Pattern Analysis

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Answer Key & Explanations

Pattern Analysis · Grade 5 · Worksheet 1

1. Liam is plotting points on a coordinate grid to show a pattern. He plots (1, 7), (3, 11), and (5, 15). If the pattern continues, what will be the y-coordinate when the x-coordinate is 9? Answer: 23 Solution: Identify the pattern in the x-coordinates: 1, 3, 5. They increase by 2 each time. Step 2: Identify the pattern in the y-coordinates: 7, 11, 15.
   Full step-by-step solution

   Step 1: Identify the pattern in the x-coordinates: 1, 3, 5. They increase by 2 each time. Step 2: Identify the pattern in the y-coordinates: 7, 11, 15. They increase by 4 each time. Step 3: Find the relationship between x and y. For x = 1, y = 7. For x = 3, y = 11. For x = 5, y = 15. Notice that y is always 2 times x plus 5? Check: 2 * 1 + 5 = 7, 2 * 3 + 5 = 11, 2 * 5 + 5 = 15. So the rule is y = 2x + 5. Step 4: Apply the rule for x = 9: y = 2 * 9 + 5 = 18 + 5 = 23. The answer is 23.

2. 12.5 × 4.2 = ? Answer: 52.5 Solution: Remove decimals temporarily to work with whole numbers. 12.5 has 1 decimal place (the .5). 4.2 has 1 decimal place (the .2).
   Full step-by-step solution

   Let's multiply 12.5 by 4.2 step by step. Step 1: Remove decimals temporarily to work with whole numbers. 12.5 has 1 decimal place (the .5). 4.2 has 1 decimal place (the .2). So together, there are 1 + 1 = 2 decimal places total in the original problem. We will put them back at the end. Step 2: Multiply the numbers as if they were whole numbers: 125 × 42. Step 3: Break it into simpler parts. First, 125 × 40 = 125 × 4 × 10 = 500 × 10 = 5000. Second, 125 × 2 = 250. Step 4: Add the results: 5000 + 250 = 5250. Step 5: Now put back the decimal point. Since we ignored 2 decimal places total earlier (from 12.5 and 4.2), we need to move the decimal point 2 places to the left in our result. 5250 becomes 52.50. Step 6: Simplify: 52.50 is the same as 52.5. Final answer: 52.5

3. (4.5 × 3.2) - (2.1 ÷ 0.7) = ? Answer: 11.4 Solution: Calculate 4.5 × 3.2 4.5 × 3.2 = 14.4 Calculate 2.1 ÷ 0.7 2.1 ÷ 0.7 = 3 14.4 - 3 = 11.4 The answer is 11.4.
   Full step-by-step solution

   Step 1: Calculate 4.5 × 3.2 4.5 × 3.2 = 14.4 Step 2: Calculate 2.1 ÷ 0.7 2.1 ÷ 0.7 = 3 Step 3: Subtract the results 14.4 - 3 = 11.4 The answer is 11.4.

4. 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

5. Emma is designing a treasure hunt for her friends using a coordinate grid. She marks the starting point at (3, 5). The first clue tells them to move 4 units to the right and 2 units up. The second clue says to move 3 units to the left and 6 units down. What are the final coordinates of the treasure? Answer: (4, 1) Solution: When working with coordinates on a grid, moving right increases the x-value, while moving left decreases it. Moving up increases the y-value, while moving down decreases it.
   Full step-by-step solution

   When working with coordinates on a grid, moving right increases the x-value, while moving left decreases it. Moving up increases the y-value, while moving down decreases it. You can track these changes step by step to find the final position.

6. Emma is designing a treasure hunt on a coordinate grid. She marks the starting point at (3, 5). Her first clue says to move 4 units to the right and 2 units up to find the next clue. The second clue says to move 3 units to the left and 6 units down to find the treasure. What are the coordinates of the treasure? Answer: (4, 1) Solution: Start at the initial point (3, 5) Move 4 units right: x-coordinate increases by 4, so 3 + 4 = 7 Move 2 units up: y-coordinate increases by 2, so 5 + 2 = 7 After first clue, position is (7, 7) Move 3 units left: x-coordinate decreases by 3, so 7 - 3 = 4 Move 6 units down: y-coordinate decreases…
   Full step-by-step solution

   Step 1: Start at the initial point (3, 5) Step 2: Move 4 units right: x-coordinate increases by 4, so 3 + 4 = 7 Step 3: Move 2 units up: y-coordinate increases by 2, so 5 + 2 = 7 Step 4: After first clue, position is (7, 7) Step 5: Move 3 units left: x-coordinate decreases by 3, so 7 - 3 = 4 Step 6: Move 6 units down: y-coordinate decreases by 6, so 7 - 6 = 1 Step 7: Final treasure location is (4, 1)

7. Mason is training for a charity run. He creates a pattern to track his training. On Day 1, he runs 8 minutes. On Day 2, he runs 16 minutes. On Day 3, he runs 24 minutes. On Day 4, he runs 32 minutes. If this pattern continues, write the coordinates for the first 5 days (day number, minutes run) and determine what the y-coordinate will be on Day 7. Answer: 56 Solution: Identify the pattern. The minutes run each day are: Day 1 = 8, Day 2 = 16, Day 3 = 24, Day 4 = 32. The minutes increase by 8 each day.
   Full step-by-step solution

   Step 1: Identify the pattern. The minutes run each day are: Day 1 = 8, Day 2 = 16, Day 3 = 24, Day 4 = 32. The minutes increase by 8 each day. Step 2: Write the rule. The number of minutes is 8 times the day number: y = 8x. Step 3: Write the coordinates for the first 5 days. Day 1: (1, 8), Day 2: (2, 16), Day 3: (3, 24), Day 4: (4, 32), Day 5: (5, 40). Step 4: Find the y-coordinate for Day 7. Using the rule y = 8x, substitute x = 7: y = 8 x 7 = 56. The answer is 56.

8. (2.5 × 4) + (1.75 × 6) = ? Answer: 20.5 Solution: Calculate 2.5 × 4 = 10 Calculate 1.75 × 6 = 10.5 Add the two results: 10 + 10.5 = 20.5 The answer is 20.5.
   Full step-by-step solution

   Step 1: Calculate 2.5 × 4 = 10 Step 2: Calculate 1.75 × 6 = 10.5 Step 3: Add the two results: 10 + 10.5 = 20.5 The answer is 20.5.