Algebraic Relationships and One-Step Equations
Warm-up
Camping trip challenge: 5 students are splitting the cost of equipment equally. Total cost: 35. Now: if they also need 2 bags of trail mix per person, how many bags total? 5x2=10. Two equations in one context.
Explore
Equation sort and solve: 12 cards with equations using all four operations (3 of each type). Students sort by operation type, solve each, verify each, and write one word problem that matches each equation. The word-problem writing requires understanding the equation structure.
Formalize
Algebraic relationship exploration: if 4 friends each bring 3 snacks, total snacks = 4x3=12. If 5 friends: 5x3=15. If n friends: nx3. Build a table: friends (1,2,3,4,5) and snacks (3,6,9,12,15). Rule: snacks = 3 x friends. This is y = 3x before the notation.
Algebraic Relationships and One-Step Equations
Four-operation fluency with the inverse: 7 x n = 84. Think: 84/7 = n = 12. Check: 7x12=84. n/8 = 9. Think: 9x8 = n = 72. Check: 72/8=9. The inverse relationship is the universal tool for solving one-step equations.
Practice
Students solve 8 one-step equations (2 per operation), verify each, write 2 word problems, and complete a camping trip planning task requiring 3 equations. Exit ticket: solve and verify n/6 = 9.
Exit ticket
Students solve 8 one-step equations (2 per operation), verify each, write 2 word problems, and complete a camping trip planning task requiring 3 equations. Exit ticket: solve and verify n/6 = 9.
Step 1: Bridge the notation gently: last year the mystery was a box (4 × ▢ = 24); this year it wears a letter: 4 × n = 24. Same mystery, dressier costume. The letter n IS a number — one we haven't identified yet, not a "thing" or a label.
Step 2: Solve by fact retrieval: 4 × 6 = 24, so n = 6.
Step 3: Substitute to confirm: 4 × 6 = 24 ✓ — "n = 6 makes the sentence true."
Step 4: Head off the two classic letter-misreadings NOW: (a) n is not an abbreviation ("n stands for nickels") — it stands for a NUMBER OF nickels; (b) different letters may hold the same value, and the same letter in one problem holds ONE value throughout.
Step 5: Fluency loop: solve n + 9 = 17, 30 ÷ n = 5, n − 8 = 8, each with the say-aloud check "…makes it true." The phrase is the concept.
Step 1: Play it live: students feed the machine (teacher) numbers; the machine answers. In 3 → out 8. In 5 → out 12. In 10 → out 22.
Step 2: Collect conjectures and TEST them against all data: "+5"? Works for 3→8, dies on 5→12. "Double plus 2"? 3→8 ✓, 5→12 ✓, 10→22 ✓. Survivor.
Step 3: Write the surviving rule in symbols: out = 2 × in + 2, or with letters: b = 2a + 2.
Step 4: Run the machine BACKWARD — the algebra move: "the machine said 30; what went in?" Undo in reverse order: 30 − 2 = 28, then 28 ÷ 2 = 14. Check forward: 2×14+2 = 30 ✓.
Step 5: The quiet payoff: forward is evaluating an expression; backward is solving an equation. Students just did both without a single grim worksheet — and the reverse-order undo (subtract THEN divide) is exactly the skill Grade 7 formalizes.
The story: stickers come in sheets of 6. Priya bought some sheets and got 42 stickers. How many sheets?
Step 1: Choose and DECLARE the variable in writing: let s = the number of sheets. (The declaration sentence is non-negotiable — it's what keeps letters meaning numbers.)
Step 2: Translate the structure: each sheet carries 6, s sheets carry 6 × s, and that's 42. Equation: 6s = 42.
Step 3: Solve: s = 42 ÷ 6 = 7. Check in the STORY: 7 sheets × 6 = 42 stickers ✓.
Step 4: The defence round — compare with a partner who wrote s = 42 × 6 (= 252 sheets?!). Reality-test both equations against the story: could 42 stickers require 252 sheets? Sense-making, not symbol-shuffling, referees.
Watch for equation-writing by keyword ("of means times, so ANY two numbers get multiplied"). The antidote is always the same question: "read your equation back as a story — is it THE story?"