In this lesson we step back and review several key arithmetic topics that are extremely relevant to this course. Before we work with algebraic expressions and equations, it is important to have a good understanding of order of operations, fractions and signed numbers. Lesson Topics Section 1.1: Order of Operations Section 1.: Fractions Improper fractions and mixed numbers Equivalent fractions Fractions in simplest form One and zero Section 1.: Operations on Fractions Addition and subtraction of fractions Multiplication and division of fractions Order of operations with fractions Section 1.4: Signed Numbers The number line Absolute value Mathematical operations with signed numbers Simplified form for a signed fraction Page 1
Lesson 1 Notes Page
Name: Date: 1 Section 1.1: Order of Operations PEMDAS If we are working with a mathematical expression that contains more than one operation, then we need to understand how to simplify. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. P Terms inside parenthesis ( ) or brackets [ ] E MD AS Exponents and roots Multiplication and division (from Left to Right). Addition and subtraction (from Left to Right). Use the order of operations to evaluate each of the following expressions. Use your calculator to check your answers. Example 1: ( 5)² = 5² = 10 7 + 1= 10 (7 + 1)= Example : 4 (4 ) = Page
Example : 4 + 5(1 + 1 6) = Example 4: 15 = 1 5 You Try Use the order of operations to evaluate each of the following expressions. Show all possible steps. Use your calculator to check your answers. 1. 11 + (7 ) =. 6 8 = 4 Page 4
Section 1.: Fractions Improper Fractions and Mixed Numbers Converting a mixed number to an improper fraction: 1. Multiply the denominator and the whole number. Add the numerator. Write the result over the denominator Example 1: Express as an improper fraction. 7 1 1 Converting an improper fraction to a mixed number: 1. Divide the numerator by the denominator. The quotient becomes the whole number part of the mixed number. Write the remainder over the denominator Example : Express an improper fraction as a mixed number. 4 5 5 9 Equivalent Fractions Example : Find two fractions equivalent to 7 Page 5
Fractions in Simplest Form Example 4: Write the following fractions in simplest form. 4 18 54 ONE and ZERO Example 5: 1 4 4 1 4 4 0 4 4 0 YOU TRY 4. Reduce the fraction to lowest terms. 6 4. Rewrite the mixed number 1 4 as an improper fraction. 5 5 5. Rewrite the improper fraction as a mixed number. 11 6. Find two fractions equivalent to 5 Page 6
Section 1.: Operations on Fractions Addition and Subtraction of Fractions Adding and Subtracting Fractions: 1. Rewrite mixed numbers and whole numbers as improper fractions.. Find a common denominator. Rewrite the fractions as equivalent fractions with the common denominator 4. Add or subtract the numerators 5. Be sure to reduce your answer to simplest form! Example 1: Perform the indicated operations 1 1 a. 11 5 b. 15 1 5 8 c. 4 1 = d. = 5 6 5 Page 7
Multiplication and Division of Fractions Multiplying Fractions: 1. Rewrite mixed numbers and whole numbers as improper fractions.. Multiply straight across (Multiply the numerators with the numerators, and the denominators with the denominators) NOTE: There is no need to find a common denominator when multiplying.. Be sure to reduce your answer to simplest form! Example : Multiply. Write your answers in simplest form a. 4 1 5 b. 5 48 7 c. 5 8 d. 1 1 1 = 5 9 Dividing Fractions: 1. Rewrite mixed numbers and whole numbers as improper fractions. NOTE: There is no need to find a common denominator when dividing.. Change the second fraction (the divisor) to its reciprocal. Multiply 4. Be sure to reduce your answer to simplest form! Example : Divide. Write your answers in simplest form. 1 4 a. b. 8 5 5 Page 8
Order of Operations with Fractions Example 4: Perform the indicated operations. 1 5 You Try 7. Perform the indicated operations. Show all possible steps. Each answer must be written as a reduced fraction. Where appropriate, write your answer as both a mixed number and an improper fraction. a. b. 5 5 c. 5 1 d. 5 e. 5 7 4 5 f. 4 5 6 Page 9
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Section 1.4: Signed Numbers The Number Line Absolute Value The ABSOLUTE VALUE of a number is the distance that number is from 0 on the number line. Example 1: Find the absolute value: a. = b. = c. = d. 0 = MATHEMATICAL OPERATIONS WITH SIGNED NUMBERS Some hints for working with signed numbers: Use ( ) to separate numbers with negative signs When two signs are given together, use these rules to resolve the signs: ( )( ) = + ( )(+) = (+)( ) = (+)(+) = + Use the number line to add and subtract Example : Perform the indicated operations. a. + ( ) = b. + = c. ( ) = d. + ( ) = Example : Multiply and divide. a. ( 5)( 6)= b. ( 4)= c. 4 8 1 = d. 5 Page 11
Example 4: Evaluate the following exponents: ( 5) = 5 = ( 5) = 5 = Example 5: Perform the indicated operations. 8 ( ) ( ) 5 = SIMPLIFIED FORM FOR A SIGNED FRACTION The following fractions are all equivalent (meaning they have the same value): 1 1 1 Notice that only the placement of the negative sign is different. HOWEVER, only the last one, 1, is considered to be in simplest form. You Try 8. Find the absolute value: 5 = 5 = 9. Perform the indicated operations. Show your work, and use your calculator to check. a. ( ) b. 6 1 4 ( ) 4 Page 1