Number Sense. Grade 7 Mathematics: Number Sense - Operations with Positive and Negative Integers Subtopic Navigator Understanding Integers Integer Addition Integer Subtraction Integer Multiplication Integer Division Order of Operations Real-World Applications Cumulative Exercises Conclusion Lesson Objectives Master addition and subtraction of positive and negative integers Understand and apply rules for multiplying and dividing integers Solve complex problems using order of operations with integers Apply integer operations to real-world situations and word problems Develop strategies for solving multi-step integer problems Understanding Integers Integers include all positive and negative whole numbers, as well as zero. Understanding how to operate with integers is fundamental to algebra and higher mathematics. Integers are used to represent real-world situations like temperatures above and below zero, elevations above and below sea level, and financial gains and losses. Integer Addition Adding integers requires understanding how positive and negative values combine. When adding integers with the same sign, we add their absolute values and keep the common sign. When adding integers with different signs, we find the difference between their absolute values and keep the sign of the number with the greater absolute value. Example 1: Adding Integers with Different Signs Calculate: [latex]-47 + 35[/latex] Solution: The numbers have different signs, so we subtract the smaller absolute value from the larger: [latex]| -47 | - | 35 | = 47 - 35 = 12[/latex] Since -47 has the larger absolute value, the result is negative: [latex]-47 + 35 = -12[/latex] Example 2: Multiple Integer Addition Evaluate: [latex]-15 + 28 + (-32) + 17[/latex] Solution: Group positive and negative numbers: Positive sum: [latex]28 + 17 = 45[/latex] Negative sum: [latex]-15 + (-32) = -47[/latex] Combine: [latex]45 + (-47) = -2[/latex] [latex]-15 + 28 + (-32) + 17 = -2[/latex] Integer Addition Problems Calculate: [latex]-89 + 156[/latex] Find the sum: [latex]-45 + (-67) + 89 + (-23)[/latex] What number must be added to -128 to get 75? Evaluate: [latex]|-56| + |34| + |-23|[/latex] The temperature was -15°C and rose by 28°C. What is the new temperature? Integer Subtraction Subtracting integers can be transformed into addition by changing the subtraction sign to addition and changing the sign of the number being subtracted. This "add the opposite" strategy simplifies integer subtraction problems. Example 1: Complex Subtraction Calculate: [latex]-85 - (-49)[/latex] Solution: Change subtraction to addition of the opposite: [latex]-85 - (-49) = -85 + 49[/latex] Now add numbers with different signs: [latex]| -85 | - | 49 | = 85 - 49 = 36[/latex] Since -85 has the larger absolute value, the result is negative: [latex]-85 + 49 = -36[/latex] Example 2: Multiple Subtractions Evaluate: [latex]64 - (-28) - 95 - (-17)[/latex] Solution: Convert all subtractions to additions: [latex]64 - (-28) - 95 - (-17) = 64 + 28 + (-95) + 17[/latex] Group positive and negative numbers: Positive sum: [latex]64 + 28 + 17 = 109[/latex] Negative: [latex]-95[/latex] Combine: [latex]109 + (-95) = 14[/latex] [latex]64 - (-28) - 95 - (-17) = 14[/latex] Integer Subtraction Problems Calculate: [latex]-156 - 89[/latex] Evaluate: [latex]75 - (-128) - 56[/latex] What number must be subtracted from -85 to get 142? Find the difference: [latex]-|-34| - |56|[/latex] A submarine was at 250 meters below sea level and ascended 180 meters. What is its new position? Integer Multiplication The product of two integers with the same sign is positive, while the product of two integers with different signs is negative. This rule extends to multiplying multiple integers, where the sign of the product depends on the number of negative factors. Example 1: Multiple Integer Multiplication Calculate: [latex](-8) times 6 times (-2)[/latex] Solution: Multiply the first two numbers: [latex](-8) times 6 = -48[/latex] (different signs = negative) Multiply the result by the third number: [latex](-48) times (-2) = 96[/latex] (same signs = positive) [latex](-8) times 6 times (-2) = 96[/latex] Example 2: Complex Multiplication with Absolute Values Evaluate: [latex]|-12| times (-7) times |5|[/latex] Solution: First, evaluate the absolute values: [latex]|-12| = 12[/latex] and [latex]|5| = 5[/latex] Now multiply: [latex]12 times (-7) times 5[/latex] Multiply the first two numbers: [latex]12 times (-7) = -84[/latex] Multiply by the third number: [latex](-84) times 5 = -420[/latex] [latex]|-12| times (-7) times |5| = -420[/latex] Integer Multiplication Problems Calculate: [latex](-15) times (-8) times (-2)[/latex] Evaluate: [latex]|-9| times 7 times (-4)[/latex] Find the product of -12, 5, and -3 If [latex]a times b = -144[/latex] and [latex]a = -12[/latex], what is the value of b? A stock lost $5 per day for 8 consecutive days. What was the total change in value? Integer Division The quotient of two integers with the same sign is positive, while the quotient of two integers with different signs is negative. Division is the inverse operation of multiplication, and the same sign rules apply. Example 1: Complex Division Calculate: [latex](-144) div (-6) div 4[/latex] Solution: Divide the first two numbers: [latex](-144) div (-6) = 24[/latex] (same signs = positive) Divide the result by the third number: [latex]24 div 4 = 6[/latex] [latex](-144) div (-6) div 4 = 6[/latex] Example 2: Division with Absolute Values Evaluate: [latex]|-84| div (-7) div |3|[/latex] Solution: First, evaluate the absolute values: [latex]|-84| = 84[/latex] and [latex]|3| = 3[/latex] Now divide: [latex]84 div (-7) div 3[/latex] Divide the first two numbers: [latex]84 div (-7) = -12[/latex] Divide by the third number: [latex](-12) div 3 = -4[/latex] [latex]|-84| div (-7) div |3| = -4[/latex] Integer Division Problems Calculate: [latex](-216) div (-9) div (-4)[/latex] Evaluate: [latex]|156| div (-12) div |-13|[/latex] Find the quotient: [latex](-480) div 16 div (-5)[/latex] If [latex]a div b = -18[/latex] and [latex]b = -6[/latex], what is the value of a? A debt of $360 was paid equally by 9 people. How much did each person pay? Order of Operations with Integers When evaluating expressions with multiple operations, we follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This rule applies to all integer operations. Example 1: Complex Order of Operations Evaluate: [latex]48 div (-6) + (-5) times 3 - (-12)[/latex] Solution: Follow PEMDAS order: Division and multiplication first (left to right): [latex]48 div (-6) = -8[/latex] [latex](-5) times 3 = -15[/latex] Now the expression becomes: [latex]-8 + (-15) - (-12)[/latex] Addition and subtraction (left to right): [latex]-8 + (-15) = -23[/latex] [latex]-23 - (-12) = -23 + 12 = -11[/latex] [latex]48 div (-6) + (-5) times 3 - (-12) = -11[/latex] Example 2: Advanced Order of Operations Evaluate: [latex](-3)^3 + 4 times [15 - (-3) times 2] div 7[/latex] Solution: Follow PEMDAS order: Parentheses first: [latex](-3) times 2 = -6[/latex] Inside brackets: [latex]15 - (-6) = 15 + 6 = 21[/latex] Exponents: [latex](-3)^3 = -27[/latex] Multiplication and division (left to right): [latex]4 times 21 = 84[/latex] [latex]84 div 7 = 12[/latex] Addition: [latex]-27 + 12 = -15[/latex] [latex](-3)^3 + 4 times [15 - (-3) times 2] div 7 = -15[/latex] Order of Operations Problems Evaluate: [latex](-8) times 3 + 15 div (-5) - (-4)[/latex] Calculate: [latex]24 - 3 times [8 + (-12) div 4][/latex] Simplify: [latex](-5)^2 - 4 times [18 div (-3) + 7][/latex] Find the value of: [latex]|-9| times 2 + 36 div (-4) - (-5)^2[/latex] Evaluate: [latex]6 times [15 - (-3)^2] div (-4) + 11[/latex] Real-World Applications Integer operations are used to solve real-world problems involving temperatures, elevations, financial transactions, sports scores, and many other situations where values can be above or below a reference point. Example 1: Temperature Changes The temperature in a city was -8°C at 6 AM. It rose 15°C by noon, then fell 22°C by 6 PM, and rose again by 7°C by midnight. What was the temperature at midnight? Solution: Start: -8°C After first change: [latex]-8 + 15 = 7°C[/latex] After second change: [latex]7 - 22 = -15°C[/latex] After third change: [latex]-15 + 7 = -8°C[/latex] The temperature at midnight was -8°C. Example 2: Financial Transactions Sarah had $350 in her bank account. She wrote checks for $85, $120, and $45. She then deposited $200. If the bank charged a $15 fee for being overdrawn, and her balance went negative at one point, what is her final balance? Solution: Start: $350 After checks: [latex]350 - 85 - 120 - 45 = 350 - 250 = 100[/latex] After deposit: [latex]100 + 200 = 300[/latex] Since her balance never went negative, no fee is charged. Final balance: $300 Real-World Application Problems A scuba diver is at 25 meters below sea level. She descends another 18 meters, then ascends 32 meters. What is her final position relative to sea level? A football team gained 8 yards, lost 15 yards, gained 22 yards, and lost 7 yards on four consecutive plays. What was their net yardage? The stock market closed at -145 points on Monday. On Tuesday, it gained 78 points. On Wednesday, it lost 112 points. What was the closing value on Wednesday? A company's profit was -$12,500 in January, $8,300 in February, -$4,200 in March, and $15,600 in April. What was their total profit for the four months? The temperature changed from -8°C to 15°C to -3°C to 22°C over four days. What was the average temperature? Cumulative Exercises Evaluate: [latex](-7) times 8 + 36 div (-4) - (-5)^2[/latex] Calculate: [latex]|-15| times (-3) + 42 div (-7) - (-8)[/latex] Simplify: [latex](-4)^3 + 5 times [24 - (-6) times 3] div 9[/latex] Find the value of: [latex]18 - 4 times [12 + (-16) div 2] + (-3)^2[/latex] Evaluate: [latex](-9) times 2 + 56 div (-8) - (-4)^2[/latex] A submarine was at 120 meters below sea level. It descended 85 meters, then ascended 150 meters. What is its final position? The temperature was -12°C and rose 25°C during the day, then fell 18°C at night. What was the final temperature? A company had a profit of -$8,500 in Quarter 1, $12,300 in Quarter 2, -$3,200 in Quarter 3, and $15,800 in Quarter 4. What was their total annual profit? A football team had these yardages on five plays: +12, -8, +25, -15, +7. What was their net yardage? Evaluate: [latex](-6)^2 - 3 times [28 div (-4) + 9] - (-5)[/latex] Show/Hide Answers Problem: Evaluate: [latex](-7) times 8 + 36 div (-4) - (-5)^2[/latex] Answer: [latex](-7) times 8 = -56[/latex], [latex]36 div (-4) = -9[/latex], [latex](-5)^2 = 25[/latex] [latex]-56 + (-9) - 25 = -65 - 25 = -90[/latex] Problem: Calculate: [latex]|-15| times (-3) + 42 div (-7) - (-8)[/latex] Answer: [latex]|-15| = 15[/latex], [latex]15 times (-3) = -45[/latex], [latex]42 div (-7) = -6[/latex] [latex]-45 + (-6) - (-8) = -51 + 8 = -43[/latex] Problem: Simplify: [latex](-4)^3 + 5 times [24 - (-6) times 3] div 9[/latex] Answer: [latex](-4)^3 = -64[/latex], [latex](-6) times 3 = -18[/latex], [latex]24 - (-18) = 42[/latex] [latex]5 times 42 = 210[/latex], [latex]210 div 9 = 23.overline{3}[/latex] [latex]-64 + 23.overline{3} = -40.overline{6}[/latex] Problem: Find the value of: [latex]18 - 4 times [12 + (-16) div 2] + (-3)^2[/latex] Answer: [latex](-16) div 2 = -8[/latex], [latex]12 + (-8) = 4[/latex], [latex]4 times 4 = 16[/latex] [latex](-3)^2 = 9[/latex] [latex]18 - 16 + 9 = 2 + 9 = 11[/latex] Problem: Evaluate: [latex](-9) times 2 + 56 div (-8) - (-4)^2[/latex] Answer: [latex](-9) times 2 = -18[/latex], [latex]56 div (-8) = -7[/latex], [latex](-4)^2 = 16[/latex] [latex]-18 + (-7) - 16 = -25 - 16 = -41[/latex] Problem: A submarine was at 120 meters below sea level. It descended 85 meters, then ascended 150 meters. What is its final position? Answer: Start: -120 m, Descend: -120 - 85 = -205 m, Ascend: -205 + 150 = -55 m Final position: 55 meters below sea level Problem: The temperature was -12°C and rose 25°C during the day, then fell 18°C at night. What was the final temperature? Answer: Start: -12°C, Rise: -12 + 25 = 13°C, Fall: 13 - 18 = -5°C Final temperature: -5°C Problem: A company had a profit of -$8,500 in Quarter 1, $12,300 in Quarter 2, -$3,200 in Quarter 3, and $15,800 in Quarter 4. What was their total annual profit? Answer: [latex]-8,500 + 12,300 + (-3,200) + 15,800 = 3,800 + 12,600 = 16,400[/latex] Total annual profit: $16,400 Problem: A football team had these yardages on five plays: +12, -8, +25, -15, +7. What was their net yardage? Answer: [latex]12 + (-8) + 25 + (-15) + 7 = 4 + 10 + 7 = 21[/latex] Net yardage: +21 yards Problem: Evaluate: [latex](-6)^2 - 3 times [28 div (-4) + 9] - (-5)[/latex] Answer: [latex](-6)^2 = 36[/latex], [latex]28 div (-4) = -7[/latex], [latex]-7 + 9 = 2[/latex] [latex]3 times 2 = 6[/latex], [latex]36 - 6 - (-5) = 30 + 5 = 35[/latex] Conclusion/Recap Mastering operations with positive and negative integers is essential for success in algebra and higher mathematics. The rules for adding, subtracting, multiplying, and dividing integers form the foundation for working with rational numbers, solving equations, and analyzing real-world situations. Regular practice with challenging problems helps develop fluency and confidence in integer operations. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c
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