Number Pattern Rules - ISEE Middle Level Mathematics Achievement

1. Which rule explains the sequence: 5, 8, 11, 14, 17?
   Answer: Add 3 each time This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence 5, 8, 11, 14, 17 shows a constant difference of 3 between consecutive terms (8-5=3, 11-8=3, 14-11=3, 17-14=3). The correct choice, 'Add 3 each time,' accurately reflects the mathematical operation that defines the sequence. A common distractor, such as 'Add 5 each time,' fails because adding 5 to 5 would give 10, not 8. Teaching strategies include having students systematically find the difference between each pair of consecutive numbers and verify consistency. Remind students that arithmetic sequences with constant differences are among the most common patterns they'll encounter.
2. Start at 11: 11, 16, 13, 18, 15. Rule?
   Answer: Add 5, then subtract 3 This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence of numbers presented follows a specific rule that students must identify from the options provided. The correct choice, 'Add 5, then subtract 3,' accurately reflects the mathematical operation that defines the sequence. A common distractor, such as 'Subtract 5 each time,' fails because it suggests a different operation than the one used in the sequence. Teaching strategies include practicing with different types of sequences and encouraging students to verbalize the rule they identify. Remind students to check their rule against each number in the sequence to ensure consistency.
3. Each number in the pattern 0.5, 0.75, 1.25, 2, 3.25, ... is determined by the numbers that come before it. What rule is used to find the next number in this pattern?
   Answer: Add the two preceding numbers. This pattern follows a Fibonacci-like rule. Check by adding the two previous terms to get the next: 0.5 + 0.75 = 1.25. 0.75 + 1.25 = 2.0. 1.25 + 2.0 = 3.25. The rule is to add the two preceding numbers.
4. How is the sequence 10, 7, 12, 9, 14 generated?
   Answer: Subtract 3, then add 5, repeat This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence 10, 7, 12, 9, 14 follows an alternating pattern: subtract 3 (10-3=7), then add 5 (7+5=12), subtract 3 (12-3=9), then add 5 (9+5=14). The correct choice, 'Subtract 3, then add 5, repeat,' accurately describes this alternating operation pattern. A common distractor, such as 'Add 2 each time,' fails because it doesn't account for the decreasing values at positions 2 and 4. Teaching strategies include having students track whether each step increases or decreases and by how much, creating a visual pattern of operations. Remind students that some sequences use alternating rules rather than a single consistent operation.
5. Start at 4: 4, 9, 14, 19, 24. What is the rule?
   Answer: Add 5 each time This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence of numbers presented follows a specific rule that students must identify from the options provided. The correct choice, 'Add 5 each time,' accurately reflects the mathematical operation that defines the sequence. A common distractor, such as 'Multiply by 2 each time,' fails because it suggests a different operation than the one used in the sequence. Teaching strategies include practicing with different types of sequences and encouraging students to verbalize the rule they identify. Remind students to check their rule against each number in the sequence to ensure consistency.
6. Start at 2: 2, 10, 50, 250, 1250. Rule?
   Answer: Multiply by 5 each time This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence of numbers presented follows a specific rule that students must identify from the options provided. The correct choice, 'Multiply by 5 each time,' accurately reflects the mathematical operation that defines the sequence. A common distractor, such as 'Multiply by 4 each time,' fails because it suggests a different operation than the one used in the sequence. Teaching strategies include practicing with different types of sequences and encouraging students to verbalize the rule they identify. Remind students to check their rule against each number in the sequence to ensure consistency.
7. Consider the sequence of fractions: $\frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, ...$. Which rule describes how to generate the next fraction in the sequence from the previous one?
   Answer: Add 1 to the numerator and 2 to the denominator of the previous fraction. Observe the pattern in the numerators and denominators separately. The numerators are 1, 2, 3, 4, ..., increasing by 1 each time. The denominators are 3, 5, 7, 9, ..., increasing by 2 each time. Thus, the rule is to add 1 to the previous numerator and add 2 to the previous denominator.
8. A sequence of numbers starts with 3. Each subsequent number is found using the same two-step rule. The first four numbers are 3, 7, 15, and 31. What is the rule used to generate the numbers in this sequence?
   Answer: Multiply the previous number by 2, then add 1. To find the rule, check the relationship between consecutive terms. From 3 to 7: (3 × 2) + 1 = 7. From 7 to 15: (7 × 2) + 1 = 15. From 15 to 31: (15 × 2) + 1 = 31. The rule is to multiply the previous number by 2 and then add 1.
9. If the pattern continues, what is next: 7, 14, 28, 56, 112?
   Answer: 224 This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence 7, 14, 28, 56, 112 follows a pattern where each term is multiplied by 2 to get the next term (7×2=14, 14×2=28, 28×2=56, 56×2=112). The correct choice, 224, is obtained by multiplying 112 by 2. A common distractor, such as 216, might come from an incorrect calculation or applying a different rule. Teaching strategies include having students verify the pattern by checking that each term divided by the previous term equals 2. Remind students that doubling patterns are common in mathematics and to carefully perform the multiplication to avoid calculation errors.
10. A sequence begins with 1 and 3. Every term after the second is found by adding the two terms that come before it. The sequence is 1, 3, 4, 7, 11, 18, ... Which rule generates any term after the second?
    Answer: Add the two previous numbers to find the next number. This is a Fibonacci-like sequence. Check the rule: 1 + 3 = 4. 3 + 4 = 7. 4 + 7 = 11. 7 + 11 = 18. The rule is to add the two preceding terms to get the next term.
11. A sequence of numbers begins 50, 48, 44, 38, 30, ... What is the rule to determine the next number in this sequence?
    Answer: Subtract the next consecutive even number, starting with 2. Examine the differences between terms: 50 - 48 = 2. 48 - 44 = 4. 44 - 38 = 6. 38 - 30 = 8. The amounts being subtracted are consecutive even numbers (2, 4, 6, 8). The rule is to subtract the next consecutive even number.
12. A special ball is dropped from a height of 128 cm. After each bounce, it reaches a peak height that is half of its previous peak height. Which rule determines the ball's peak height after any given bounce?
    Answer: Multiply the previous peak height by $\frac{1}{2}$. The problem describes a geometric decay. The heights are 128, 64, 32, 16, ... Each term is half of the previous term. This is equivalent to multiplying by $\frac{1}{2}$ or dividing by 2. Choice A correctly states this rule.
13. Start at 2: 2, 7, 12, 17, 22. Next number?
    Answer: 27 This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence of numbers presented follows a specific rule, and students must find the next number from the options provided. The correct choice, '27,' is found by adding 5 to the previous term each time. A common distractor, such as '25,' fails because it does not follow the consistent addition of 5. Teaching strategies include practicing with different types of sequences and encouraging students to verbalize the rule they identify. Remind students to check their rule against each number in the sequence to ensure consistency.
14. Start with 2, 5: 2, 5, 7, 12, 19. Rule?
    Answer: Add the previous two numbers This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence of numbers presented follows a specific rule that students must identify from the options provided. The correct choice, 'Add the previous two numbers,' accurately reflects the mathematical operation that defines the sequence. A common distractor, such as 'Multiply by 2 each time,' fails because it suggests a different operation than the one used in the sequence. Teaching strategies include practicing with different types of sequences and encouraging students to verbalize the rule they identify. Remind students to check their rule against each number in the sequence to ensure consistency.
15. Start at 20: 20, 17, 21, 18, 22. Rule?
    Answer: Subtract 3, then add 4 This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence of numbers presented follows a specific rule that students must identify from the options provided. The correct choice, 'Subtract 3, then add 4,' accurately reflects the mathematical operation that defines the sequence. A common distractor, such as 'Multiply by 2, then subtract 1,' fails because it suggests a different operation than the one used in the sequence. Teaching strategies include practicing with different types of sequences and encouraging students to verbalize the rule they identify. Remind students to check their rule against each number in the sequence to ensure consistency.
16. A sequence starts with 4. Each subsequent number is found by applying the same two operations in the same order to the previous number. The sequence is 4, 9, 19, 39, 79, ... What is the rule?
    Answer: Multiply the previous number by 2, then add 1. The stem specifies that the same two operations are applied each time. Let's test choice C. For 4 to 9: (4 × 2) + 1 = 9. For 9 to 19: (9 × 2) + 1 = 19. For 19 to 39: (19 × 2) + 1 = 39. For 39 to 79: (39 × 2) + 1 = 79. This rule works for the entire sequence. Choice B also describes the pattern correctly (differences are 5, 10, 20, 40), but it does not use the 'same two operations' each time, as the number being added changes.
17. What is the rule for the sequence: 4, 12, 36, 108, 324?
    Answer: Multiply by 3 each time This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence 4, 12, 36, 108, 324 shows that each term is multiplied by 3 to get the next term (4×3=12, 12×3=36, 36×3=108, 108×3=324). The correct choice, 'Multiply by 3 each time,' accurately reflects the mathematical operation that defines the sequence. A common distractor, such as 'Multiply by 2 each time,' fails because 4×2=8, not 12. Teaching strategies include having students check the ratio between consecutive terms by dividing each term by the previous one. Remind students that multiplication patterns create sequences that grow rapidly and to verify their rule works for all given consecutive pairs.
18. A pattern of numbers is shown: 5, 6, 8, 11, 15, ... Which rule explains how to get the next number in the pattern based on the previous number?
    Answer: Add a number that is 1 greater than the number added to make the previous term. Look at the differences between the numbers: 6 - 5 = 1. 8 - 6 = 2. 11 - 8 = 3. 15 - 11 = 4. The number being added increases by 1 each time. So, the rule is to add a number that is 1 greater than the number added to the previous term.
19. Start at 3: 3, 6, 12, 24, 48. Next number?
    Answer: 96 This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence of numbers presented follows a specific rule, and students must find the next number from the options provided. The correct choice, '96,' is found by multiplying the previous term by 2 each time. A common distractor, such as '72,' fails because it does not follow the consistent multiplication by 2. Teaching strategies include practicing with different types of sequences and encouraging students to verbalize the rule they identify. Remind students to check their rule against each number in the sequence to ensure consistency.
20. In a video game, a player starts with 100 points. For the first level completed, 20 points are added. For each subsequent level, the number of points added is 10 more than the points added for the previous level. What rule determines the player's new score after completing a level?
    Answer: Add 10 more than the point value of the previous level's reward to the current score. The sequence of scores is 100, 120, 150, 190, ... The points added are +20, +30, +40, ... Each time, the reward increases by 10. So, to find the new score, you add an amount to the current score that is 10 greater than the amount added for the prior level.

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