Pattern Recognition

A **pattern** is a list of numbers or shapes that follows a **rule**. Each item in the list is called a **term**. To recognize a pattern, look at how each term changes into the next one: - **Adding pattern**: the same number is added each time. $5, 9, 13, 17$ (add $4$) - **Subtracting pattern**: the same number is taken away each time. $30, 25, 20, 15$ (subtract $5$) - **Multiplying pattern**: each term is multiplied by the same number. $3, 6, 12, 24$ (multiply by $2$) - **Alternating pattern**: two steps take turns. $10, 13, 11, 14, 12$ (add $3$, subtract $2$) - **Growing pattern**: the step itself grows. $1, 2, 4, 7, 11$ (add $1$, then $2$, then $3$, then $4$) - **Repeating pattern**: a block repeats again and again, like circle, square, triangle, circle, square, triangle A rule only counts if it works for **every** term, not just the first two. Finding a rule from examples is **inductive reasoning**; checking it on all terms keeps you honest.

An Adding Pattern

Find the rule and the next two terms: $5, 9, 13, 17, \ldots$

Find the change between neighbors: $9 - 5 = 4$, $13 - 9 = 4$, $17 - 13 = 4$ = The difference is always $4$

State the rule: Every term is $4$ more than the one before = Rule: start at $5$, add $4$

Use the rule to continue: $17 + 4 = 21$, $21 + 4 = 25$ = $21$ and $25$

Answer: The rule is add $4$. The next two terms are $21$ and $25$.

A Multiplying Pattern

Find the rule and the next two terms: $3, 6, 12, 24, \ldots$

Try differences first: $6 - 3 = 3$, $12 - 6 = 6$, $24 - 12 = 12$ = Not the same, so it is not an adding pattern

Try multiplying: $3 \times 2 = 6$, $6 \times 2 = 12$, $12 \times 2 = 24$ = Each term is multiplied by $2$

Use the rule to continue: $24 \times 2 = 48$, $48 \times 2 = 96$ = $48$ and $96$

Answer: The rule is multiply by $2$. The next two terms are $48$ and $96$.

A Growing Pattern

Find the next two terms: $1, 2, 4, 7, 11, \ldots$

Find the differences: $2 - 1 = 1$, $4 - 2 = 2$, $7 - 4 = 3$, $11 - 7 = 4$ = Differences: $1, 2, 3, 4$

Find the pattern in the differences: The differences go up by $1$ each time = Next differences: $5$, then $6$

Use the differences to continue: $11 + 5 = 16$, $16 + 6 = 22$ = $16$ and $22$

Answer: The step grows by $1$ each time. The next two terms are $16$ and $22$.

An Alternating Pattern

Find the rule and the next two terms: $10, 13, 11, 14, 12, 15, \ldots$

Write down every change: $+3, -2, +3, -2, +3$ = Two steps take turns

State the rule: Add $3$, then subtract $2$, again and again = Rule: $+3$, $-2$, $+3$, $-2$, $\ldots$

Decide which step comes next: The last change was $+3$, so now it is $-2$: $15 - 2 = 13$, then $13 + 3 = 16$ = $13$ and $16$

Answer: The rule is add $3$, subtract $2$, taking turns. The next two terms are $13$ and $16$.

The 10th Figure of a Tile Pattern

Figure 1 has $4$ tiles, Figure 2 has $7$ tiles, Figure 3 has $10$ tiles. Each new figure adds the same number of tiles. How many tiles are in Figure 10?

Find the step: $7 - 4 = 3$, $10 - 7 = 3$ = Each figure adds $3$ tiles

Count the jumps from Figure 1 to Figure 10: From $1$ to $10$ there are $10 - 1 = 9$ jumps = $9$ jumps

Add all the jumps to the first term: $4 + 9 \times 3 = 4 + 27$ = $31$

Check with the known figures: Figure 3: $4 + 2 \times 3 = 10$ ✓ = The method works

Answer: Figure 10 has $31$ tiles.

Patterns are everywhere, and spotting them saves time: - **Calendars**: every Monday of a month is $7$ days after the last one, so you can find any date without counting day by day - **Tiles and building**: if each new row of a floor needs $3$ more tiles, you can plan how many tiles to buy - **Saving money**: putting the same amount in a piggy bank each week lets you predict when you reach your goal - **Music and art**: rhythms and borders are repeating patterns - **Science and computers**: scientists look for patterns in data, and computers are trained to recognize patterns in pictures and sounds Pattern recognition is the first step of logical thinking: you notice a rule, test it, and then use it to predict.

Mistake: Choosing a rule after looking only at the first two terms

Why: In $2, 4, 8, 16$ the first jump is $+2$, but $4 + 2 = 6$, not $8$. Many rules fit two numbers.

Correct: Test the rule on every pair of neighbors. Here $\times 2$ works for all of them: $2, 4, 8, 16$.

Mistake: Using $10$ jumps to reach the 10th term

Why: The first term is already there. From term $1$ to term $10$ there are only $9$ jumps.

Correct: 10th term $=$ first term $+ 9 \times$ step. For $4, 7, 10, \ldots$: $4 + 9 \times 3 = 31$, not $4 + 10 \times 3 = 34$.

Mistake: Thinking every pattern adds the same number

Why: In $2, 6, 18, 54$ the differences are $4, 12, 36$, so it is not an adding pattern.

Correct: If the differences are not equal, try multiplying ($\times 3$ here), alternating steps, or growing differences.

Mistake: Forgetting where you are in an alternating pattern

Why: In $10, 13, 11, 14, 12, 15$ the next step depends on the last one. Using $+3$ again gives $18$, which is wrong.

Correct: Write the changes under the terms ($+3, -2, +3, \ldots$) and continue the turn: $15 - 2 = 13$.

Calendars

The same weekday comes back every $7$ days, so dates of that weekday form an adding pattern.

If the first Monday of a month is the $3$rd, the Mondays are the $3$rd, $10$th, $17$th, $24$th and $31$st.

Tiles and Patterns in Building

Builders and designers use tile patterns for floors, paths and walls, and they need to know how many tiles each part uses.

A path where each new section adds $2$ tiles uses $6, 8, 10, 12, \ldots$ tiles. The builder can predict any section without drawing it.

Saving Money

Saving the same amount every week makes the total in a piggy bank grow in an adding pattern.

With 10 dollars at the start and 4 dollars saved each week, the totals are 14, 18, 22, 26 dollars after weeks 1 to 4.

A pattern follows a rule, and each item in it is called a term

First look at the differences between neighbors: equal differences mean an adding or subtracting pattern

If the differences are not equal, try multiplying, alternating steps or growing differences

A rule is only correct if it works for every term, not just the first two

To get the 10th term of an adding pattern, add $9$ steps to the first term

In a repeating pattern, divide the position by the length of the block and use the remainder

Q: How do I start when I see a new pattern?

A: Write the difference between each pair of neighboring terms under the list. If they are all the same, you have an adding or subtracting pattern. If not, try multiplying or look for a pattern in the differences.

Q: Can a pattern have more than one rule?

A: With only two or three terms, yes, several rules can fit. That is why you need enough terms and must check your rule on all of them. With $2, 4$ the rule could be $+2$ or $\times 2$, but $2, 4, 8$ fits only $\times 2$.

Q: Why do I add 9 steps and not 10 to get the 10th term?

A: The first term is already in place. To go from term $1$ to term $10$ you make $9$ jumps, just like walking from the 1st to the 10th step of a ladder.

Q: How do I find the 20th shape in a repeating pattern?

A: Count the shapes in one block. For circle, square, triangle the block has $3$ shapes. $20 \div 3 = 6$ remainder $2$, so the 20th shape is the 2nd shape of the block: a square.

Exit Presentation

Pattern Recognition

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Logic & Sets / Logical Reasoning

Pattern Recognition

Learn to spot the rule behind number and shape patterns, check it against every term, and predict what comes next.

Elementary20 min

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