Graph Points

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Answer Key & Explanations

Graph Points · Grade 5 · Worksheet 3

1. Emma is creating a map of her neighborhood on a coordinate plane where each unit represents 100 meters. Her house is located at point (5, 8) and the community center is located at point (12, 3). If Emma walks from her house directly to the community center following the grid lines (first east/west, then north/south), what is the total distance she walks in meters? Answer: 1200 Solution: Emma's house is at (5, 8) and community center is at (12, 3) Horizontal distance = |12 - 5| = |7| = 7 units Vertical distance = |3 - 8| = |-5| = 5 units Total units = 7 + 5 = 12 units Each unit represents 100 meters Total distance = 12 × 100 = 1200 meters The answer is 1200 meters.
   Full step-by-step solution

   Step 1: Find the horizontal distance between the points Emma's house is at (5, 8) and community center is at (12, 3) Horizontal distance = |12 - 5| = |7| = 7 units Step 2: Find the vertical distance between the points Vertical distance = |3 - 8| = |-5| = 5 units Step 3: Calculate the total distance along grid lines Total units = 7 + 5 = 12 units Step 4: Convert units to meters Each unit represents 100 meters Total distance = 12 × 100 = 1200 meters The answer is 1200 meters.

2. Lily is helping her teacher set up desks in the classroom. She needs to arrange them in rows and columns. If she places the first desk at point (3, 4) on the coordinate grid and the second desk at point (7, 9), how many units apart are the centers of these two desks? Answer: 6.4 Solution: To find the distance between the centers of the two desks at points (3, 4) and (7, 9), we use the distance formula. distance = square root of ( (x2 - x1)^2 + (y2 - y1)^2 ) Identify the coordinates.
   Full step-by-step solution

   To find the distance between the centers of the two desks at points (3, 4) and (7, 9), we use the distance formula. The distance formula is: distance = square root of ( (x2 - x1)^2 + (y2 - y1)^2 ) Step 1: Identify the coordinates. First desk: (x1, y1) = (3, 4) Second desk: (x2, y2) = (7, 9) Step 2: Calculate the difference in the x-coordinates. x2 - x1 = 7 - 3 = 4 Step 3: Calculate the difference in the y-coordinates. y2 - y1 = 9 - 4 = 5 Step 4: Square these differences. (4)^2 = 16 (5)^2 = 25 Step 5: Add the squared differences. 16 + 25 = 41 Step 6: Take the square root of the sum. square root of 41 Step 7: Calculate the numerical value. We know that 6^2 = 36 and 7^2 = 49. Since 41 is between 36 and 49, the square root is between 6 and 7. 6.4^2 = 40.96, which is very close to 41. Therefore, the distance is the square root of 41, which is approximately 6.4 units. Final Answer: 6.4

3. Plot point (9, 8) on the coordinate plane. What is the y-coordinate? Answer: 8 Solution: Look at the ordered pair (9, 8). The first number is the x-coordinate (9), and the second number is the y-coordinate (8).
   Full step-by-step solution

   Step 1: Look at the ordered pair (9, 8). The first number is the x-coordinate (9), and the second number is the y-coordinate (8). Step 2: The y-coordinate is 8, which means you move 8 units up from the origin (0,0) after moving 9 units right. Step 3: The question asks for the y-coordinate only, which is 8. The answer is 8.

4. Lily is helping her teacher set up a coordinate grid for a class treasure hunt. She places the treasure chest at point (4, 7) on the grid. Then, she needs to place a clue card exactly 3 units to the right and 2 units down from the treasure chest. What are the coordinates of the point where Lily should place the clue card? Answer: (7, 5) Solution: 1. The treasure chest is at point (4, 7). In coordinates, the first number is the x-coordinate (horizontal position) and the second number is the y-coordinate (vertical position).
   Full step-by-step solution

   Let's go step-by-step. 1. The treasure chest is at point (4, 7). In coordinates, the first number is the x-coordinate (horizontal position) and the second number is the y-coordinate (vertical position). 2. Moving 3 units to the right means we add 3 to the x-coordinate. New x = 4 + 3 = 7. 3. Moving 2 units down means we subtract 2 from the y-coordinate (because down means decreasing y). New y = 7 - 2 = 5. 4. So the new coordinates are (7, 5). Therefore, Lily should place the clue card at (7, 5).

5. Emma is designing a treasure hunt for her friends. She places the starting point at (2, 5) on a coordinate grid where each unit represents 10 feet. The first clue tells players to move 4 units east and 3 units north. What are the coordinates of the point where players will find the first clue? Answer: (6, 8) Solution: The starting point is at (2, 5). Moving 4 units east means adding 4 to the x-coordinate: 2 + 4 = 6. Moving 3 units north means adding 3 to the y-coordinate: 5 + 3 = 8.
   Full step-by-step solution

   Step 1: The starting point is at (2, 5). Step 2: Moving 4 units east means adding 4 to the x-coordinate: 2 + 4 = 6. Step 3: Moving 3 units north means adding 3 to the y-coordinate: 5 + 3 = 8. Step 4: The new coordinates are (6, 8). The answer is (6, 8).

6. Liam is helping his teacher arrange desks in their classroom. He places the first desk at coordinate (3, 4) on the grid map. Then he places another desk 2 units to the right and 3 units down from the first desk. What are the coordinates of the second desk? Answer: (5, 1) Solution: Identify the starting coordinates of the first desk. The first desk is at (3, 4). "2 units to the right" means we add 2 to the x-coordinate.
   Full step-by-step solution

   Let's go step-by-step. Step 1: Identify the starting coordinates of the first desk. The first desk is at (3, 4). Step 2: Understand the movement for the second desk. "2 units to the right" means we add 2 to the x-coordinate. "3 units down" means we subtract 3 from the y-coordinate (because moving down decreases the y-value on a grid). Step 3: Apply the movement to the x-coordinate. Starting x = 3 Add 2 → 3 + 2 = 5 So new x = 5. Step 4: Apply the movement to the y-coordinate. Starting y = 4 Subtract 3 → 4 - 3 = 1 So new y = 1. Step 5: Write the new coordinates. The second desk is at (5, 1). Final answer: (5, 1)

7. Sophia is drawing a map of her neighborhood on a coordinate plane. Her house is at point (6, 1), the library is at point (1, 6), and the park is at point (6, 6). If she connects these three points to form a triangle, what is the area of the triangle in square units? Answer: 12.5 Solution: Identify the points: House (6, 1), Library (1, 6), Park (6, 6). Notice that the Park (6, 6) and House (6, 1) share the same x-coordinate (6), so the vertical distance between them is 6 - 1 = 5 units.
   Full step-by-step solution

   Step 1: Identify the points: House (6, 1), Library (1, 6), Park (6, 6). Step 2: Notice that the Park (6, 6) and House (6, 1) share the same x-coordinate (6), so the vertical distance between them is 6 - 1 = 5 units. This is one leg of the right triangle. Step 3: Notice that the Park (6, 6) and Library (1, 6) share the same y-coordinate (6), so the horizontal distance between them is 6 - 1 = 5 units. This is the other leg of the right triangle. Step 4: The triangle is a right triangle with legs of 5 units and 5 units. The area of a triangle is (1/2) x base x height. Step 5: Area = (1/2) x 5 x 5 = (1/2) x 25 = 12.5 square units. The area of the triangle is 12.5 square units.