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Math 90

Math 90. Section 1.5 and 1.6 Adding and subtracting Integers September 4 th is the chapter 1 test. Integers. Look at the root of the word. Warm up SIMPLIFY. 3/5 - 2/3. 3/5(2/3). 3/5/2/3. In Math it pays to remember the rules. Learning Objective Name _____________________

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Math 90

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  1. Math 90 • Section 1.5 and 1.6 • Adding and subtracting Integers • September 4th is the chapter 1 test

  2. Integers Look at the root of the word.

  3. Warm up SIMPLIFY 3/5 - 2/3 3/5(2/3) 3/5/2/3 In Math it pays to remember the rules.

  4. Learning Objective Name _____________________ Today, we will add and subtract positive and negative integers. CFU What are we going to do today? What are we going to add and subtract? Activate (or provide) Prior Knowledge If lunch was $6.00, and you only had $5.00, could you buy lunch? Why or why not? CFU Sometimes we don’t have enough money to buy things. If we were allowed to buy things without having the right amount of money, we would owe someone money, which could cause problems. In math, we use specific numbers to represent money that you have—numbers greater than 0—and money you owe—numbers less than 0. We call these positive and negative numbers. Today, we will add and subtract positive and negative integers. Today, we are going to add and subtract positive and negative integers. We are going to add and subtract positive and negative integers. 6th Grade Number Sense 2.3 (6Q) Solve addition, subtraction, multiplication, and division problems including those arising in concrete situations that use positive and negative integers and combinations of these operations. Lesson to be used by EDI–trained teachers only.

  5. Concept Development • The Integers are the whole numbers and their opposites. • Examples: –5 0 5 • Non–examples: 1.33 –1 ½ 927.01 • Positive integers are whole numbers greater than zero. • Positive integers are located on the right side of a number line. • Positive integers are written with a positive symbol (+), but the symbol is not always shown. • Examples: 1 2 3 2,394 1,590,903 • Negative integers are whole numbers less than zero. • Negative integers are located on the left side of a number line. • Negative integers are written with a negative symbol (–), and the negative symbol is always shown. • Examples: –1,694,617 –12 –2 –1 Negative Positive -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 CFU On your whiteboard, write a positive integer. On your whiteboard, write a negative integer. On which side of 0 on the number line is a positive integer located? On which side of 0 on the number line is a negative integer located? In your own words, what are integers? 6th Grade Number Sense 2.3 (6Q) Solve addition, subtraction, multiplication, and division problems including those arising in concrete situations that use positive and negative integers and combinations of these operations. Lesson to be used by EDI–trained teachers only.

  6. Concept Development (continued) Addition Integer Rules If the numbers have the same sign, add the numbers and keep the sign. (Positive) + (Positive) = Positive (Negative) + (Negative) = Negative -4 + -3 = -7 -6 + -8 = -14 -3 + -4 = -7 -8 + -6 = -14 4 + 3 = 7 6 + 8 = 14 3 + 4 = 7 8 + 6 = 14 If the numbers have different signs, subtract the numbers and use the sign of the larger number. (Positive) + (Negative) = Sign of the Larger Number 4 + -3 = 1 Because 4 – 3 = 1, and the larger number is positive, then the answer is positive. 6 + -8 = -2 Because 8 – 6 = 2, and the larger number is negative, then the answer is negative. CFU Will the answer to 5 + 3 be positive or negative? How do you know? Will the answer to -5 + -3 be positive or negative? How do you know? Will the answer to 5 + -3 be positive or negative? How do you know? Will the answer to -5 + 3 be positive or negative? How do you know? In your own words, what is the rule for adding two positive integers? In your own words, what is the rule for adding two negative integers? In your own words, what is the rule for adding integers with different signs? (Negative) + (Positive) = Sign of the Larger Number -4 + 3 = -1 Because 4 – 3 = 1, and the larger number is negative, then the answer is negative. -6 + 8 = 2 Because 8 – 6 = 2, and the larger number is positive, then the answer is positive. 6th Grade Number Sense 2.3 (6Q) Solve addition, subtraction, multiplication, and division problems including those arising in concrete situations that use positive and negative integers and combinations of these operations. Lesson to be used by EDI–trained teachers only.

  7. Skill Development/Guided Practice Negative Positive -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 + + – – – – + – + – – + + + – – – + Integers are any positive or negative whole number or zero. Positive integers are whole numbers greater than zero. Negative integers are whole numbers less than zero. Add positive and negative integers. + + (Pos) + (Pos) = Pos (Pos) + (Pos) = Pos 38 40 (Neg) + (Neg) = Neg (Neg) + (Neg) = Neg –9 –9 (Pos) + (Neg) (Pos) + (Neg) = Pos = Pos 7 3 = Neg (Neg) + (Pos) = Neg (Neg) + (Pos) –11 –6 = Neg (Pos) + (Neg) = Neg (Pos) + (Neg) –3 –15 CFU Why did I place the sign above each number? How did I decide which rule to use? How did I solve the problem? Why did you place the sign above each number? How did you decide which rule to use? How did you solve the problem? 6th Grade Number Sense 2.3 (6Q) Solve addition, subtraction, multiplication, and division problems including those arising in concrete situations that use positive and negative integers and combinations of these operations. Lesson to be used by EDI–trained teachers only.

  8. Concept Development Subtraction Integer Rules To subtract integers, add the opposite of the second integer. (Positive) – (Positive) Add the opposite of the second integer. 9 – 4 = 5 is equivalent to 9 + (-4) = 5 4 – 9 = -5 is equivalent to 4 + (-9) = -5 (Negative) – (Negative) Add the opposite of the second integer. -9 – (-4) = -5 is the same as -9 + (4) = 5 -4 – (-9) = -5 is the same as -4 + (9) = 5 (Positive) – (Negative) Add the opposite of the second integer. 9 – (-4) = 13 is equivalent to 9 + (4) = 13 4 – (-9) = 13 is equivalent to 4 + (9) = 13 (Negative) – (Positive) CFU What is the rule for subtracting any two integers? Which expression below is equivalent to -10 – (-4)? How do you know? A. -10 – (4) B. –10 + (4) Add the opposite of the second integer. -9 – 4 = -13 is the same as -9 + (-4) = -13 -4 – 9 = -13 is the same as -4 + (-9) = -13 6th Grade Number Sense 2.3 (6Q) Solve addition, subtraction, multiplication, and division problems including those arising in concrete situations that use positive and negative integers and combinations of these operations. Lesson to be used by EDI–trained teachers only.

  9. Skill Development/Guided Practice Negative Positive -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 + – + + + + – – – – – + – + Integers are any positive or negative whole number or zero. Positive integers are whole numbers greater than zero. Negative integers are whole numbers less than zero. Subtract positive and negative integers. = Pos (Pos) + (Neg) (Pos) + (Neg) = Pos + – 6 7 13 + (-7) = 15 + (-8)= (Pos) + (Pos) = Pos (Pos) + (Pos) = Pos 16 12 + 4 = 12 + 7 = 19 (Neg) + (Neg) = Neg (Neg) + (Neg) = Neg –16 –17 -14 + (-3) = -11 + (-5) = (Neg) + (Pos) = Neg (Neg) + (Pos) = Neg –5 -8 + 3 = –3 -7 + 4 = CFU Why did I place the sign I did above each number? How did I decide which rule to use? How did I solve the problem? Why did you place the sign you did above each number? How did you decide which rule to use? How did you solve the problem? 6th Grade Number Sense 2.3 (6Q) Solve addition, subtraction, multiplication, and division problems including those arising in concrete situations that use positive and negative integers and combinations of these operations. Lesson to be used by EDI–trained teachers only.

  10. Negative Positive -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 + + + – + – + – Closure 1. In your own words, what is an integer? 2. Add or subtract the positive and negative integers below. 3. What did you learn today about adding and subtracting positive and negative integers? Why is that important to you? (pair–share) 3. 2. 7 –10 4. 1. 18 77 (Neg) + (Neg) = Neg (Pos) + (Pos) = Pos 13 + (-6) = 12 + (6) = (Pos) + (Neg) (Pos) + (Pos) = Pos = Pos 6th Grade Number Sense 2.3 (6Q) Solve addition, subtraction, multiplication, and division problems including those arising in concrete situations that use positive and negative integers and combinations of these operations. Lesson to be used by EDI–trained teachers only.

  11. Name __________________________ Independent Practice Negative Positive -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 + + – – + – – – – + – + Integers are any positive or negative whole number or zero. Positive integers are whole numbers greater than zero. Negative integers are whole numbers less than zero. Add and subtract positive and negative integers. + + (Neg) + (Neg) = Neg (Pos) + (Pos) = Pos -10 25 + – (Pos) + (Neg) (Neg) + (Pos) = Pos = Neg 13 –6 (Pos) + (Neg) = Neg (Pos) + (Pos) = Pos 29 -1 21 + (8) = 8 + (-9) = (Neg) + (Pos) = Neg (Neg) + (Neg) = Neg –11 –42 -30 + (-12) = -17 + (6) = 6th Grade Number Sense 2.3 (6Q) Solve addition, subtraction, multiplication, and division problems including those arising in concrete situations that use positive and negative integers and combinations of these operations. Lesson to be used by EDI–trained teachers only.

  12. Periodic Review 1 Name ____________________________ Negative Positive -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 + + – – + – – – – + – + Integers are any positive or negative whole number or zero. Positive integers are whole numbers greater than zero. Negative integers are whole numbers less than zero. Add and subtract positive and negative integers. + + (Neg) + (Neg) = Neg (Pos) + (Pos) = Pos -8 25 + – (Pos) + (Neg) (Neg) + (Pos) = Pos = Neg 15 –6 (Pos) + (Neg) = Neg (Pos) + (Pos) = Pos -6 38 25 + (13) = 5 + (-11) = (Neg) + (Pos) = Pos (Neg) + (Neg) = Neg 5 –24 -10 + (-14) = -4 + (9) = 6th Grade Number Sense 2.3 (6Q) Solve addition, subtraction, multiplication, and division problems including those arising in concrete situations that use positive and negative integers and combinations of these operations. Lesson to be used by EDI–trained teachers only.

  13. Today we have talked about addition of real numbers. We will learn two ways to do this. Two ways

  14. Method 1: Add numbers using the rules. We just did this. Who can state the three rules?

  15. When adding numbers1) If the signs are the same you add the numbers and keep the sign.2) If the signs are different you subtract the smaller number from the larger and attach the sign of the larger number.3) Change subtracting into adding the opposite.

  16. To add two numbers using a number line you start by plotting the first and then move the second. Method 2 Use A Number Line

  17. Example 1: Add -4 + 9To perform this addition on the number line we first plot the number -4. Then since 9 is positive we move 9 units to the right of -4. Since we end up at 5, that is our answer.

  18. Example 2: Add 3 + -5To perform the addition on a number line we first plot the number 3. Then since -5 is negative we move 5 units to the left of 3. Since we end up at - 2 this is our answer.

  19. Example 3: Add -4 + -3To perform the addition on a number line we start by plotting -4. Then since -3 is negative we move 3 units to the left of -4. Since we end up at -7 this is our answer.

  20. Example 4 Add -5.2 + 0.To perform the addition on a number line we start by plotting -5.2 then since 0 is neither positive nor negative we simply stay at - 5.2 this is our answer.

  21. A) Add -12 + -7Since -12 and -7 have the same sign we add their absolute values and keep the sign. The answer is -19.

  22. B) Add -1.4 + 8.5. Since -1.4 and 8.5 have different signs we subtract 1.4 from 8.5 and attach the sign of the bigger number. Now 8.5 - 1.4 is 7.1 and we leave it positive.

  23. C) Add -36 + 21. Since the numbers have different signs we subtract 21 from 36 and attach a - symbol which is the sign of the larger absolute value. 36 - 21 = 15. Our answer is - 15.

  24. D) Add 1.5 + - 1.5. Notice that these numbers are opposites of each other so their sum is 0. This shows the property of opposites.

  25. For all real numbers a, a + 0 = 0 = 0 + a. 0 is called the additive identity. The identity property of 0

  26. E) Add – 7/8 + 0. Any number plus 0 is itself so are answer is – 7/8.

  27. Performing addition with several terms

  28. When adding multiple terms of different signs we first add all of the positive numbers and then I add all of the negative numbers. Then we add those 2 numbers. The Rule

  29. Example 6 Add 15 + - 2 + 7 + 14 + - 5 + - 12. Start by adding all of the positive numbers. 15 + 7 + 14 is 15 + 21 is 36. - 2 + - 5 + - 12 is - 7 + - 12 is - 19. Now 36 + - 19 must be performed. Since the numbers have different signs we subtract 19 from 36 and make our answer positive. 36 - 19 is 17 and we leave it positive since the 36 was positive.

  30. Example 7 Interest rates. Between 1994 and 2007 the average interest rate for a 30 year fixed rate mortgage dropped 2.5 percent rose 1.75 percent dropped 3.25 percent and rose 1 percent. By how much did the average interest rate change?

  31. The first thing we must do is to translate this problem into a number sentence. Note that each drop represents a - number and each raise represents a positive number. After translating we get the number sentence - 2.5 + 1.75 + - 3.25 + 1. Note that this represents the total change because we accounted for every drop and rise. Start by adding the positive numbers 1.75 + 1 equals 2.75. - 2.5 + - 3.25 = - 5.75. Sinse our numbers have different signs we subtract the smaller from the larger and attach the sign of the larger number. 5.75 - 2.75 = 3. Since the larger number was - are answer is -, -3 to be exact.

  32. Combining like terms

  33. Recall that like terms have the same variable part. When two numbers are like terms we may combine them. Recall that the coefficient of a term is the number multiplying the term. To combine like terms you simply add the coefficients of the terms.

  34. ExampleSimplify -4 + -10 + 8 + 1-6+-7+-12+3

  35. Like terms have the same variable part. The number on the side is the coefficient. To combine like terms means add the coefficients of the like terms. Terms with no variable are called constants. Combining Like Terms

  36. Combine like terms. • A) Add – 7x + 9xB) 2a + - 3B + - 5A + 9BC) 6 + y + - 3.5y + 2

  37. HW and IC • Italics mean 5th edition • HW 42/13-59 odds 43/71-83 odds • HW 41/7-57 ODDS 69-79 ODDS HW 49/21-123 odds HW 48/19-99, 109-115 ODDS IC 42/ 26,28,30,52,54,56,80,82 41/8,16,26,32,34,44,48,52 IC 49/31,36,42,50,88,104,114,124 IC 48/22,28,32,38,46,4858,70,80,92,96,110,114

  38. Quiz

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