If you have $5 and you owe me $8, do you owe me money? Why? How do we use algebra to write this? Why do we need negative numbers?

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1 Basic Things You Absolutely Need to Know: Integers Vocabulary and Algebraic Symbols: What are integers? Give some examples and non-examples. If you have $5 and you owe me $8, do you owe me money? Why? How do we use algebra to write this? Why do we need negative numbers? If you owe me $3 and I take away $7 more, do you owe me money? Why? How do we use algebra to write this? If you owe $8 and I take away the fact that you also owe me $3, do you owe me money? Why? How do we use algebra to write this? KISS Keep it Simple Summary: Addition of integers: 5+ 7= 12 positive 5 and positive 7 is lots of positives, so ( 7) = 12 negative 5 and negative 7 is lots of negatives, so = 2 negative 5 and positive 7 is more positives than negatives, so ( 7) = 2 positive 5 and negative 7 is more negatives than positives so 2. Subtraction of integers: Rewrite subtraction as addition, then add! 3 8 positive 3 take away positive 8 as addition (take away a positive is adding a negative) 3 + ( 8) = 5 3 ( 8) positive 3 take away negative 8 as addition (take away a negative is adding a postive) 3+ 8= 11 Worksheet: Adding/Subtracting Integers S. Stirling Page 1 of 8

2 Multiplication and division of integers is much easier. Just remember the rules below and whatever you do DO NOT get the rules for addition and subtraction confused with multiplication and division! Addition & Subtraction Integers Only add or subtract like things. Write subtraction as addition. Think money! = 5 have 3 and owe 8, so owe 5 ( ) ( ) ( 5) ( 2) + = 7 owe 5 and owe 2, so owe 7 ( 5) ( 2) ( 5) ( 2) = + + = 3 taking away owing 2 is like adding 2 owe 5 and have 2, so owe only 3 Multiplication & Division Integers Signs same positive Signs different negative Worksheet: Multiplying Integers Worksheet: Dividing Integers Basic Things You Absolutely Need to Know: Decimals You will basically follow the same procedures you used with integers. Use an estimate with integers to decide on your procedure and what the sign of your answer should be. Main Idea: You can only add (or subtract) like things. So to add decimals, you must line up the decimal places first. This way you are adding tens to tens and hundredths to hundredths, etc. Now put these two ideas together. Ex Write as addition: ( 5.06) 3 + ( 5) 2. Estimate. You will use the same procedures with the decimals. 3. Line up the decimals and repeat your procedure you established in step two. 4. Check your answer with the estimate Procedure: You are actually subtracting 3 from 5 and you have more negatives than positives, so the answer is negative. Procedure: Subtract 3.2 from 5.06 and the answer is negative is close to 2. Yeah! Ex Write as addition: ( 2.05) 15 + ( 2) 2. Estimate. You will use the same procedures with the decimals. 3. Line up the decimals and repeat your procedure you established in step two Procedure: You are actually adding 15 and 2 and you have only negatives, so the answer is negative. Procedure: Add 14.6 and 2.05 and the answer is negative. 4. Check your answer with the estimate is close to 17. Yeah! Worksheet: Adding/Subtracting Decimals S. Stirling Page 2 of 8

3 Since the rules for multiplying and dividing integers are really easy, we will simply review the decimal computation rules. Use the rules for integers to determine the sign of your answers. Vocabulary Review: factor factor = product dividend divisor = quotient As a long division problem: quotient divisor dividend Multiplication Ex Multiply as if the factors were whole numbers. 2. Count up the total number of decimal places in the factors. 3. Your answer, the product, should have that number of decimal places. The answer should have three decimal places. So the answer is Multiply 2 by 401. Multiply 30 by 401. Division Ex Set up to problem as a long division problem. 2. Move the decimal in the divisor to make it a whole number. Move the decimal in the dividend the same number of places and then bring it up to the quotient. 3. Divide as you do with whole numbers.?.? Multiply 4 by 201. Subtract 864 and 804, bring down 3. Multiply 3 by 201. Subtract. Worksheet: Multiplying and Dividing Decimals S. Stirling Page 3 of 8

4 Basic Things You Absolutely Need to Know: Fractions Equivalent Fractions Multiply the numerator and denominator by the same number = They are equivalent because you are simply multiplying the fraction by one, 3/3. Improper Fractions means 2 and or You are adding, so you need like denominators = = + = So 2 = 9 9 Addition & Subtraction Fractions Only add or subtract like things. Must make like fractions to be able to add! unlike fractions, make denominators the same equivalent fraction with denominator = add numerators the denominator is the same. Note: means Multiplication & Division Fractions Need improper fractions to be able to multiply! You will multiply the numerators together and the denominators together = = = now simplify = You may also simplify by making ones before multiplying! = = = or now rearrange and make like then add like things = For division, just rewrite as multiply by the reciprocal, then multiply = = now make ones = or Reciprocal Just flip an improper fraction. See Skills Handbook The reciprocal of 2 9 is 9 2. page 871. Do the odds then check 2 20 The reciprocal of 2 = is 9 your answers. Then do the evens. 5 The reciprocal of 5 = is 1 SHOW ALL STEPS!! 1 5. S. Stirling Page 4 of 8

5 Order of Operations PEMDAS You must use the order of operations when simplifying. Parenthesis (Innermost first) Exponents and Roots Multiplication and Division (Left to right) Addition and Subtraction (Left to right) The hardest part of learning algebra is trying to read it. Vocabulary and Algebraic Symbols: What does simplify mean? Is the meaning different in algebra than it is in other contexts? Give an example of simplifying in algebra. What is an expression in math? Is the meaning different in algebra than it is in other contexts? Give an example of an expression in algebra. Parenthesis [and brackets] are used to group expressions. Write some examples. What math operation is also indicated by parenthesis? Write some examples. Compare these uses. S. Stirling Page 5 of 8

6 Give an example of an algebraic expression that involves exponents. What does the exponent tell you to do? What is the value of your expression? Give an example of an algebraic expression that involves square roots. What does the square root tell you to do? What is the value of your expression? What symbols are used to represent multiplication in algebra? Write three times five in as many ways possible. Why is using the x symbol a bad idea in algebra? What symbols are used to represent division in algebra? Write ten divided by two in as many ways possible. Why is this confusing? Why do we even need an Order of Operations? Simplify the following expressions. Compare your answers. Is it possible to get two different values (answers) for the same expression? Ex Ex Ex S. Stirling Page 6 of 8

7 Now you need to put it all together. Remember, you need to follow the order of operations and use your rules for integers. Ex ( 1+ 5) + 2 ( 10) 3 3( 1+ 5) + 2 ( 10) copy problem 3(4) + 8( 10) parenthesis then exponents multiplication 92 addition Why do they write the parenthesis in the expression 4 ( 2)? Ex (4 ( 2)) (4 ( 2)) division 10 9 copy problem parenthesis then exponents addition and subtraction left 1 to right Why do you have to go from left to right? Try simplifying going left to right and then right to left. Use what you know about integers, operations and order of operations to simplify the expressions on page 873. Check as you go. SHOW ALL STEPS as in the example! Write the problem and the resulting value. When there is more than one operation, show how you re using the order of operations to get to the simplified values. S. Stirling Page 7 of 8

8 Reflect on what you know and don t know. I know how to really well. I m pretty good at, but may need some additional practice. I have NO clue about, need further instruction and need lots of additional practice. Warning: You really need to make sure you understand these concepts, procedures and skills really well before we move on to Chapter 1. Seek help NOW if you are not confident!! Complete Check Your Readiness for Chapter 1. Complete the problems on page 2 #1 26. Check your answers and seek help if needed. S. Stirling Page 8 of 8