Arithmetic patterns are regular numerical sequences formed by adding or multiplying the same amount each step. Students identify and extend patterns in addition tables (each row increases by the same amount) and multiplication tables (products in a column are multiples of the column number). They explain patterns using properties of operations.
Use color-coding on multiplication tables to highlight patterns: the 9s column has digits that sum to 9, every second multiple of 2 is even, and so on. Have students write rules in their own words before formalizing.
A pattern is a sequence with a rule. Once you know the rule, you can predict any term in the sequence — even ones far down the list — without writing them all out. In arithmetic, there are two main kinds of rules: additive patterns (add the same amount each step) and multiplicative patterns (multiply by the same amount each step). These look different and grow at very different rates.
In an additive pattern like 5, 8, 11, 14, ..., you're adding 3 each time. The gap between any two neighbors is always 3. This is the same structure as skip-counting, which you've practiced before. In a multiplication table, each row is also an additive pattern: the row for 4 goes 4, 8, 12, 16, ... — you add 4 each time. The row for 7 adds 7 each time. Recognizing this helps you see multiplication tables not as a list to memorize but as a system with built-in structure.
When you look at a multiplication table, more patterns emerge. Every number in the 2s column is even. The 5s column alternates between 5 and 0 in the ones place. The 9s column has a special property: the two digits of each product always sum to 9 (9, 18, 27, 36 — check: 1+8=9, 2+7=9, 3+6=9). These aren't coincidences; they're consequences of how multiplication works with our base-ten number system.
The key skill is being able to describe a pattern with a rule — not just the next few terms. "Add 6 each time" is a rule. "Each number is three times the position number" is a rule. Once you have the rule, you can extend the pattern confidently, check whether a number belongs to it, and explain to someone else why the pattern works. This is early algebraic thinking: you're treating a rule as an object you can describe and use, not just a sequence you happen to notice.