Four Quadrants

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Answer Key & Explanations

Four Quadrants · Grade 6 · Worksheet 2

1. Liam is tracking the temperature changes in his city over a week. On Monday morning, the temperature was 12°C. By Monday evening, it dropped 8°C. On Tuesday morning, it rose 5°C from Monday evening's temperature, but then a cold front came through and dropped it 15°C by Tuesday night. What was the final temperature on Tuesday night? Answer: -6°C Solution: Monday morning: 12°C. It dropped 8°C from Monday morning: 12°C − 8°C = 4°C. So Monday evening temperature = 4°C.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Monday morning temperature** Monday morning: 12°C. --- **Step 2: Monday evening temperature** It dropped 8°C from Monday morning: 12°C − 8°C = 4°C. So Monday evening temperature = 4°C. --- **Step 3: Tuesday morning temperature** It rose 5°C from Monday evening: 4°C + 5°C = 9°C. So Tuesday morning temperature = 9°C. --- **Step 4: Tuesday night temperature** A cold front dropped it 15°C from Tuesday morning: 9°C − 15°C = −6°C. --- **Step 5: Final answer** The temperature on Tuesday night was −6°C.

2. Ava is designing a city map on a coordinate plane for her social studies project. She places the town hall at coordinates (7, -8). The library is located 15 units to the left and 12 units up from the town hall. What are the coordinates of the library? Answer: (-8, 4) Solution: The town hall is at (7, -8). Moving 15 units to the left means subtracting 15 from the x-coordinate: 7 - 15 = -8. Moving 12 units up means adding 12 to the y-coordinate: -8 + 12 = 4.
   Full step-by-step solution

   Step 1: The town hall is at (7, -8). Step 2: Moving 15 units to the left means subtracting 15 from the x-coordinate: 7 - 15 = -8. Step 3: Moving 12 units up means adding 12 to the y-coordinate: -8 + 12 = 4. Step 4: The library is at (-8, 4). The answer is (-8, 4).

3. Mere is designing a rectangular garden on a coordinate plane. The corners of the garden are at (-6, 4), (2, 4), (2, -8), and (-6, -8). She wants to place a water fountain at the midpoint of the diagonal from (-6, 4) to (2, -8). What are the coordinates of the fountain? Answer: (-2, -2) Solution: Identify the endpoints of the diagonal: (-6, 4) and (2, -8). Find the midpoint by averaging the x-coordinates: (-6 + 2) / 2 = (-4) / 2 = -2. Average the y-coordinates: (4 + (-8)) / 2 = (-4) / 2 = -2.
   Full step-by-step solution

   Step 1: Identify the endpoints of the diagonal: (-6, 4) and (2, -8). Step 2: Find the midpoint by averaging the x-coordinates: (-6 + 2) / 2 = (-4) / 2 = -2. Step 3: Average the y-coordinates: (4 + (-8)) / 2 = (-4) / 2 = -2. Step 4: The midpoint is (-2, -2). The fountain should be placed at (-2, -2).

4. Emma is designing a city park on a coordinate plane. She places a large fountain at point (5, -10). A walking path goes from the fountain to a picnic area that is located 20 units directly west and 15 units directly north of the fountain. What are the coordinates of the picnic area? Answer: (-15, 5) Solution: The fountain is at (5, -10). Moving 20 units west means subtracting 20 from the x-coordinate: 5 - 20 = -15. Moving 15 units north means adding 15 to the y-coordinate: -10 + 15 = 5.
   Full step-by-step solution

   Step 1: The fountain is at (5, -10). Moving 20 units west means subtracting 20 from the x-coordinate: 5 - 20 = -15. Step 2: Moving 15 units north means adding 15 to the y-coordinate: -10 + 15 = 5. Step 3: The picnic area is at (-15, 5). The answer is (-15, 5).

5. Olivia is drawing a geometric design on a coordinate plane. She plots three points: A at (-9, 5), B at (7, 5), and C at (7, -11). She then plots point D so that the four points form a rectangle. What are the coordinates of point D? Answer: (-9, -11) Solution: Identify the given points: A = (-9, 5), B = (7, 5), C = (7, -11). Notice that A and B share the same y-coordinate (5), so they form the top side of the rectangle.
   Full step-by-step solution

   Step 1: Identify the given points: A = (-9, 5), B = (7, 5), C = (7, -11). Step 2: Notice that A and B share the same y-coordinate (5), so they form the top side of the rectangle. Step 3: B and C share the same x-coordinate (7), so they form the right side of the rectangle. Step 4: For a rectangle, opposite sides are parallel and equal. The missing point D must have the same x-coordinate as A (since D is directly below A) and the same y-coordinate as C (since D is directly to the left of C). Step 5: So D's x-coordinate is -9 (same as A), and D's y-coordinate is -11 (same as C). Step 6: Therefore, point D is at (-9, -11).

6. (-3, 4) + (2, -7) = ? Answer: (-1, -3) Solution: We are adding two points in coordinate form: (-3, 4) and (2, -7). Identify the x-coordinates and y-coordinates separately. First point: x1 = -3, y1 = 4 Second point: x2 = 2, y2 = -7 Add the x-coordinates together.
   Full step-by-step solution

   We are adding two points in coordinate form: (-3, 4) and (2, -7). Step 1: Identify the x-coordinates and y-coordinates separately. First point: x1 = -3, y1 = 4 Second point: x2 = 2, y2 = -7 Step 2: Add the x-coordinates together. x1 + x2 = (-3) + 2 This equals -1. Step 3: Add the y-coordinates together. y1 + y2 = 4 + (-7) This equals 4 - 7 = -3. Step 4: Combine the results into a new point. The resulting point is (-1, -3). Final answer: (-1, -3)

7. (-3, 4) + (7, -2) = ? Answer: (4, 2) Solution: We are adding two vectors: (-3, 4) and (7, -2). Identify the components of each vector. The first vector has x-component = -3 and y-component = 4.
   Full step-by-step solution

   We are adding two vectors: (-3, 4) and (7, -2). Step 1: Identify the components of each vector. The first vector has x-component = -3 and y-component = 4. The second vector has x-component = 7 and y-component = -2. Step 2: Add the x-components together. x_total = (-3) + 7 x_total = 4 Step 3: Add the y-components together. y_total = 4 + (-2) y_total = 4 - 2 y_total = 2 Step 4: Combine the results into a single vector. The resulting vector is (x_total, y_total) = (4, 2). Final Answer: (4, 2)

8. Charlotte is helping her family plan a new garden layout on a coordinate grid. They decide to place a birdbath at point (-7, 2). The rose bush is located 12 units directly to the right and 9 units directly down from the birdbath. What are the coordinates of the rose bush? Answer: (5, -7) Solution: Start with the birdbath's coordinates: (-7, 2). Moving 12 units to the right means adding 12 to the x-coordinate: -7 + 12 = 5. Moving 9 units down means subtracting 9 from the y-coordinate: 2 - 9 = -7.
   Full step-by-step solution

   Step 1: Start with the birdbath's coordinates: (-7, 2). Step 2: Moving 12 units to the right means adding 12 to the x-coordinate: -7 + 12 = 5. Step 3: Moving 9 units down means subtracting 9 from the y-coordinate: 2 - 9 = -7. Step 4: The rose bush is located at (5, -7). The answer is (5, -7).