Negative Numbers

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Answer Key & Explanations

Negative Numbers · Grade 6 · Worksheet 2

1. A rectangular garden is drawn on a coordinate plane with vertices at (-3, 2), (5, 2), (5, -4), and (-3, -4). What is the perimeter of this garden? Answer: 28 Solution: A = (-3, 2) B = (5, 2) C = (5, -4) D = (-3, -4) Plotting them in order: A → B → C → D → A. A = (-3, 2), B = (5, 2) These points have the same y-coordinate (y = 2), so AB is horizontal.
   Full step-by-step solution

   Let's find the perimeter step by step. --- **Step 1: Identify the vertices in order** The vertices are: A = (-3, 2) B = (5, 2) C = (5, -4) D = (-3, -4) Plotting them in order: A → B → C → D → A. --- **Step 2: Determine the length of side AB** A = (-3, 2), B = (5, 2) These points have the same y-coordinate (y = 2), so AB is horizontal. Length AB = |5 - (-3)| = |5 + 3| = 8 --- **Step 3: Determine the length of side BC** B = (5, 2), C = (5, -4) These points have the same x-coordinate (x = 5), so BC is vertical. Length BC = | -4 - 2 | = | -6 | = 6 --- **Step 4: Determine the length of side CD** C = (5, -4), D = (-3, -4) These points have the same y-coordinate (y = -4), so CD is horizontal. Length CD = | -3 - 5 | = | -8 | = 8 --- **Step 5: Determine the length of side DA** D = (-3, -4), A = (-3, 2) These points have the same x-coordinate (x = -3), so DA is vertical. Length DA = | 2 - (-4) | = | 2 + 4 | = 6 --- **Step 6: Calculate the perimeter** Perimeter = AB + BC + CD + DA Perimeter = 8 + 6 + 8 + 6 Perimeter = 28 --- **Step 7: Conclusion** The garden is a rectangle with length 8 and width 6. Perimeter = 2 × (length + width) = 2 × (8 + 6) = 2 × 14 = 28. **Final answer: 28**

2. A triangle is drawn on a coordinate plane with vertices at (-4, 3), (2, 3), and (2, -5). What is the area of this triangle in square units? Answer: 24 Solution: Identify the coordinates: A(-4, 3), B(2, 3), C(2, -5) Notice that points A and B have the same y-coordinate (3), so AB is horizontal Points B and C have the same x-coordinate (2), so BC is vertical This means AB and BC are perpendicular, forming a right triangle with the right angle at point B…
   Full step-by-step solution

   Step 1: Identify the coordinates: A(-4, 3), B(2, 3), C(2, -5) Step 2: Notice that points A and B have the same y-coordinate (3), so AB is horizontal Step 3: Points B and C have the same x-coordinate (2), so BC is vertical Step 4: This means AB and BC are perpendicular, forming a right triangle with the right angle at point B Step 5: Calculate the length of base AB: |2 - (-4)| = |6| = 6 units Step 6: Calculate the length of height BC: |3 - (-5)| = |8| = 8 units Step 7: Apply the triangle area formula: Area = (1/2) × base × height Step 8: Area = (1/2) × 6 × 8 = (1/2) × 48 = 24 Step 9: The area of the triangle is 24 square units.

3. -15 + 8 - (-3) = ? Answer: -4 Solution: Start with the expression: -15 + 8 - (-3) Simplify the subtraction of the negative number: -(-3) becomes +3 Rewrite the expression: -15 + 8 + 3 Add the numbers from left to right: -15 + 8 = -7 Continue adding: -7 + 3 = -4 The final result is -4.
   Full step-by-step solution

   Step 1: Start with the expression: -15 + 8 - (-3) Step 2: Simplify the subtraction of the negative number: -(-3) becomes +3 Step 3: Rewrite the expression: -15 + 8 + 3 Step 4: Add the numbers from left to right: -15 + 8 = -7 Step 5: Continue adding: -7 + 3 = -4 Step 6: The final result is -4.

4. Ava is tracking the altitude of a hot air balloon during a flight. The balloon starts at 1,250 meters above sea level. It first ascends 850 meters to catch a favorable wind current. Then it descends 1,100 meters to fly through a valley. Finally, it ascends 475 meters to reach its landing zone. What is the balloon's final altitude relative to sea level? Answer: 1475 Solution: The balloon starts at 1,250 meters. It ascends 850 meters. This is a positive change: 1,250 + 850 = 2,100 meters.
   Full step-by-step solution

   Step 1: The balloon starts at 1,250 meters. Step 2: It ascends 850 meters. This is a positive change: 1,250 + 850 = 2,100 meters. Step 3: It descends 1,100 meters. This is a negative change: 2,100 - 1,100 = 1,000 meters. Step 4: It ascends 475 meters. This is a positive change: 1,000 + 475 = 1,475 meters. The final altitude is 1,475 meters above sea level.

5. Liam is monitoring the temperature in a science experiment. At 9:00 AM, the temperature was -5°C. Over the next 3 hours, it rose by 8 degrees. Then, a chemical reaction caused it to drop by 11 degrees. What was the final temperature? Answer: -8 Solution: At 9:00 AM, the temperature was -5°C. First change — temperature rises by 8 degrees -5 + 8 = 3 So after 3 hours, the temperature is 3°C.
   Full step-by-step solution

   Let's go step-by-step. **Step 1: Initial temperature** At 9:00 AM, the temperature was -5°C. **Step 2: First change — temperature rises by 8 degrees** Rising means we add: -5 + 8 = 3 So after 3 hours, the temperature is 3°C. **Step 3: Second change — temperature drops by 11 degrees** Dropping means we subtract: 3 - 11 = -8 So the final temperature is -8°C. **Final answer:** -8

6. Noah is tracking the depth of a submarine during a research dive. The submarine starts at sea level (0 meters). It first descends 125 meters to observe a coral reef. Then it ascends 85 meters to study a school of fish. Finally, it descends 215 meters to reach the ocean floor. What is the submarine's final depth relative to sea level? Answer: -255 meters Solution: Start at sea level: 0 meters Descend 125 meters: 0 - 125 = -125 meters Ascend 85 meters: -125 + 85 = -40 meters Descend 215 meters: -40 - 215 = -255 meters The submarine's final depth is 255 meters below sea level, which is represented as -255 meters.
   Full step-by-step solution

   Step 1: Start at sea level: 0 meters Step 2: Descend 125 meters: 0 - 125 = -125 meters Step 3: Ascend 85 meters: -125 + 85 = -40 meters Step 4: Descend 215 meters: -40 - 215 = -255 meters The submarine's final depth is 255 meters below sea level, which is represented as -255 meters.

7. -4 × (3 + (-7)) = ? Answer: 16 Solution: First, solve the operation inside the parentheses: 3 + (-7) = -4 Now the expression becomes: -4 × (-4) When multiplying two negative numbers, the result is positive: -4 × (-4) = 16 The answer is 16.
   Full step-by-step solution

   Step 1: First, solve the operation inside the parentheses: 3 + (-7) = -4 Step 2: Now the expression becomes: -4 × (-4) Step 3: When multiplying two negative numbers, the result is positive: -4 × (-4) = 16 The answer is 16.

8. -3 × (4 - 9) + (-12) ÷ (-2) = ? Answer: 21 Solution: Solve the operation inside the parentheses: 4 - 9 = -5 The expression becomes: -3 × (-5) + (-12) ÷ (-2) Perform multiplication and division from left to right: -3 × (-5) = 15 (since a negative times a negative is positive) (-12) ÷ (-2) = 6 (since a negative divided by a negative is positive) The…
   Full step-by-step solution

   Step 1: Solve the operation inside the parentheses: 4 - 9 = -5 Step 2: The expression becomes: -3 × (-5) + (-12) ÷ (-2) Step 3: Perform multiplication and division from left to right: -3 × (-5) = 15 (since a negative times a negative is positive) Step 4: (-12) ÷ (-2) = 6 (since a negative divided by a negative is positive) Step 5: The expression becomes: 15 + 6 Step 6: Perform the addition: 15 + 6 = 21 The answer is 21.