Algebraic expressions
The idea
When a quantity is unknown or keeps changing, you let a letter hold its place: if each box holds s shirts, then 3 boxes hold 3s. An algebraic expression is a recipe built from such letters and numbers — 2s + 3 says double the amount, then add 3. You already use the properties of operations and spot number patterns; expressions just give the pattern a name so you can compute with it before knowing the actual value.
Two skills carry most of the weight: combining like terms, where s + 2s = 3s because one batch plus two batches is three batches of the same thing, and the distributive property, where 3(2x + 5) = 6x + 15. The trap to avoid is treating the letter as an object instead of a number — reading 3s as 'three shirts' invites mistakes like 2x + 3 = 5x, which wrongly merges a number of x-batches with a plain number. The letter always stands for a value, not a thing.
Worked example
A student council sells s shirts on Saturday. On Sunday they sell 3 more than twice Saturday's count. Write a simplified expression for the weekend total, then evaluate it when s = 12.
- Translate Sunday into symbols: 3 more than twice Saturday is 2s + 3.
- Add the two days and combine like terms: s + (2s + 3) = 3s + 3, since one batch of s plus two batches of s makes three batches.
- Substitute s = 12 into the simplified form: 3 × 12 + 3 = 36 + 3 = 39.
- Verify against the unsimplified story: Sunday alone is 2 × 12 + 3 = 27, and 12 + 27 = 39, so the simplification preserved the meaning.
Answer. The weekend total is 3s + 3 shirts, which equals 39 shirts when s = 12.
Check your understanding
- Why is it legal to combine s and 2s but not 2s and 3 — what exactly makes terms 'like'?
- How could you convince someone that 3s + 3 and s + 2s + 3 give the same value for every choice of s?
- What does the distributive property let you do to an expression, and when does using it backwards help?
- How would the expression change if Sunday were 3 fewer than twice Saturday, and which words signaled the change?
Build the foundations first
Algebraic expressions builds on these concepts. If any feel shaky, start there.