Pattern and Function - Basic Algebraic Thinking

Patterns and Sequences: A pattern is a sequence of numbers, shapes, or objects that follow a specific rule. For example, a repeating shape pattern could be described as circle, square, circle, square. In math, we look for the core of the pattern that repeats over and over.

Growing and Shrinking Patterns: These are number patterns that increase or decrease by a constant amount. Visualize a staircase where each step is $3$ units higher than the last; this represents a pattern of $3, 6, 9, 12, \dots$ where the rule is 'add $3$'.

Equality as Balance: The equal sign $=$ acts like the center of a balance scale. It means that the value on the left side must be exactly the same as the value on the right side. Picture a scale with $5 + 5$ on one side and $10$ on the other; the scale stays perfectly level because they are equal.

Unknowns and Placeholders: When a number is missing in an equation, we use a symbol like $\Box$, $?$, or a letter to represent it. For instance, in the equation $8 + \Box = 12$, the box is a placeholder for the number we need to find to make the equation true.

Function Machines (Input and Output): Think of a function as a machine with an 'in' slot and an 'out' slot. When you put a number (Input) into the machine, it applies a rule (like $+ 5$) and spits out a new number (Output). If you put $10$ into a $+5$ machine, the output is $15$.

Inverse Operations: This is the idea that addition and subtraction are opposites. If you have a missing number in an addition problem, like $\Box + 7 = 20$, you can 'undo' the addition by subtracting: $20 - 7 = 13$.

Commutative Property: The order in which numbers are added does not change the sum. For example, $4 + 2 = 6$ is the same as $2 + 4 = 6$. Visualizing two groups of blocks, one of $4$ and one of $2$, helps show that the total remains $6$ no matter which group you count first.