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Pattern Analysis

Grade 5 · Mathematics · Worksheet 1

  1. Emma is plotting points on a coordinate grid based on a pattern she discovered. The first point is at (1, 7), the second point is at (3, 21), and the third point is at (5, 35). If the pattern continues where the x-coordinate increases by 2 each time and the y-coordinate follows the rule y = 7 × x, what will be the coordinates of the 7th point in this pattern? Answer: ______________
  2. A pattern of points on a coordinate plane follows this rule: Start at (750, 900), then add 8 to the x-coordinate and subtract 18 from the y-coordinate for each subsequent point. What is the y-coordinate of the 7th point in this pattern? Answer: ______________
  3. (3.5 × 4) + (2.25 × 2) = ? Answer: ______________
  4. Sophia is tracking the number of pages she reads each day for a week. She notices a pattern: on day 1 she reads 6 pages, on day 2 she reads 11 pages, on day 3 she reads 16 pages, and on day 4 she reads 21 pages. If this pattern continues, write the ordered pairs (day, pages) for days 1 through 6. Then, what would be the coordinates for day 6? Answer: ______________
  5. 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: ______________
  6. (3.75 × 4) + (2.5 × 3) = ? Answer: ______________
  7. Emma is saving money to buy a new bicycle. She starts with $5 saved. Each week, she adds $3 more to her savings. If she continues this pattern, what will be her savings after 6 weeks? Plot the points (week, savings) for weeks 1 through 6 on a coordinate plane. What is the savings amount for week 6? Answer: ______________
  8. Pattern: (12, 24), (15, 30), (18, 36). Find the next point. Answer: ______________
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Answer Key & Explanations

Pattern Analysis · Grade 5 · Worksheet 1

  1. Emma is plotting points on a coordinate grid based on a pattern she discovered. The first point is at (1, 7), the second point is at (3, 21), and the third point is at (5, 35). If the pattern continues where the x-coordinate increases by 2 each time and the y-coordinate follows the rule y = 7 × x, what will be the coordinates of the 7th point in this pattern? Answer: (13, 91) Solution: Identify the pattern in x-coordinates: 1, 3, 5, ... Each time, x increases by 2. To find the 7th x-coordinate: The first x is 1, second is 3, third is 5.
    Full step-by-step solution

    Step 1: Identify the pattern in x-coordinates: 1, 3, 5, ... Each time, x increases by 2. So the x-values are odd numbers starting at 1. Step 2: To find the 7th x-coordinate: The first x is 1, second is 3, third is 5. Notice that the nth x = 2n - 1. For n = 7: x = 2(7) - 1 = 14 - 1 = 13. Step 3: The rule is y = 7 × x. For x = 13: y = 7 × 13 = 91. Step 4: The 7th point is (13, 91). The answer is (13, 91).

  2. A pattern of points on a coordinate plane follows this rule: Start at (750, 900), then add 8 to the x-coordinate and subtract 18 from the y-coordinate for each subsequent point. What is the y-coordinate of the 7th point in this pattern? Answer: 792 Solution: Identify the starting point: (750, 900) The pattern rule: x increases by 8, y decreases by 18 each time To find the 7th point, we apply the pattern rule 6 times (from 1st to 2nd, 2nd to 3rd, and so on up to 6th to 7th) Focus on the y-coordinate changes: 6 decreases of 18 each Calculate total…
    Full step-by-step solution

    Step 1: Identify the starting point: (750, 900) Step 2: The pattern rule: x increases by 8, y decreases by 18 each time Step 3: To find the 7th point, we apply the pattern rule 6 times (from 1st to 2nd, 2nd to 3rd, and so on up to 6th to 7th) Step 4: Focus on the y-coordinate changes: 6 decreases of 18 each Step 5: Calculate total decrease in y: 6 × 18 = 108 Step 6: Apply the decrease to the starting y-value: 900 - 108 = 792 Step 7: The y-coordinate of the 7th point is 792

  3. (3.5 × 4) + (2.25 × 2) = ? Answer: 18.5 Solution: Calculate 3.5 × 4 3.5 is the same as 3 and 1/2, or 7/2. So 3.5 × 4 = (7/2) × 4 = (7 × 4) / 2 = 28 / 2 = 14. Calculate 2.25 × 2 2.25 is the same as 2 and 1/4, or 9/4.
    Full step-by-step solution

    Let's solve step-by-step: Step 1: Calculate 3.5 × 4 3.5 is the same as 3 and 1/2, or 7/2. So 3.5 × 4 = (7/2) × 4 = (7 × 4) / 2 = 28 / 2 = 14. Thus, 3.5 × 4 = 14. Step 2: Calculate 2.25 × 2 2.25 is the same as 2 and 1/4, or 9/4. So 2.25 × 2 = (9/4) × 2 = (9 × 2) / 4 = 18 / 4 = 4.5. Thus, 2.25 × 2 = 4.5. Step 3: Add the two results (3.5 × 4) + (2.25 × 2) = 14 + 4.5 = 18.5. Final Answer: 18.5

  4. Sophia is tracking the number of pages she reads each day for a week. She notices a pattern: on day 1 she reads 6 pages, on day 2 she reads 11 pages, on day 3 she reads 16 pages, and on day 4 she reads 21 pages. If this pattern continues, write the ordered pairs (day, pages) for days 1 through 6. Then, what would be the coordinates for day 6? Answer: (6, 31) Solution: Identify the pattern. From day 1 to day 2: 11 - 6 = 5 pages increase. From day 2 to day 3: 16 - 11 = 5 pages increase.
    Full step-by-step solution

    Step 1: Identify the pattern. From day 1 to day 2: 11 - 6 = 5 pages increase. From day 2 to day 3: 16 - 11 = 5 pages increase. From day 3 to day 4: 21 - 16 = 5 pages increase. So the pattern adds 5 pages each day. Step 2: Write ordered pairs (day, pages): (1, 6), (2, 11), (3, 16), (4, 21). Step 3: Find day 5: 21 + 5 = 26 pages. Ordered pair: (5, 26). Step 4: Find day 6: 26 + 5 = 31 pages. Ordered pair: (6, 31). The coordinates for day 6 are (6, 31).

  5. 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

  6. (3.75 × 4) + (2.5 × 3) = ? Answer: 22.5 Solution: Calculate the first multiplication: 3.75 × 4 = 15 Calculate the second multiplication: 2.5 × 3 = 7.5 Add the two results: 15 + 7.5 = 22.5 The answer is 22.5.
    Full step-by-step solution

    Step 1: Calculate the first multiplication: 3.75 × 4 = 15 Step 2: Calculate the second multiplication: 2.5 × 3 = 7.5 Step 3: Add the two results: 15 + 7.5 = 22.5 The answer is 22.5.

  7. Emma is saving money to buy a new bicycle. She starts with $5 saved. Each week, she adds $3 more to her savings. If she continues this pattern, what will be her savings after 6 weeks? Plot the points (week, savings) for weeks 1 through 6 on a coordinate plane. What is the savings amount for week 6? Answer: 23 Solution: Identify the pattern. Emma starts with $5 and adds $3 each week. So for week 1, savings = 5 + 3 = 8.
    Full step-by-step solution

    Step 1: Identify the pattern. Emma starts with $5 and adds $3 each week. So for week 1, savings = 5 + 3 = 8. Week 2: 8 + 3 = 11. Week 3: 11 + 3 = 14. Week 4: 14 + 3 = 17. Week 5: 17 + 3 = 20. Week 6: 20 + 3 = 23. Step 2: The relationship can be written as savings = 3 * week + 5. For week 6: savings = 3 * 6 + 5 = 18 + 5 = 23. Step 3: The coordinates to plot are: (1, 8), (2, 11), (3, 14), (4, 17), (5, 20), (6, 23). The answer is 23.

  8. 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).