Arithmetic asks for an answer. Algebra asks for a rule. Everything on this poster rests on one promise: simplifying changes how an expression looks — never what it is worth.
🧩Read an Expression Before You Touch It
An expression is a math phrase — it names a value but makes no claim (no equals sign). Parse it first: cut at every + and − sign, and let each minus sign ride along with the term on its right.
⚠️Invisible coefficients bite. In a bare x the coefficient is 1; in −x it is −1; in −x²y it is −1 as well. More Algebra I points are lost to that one detail than to any other.
🔧Evaluate = Substitute, Then Obey PEMDAS
Evaluating is mechanical, not creative. Drop each value in wearing parentheses, then follow the order of operations. Those parentheses are the fence that keeps a negative sign where it belongs.
✅With parentheses
Evaluate 2x² − 7 when x = −3
2(−3)² − 7
= 2(9) − 7
= 18 − 7 = 11
The exponent grabs the whole −3, so (−3)² = 9.
❌Without them
Same instruction, written carelessly
2 · −3² − 7
= 2(−9) − 7
= −18 − 7 = −25
Here the exponent grabs only the 3. 36 units of error.
🧮Try a two-variable one: evaluate ab + (a − b) ÷ 2 at a = −3, b = 5. → (−3)(5) + (−3 − 5) ÷ 2 = −15 + (−8) ÷ 2 = −15 − 4 = −19. Division outranks the final subtraction — do it first.
📦Only Like Terms Combine
Like terms have identical variable parts — same letters, same exponents. Combining them changes only the coefficients, because you are counting, exactly the way 5 nickels + 3 nickels = 8 nickels.
💡When two terms are unlike, the expression is already fully simplified. Pushing further doesn’t tidy it — it breaks it. 5x + 3x² is a finished answer.
📐The Distributive Property Runs Both Ways
Picture a rectangle a tall and (b + c) wide. Measure it as one big rectangle or as two glued together — the two measurements have no choice but to match. That is the whole proof.
➕Expand carefully: −4(3 − 2m)−4 × 3 = −12, and −4 × (−2m) = +8m. So the answer is 8m − 12. That second sign flip is where most errors live — slow down there deliberately, every single time.
🧮Why mental math works7 × 103 head-on is unpleasant. But 7(100 + 3) = 700 + 21 = 721 takes two seconds. Distribution isn’t a classroom rule — it describes how × and + have always behaved toward each other.
✏️Full Worked Example
Simplify 6(2x − 5) + 3x
1
Distribute the 6 across both terms inside: 12x − 30 + 3x(6 × 2x = 12x, 6 × −5 = −30)
2
Find the like terms. 12x and 3x share the variable part x. The −30 is a lone constant with nothing to pair with.
3
Combine:12x + 3x = 15x, so the expression becomes 15x − 30.
4
Spot-check at x = 2. Original: 6(4 − 5) + 6 = 6(−1) + 6 = 0. Simplified: 15(2) − 30 = 0. They match.
⭐ 6(2x − 5) + 3x = 15x − 30
!
One matching value is reassuring, not proof. Two expressions are equivalent only when they agree for every input — and here that is guaranteed, because every move was a named property, not a hopeful guess.
💭English → Algebra
This is the switch that makes every word problem after this one solvable. Watch the order: “less than” and “subtracted from” both flip it.
The sentence
The algebra
five more than twice a number n
2n + 5
five less than twice a number n — not 5 − 2n!
2n − 5
twice the quantity “n minus five”
2(n − 5)
the quotient of n and 4, decreased by 7
n/4 − 7
a rectangle’s length is 3 more than twice its width w
ℓ = 2w + 3
…so that rectangle’s perimeter is
2(w + 2w + 3) = 6w + 6
🔑Key Terms
❓VariableA letter holding a place where a number will stand. In 40 + 25m, m is however many months have passed.
📝ExpressionNumbers, variables and operations that name a value but make no claim. 2n − 5 is an expression; 2n − 5 = 11 is an equation.
✂️TermA number, a variable, or a product of them, separated from its neighbours by + and −. 3x² − 5x + 2 has exactly three.
✕️CoefficientThe number multiplying the variable part. In −7ab it is −7; in a bare x it is an invisible 1.
⚓ConstantA term with a fixed value and no variable. In 40 + 25m the constant is 40.
📦Like TermsTerms with identical variable parts — same letters, same powers. 4x²y and −9x²y qualify; 3x² and 3x do not.
🔧Evaluate / SubstituteReplace each variable with its given number (in parentheses) and simplify by the order of operations until one value remains.
📐Distributive Propertya(b + c) = ab + ac. Forwards it expands; backwards it factors: 12x + 18 = 6(2x + 3).
🔄Equivalent ExpressionsTwo expressions producing the same value for every legal input. Agreeing at one lucky value proves nothing.
🔢JuxtapositionWriting things side by side to mean multiply. Algebra dropped the × sign because it collides with the variable x — so 3x is 3 · x.
🌎Where This Shows Up
📊Spreadsheets. Typing =B2*1.0825 into a cell is really writing 1.0825p, where p is whatever price lands in B2. Write the expression once and it evaluates itself for all 4,000 rows — the same compression as 40 + 25m instead of listing every month by hand.
🎮Game design. Damage is an expression like 2·atk + 5·lvl − def. Designers balance the game by nudging coefficients — bump the 5 to a 7 and levelling up suddenly matters more. Engineers then combine and factor so it computes the same result with fewer operations, sixty times a second.
📜The word itself. “Algebra” comes from al-jabr, from a book written around 820 CE by al-Khwarizmi in Baghdad — written entirely in ordinary sentences. There were no x’s yet; every equation was a paragraph. The habit of x, y, z for unknowns was popularised by René Descartes in 1637.
📌Remember This
1Evaluating is mechanical: substitute inside parentheses, then follow the order of operations. Those parentheses are what keep (−3)² = 9 from collapsing into −9.
2Only like terms combine, and only their coefficients change. Same letters, same exponents, or hands off.
3Distribution runs both directions — expand a(b + c), or factor it back. Every legal step preserves value, so simplifying changes an expression’s appearance and never its worth.
🤔 Think about it
We say 5x + 3x = 8x but 5x + 3x² is NOT 8x². Test both claims at x = 2 and again at x = 3, write down all four values, and pinpoint exactly where the second claim breaks. What does that failure reveal about what the word “like” is really demanding?
The expressions 2 + 3x and 5x agree at exactly one value of x. Find it. Then explain why one matching value is nowhere near enough evidence to call two expressions equivalent — and what would count as enough.
⭐Remember: a variable is a slot, not a secret. Cut at the signs, count the like terms, distribute both ways — and never let a minus sign wander off from the term it belongs to.
✏️ ClickClass Stretch Anchor Chart · Algebra I · Variables, Expressions & Like Terms