Operations with Integers
Warm-up
Open with the money ledger: "Your account: deposit $8, withdraw $5, withdraw $7, deposit $3. Where do you stand?" Students track it however they like — most land at and can explain the debt.
They've just added integers with mixed signs by instinct. Today turns that instinct into rules they can trust when the numbers stop being friendly — and gives multiplication and division of integers their reasons, not just their sign-chants.
Explore
Two-model workshop for addition/subtraction: (1) Counters — red for negative, yellow for positive; a red-yellow pair is a ZERO PAIR and vanishes. Build (two reds survive: ), then by adding zero pairs until seven yellows exist to remove. (2) Number line — subtraction as distance/direction; read as "how far, and which way, from to ": 7 rightward.
For multiplication, build the pattern ladder and let it force the rule: , , , , then keep the pattern marching: . The ladder's steady steps make negative-times-negative-is-positive a CONSEQUENCE, not a decree.
Formalize
Formalize the operations. Subtraction is addition of the opposite; multiplication and division follow the sign rules the ladder forced:
Absolute value organizes addition: same signs — add the sizes, keep the sign; opposite signs — subtract the sizes, keep the LARGER size's sign. Say it with the counters in view so the rule stays attached to the zero-pair picture.
Practice
Practice: a mixed set of 12 computations spanning all four operations; two ledger stories; one pattern-ladder construction for times the integers; one error-autopsy on the classic (read as — the sign of the 6 was dropped).
Exit ticket: compute , , , and , with a one-line reason for the third.
Exit ticket
Practice: a mixed set of 12 computations spanning all four operations; two ledger stories; one pattern-ladder construction for times the integers; one error-autopsy on the classic (read as — the sign of the 6 was dropped).
Exit ticket: compute , , , and , with a one-line reason for the third.
The story: dawn temperature C; by afternoon it rose . Afternoon temperature?
Way 1 — number line: start at , travel 23 right: 14 steps reach 0, the remaining 9 land at .
Way 2 — absolute-value rule: opposite signs → subtract sizes (), keep the larger size's sign (positive): .
Way 3 — counters: 14 reds meet 23 yellows; 14 zero pairs vanish; 9 yellows remain.
All three agree: . The split-at-zero move (Way 1) is the mental-math winner; the rule (Way 2) is its formalization; the counters (Way 3) are the proof you can touch.
Way 1 — distance and direction: asks "from to : how far, which way?" From up to 0 is 3; from 0 to 5 is 5 more: 8, rightward → .
Way 2 — the debt story: your balance includes a $3 debt. Someone REMOVES the debt. Removing a enriches you by — so subtracting is adding 3: .
Step 3: Unify with the formal rule: , and the opposite of is .
Step 4: Immediate application with no comfort numbers: . The rewrite-then-add pipeline handles every case; the two stories explain why it's legal.
Step 1: Exponent first: . (Flag the landmine: — the square binds tighter than the bare minus. Parentheses decide.)
Step 2: Multiplication: . Numerator: .
Step 3: Division: (negative over negative).
Step 4: Addition: .
Step 5: Audit trail — sign decisions were made at four separate moments (square, subtraction, division, none at the final add). Each was local and small. Students who defer all sign thinking to the end are doing four decisions at once from memory; that's where the wheels come off.