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Chapter 1 Notes 1-1 Variables and Expressions 1) Identifying Numerical and Algebraic Expressions Vocabulary – What do these words mean? numerical – a number ex) 4 consisting of only numbers algebraic expression – a math phrase, with variables, numbers, and operation symbols. ex) x+4 variable – a variable is a letter that represents something we do not know ex) x Write words that mean these operations. Addition (+) Subtraction () more than less than total fewer increase decrease plus difference show addition n+7 7+n These are the same. show subtraction n−7 7−n Are these the same? Multiplication (x) times product Division (÷) each divided by quotient show multiplication 4 ( 7 ) or (4)(7) 4•7 4b show division ½v Do not use x for multiplication means divide v by two ¼v means divide v by four 2) Writing Algebraic Expressions To write an algebraic expression, write the letter (variable) to stand for an unknown number, write the operation needed, Test it, to make sure the expression makes sense. example) a box of books, and four more is x + 4 Notes 1-2 Order of Operations Using the Order of Operations 1st do Grouping Symbols 2nd do Exponents 3rd do Multiplication and Division ex) (2 + 2) ex) x³ ex) 12 ÷ 6 • 2 ex) [3(2 + 2)] ex) 4² ex) 12 • 6 ÷ 2 P E MD ex) 12 + 6 − 2 ex) 12 − 6 + 2 AS from left to right 4th do Addition and subtraction from left to right example) grouping symbols exponents a multiplication, division addition, subtraction 30 ÷ (2 + 3) − 1² 30 ÷ 5 − 1² 30 ÷ 5 − 1² 30 ÷ 5 − 1 30 ÷ 5 − 1 6−1 6−1 = 5 P E MD AS Notes 1-3 Writing and Evaluating Expressions Evaluating Algebraic Expressions Vocabulary – What do these words mean? ex) 4 evaluate – solve, give the answer, usually it is a number variable – a letter that represents a number we do not know ex) x grouping symbols – ( ) parenthesis, [ ] brackets ex) { } simplify – replace with an expression having as few terms as possible To evaluate an expression, solve for the variable, simplify by performing operations, and give an answer in the fewest terms possible. 4y − 15 ex) Evaluate 4y − 15, when y=9. copy the expression 4(9) − 15 replace the y with 9 36 − 15 multiply = 21 subtract Sometimes you have to replace more than one variable with a number. Notes 1-4 Integers and Absolute Value 1) Comparing Integers Vocabulary – What do these words mean? Integers – positive numbers, negative numbers, and zero Opposites – numbers the same distance from zero in opposite directions ex) -2 and 2 Absolute Value – the distance from zero to this number IT IS ALWAYS POSITIVE. _____|________|_______|_______|_______|___ Integers get bigger from left to right. -2 -1 0 1 2 1 is bigger than -2 (it is more) 1 > -2 and -2 < 1 (large side goes to the large number) 2) Finding Absolute Value To find the absolute value, count how many spaces from zero the number is. It is the same number – without a negative sign. IT IS ALWAYS POSITIVE. ex) /-4/ = 4 ex) /4/ = 4 ex) /-55/ = 55 ex) /55/ = 55 Notes 1-5 Adding Integers 1) Adding Integers with the same sign When adding integers with the same sign, just add the two numbers and KEEP the same sign. ex) -5 + -2 = -7 ex) 13 + 5 = 18 ex) -13 + -5 = -18 2) Adding Integers with different signs When adding integers with different signs, subtract them and keep the sign of the bigger absolute value. ex) 5 + -2 = 3 ex) -13 + 5 = -8 ex) 18 + -5 = 13 5 was positive 13 was negative 18 was positive answer is positive answer is negative answer is positive ex) 5 + 2 = 7 ex) -9 + 1 = -8 9 was negative answer is negative Notes 1-6 Subtracting Integers Subtracting Integers We DO NOT subtract integers We ADD THE OPPOSITE NUMBER. To subtract integers, 1) Copy the problem. 2) Circle the subtraction sign AND the NEXT number. 3) CHANGE the subtraction to addition AND the NEXT number to its OPPOSITE. 4) Then add. ex) 13 − 6 = 13 − 6 = 13 + -6 = 7 Copy the problem. Circle. (The first number DOES NOT CHANGE.) CHANGE, then add. + + + + + + + + + + + + + You can draw positive / negative signs to help you add, and circle zero pairs. The answer will be what is left. − − − − − − Notes 1-9 Multiplying and Dividing Integers Rules to Multiply and Divide Integers The number of negatives in problem . . . tell if the answer will be positive or negative. negatives in ANSWER problem ? odd # = negative answer even # = positive answer 0 + 1 − 2 + 3 − 4 + 5 − 6 +