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TOPIC 8: Writing and Interpreting Numerical Expressions Why do we need a standard method for evaluating expressions? When and how do we use the order of operations? How can we model real life situations mathematically? Expression: a mathematical phrase containing numbers, operation symbols, grouping symbols, and sometimes variables. Example: 6 + 2 * 9 Evaluate an Expression: to carry out the operations in an expression in the proper order in order to find the value of it. Example: 6 + 2 * 9 6 + 18 24 Order of Operations: A procedure for correctly evaluating an expression. The sequence in which you must carry out the operations is: 1. Parentheses: Simplify the operations inside parentheses first! 2. Exponents: find the value of the powers in the expression. 3. Multiplication & Division: OR All in one step from left to right. Division & Multiplication: 4. Addition & Subtraction: OR All in one step from Left to Right Subtraction & Addition: We abbreviate this procedure by using the acronym: P Please E Excuse MD My Dear AS Aunt Sally Funnel Method: The way Mrs. Gerould requires students to show the steps used to evaluate an expression. 1. Write out an expression horizontally. 2. Determine which operation should be carried out first according to the order of operations. UNDERLINE IT. 3. Carry out that operation and rewrite the rest of the expression exactly the way it appeared before under the original expression. 4. Underline the next operation. 5. Continue carrying out one operation at a time and rewriting the expression after each step until you’ve completed all operations. 6. Circle your final value. 1. 3 + 5 * (5 – 2) + 8 3. 3 + 5 * (5 – 2) + 8 4. 3+5*3+8 5. 6. 3+5*3+8 3 + 15 + 8 18 + 8 2. 26 Grouping Symbols: symbols used to increase the priority of a specific operation. (Parentheses)- 3 * (2 + 7) In this example, they require that the addition is carried out before the multiplication. [Brackets]- 54 [3 * (2 + 7)] Used when there are parentheses inside another set of grouping symbols. {Braces}- 10 – {54 [3 * (2 + 7)] + 3} Grouping symbols inside grouping symbols, inside grouping symbols. Always start carrying out operations within the most inside set of grouping symbols and work your way out!!! As you eliminate the parentheses, change the brackets to parentheses and the braces to brackets. EXAMPLE (using the funnel method): 10 – {54 [3 * (2 + 7)] + 3} 10 – [54 (3 * 9) + 3] 10 – (54 27 + 3) 10 – (2 + 3) 10 – 5 5 Translating Verbal Phrases into Mathematical Expressions Translate: to change words from one language to another. Example: English -“How are you?” Spanish – “Como estas?” The sentences look different, but mean the same thing. We will be translating from English words to Math symbols. You must identify the key word(s) in the phrase that indicate which operation to use. Examples of Numerical Expressions: the product of 3 and 5 …………….……………3 • 5 6 more than 2 …………………………………..2 + 6 15 decreased by 7 ……..………………………15 – 7 the quotient of 12 and 3, decreased by 1………12 ÷ 3 – 1 add 8 and 7, then multiply by 3……………….(8 + 7) • 3 **Be sure to add grouping symbols if you are supposed to add or subtract before you multiply or divide.** Examples of Algebraic Expressions: Anytime there is an unknown number in the phrase, use a variable (letter) in its place. the product of a number and 3, increased by 5……….3 • n + 5 add 6 and a number, then multiply by 4 ……………..(6 + n) • 4