Pattern Analysis

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Answer Key & Explanations

Pattern Analysis · Grade 5 · Worksheet 2

1. 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).

2. Liam is designing a treasure map on a coordinate plane. He marks the starting point at (2, 3) and the treasure at (8, 7). If Liam moves 3 units to the right and 2 units up from the starting point, what are the coordinates of his new location? Answer: (5, 5) Solution: Liam starts at (2, 3). On a coordinate plane, the first number is the x-coordinate (horizontal position), and the second number is the y-coordinate (vertical position). Moving right means increasing the x-coordinate.
   Full step-by-step solution

   Let's go step-by-step. **Step 1: Understand the starting point** Liam starts at (2, 3). On a coordinate plane, the first number is the x-coordinate (horizontal position), and the second number is the y-coordinate (vertical position). **Step 2: Moving right** Moving right means increasing the x-coordinate. He moves 3 units to the right, so: New x-coordinate = old x-coordinate + 3 New x-coordinate = 2 + 3 = 5 **Step 3: Moving up** Moving up means increasing the y-coordinate. He moves 2 units up, so: New y-coordinate = old y-coordinate + 2 New y-coordinate = 3 + 2 = 5 **Step 4: Combine the new coordinates** New location = (new x-coordinate, new y-coordinate) = (5, 5) **Final Answer:** (5, 5)

3. Pattern: (7, 21), (8, 24), (9, 27). Find the next point. Answer: (10, 30) Solution: Look at the pattern in the x-coordinates: 7, 8, 9. They increase by 1 each time, so the next x is 10. Look at the relationship between x and y.
   Full step-by-step solution

   Step 1: Look at the pattern in the x-coordinates: 7, 8, 9. They increase by 1 each time, so the next x is 10. Step 2: Look at the relationship between x and y. For (7, 21): 7 × 3 = 21. For (8, 24): 8 × 3 = 24. For (9, 27): 9 × 3 = 27. Step 3: The rule is y = x × 3. Step 4: Apply the rule to the next x: 10 × 3 = 30. Step 5: The next point is (10, 30). The answer is (10, 30).

4. Matiu is plotting points on a coordinate grid to show the relationship between the number of hours he works and the money he earns. He earns $12 per hour. He plots the point for 1 hour as (1, 12), for 2 hours as (2, 24), and for 3 hours as (3, 36). If he continues this pattern, what will be the y-coordinate of the point when x = 8? Answer: 96 Solution: Identify the pattern. For each point (x, y), the y-coordinate is 12 times the x-coordinate. Check: 1 x 12 = 12, 2 x 12 = 24, 3 x 12 = 36.
   Full step-by-step solution

   Step 1: Identify the pattern. For each point (x, y), the y-coordinate is 12 times the x-coordinate. Check: 1 x 12 = 12, 2 x 12 = 24, 3 x 12 = 36. Step 2: The rule is y = 12x. Step 3: Substitute x = 8 into the rule: y = 12 x 8. Step 4: Multiply: 12 x 8 = 96. The y-coordinate for x = 8 is 96.

5. Pattern: (2,7), (7,17), (12,27). Find the next point. Answer: (17,37) Solution: Look at the x-coordinates: 2, 7, 12. The pattern is adding 5 each time. Next x = 12 + 5 = 17.
   Full step-by-step solution

   Step 1: Look at the x-coordinates: 2, 7, 12. The pattern is adding 5 each time. Next x = 12 + 5 = 17. Step 2: Look at the y-coordinates: 7, 17, 27. The pattern is adding 10 each time. Next y = 27 + 10 = 37. Step 3: Check the relationship between x and y in each point: (2,7): 2 × 3 + 1 = 7; (7,17): 7 × 2 + 3 = 17; (12,27): 12 × 2 + 3 = 27. The rule is y = 2x + 3 (since 2×2+3=7, 2×7+3=17, 2×12+3=27). Step 4: Apply the rule to the next x: y = 2 × 17 + 3 = 34 + 3 = 37. The next point is (17,37).

6. (3/4 + 1/2) × 12 = ? Answer: 15 Solution: Add the fractions 3/4 + 1/2 Find a common denominator for 4 and 2, which is 4 Convert 1/2 to 2/4 Add 3/4 + 2/4 = 5/4 Multiply 5/4 × 12 5/4 × 12 = (5 × 12)/4 = 60/4 Simplify 60/4 = 15 The answer is 15.
   Full step-by-step solution

   Step 1: Add the fractions 3/4 + 1/2 Step 2: Find a common denominator for 4 and 2, which is 4 Step 3: Convert 1/2 to 2/4 Step 4: Add 3/4 + 2/4 = 5/4 Step 5: Multiply 5/4 × 12 Step 6: 5/4 × 12 = (5 × 12)/4 = 60/4 Step 7: Simplify 60/4 = 15 The answer is 15.

7. (3/4 × 12) + (2.5 × 4) = ? Answer: 19 Solution: Calculate 3/4 × 12 3/4 × 12 = (3 × 12)/4 = 36/4 = 9 Calculate 2.5 × 4 2.5 × 4 = 10 9 + 10 = 19 The answer is 19.
   Full step-by-step solution

   Step 1: Calculate 3/4 × 12 3/4 × 12 = (3 × 12)/4 = 36/4 = 9 Step 2: Calculate 2.5 × 4 2.5 × 4 = 10 Step 3: Add the results 9 + 10 = 19 The answer is 19.

8. Mason is analyzing a pattern on a coordinate grid. The first three points in the pattern are: (8, 16), (9, 18), (10, 20). If the pattern continues with the same relationship, what will be the y-coordinate of the next point when the x-coordinate is 11? Answer: 22 Solution: Examine the given points: (8, 16), (9, 18), (10, 20). For each point, notice that the y-coordinate is twice the x-coordinate: 16 = 2 x 8, 18 = 2 x 9, 20 = 2 x 10. The pattern rule is y = 2x.
   Full step-by-step solution

   Step 1: Examine the given points: (8, 16), (9, 18), (10, 20). For each point, notice that the y-coordinate is twice the x-coordinate: 16 = 2 x 8, 18 = 2 x 9, 20 = 2 x 10. Step 2: The pattern rule is y = 2x. This means for any x, the y is double that number. Step 3: For the next point, x = 11. Apply the rule: y = 2 x 11 = 22. Step 4: Therefore, the next point in the pattern is (11, 22). The answer is 22.