Pattern Analysis

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Answer Key & Explanations

Pattern Analysis · Grade 5 · Worksheet 2

1. Liam is creating a treasure map on a coordinate plane. He marks his starting point at (2, 3). His first clue says to move 4 units right and 2 units up. His second clue says to move 1 unit left and 5 units up. What are the final coordinates of the treasure? Answer: (5, 10) Solution: Start at the initial point. Liam starts at (2, 3). First clue says: move 4 units right and 2 units up.
   Full step-by-step solution

   Let's go step-by-step. **Step 1: Start at the initial point.** Liam starts at (2, 3). **Step 2: Apply the first clue.** First clue says: move 4 units right and 2 units up. - Moving right means add to the x-coordinate: 2 + 4 = 6 - Moving up means add to the y-coordinate: 3 + 2 = 5 After first clue, position is (6, 5). **Step 3: Apply the second clue.** Second clue says: move 1 unit left and 5 units up. - Moving left means subtract from the x-coordinate: 6 - 1 = 5 - Moving up means add to the y-coordinate: 5 + 5 = 10 After second clue, position is (5, 10). **Step 4: Final answer.** The treasure is at (5, 10).

2. Pattern: (12, 36), (15, 45), (18, 54), (21, ?) Answer: 63 Solution: Examine the pattern. For (12, 36), 12 × 3 = 36. For (15, 45), 15 × 3 = 45.
   Full step-by-step solution

   Step 1: Examine the pattern. For (12, 36), 12 × 3 = 36. For (15, 45), 15 × 3 = 45. For (18, 54), 18 × 3 = 54. Step 2: The rule is y = 3 × x. Step 3: For x = 21, y = 3 × 21 = 63. The answer is 63.

3. Pattern: (7,14), (8,16), (9,18). Find the next point. Answer: (10,20) Solution: Look at the pattern in the points. For (7,14): 14 = 7 × 2. For (8,16): 16 = 8 × 2.
   Full step-by-step solution

   Step 1: Look at the pattern in the points. For (7,14): 14 = 7 × 2. For (8,16): 16 = 8 × 2. For (9,18): 18 = 9 × 2. So the rule is y = 2x. Step 2: The x-coordinates increase by 1 each time: 7, 8, 9. The next x-coordinate is 10. Step 3: Apply the rule: y = 2 × 10 = 20. The next point is (10,20).

4. Pattern: (12, 24), (15, 30), (18, 36). Find the next point. Answer: (21, 42) Solution: Identify the pattern in the x-coordinates: 12, 15, 18. Each increases by 3. Identify the relationship between x and y in each point: For (12, 24), y = 2 × 12.
   Full step-by-step solution

   Step 1: Identify the pattern in the x-coordinates: 12, 15, 18. Each increases by 3. So the next x-coordinate is 18 + 3 = 21. Step 2: Identify the relationship between x and y in each point: For (12, 24), y = 2 × 12. For (15, 30), y = 2 × 15. For (18, 36), y = 2 × 18. So y is always twice x. Step 3: Apply the rule to the next x-coordinate: y = 2 × 21 = 42. The next point is (21, 42).

5. (3/4 × 8) + (2.5 × 4) = ? Answer: 16 Solution: Calculate 3/4 × 8 3/4 × 8 = 24/4 = 6 Calculate 2.5 × 4 2.5 × 4 = 10 Add the results from Step 1 and Step 2 6 + 10 = 16 The answer is 16.
   Full step-by-step solution

   Step 1: Calculate 3/4 × 8 3/4 × 8 = 24/4 = 6 Step 2: Calculate 2.5 × 4 2.5 × 4 = 10 Step 3: Add the results from Step 1 and Step 2 6 + 10 = 16 The answer is 16.

6. Liam is designing a treasure map using a coordinate grid. He marks the treasure at point (3, 7) and a clue at point (8, 7). If Liam walks in a straight line from the treasure to the clue, how many units does he walk? Answer: 5 Solution: Treasure at (3, 7) Clue at (8, 7) Observe the coordinates. The y-coordinate is the same for both points (y = 7). That means the two points are on a horizontal line.
   Full step-by-step solution

   We are given two points: Treasure at (3, 7) Clue at (8, 7) Step 1: Observe the coordinates. The y-coordinate is the same for both points (y = 7). That means the two points are on a horizontal line. Step 2: To find the distance between two points on a horizontal line, we subtract the x-coordinates and take the absolute value. Distance = |x2 - x1| = |8 - 3| = |5| = 5. Step 3: Conclusion. Liam walks 5 units in a straight line from the treasure to the clue. Answer: 5

7. Charlotte is tracking the number of flowers blooming in her garden each week. She notices a pattern: in Week 1, there are 2 flowers; in Week 2, there are 7 flowers; in Week 3, there are 12 flowers; in Week 4, there are 17 flowers. If this pattern continues, plot the points on a coordinate grid where the x-coordinate is the week number and the y-coordinate is the number of flowers. What will be the coordinates of the point for Week 7? Answer: (7, 32) Solution: Identify the pattern. From Week 1 to Week 2: 2 to 7, increase of 5. From Week 2 to Week 3: 7 to 12, increase of 5.
   Full step-by-step solution

   Step 1: Identify the pattern. From Week 1 to Week 2: 2 to 7, increase of 5. From Week 2 to Week 3: 7 to 12, increase of 5. From Week 3 to Week 4: 12 to 17, increase of 5. The pattern is adding 5 flowers each week. Step 2: Write the rule. For week number x, the number of flowers y is: y = 5x - 3. Check: Week 1: 5(1) - 3 = 2; Week 2: 5(2) - 3 = 7; Week 3: 5(3) - 3 = 12; Week 4: 5(4) - 3 = 17. Correct. Step 3: Find Week 7. y = 5(7) - 3 = 35 - 3 = 32. Step 4: The coordinates are (7, 32). The answer is (7, 32).

8. Oliver is plotting points for a treasure map. The points follow a rule where the y-coordinate is 4 times the x-coordinate. If Oliver plots a point at x = 7, what will be the y-coordinate? Answer: 28 Solution: The rule is that the y-coordinate equals 4 times the x-coordinate. For x = 7, the y-coordinate is 4 × 7 = 28.
   Full step-by-step solution

   Step 1: The rule is that the y-coordinate equals 4 times the x-coordinate. Step 2: For x = 7, the y-coordinate is 4 × 7 = 28. The answer is 28.