Columbus State Community College

1. Columbus State Community College Developmental Education Department Dev 031: Pre-Algebra An Alternative Method for Sections 1.3 & 1.4

2. Chapter 1 – Objectives for Sections 1.3 and 1.4 • Add and subtract integers. • Identify properties of addition. • Combine adding and subtracting of integers. • Identify properties of addition.

3. Alternative Method The following slides will explain how to add and subtract signed numbers. This material is explained in Sections 1.3 and 1.4 of your textbook. However, we will be using an alternative approach. You can decide which method works best for you. In our explanation here, we will learn how to eliminate “double signs” in order to simplify mathematical expressions. We will use this method throughout the rest of this course.

4. Alternative Method • In our explanation, we will recognize the following: • The “+” symbol can be used for two different purposes. • “+” can identify the addition operator“+” can identify the positive sign of a number • The “–” symbol can be used for two different purposes. • “–” can identify the subtraction operator “–” can identify the negative sign of a number

5. Alternative Method Here’s what we mean by the symbols looking the same. +3 – +4 + –2 Here’s what we mean by the symbols looking the same. +3 –+4 +–2 Signs Operators Notice the “signs” and the “operators” look the same. We will refer to the highlighted areas as “double signs”.

6. Eliminating “Double Signs” We can demonstrate how to eliminate “double signs” using the following examples. With Double Signs Without Double Signs match +3 + +4 3 + 4 don’t match +3 + –4 3 – 4 don’t match +3 – +4 3 – 4 match +3 – –4 3 + 4

7. Eliminating “Double Signs” Eliminate all double signs. Do not simplify the expressions. You try… With Double Signs Without Double Signs –8 – +5 – –2 -8 – 5 + 2 1) 15 + –3 – 20 + –2 15 – 3 – 20 – 2 2) 3) –2 + +3 + 1 – –9 – +7 –2 + 3 + 1 + 9 – 7 Not a double sign Not a double sign

8. -5 -4 -3 -2 -1 0 1 2 3 4 5 Addition Using a Number Line 2 + 3 = 5 Question: Without double signs, adding moves us which direction on the number line? 1 + 2 = 3 -4 + 3 = -1 -5 + 8 = 3

9. -5 -4 -3 -2 -1 0 1 2 3 4 5 Subtraction Using a Number Line 5 – 1 = 4 Question: Without double signs, subtracting moves us which direction on the number line? 4 – 2 = 2 5 – 8 = -3 -3 – 2 = -5

10. -5 -4 -3 -2 -1 0 1 2 3 4 5 Addition & Subtraction Using a Number Line Use the number line to simplify the following expression. 3 – 7 + 6 -4 + 6 2

11. -5 -4 -3 -2 -1 0 1 2 3 4 5 Addition & Subtraction Using a Number Line Use the number line to simplify the following expression. -1 – 4 + 8 -5 + 8 3

12. -5 -4 -3 -2 -1 0 1 2 3 4 5 Addition & Subtraction Using a Number Line Use the number line to simplify the following expression. -3 + 7 – 5 – 4 + 6 4 – 5 – 4 + 6 -1 – 4 + 6 -5 + 6 1

13. Addition & Subtraction Using Money Does Johnny have enough money to buy his motorcycle supplies? Now, add his expenses and his money. First, organize the expenses and Johnny’s money. Does he have enough money? No. $195 – $185 = $10 (He needs $10) $155 Savings $75 Helmet $155 Savings $75 Helmet $30 Allowance $120 Boots $195 Total Cost $185 Total Money $120 Boots $30 Allowance

14. Addition & Subtraction Using Money Does Edward have enough money to pay his bills? Yes. $980 – $860 = $120 (left over) Determine if the income will “cover” the bills. First, organize the bills and the income. Now add the total bills and the total income. $40 Gas Bill $50 Gift $30 Rebate Check $40 Gas Bill $650 Rent $900 Paycheck $20 Life Insurance $50 Gift $650 Rent $30 Rebate Check $60 Electric Bill $980 Total Income $30 Water Bill $60 Electric Bill $30 Water Bill $20 Life Insurance $900 Paycheck $60 Auto Insurance $60 Auto Insurance $860 Total Bills

15. Addition & Subtraction by Organizing Terms –3 + 7 – 5 – 4 + 6 Now, organize the positive and negative terms. Add each type of term (positive and negative). Lastly, determine the result. First, identify the positive and negative terms. –3+ 7– 5 – 4 + 6 –3 – 5 – 4 + 7+ 6 –12+ 13 The positive term is “larger than” the negative term by 1. 1

16. Addition & Subtraction by Organizing Terms 1 – 8 + 15 – 10 + 24 – 30 First, identify the positive and negative terms. Now, organize the positive and negative terms. Add each type of term (positive and negative). Lastly, determine the result. 1 – 8 + 15 – 10 + 24 – 30 1 + 15 + 24– 8 – 10 – 30 Assumed positive 40 – 48 The negative term is “larger than” the positive term by 8. –8

17. Addition & Subtraction by Organizing Terms –12 + 4 + 6 – 15 + 17 – 24 Now, organize the positive and negative terms. Add each type of term (positive and negative). Lastly, determine the result. First, identify the positive and negative terms. –12+ 4 + 6 – 15 + 17 – 24 –12 – 24 – 15+ 4 + 6 + 17 –51+ 27 The negative term is “larger than” the positive term by 24. –24

18. Addition & Subtraction by Organizing Terms –3 + 15 + –5 – –9 + +8 – +1 – 7 Now, identify the positive and negative terms. Organize the positive and negative terms. Add each type of term (positive and negative). Lastly, determine the result. First, eliminate all double signs. –3 + 15 – 5 + 9 + 8 – 1 – 7 –3+ 15 – 5 + 9 + 8 – 1 – 7 –3 – 5 – 1 – 7+ 15 + 9 + 8 –16 + 32 The positive term is “larger than” the negative term by 16. 16

19. Addition & Subtraction by Organizing Terms 5 – 10 – –15 + 20 – 30 + –35 – +40 Lastly, determine the result. First, eliminate all double signs. Now, identify the positive and negative terms. Organize the positive and negative terms. Add each type of term (positive and negative). 5 – 10 + 15 + 20 – 30 – 35 – 40 5– 10 + 15 + 20 – 30 – 35 – 40 5+ 15 + 20 – 10 – 30 – 35 – 40 40 – 115 The negative term is “larger than” the positive term by 75. –75

20. Addition & Subtraction by Organizing Terms –12 – –7 + –6 + 9 – +1 – 8 + +20 Lastly, determine the result. First, eliminate all double signs. Now, identify the positive and negative terms. Organize the positive and negative terms. Add each type of term (positive and negative). –12 + 7 – 6 + 9 – 1 – 8 + 20 –12+ 7 – 6 + 9 – 1 – 8 + 20 –12 – 6 – 1 – 8+ 7 + 9 + 20 –27 + 36 The positive term is “larger than” the negative term by 9. 9

21. Addition Property of 0 Addition Property of 0 Adding 0 to any number leaves the number unchanged. Some examples are shown below. 0 + 9 = 9 –75 + 0 = –75 18,345 + 0 = 18,345

22. Commutative Property of Addition Commutative Property of Addition Changing the order of two addends does not change the sum. Here are some examples. 14 + 26 = 26 + 14 Both sums are 40. –24 + 8 = 8 + –24 Both sums are –16.

23. Associative Property of Addition Associative Property of Addition Changing the grouping of addends does not change the sum. Here are some examples. (3 + 2) + 8 = 3 + (2 + 8) (–1 + –9) + 5 = –1 + (–9 + 5) = = 5 + 8 3 + 10 –10 + 5 –1 + –4 13 13 –5 –5 = =

24. Note on Using the Associative Property of Addition NOTE When using the associative property to make the addition of a group of numbers easier: • Look for two numbers whose sum is 0. • Look for two numbers whose sum is a multiple of 10 (the sum ends in 0, such as 10, 20, 30, or –100, –200, etc.). If neither of these occurs, look for two numbers that are easier for you to add. For example, in 39 + 18 + 6, you may find that adding 18 + 6 is easier than adding 39 + 18.

25. Chapter 1 Sections 3 and 4 Alternative Method for Chapter 1; Sections 3 and 4 – End Written by John T. Wallace