Summary of Understanding Numeric and Geometric Patterns

## Introduction Numeric patterns are sequences of numbers that follow a rule. Learning about numeric patterns helps you predict what comes next and see how numbers grow or change. In this guide we will learn simple ways patterns can change and how to record them using tables and flow ideas. > **Definition:** A numeric pattern is a list of numbers that follow a rule to get the next number. ## Two main kinds of numeric patterns ### Arithmetic patterns (adding or subtracting) An **arithmetic pattern** changes by the same amount each time. You add or subtract a constant number to go from one term to the next. > **Definition:** An arithmetic sequence is a list of numbers where each term after the first is found by adding the same number. Example: 1, 4, 7, 10, 13 Each term adds $3$ to the previous term, so the rule is "add $3$". Display rule as a simple step: $$\text{start at }1$$ $$\text{add }3\text{ each time}$$ Real-world example: - Counting chairs added to a row if you always put 3 more chairs each time: $1$, $4$, $7$, $10$. ### Geometric patterns (multiplying) A **geometric pattern** changes by multiplying by the same number each time. Each term is the previous term times the constant factor. > **Definition:** A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by the same number. Example: 2, 4, 8, 16, 32 Each term is multiplied by $2$, so the rule is "times $2$". Display rule: $$\text{start at }2$$ $$\times 2\text{ each time}$$ Real-world example: - Doubling the number of seeds each day in a game: $2$, $4$, $8$, $16$. ## Patterns that change in other ways Not all patterns are simple add or multiply. Some patterns add increasing amounts or follow another rule. Example where the amount added grows: - Term values: $2$, $5$, $9$, $14$ ... - Differences: $+3$, $+4$, $+5$ ... so the amount added increases by $1$ each time. > **Definition:** A pattern formed in other ways is when the rule for the next term changes each time, such as adding a growing number. ## Using flow diagrams (idea) A flow diagram shows how each term becomes the next term. Think of it as an input number that goes through a rule and becomes the output, and that output becomes the input for the next step. - Start at Term $1$ (input). - Apply the rule (add, multiply, or another rule). - The output becomes the next input. Example flow for arithmetic rule "add $3$": $$\text{Input }1\rightarrow \text{Add }3\rightarrow \text{Output }4$$ Then repeat using $4$ as the next input. ## Pattern tables Pattern tables record terms and changes. They help you see the rule clearly. Simple two-line table example for a geometric pattern (multiply by $2$): | Term number | 1 | 2 | 3 | 4 | 5 | | --- | --- | --- | --- | --- | --- | | Counter in term | $1$ | $2$ | $4$ | $8$ | $16$ | Arithmetic pattern (add $3$) two-line table: | Term number | 1 | 2 | 3 | 4 | 5 | | --- | --- | --- | --- | --- | --- | | Counters in term | $3$ | $6$ | $9$ | $12$ | $15$ | | Counters added to get next | +$3$ | +$3$ | +$3$ | +$3$ | +$3$ | Four-line tables can show term number, value, the change (add or multiply), and the next term value. Use them when you need more steps. ## How to find the rule 1. Look at how the numbers change from one term to the next. 2. If the change is the same each time, it is arithmetic (add or subtract). Example: differences $+4$, $+4$, $+4$. 3. If each term is a fixed multiple of the previous term, it is geometric (multiply). Example: ratios $\times 3$, $\times 3$, $\times 3$. 4. If neither differences nor ratios are constant, check if the change itself follows a pattern (like increasing by $1$ each time). ## Quick check examples 1. Sequence: $5$, $10$, $20$, $40$ . Rule? Each term is times $2$ so geometric with factor $2$. 2. Sequence: $7$, $10$, $13$, $16$ . Rule? Add $3$ each time so arithmetic with difference $3$. 3. Sequence: $2$, $5$, $9$, $14$ . Rule? Differences $+3$, $+4$, $+5$ so the a