Pattern and Function - Number Patterns and Sequences

A number pattern is a sequence of numbers that follows a specific rule. For example, in the sequence $2, 4, 6, 8$, the rule is to add $2$ to the previous number. You can visualize this as a growing tower of blocks where each new tower is $2$ blocks taller than the one before it.

Ascending patterns are sequences where the numbers get larger. This usually involves addition or multiplication. Think of this like climbing up a ladder where each rung is a fixed distance higher than the last, such as $+5$ or $+10$.

Descending patterns are sequences where the numbers get smaller. This usually involves subtraction or division. Imagine a countdown timer or a cooling thermometer where the value drops by a consistent amount like $-3$ at every step.

The 'Rule' is the secret formula that connects one number to the next. If you have an Input/Output table, the rule describes what happens to the 'Input' number to turn it into the 'Output' number. For example, if the rule is $\times 2$, then an input of $5$ results in an output of $10$.

Repeating patterns involve a sequence of numbers or shapes that repeat in the same order. For instance, the sequence $1, 3, 5, 1, 3, 5$ has a core of $1, 3, 5$ that repeats. You can visualize this as a pattern of colored beads on a string that follows a cycle.

Skip counting is a basic form of number patterns. Counting by $2$s ($2, 4, 6, 8$), $5$s ($5, 10, 15, 20$), or $10$s ($10, 20, 30, 40$) helps in identifying the common difference in arithmetic sequences.

Missing terms are found by first identifying the rule between the known numbers and then applying it to the empty spaces. If a pattern is $25, 30, \_, 40$, we see the difference between $25$ and $30$ is $+5$, so we add $5$ to $30$ to find the missing number $35$.