Patterns & relationships
The idea
Mathematicians are pattern hunters. A pattern is a list that follows a rule — 4, 7, 10, 13 grows by 3 each time — and once you catch the rule, you can predict numbers far down the list without writing every step. Some patterns repeat, like the days of the week; others grow, like stacking one more block on a tower each day. Spotting the rule, saying it clearly, and using it to predict is the start of real algebraic thinking.
The reliable hunting method is to compare neighbors: subtract each number from the next and see whether the jumps match. Here is the trap, though — checking only the first jump. The list 2, 4 could be adding 2, or it could be doubling, and the two rules disagree from the third number on. A rule is only trustworthy after it works for every number you were given. Once it survives that test, ride it forward, or use a shortcut: many equal jumps at once is just multiplication.
Worked example
A number pattern starts 4, 7, 10, 13, and keeps following the same rule. What is the 7th number in the pattern?
- Hunt for the rule by comparing neighbors: 7 − 4 = 3, then 10 − 7 = 3, then 13 − 10 = 3. Every jump adds 3, and the rule works for every pair given, not just the first.
- Ride the rule forward one step at a time: the 5th number is 13 + 3 = 16, the 6th is 16 + 3 = 19, and the 7th is 19 + 3 = 22.
- Confirm with the jump shortcut: getting from the 1st number to the 7th takes 6 jumps of 3, which is 6 × 3 = 18 in total, and 4 + 18 = 22.
- Both roads — stepping one jump at a time and jumping all at once — land on 22, so the answer is solid.
Answer. The 7th number in the pattern is 22.
Check your understanding
- Why should you test a pattern rule against every number you are given, not just the first two?
- How does the jump-counting shortcut save work when you need the 50th number instead of the 7th?
- What different rules could begin a pattern with 2 and then 4, and how would seeing more numbers settle which rule is right?
- Where do repeating or growing patterns show up in calendars, songs, or buildings around you?