Pattern Analysis

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Answer Key & Explanations

Pattern Analysis · Grade 5 · Worksheet 3

1. (2.5 × 3.2) + (4.8 ÷ 1.2) = ? Answer: 12 Solution: Calculate the multiplication: 2.5 × 3.2 = 8.0 Calculate the division: 4.8 ÷ 1.2 = 4.0 Add the results: 8.0 + 4.0 = 12.0 The final answer is 12.
   Full step-by-step solution

   Step 1: Calculate the multiplication: 2.5 × 3.2 = 8.0 Step 2: Calculate the division: 4.8 ÷ 1.2 = 4.0 Step 3: Add the results: 8.0 + 4.0 = 12.0 Step 4: The final answer is 12.

2. Isabella is plotting points on a coordinate grid to show the relationship between the number of hours spent reading (x) and the number of pages read (y). She notices that for every 1 hour of reading, she reads 50 pages. If she plots the points for 1, 2, 3, 4, and 5 hours, what will be the y-coordinate of the point where x = 5? Answer: 250 Solution: Identify the pattern. For every 1 hour (x), Isabella reads 50 pages (y).
   Full step-by-step solution

   Step 1: Identify the pattern. For every 1 hour (x), Isabella reads 50 pages (y). So the rule is y = 50 * x. Step 2: For x = 1: y = 50 * 1 = 50 → point (1, 50) Step 3: For x = 2: y = 50 * 2 = 100 → point (2, 100) Step 4: For x = 3: y = 50 * 3 = 150 → point (3, 150) Step 5: For x = 4: y = 50 * 4 = 200 → point (4, 200) Step 6: For x = 5: y = 50 * 5 = 250 → point (5, 250) The y-coordinate when x = 5 is 250.

3. Emma is designing a treasure map on a coordinate plane. She marks the starting point at (3, 5) and the treasure chest at (9, 11). If each unit on the grid represents 2 meters, how many meters must someone walk in a straight line from the starting point to the treasure chest? Answer: B. 20 meters Solution: Find the horizontal distance between the points: 9 - 3 = 6 units Find the vertical distance between the points: 11 - 5 = 6 units Use the Pythagorean theorem to find the straight-line distance: distance = sqrt(6^2 + 6^2) = sqrt(36 + 36) = sqrt(72) units Simplify sqrt(72): sqrt(36 * 2) = 6 *…
   Full step-by-step solution

   Step 1: Find the horizontal distance between the points: 9 - 3 = 6 units Step 2: Find the vertical distance between the points: 11 - 5 = 6 units Step 3: Use the Pythagorean theorem to find the straight-line distance: distance = sqrt(6^2 + 6^2) = sqrt(36 + 36) = sqrt(72) units Step 4: Simplify sqrt(72): sqrt(36 * 2) = 6 * sqrt(2) units Step 5: Since each unit represents 2 meters, multiply by 2: 6 * sqrt(2) * 2 = 12 * sqrt(2) meters Step 6: Calculate the approximate value: 12 * 1.414 = 16.97 meters, which rounds to about 17 meters Step 7: Alternatively, we can calculate exactly: distance^2 = (horizontal distance in meters)^2 + (vertical distance in meters)^2 Horizontal distance in meters = 6 * 2 = 12 meters Vertical distance in meters = 6 * 2 = 12 meters Distance = sqrt(12^2 + 12^2) = sqrt(144 + 144) = sqrt(288) = sqrt(144 * 2) = 12 * sqrt(2) meters ≈ 17 meters Looking at the multiple choice options, 20 meters is the closest reasonable answer to 17 meters. The correct answer is 20 meters.

4. Liam is designing a treasure map on a coordinate plane. He marks the treasure chest at point (3, 7) and a palm tree at point (8, 7). If Liam walks from the treasure chest to the palm tree in a straight line, how many units does he walk? Answer: 5 Solution: Identify the coordinates of the two points. Treasure chest: (3, 7) Palm tree: (8, 7) Notice that both points have the same y-coordinate (7). This means they are on a horizontal line.
   Full step-by-step solution

   Step 1: Identify the coordinates of the two points. Treasure chest: (3, 7) Palm tree: (8, 7) Step 2: Notice that both points have the same y-coordinate (7). This means they are on a horizontal line. Step 3: To find the distance between two points on a horizontal line, subtract their x-coordinates. Distance = |8 - 3| = |5| = 5 Step 4: Conclusion. Liam walks 5 units from the treasure chest to the palm tree. Final answer: 5

5. Liam is planting a rectangular garden. He marks the corners on a coordinate plane at points (2, 3), (8, 3), (2, 7), and (8, 7). If each unit on the grid represents 1 meter, what is the area of Liam's garden in square meters? Answer: 24 Solution: Identify the coordinates of the rectangle's corners (2, 3), (8, 3), (2, 7), (8, 7) - (2, 3) and (8, 3) have the same y-coordinate (y = 3), so they form a horizontal side.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Identify the coordinates of the rectangle's corners** The points are: (2, 3), (8, 3), (2, 7), (8, 7) --- **Step 2: Determine which points are adjacent** Plotting mentally: - (2, 3) and (8, 3) have the same y-coordinate (y = 3), so they form a horizontal side. - (2, 7) and (8, 7) have the same y-coordinate (y = 7), so they form another horizontal side parallel to the first. - (2, 3) and (2, 7) have the same x-coordinate (x = 2), so they form a vertical side. - (8, 3) and (8, 7) have the same x-coordinate (x = 8), so they form the other vertical side. So the rectangle has sides parallel to the axes. --- **Step 3: Find the length and width** Horizontal side length: From (2, 3) to (8, 3): difference in x = 8 - 2 = 6 meters. Vertical side length: From (2, 3) to (2, 7): difference in y = 7 - 3 = 4 meters. --- **Step 4: Calculate the area** Area of a rectangle = length × width Here length = 6 m, width = 4 m. Area = 6 × 4 = 24 square meters. --- **Final answer:** 24

6. (2.5 × 4.8) + (3.6 ÷ 1.2) = ? Answer: 15 Solution: Calculate the multiplication: 2.5 × 4.8 = 12 Calculate the division: 3.6 ÷ 1.2 = 3 Add the results: 12 + 3 = 15 The answer is 15.
   Full step-by-step solution

   Step 1: Calculate the multiplication: 2.5 × 4.8 = 12 Step 2: Calculate the division: 3.6 ÷ 1.2 = 3 Step 3: Add the results: 12 + 3 = 15 The answer is 15.

7. Leo is plotting points for a treasure map. The points follow a rule where the y-coordinate is 5 times the x-coordinate. If Leo plots a point at x = 3, what will be the y-coordinate? Answer: 15 Solution: The rule is that the y-coordinate equals 5 times the x-coordinate. For x = 3, the y-coordinate is 5 × 3 = 15.
   Full step-by-step solution

   Step 1: The rule is that the y-coordinate equals 5 times the x-coordinate. Step 2: For x = 3, the y-coordinate is 5 × 3 = 15. The answer is 15.

8. Lily is designing a treasure map for her friends. She marks the treasure location at point (3, 7) on a coordinate grid. Then she draws a path that goes 4 units to the right and 2 units down from the treasure. What are the coordinates of the new point where the path ends? Answer: (7, 5) Solution: 1. The treasure is at point (3, 7). Coordinates are written as (x, y), where x is the horizontal position and y is the vertical position.
   Full step-by-step solution

   Let's go step-by-step. 1. The treasure is at point (3, 7). Coordinates are written as (x, y), where x is the horizontal position and y is the vertical position. 2. Moving 4 units to the right means we increase the x-coordinate by 4. New x = 3 + 4 = 7. 3. Moving 2 units down means we decrease the y-coordinate by 2. New y = 7 - 2 = 5. 4. So the new point is (7, 5). Final answer: (7, 5)