations can be solved. The text also covers verbal translations of algebraic expressions, introduces patterns and sequences, and outlines key algebraic properties, which are essential for understanding problem-solving techniques. This resource is designed for learners to grasp algebraic fundamentals effectively.

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Understanding Expressions, Equations, and Functions: An Introductory Guide

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  1. Expressions, Equations, and Functions Chapter 1

  2. Introductory terms and symbols: • Variable • A letter or symbol to represent an unknown • Examples: • Algebraic expression • One or more numbers or variables along with one or more arithmetic operations • Examples: • You may evaluate and simplify expressions, but you cannot solve expressions…you solve equations!

  3. Verbal Translations

  4. Translate verbal expressions to algebraic expressions • Examples: • 7 less than the product of 3 and a number • The product of 7 and a number divided by the product of 8 and a number • 5 more than half a number • The quotient of 3 and the square of a number • Twice the sum of 15 and a number

  5. Patterns and Sequences • 7, 13, 19, 25, … • 51, 43, 35, 27, … • 343, 81, 27, 9, … • 2, 3.5, 5, 6.5, … • 1, 4, 16, 64, …. • Find the next three terms in each sequence • Group the sequences according to similarities

  6. Patterns and Sequences Arithmetic Geometric 343, 81, 27, 9, … 1, 4, 16, 64, …. 3, 9, 27, 81, …. • 7, 13, 19, 25, … • 51, 43, 35, 27, … • 2, 3.5, 5, 6.5, … • 3, 6, 9, 12, …

  7. Which Changes Faster?? • Geometric! • How does it change when: • multiply by a whole number • multiply by a fraction • WHY?

  8. Writing expressions for patterns • Draw a picture • Make a table and look for a pattern • Model using manipulatives

  9. Open SentencesVocabulary • Set • Element • Replacement set • Solution set • Solution • Equation • inequality

  10. Symbols • = • = • < • > • < • > • 0

  11. Find the solution set. The replacement set is {0,1,2,3,4,5} • 6b + 7= 37 • y + 5 < 7 • 8 – x > 7 • t + 3 = 3 4

  12. Algebraic Properties • Additive Identity • For any number a , a + 0 = 0 + a = a • Multiplicative Identity • For any number a, (a)(1) = 1a = a • Multiplicative Inverse Property (reciprocal) For any non-zero number a/b, where a, b don’t = 0, (a/b)(b/a) = 1

  13. More Algebraic Properties • Distributive Property • For any numbers a, b, and c, a(b + c) = ab + ac and (b + c)a = ba + ca a(b - c) = ab - ac and (b - c)a = ba - ca • Commutative Property • For any numbers a and b, a + b = b + a and ab = ba • Associative Property • For any numbers a, b, c, ( a + b ) + c = a = ( b + c ) and (ab)c = a(bc)

  14. These properties along with some others allow algebra to work!

  15. ExpressionsVocabulary • Equivalent expression • denote the same number • Simplify expressions • Write an expression with the least amount of symbols, numbers, and variables

  16. Termsvocabulary • Term • a number or variable or the product of a number and variable • Like terms • Terms that contain the same variable • Like terms can be grouped (combined) • Constant • A numerical term containing NO variables • Coefficient • The numerical factor of a term

  17. Terms 8m a 9 -7j² -4a 8 2cd x/8 7g ¼ b 3xy j 9b 5x –y 2d 4g m 6y 6a³ -9a³

  18. Terms Like Terms Non Like Terms a and 9 -4a and 8 2x and 3xy 5j and -7j² 2d and 2cd • 8m and m • 4g and 7g • 9b and ¼ b • 5x and x/8 • 6y and –y • 6a³ and -9a³

  19. Equivalent Expressions Expression Simplified expression • 8m - m • 4g + 7g • 9b + ¼ b • 5x + x/8 • 6y + (–y) • 6a³ - 9a³

  20. Coefficients Term Coefficient • 2b • 8c² • K • -5t³ • 9

  21. A Preview to Functions • A function is a relationship between input and output values • With a function, there is exactly one output for each input! • A function (relation) can be expressed as ordered pairs

  22. Relation Input Output Dependent variable Y-coordinate range • Independent variable • X - coordinate • domain

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