Connect recognition, extension, tables, flow diagrams, formulas, applications, checking, and justification.
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\[\text{Observe}\to\text{classify}\to\text{represent}\to\text{apply}\to\text{check}\to\text{justify}\]
Introduction
This lesson brings every number-pattern idea together. Each unfamiliar pattern can be approached through one dependable chain of reasoning.
What makes an answer complete?
\[8,\ 13,\ 18,\ 23,\ldots\]
Topic 1 · Recognise and classify
The first term is 8, the third term is 18, and 23 is in position 4. Clear language prevents confusion later.
\[14,20,26,32,\ldots\qquad d=+6\]
Topic 1 · Recognise and classify
Every consecutive difference is 6, so the pattern is arithmetic.
\[4,12,36,108,\ldots\qquad\times3\]
Topic 1 · Recognise and classify
Each term is three times the previous term, so the pattern is geometric.
\[1,4,9,16,\ldots\qquad T_n=n^2\]
Topic 1 · Recognise and classify
The differences are not constant, but the terms are square numbers. Classification is about the relationship, not whether the list rises.
\[\boxed{\text{difference? multiplier? alternating? square? power?}}\]
Topic 1 · Checkpoint
You can identify terms and positions and classify arithmetic, geometric and other structured patterns.
\[-10,-4,2,8,\boxed{14},\boxed{20}\]
Topic 2 · Extend patterns
The constant difference is (+6). Add 6 twice to find the next two terms.
\[81,27,9,3,\boxed{1},\boxed{\tfrac13}\]
Topic 2 · Extend patterns
Divide each term by 3, or multiply by ( frac13).
\[2,5,10,17,26,\boxed{37},\boxed{50}\]
Topic 2 · Extend patterns
The rule is (T_n=n^2+1). Equivalently, add consecutive odd numbers 3, 5, 7, 9, 11, 13, ....
\[2,5,4,10,8,20,\boxed{16},\boxed{40}\]
Topic 2 · Extend patterns
Separate odd and even positions. Both linked sequences double independently.
\[\boxed{\text{Extend only after the rule fits every term}}\]
Topic 2 · Checkpoint
You can extend arithmetic, geometric, square and alternating patterns and explain the operation used.
\[7,10,13,16,\ldots\]
Topic 3 · Represent rules
The verbal rule is “start at 7 and add 3 each time”. To connect position directly to term, use (T_n=3n+4).
\[n\longrightarrow\times3\longrightarrow+4\longrightarrow T_n\]
Topic 3 · Represent rules
The flow diagram reveals the operation order and supports substitution and reverse operations.
\[\begin{array}{c|ccccc}n&1&2&3&4&5\\\hline T_n&7&10&13&16&19\end{array}\]
Topic 3 · Represent rules
Each column is an ordered pair ((n,T_n)), such as ((1,7)) and ((5,19)).
\[\text{words}\leftrightarrow\text{flow}\leftrightarrow\text{table}\leftrightarrow\text{formula}\]
Topic 3 · Represent rules
Every representation must generate the same outputs. A mismatch reveals an operation-order or adjustment error.
\[\boxed{\text{Four representations, one relationship}}\]
Topic 3 · Checkpoint
You can express and compare a rule in words, a flow diagram, a table and algebraic notation.
\[6,10,14,18,22,\ldots\Rightarrow T_n=4n+2\]
Topic 4 · Use formulas
The difference is 4. Since (4n) is 2 below every target term, add 2.
\[T_{25}=4(25)+2=102\]
Topic 4 · Use formulas
A formula avoids listing the first 25 terms. Substitute the required position and simplify.
\[4n+3=51\Rightarrow4n=48\Rightarrow n=12\]
Topic 4 · Use formulas
Set the formula equal to the known output and reverse the operations.
\[T_1=6,\quad T_2=10,\quad T_5=22\]
Topic 4 · Use formulas
The formula (4n+2) reproduces the first, second and fifth terms. This is stronger than checking only one value.
\[\boxed{\text{formula}\Rightarrow\text{distant term or unknown position}}\]
Topic 4 · Checkpoint
You can find a rule, calculate a distant term, solve for a position, and verify the formula.
1 2 3
Topic 5 · Diagrams and contexts
A diagram with 4, 7, 10, 13, ... objects adds 3 each time. The table turns the visual growth into a numeric sequence.
\[4,7,10,13,\ldots\Rightarrow T_n=3n+1\]
Topic 5 · Diagrams and contexts
The difference is 3. The adjustment is +1 because (3(1)+1=4).
\[T_{12}=3(12)+1=37\text{ objects}\]
Topic 5 · Diagrams and contexts
Substitution gives the object count for any figure number.
\[\text{Chairs: }10,13,16,19,\ldots\Rightarrow T_n=3n+7\]
Topic 5 · Diagrams and contexts
Row 1 has 10 chairs and each new row adds 3. Row 15 has (3(15)+7=52) chairs; Row 14 has 49.
\[\boxed{\text{context}\to\text{table}\to\text{formula}\to\text{answer with units}}\]
Topic 5 · Checkpoint
You can turn a diagram or real context into a pattern rule and use it to answer practical questions.
\[20,17,14,11,8\qquad d=-3\]
Topic 6 · Create and justify
Choose a first term and repeatedly apply the required difference. Then verify every pair of consecutive terms.
\[1,4,16,64,256\qquad\times4\]
Topic 6 · Create and justify
Multiplying by 4 each time creates a geometric pattern. State both the terms and multiplier.
\[1,4,9,16,25\qquad T_n=n^2\]
Topic 6 · Create and justify
This sequence is based on position squared. Its differences change, so it is not a constant-difference pattern.
Which evidence justifies the square rule?
\[\text{Claim}\rightarrow\text{rule}\rightarrow\text{substitution checks}\rightarrow\text{conclusion}\]
Topic 6 · Create and justify
Name the structure, state the rule, show that it reproduces several terms, and conclude that it fits the given pattern.
\[\boxed{\text{Create}\ +\ \text{state}\ +\ \text{check}\ +\ \text{justify}}\]
Topic 6 · Checkpoint
You can design arithmetic, geometric and non-linear patterns and defend the rules you chose.
\[\begin{gathered}\text{Observe}\to\text{compare}\to\text{classify}\to\text{represent}\\\to\text{apply}\to\text{check}\to\text{justify}\end{gathered}\]
Summary
You have completed the three-lesson number-pattern progression.