A student looks at the sequence 2, 6, 18, 54 and says 'Add 4 each time to get the next number.' What is wrong with this analysis?
AThe sequence doesn't have a rule
BThe student confused a multiplicative pattern for an additive one; each term is multiplied by 3, not increased by 4
CThe student is correct; 2 + 4 = 6
DThe student used the right type of rule but made an arithmetic mistake
2, 6, 18, 54 is a multiplicative pattern: each term is multiplied by 3 (2×3=6, 6×3=18, 18×3=54). It is not an additive pattern — the gaps between consecutive terms are 4, 12, 36, which are not equal. The student noticed that 6−2=4 and assumed that gap repeats, but additive patterns require a *constant* difference. This is the core distinction: additive patterns grow by adding the same amount; multiplicative patterns grow by multiplying by the same factor.
Question 2 Multiple Choice
In a multiplication table, what is always true of every product in the 5s column?
AEvery number is odd
BEvery number ends in 5 or 0
CEvery number is greater than 10
DEvery number is a multiple of 2
Multiples of 5 always end in 5 or 0, alternating as you go down the column: 5, 10, 15, 20, 25... This is a consequence of how base-ten place value interacts with multiplication by 5. Noticing this pattern is an example of using arithmetic structure — rather than memorizing each product separately, you can check your work using the 'ends in 5 or 0' rule. Similar patterns exist for 2s (always even), 9s (digits sum to 9), and others.
Question 3 True / False
Two number sequences can start with the same first two terms but diverge dramatically if one follows an additive rule and the other follows a multiplicative rule.
TTrue
FFalse
Answer: True
True. For example, both '+3' and '×3' sequences starting from 3 would begin 3, 6... but then diverge: the additive sequence continues 9, 12, 15, 18..., while the multiplicative sequence continues 18, 54, 162, 486... By the tenth term, the additive sequence is around 30 while the multiplicative sequence is over 19,000. Additive growth is linear; multiplicative growth is exponential. The distinction is not just academic — it changes every prediction you make.
Question 4 True / False
Finding the next two terms of a pattern is sufficient evidence that you understand the pattern's rule.
TTrue
FFalse
Answer: False
False. You can get the next few terms right by trial-and-error or by noticing the immediate difference between adjacent terms — without grasping the underlying rule. The rule is a more powerful thing: 'add 6 each time' or 'each term is 4 times its position number.' With the rule, you can find the 50th term, check whether 148 belongs to the pattern, or explain why the pattern works. Extending a few terms shows recognition; articulating the rule shows understanding.
Question 5 Short Answer
What is the difference between an additive pattern and a multiplicative pattern, and why does the distinction matter?
Think about your answer, then reveal below.
Model answer: An additive pattern is formed by adding the same amount each step (e.g., 3, 7, 11, 15 — add 4 each time). A multiplicative pattern is formed by multiplying by the same factor each step (e.g., 2, 6, 18, 54 — multiply by 3 each time). The distinction matters because they grow at completely different rates: additive patterns grow linearly while multiplicative patterns grow exponentially, and confusing the two rule-type leads to badly wrong predictions for later terms.
Recognizing which type of rule governs a pattern is the key analytical step — before you can extend a pattern accurately, you have to know whether to add or multiply. This distinction also builds the groundwork for algebra: additive patterns become linear equations, multiplicative patterns become exponential ones. Developing the habit of asking 'is this a constant difference or a constant ratio?' is the beginning of algebraic thinking.