Variables, Constants, and Expressions
How do symbols turn a numerical pattern into an expression, equation, and function rule?
How this lesson unfolds.
Begin with Why does maths use letters? The sequence then connects Variables and constants and builds toward Substitute and evaluate.
Why does maths use letters?
Letters express a repeated numerical pattern with one reusable rule.
- Helpful foundationswhole-number multiplication
- Next connectionDistinguish the changing count from fixed prices.
More context
A notebook costs 20 rupees. Two cost 40 and three cost 60. Define n as the number of notebooks. The expression 20 × n gives the cost in rupees for any allowed count. The number changes while the multiplication rule stays fixed. A letter must be defined in context.
What changes, and what stays fixed?
In 20n + 5, n is a variable, 20 is a constant coefficient, and 5 is a constant term.
- Helpful foundationsmeaning of mathematical symbols
- Next connectionRead the expression as a sequence of operations.
More context
At a shop, notebooks cost 20 rupees each with a fixed 5-rupee fee per purchase. In 20n + 5, n is the varying notebook count, 20 is its fixed coefficient, and 5 is a constant term. The count can be a positive whole number. Constants remain fixed within this model; a different shop could use different constants.
How can an expression be a recipe?
An expression specifies operations; 20n + 5 means multiply n by 20, then add 5.
- Helpful foundationsvariables and constants · multiplication and addition
- Next connectionAdd an equals sign to make a claim about the total.
More context
The expression 20n + 5 describes the total notebook cost. Start with n notebooks, multiply by 20 to obtain the base cost, then add a fee of 5. Multiplication is performed before addition. An expression does not claim that two values are equal. It becomes a numerical value when n is given.
What does an equals sign claim?
An equation claims that both sides have the same value; a solution makes that claim true.
- Helpful foundationsalgebraic expressions · multiplication and addition
- Next connectionSubstitute a chosen count and evaluate the expression.
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The equation 20n + 5 = 65 asks which notebook count produces a total of 65 rupees. If n is 2, the left side equals 45, so the claim is false. If n is 3, the left side equals 65, so the claim is true. Thus n = 3 is a solution. The equals sign states equal values in either direction.
How do we turn a recipe into a value?
Substitution replaces a variable with a value; evaluation performs the operations and leads to a function rule.
- Helpful foundationsalgebraic expressions · order of operations
- Next connectionExplore functions as rules that assign one output to each allowed input.
More context
For four notebooks, substitute n = 4 in 20n + 5. This gives 20 × 4 + 5, then 80 + 5, then 85 rupees. Multiply before adding. The rule C(n) = 20n + 5 names the total cost as a function of the input count n. Each allowed notebook count gives one total cost.