Writing and Evaluating Expressions
Warm-up
The phone-plan translation race: "$30 base plus $5 per gigabyte" — write it algebraically. (.) Then reverse: hand them and ask for a STORY that fits. (Twelve dollars a lawn, minus the $8 broken-rake fine…)
Both directions matter: Grade 8 expressions are about translating fluently between situations and symbols, then evaluating and simplifying with negative numbers fully in play.
Explore
Translation gymnasium: (1) situation→expression cards with near-miss decoys ("3 less than double n": , NOT — order words are traps); (2) expression→situation invention with peer audit (does the story really produce ?); (3) like-terms consolidation with negatives: , sorted physically with term-tiles first.
Evaluation follows with the parentheses ritual: at : .
Formalize
Formalize the anatomy and the two core skills:
Like terms share the same variable part exactly ( and are strangers); the sign travels WITH its term when sorting. Translation order-words table: "less than" and "subtracted from" flip the order; "quotient of a and b" is in the stated order.
Practice
Practice: eight translations (both directions, decoys included); three simplifications with negatives and squared terms; three evaluations at negative inputs; one story-audit of a partner's invention.
Exit ticket: simplify and evaluate at . (; at : .)
Exit ticket
Practice: eight translations (both directions, decoys included); three simplifications with negatives and squared terms; three evaluations at negative inputs; one story-audit of a partner's invention.
Exit ticket: simplify and evaluate at . (; at : .)
The setup: shirts cost \$6 each to print; the team sells at \$15 each; fixed table fee \$40.
Step 1: Build the profit expression for shirts: income , costs → profit .
Step 2: Simplify — distribute the minus across BOTH cost terms: .
Step 3: Evaluate at : dollars.
Step 4: Interpret the simplified form's anatomy: the 9 is profit-per-shirt (margin), the the fee's drag; break-even where → about 5 shirts. The SIMPLIFIED expression carried business meaning the raw one hid — simplification is translation into a more honest dialect.
Step 1: The minus-two distributes to BOTH terms inside: and .
Step 2: Rewrite fully: .
Step 3: Collect: .
Step 4: Autopsy the two standard wrecks: (a) distributing only to the first term ( ✗); (b) losing the sign on the second ( ✗). Both die at the same checkpoint: numerical spot-check at — original: ; our answer: ✓; wreck (a): 0 ✗; wreck (b): ✗.
Step 5: The spot-check habit: one cheap substitution referees any simplification dispute, permanently.
The design: a square pool of side metres gets a 1-metre tile border. How many 1×1 tiles?
Step 1: Student A counts by strips: four side-strips of tiles plus four corner tiles: .
Step 2: Student B counts by subtraction: big square minus pool: .
Step 3: Are they the same? Test : A gives 16; B gives ✓. Test : 44 and ✓.
Step 4: Show it symbolically (a preview of expanding, kept gentle): , so B = A. Two honest countings of one border MUST agree — and the algebra just proved it for every pool at once.
Step 5: The takeaway that fuels all of algebra: equivalent expressions are different STORIES about the same quantity, and simplification is how you find out two stories match.