Find the number of seats for 12 tables
- Define n as the table count. The rule is s = 4n + 2.
- Substitute 12: s = 4(12) + 2.
- Multiply, then add: s = 48 + 2 = 50 seats.
Patterns, variables & expressions
Turn a growing pattern into a rule that works beyond the next term.
Lesson 01 · Start where you need
Your learning goal
Define a variable, distinguish an expression from an equation, and check a rule against a table.
Start with quantities, not letters
A variable represents a quantity whose value can change or is not yet known. Define what it means before using it. If n counts the number of tables in a row, n can take whole-number values such as 1, 2 and 3. A coefficient multiplies a variable: 4n means four times n. A constant such as 2 does not change with n.
Words worth knowing
- Variable
- A quantity that can change or whose value is unknown.
- Coefficient
- The number multiplying a variable.
- Constant
- A term whose value does not change.
- Equation
- A statement that two expressions have equal value.
1
Make the idea visible.
Predict a change, adjust one value, then explain what stayed the same.
See the pattern grow
Four more seats. The same two ends.
Each joined table adds two seats on each long side. The two end seats stay fixed. Predict the next total before changing the stage.
Blue: four seats per table. Purple: two fixed end seats. For this arrangement, n is a positive whole number.
Now connect the pattern to a graph.
Set m = 4 and b = 2. Only positive whole-number inputs describe complete table arrangements.
The line represents a rule. Use the context to decide which inputs are meaningful.
y = 3x + 8
The same rule in a table
| x | y |
|---|
Describe the structure
Imagine tables joined in a row, with four seats added for each table and two seats at the ends. The expression 4n + 2 represents the number of seats. It is more useful than “add four” because it can produce the output for any permitted input directly. The equation s = 4n + 2 names the output s and states an equality.
Check more than the next value
For n = 1, 2, 3 and 4, the rule gives 6, 10, 14 and 18 seats. Substitute the input, multiply first, then add. Several rules can agree on a short list, so explain why the structure of the situation supports your rule. A numerical pattern alone does not prove the physical situation continues forever.
Use the graph thoughtfully
The graph model represents y = mx + b. To represent the seat pattern, set m to 4 and b to 2. The straight line shows a linear rule, but only whole-number table counts describe this particular context. A point at n = 2.5 belongs to the drawn line; it is not an arrangement of two and a half complete tables in this model.
Recursive or direct?
“Start at 6 and add 4 each time” is a recursive rule: it uses the previous term. The direct rule s = 4n + 2 uses the term number n. At the 100th term, the direct rule gives 402 without listing 99 earlier terms. Both descriptions fit this pattern, but they answer different questions.
Worked examples
Follow the reasoning.
Does 4n + 2 match the pair (5, 22)?
- Use the input n = 5, not the output 22.
- 4(5) + 2 = 22. The calculated output agrees.
- This pair fits the rule. One matching pair alone cannot establish the whole rule.
Try it with support
Show your thinking.
Write an expression for a $5 entry fee plus $3 per game.
Give me a hint
Let g count the games. Which amount repeats?
Compare a worked response
3g + 5
The $3 charge repeats once per game; the $5 fee is paid once.
Explain why 4n + 2 is not the same as 4(n + 2).
Give me a hint
Try n = 1 or expand the brackets.
Compare a worked response
4(n + 2) = 4n + 8.
In the bracketed expression, both terms are multiplied by 4.
Independent practice
Check your understanding.
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
0 of 8 questions completed
This round saves on this device. It is practice, not a school mark.
Check the concepts, too.
Choose a response, then use the explanation to check your reasoning. This is practice, not a school mark.
Go further
Keep the connections close.
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