Algorithms and Number Patterns
Steps in order
An algorithm is a set of steps carried out in a fixed order to accomplish a task. A recipe is an everyday algorithm, and so is a rule for building numbers: start at 2, then add 3, and add 3 again. The defining feature is that the steps are precise and ordered, so that following them always produces the same result from the same starting point. In mathematics, algorithms are used to generate sets of numbers by repeating steps, and learning to follow and write them carefully is the heart of this unit.
A rule builds a sequence
When a step is repeated, a single rule generates a whole sequence of numbers. The rule "start at 1 and add 2 each time" produces 1, 3, 5, 7, 9, where every term comes from the one before by the same step. The starting value and the rule together fix the entire sequence. Rules can add, subtract, or multiply, and a subtracting rule produces a decreasing sequence. Seeing a sequence as the output of a rule, rather than a random list, is what lets you continue it, describe it, and reason about it.
Predicting the next term
Once you know the rule behind a sequence, you can predict what comes next. In 2, 4, 6, 8 the step between terms is 2, so the next term is 10. The method is always the same: find the change from one term to the next, check it stays constant, then apply that change once more. This is really running the algorithm forward by one more step. Predicting terms is a first test of whether you have correctly understood the rule, and it is the skill that makes a sequence useful for solving problems.
A decision step
Algorithms are not limited to doing the same step every time; they can also make decisions. A decision step checks a condition and then chooses which step to do next, so the path through the algorithm can branch. A rule might say "if the number is even, halve it; if it is odd, multiply by 3 and add 1". For the input 8, which is even, the algorithm halves it to 4. Decisions let one algorithm respond differently to different inputs, which makes the sets of numbers it generates far richer than a single repeated step alone.
Finding the rule
Often a sequence is given and the task is to work backwards to its rule. The method is to look at how each term changes into the next: in 3, 6, 9, 12 each term is 3 more than the last, so the rule is "add 3 each time". If the terms instead double, like 2, 4, 8, 16, the rule is multiplication. Finding the rule means spotting the repeated operation that links consecutive terms, and it turns a bare list of numbers back into the algorithm that produced it, ready to be continued or explained.
Explaining the pattern
Generating a sequence with a rule often makes a pattern emerge, and the final skill is to identify, interpret and explain it. The rule "start at 2, add 2" produces the even numbers; a rule that adds one more each time, 1, 3, 6, 10, produces the triangular numbers. Explaining a pattern means saying clearly what kind of numbers appear and why the rule produces them. A good explanation does more than list terms: it lets you predict terms far ahead and connects the sequence to the wider patterns of number that run through mathematics.
Where algorithms lead
Algorithms and number patterns are a doorway into both higher mathematics and computing. The idea of a precise sequence of steps, with decisions that branch on a condition, is exactly how computer programs work, and writing rules to generate numbers is a first taste of programming and of algebraic thinking. In later mathematics these grow into formulas for the nth term, functions, and the study of sequences and series. Learning to create, follow, and explain algorithms builds the clear, step-by-step reasoning that underpins mathematics, science and technology.