A sequence is an ordered list where position matters: the first element, the second element, the third element, and so on. Unlike a set (where order does not matter), the sequence 3, 5, 7 is different from 7, 5, 3. Sequences are governed by rules that determine what element appears at each position. A series is the sum of the elements of a sequence. Understanding sequences means understanding that order carries information — changing the order changes the meaning. This is a foundational concept for algorithms, instructions, and mathematical reasoning.
Compare sequences to sets: {3, 5, 7} is the same set as {7, 5, 3}, but the sequence 3, 5, 7 is different from 7, 5, 3. Use everyday examples: the alphabet is a sequence (A comes before B), days of the week are a sequence, steps in a recipe are a sequence. Practice identifying position: "What is the 4th term?" Have students create sequences from rules and rules from sequences. Introduce the idea that position matters by rearranging steps in a recipe and asking what would go wrong.
You have learned about patterns and their rules. Now you are going to study the sequence — the mathematical structure that patterns live in. A sequence is an ordered list. The key word is ordered: position matters.
Consider the difference between a bag of blocks and a line of blocks. In a bag, the blocks are just mixed together — there is no first or last. In a line, there is a clear order: first block, second block, third block. A sequence is like the line, not the bag. The sequence 1, 2, 3 is different from 3, 2, 1 because the elements are in different positions, even though the same numbers appear.
This might seem obvious, but the distinction is important. In a set (which you may study later), {1, 2, 3} and {3, 2, 1} are the same thing — sets only care about *what* is in the collection, not the order. In a sequence, position is part of the identity. Think about it this way: the word "TAR" and the word "RAT" use the same letters, but they are completely different words because the order of letters changes the meaning. Sequences work the same way.
Many things in life are naturally sequences: the steps of a recipe (mix, then bake, then cool — not cool, then mix, then bake), the digits of a phone number (changing the order gives a different number), the notes of a melody (same notes in a different order make a different song). Recognizing that order carries information is a fundamental insight.
When a sequence follows a rule — like "start at 5, add 3 each time" — you get a pattern. But sequences do not require rules. A random list of numbers is still a sequence; it just is not a very interesting one. The sequences worth studying are the ones with rules, because rules let you predict, extend, and understand them. From here, you will explore sequences where order matters in new ways: ordinal reasoning (first, second, third) and algorithmic thinking (steps that must happen in a specific order).