LESSON PLAN

Operations with Integers

Apothem Team

Grade 8 · Computational Fluency

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Post a rapid-fire ladder on the board, answers on whiteboards: $-7 + 12$ , $-3 - 9$ , $(-6)(4)$ , $\frac{-24}{-8}$ , $(-2)^3$ , $-2^4$ . The last pair splits the room — $(-2)^3 = -8$ but $-2^4 = -16$ — and the debrief of THAT pair is the day's hook: notation details carry sign decisions.

Grade 8 integers are about fluency under fire: all four operations, powers, and order of operations, with signs tracked like money.

Explore

Fluency circuits with a twist — every station pairs computation with JUSTIFICATION: (1) mixed four-operations ladder where each answer feeds forward; (2) the powers-of-negatives table — $(-1)^n$ for $n = 1..8$ exposing the odd/even alternation; (3) expression evaluation: $3a^2 - ab$ at $a = -2, b = 5$ (the $a^2$ vs $-a^2$ landmine defused with substitution parentheses: $3(-2)^2 - (-2)(5) = 12 + 10 = 22$ ).

Station 3's rule earns a poster: SUBSTITUTE IN PARENTHESES, always — the single habit that prevents most sign carnage in algebra.

Formalize

Formalize the sign laws compactly, including powers:

$(-)(-) = (+) \qquad (-a)^{\text{even}} > 0, \;\; (-a)^{\text{odd}} < 0 \qquad -a^2 \ne (-a)^2$

The $-a^2$ convention settled once: exponents bind before the bare negation, so $-3^2 = -(3^2) = -9$ , while $(-3)^2 = 9$ . Parentheses are the only thing separating the two — reading them is not optional.

Practice

Practice: 15 mixed computations under light time pressure; three expression evaluations with negative substitutions; one order-of-operations gauntlet with nested signs; one error autopsy on a dropped parenthesis.

Exit ticket: evaluate $x^2 - 3x$ at $x = -4$ , with the substitution shown in parentheses. ( $(-4)^2 - 3(-4) = 16 + 12 = 28$ .)

Exit ticket

Exit ticket: evaluate $x^2 - 3x$ at $x = -4$ , with the substitution shown in parentheses. ( $(-4)^2 - 3(-4) = 16 + 12 = 28$ .)

TIP The parentheses-on-substitution rule is worth a week of nagging: it converts the hardest recurring algebra error into a mechanical habit that costs two pencil strokes.

WORKED EXAMPLES

Example 1 — The account ledger, now with powers: a week of transactions

The week: start at $-\$35$ (overdrawn). Deposit $60. Three withdrawals of $12 each. A doubling penalty on the remaining overdraft… let's keep it honest: then withdraw $25.

Step 1: Deposit: $-35 + 60 = 25$ .

Step 2: Three withdrawals as multiplication: $3 \times (-12) = -36$ ; balance $25 + (-36) = -11$ .

Step 3: Final withdrawal: $-11 - 25 = -36$ .

Step 4: Read the story: the week ends $36 overdrawn. The multiplication in step 2 is why integer products matter: repeated negative transactions are exactly $n \times (-a)$ , and the ledger IS the number line.

Example 2 — Evaluate like a professional:

\frac{b^2 - 4ac}{2a}

a = -1, b = -6, c = 5

(Yes — this is the quadratic formula's engine room, visited early on purpose.)

Step 1: Substitute IN PARENTHESES: $\frac{(-6)^2 - 4(-1)(5)}{2(-1)}$ .

Step 2: Numerator, piece by piece: $(-6)^2 = 36$ ; $4(-1)(5) = -20$ ; so $36 - (-20) = 36 + 20 = 56$ .

Step 3: Denominator: $2(-1) = -2$ .

Step 4: Divide: $\frac{56}{-2} = -28$ .

Step 5: Count the sign decisions: five, each local and small (square, product, subtract-a-negative, denominator, final quotient). The parentheses habit made each one visible. Students who skip the parentheses face all five signs at once, from memory, and the casualty rate shows it.

Example 3 — The alternating tower: what is

(-1)^{100}

, and who cares?

Step 1: Build the pattern: $(-1)^1 = -1$ , $(-1)^2 = 1$ , $(-1)^3 = -1$ , $(-1)^4 = 1$ — flip, flip, flip. Even exponents land on $+1$ , odd on $-1$ .

Step 2: So $(-1)^{100} = 1$ (100 is even) — no calculator, no hundred multiplications; PARITY answers it.

Step 3: Apply the shortcut to a monster: $(-2)^{9}$ — sign from parity (odd → negative), size from $2^9 = 512$ : answer $-512$ . Sign and size are separable questions.

Step 4: Who cares: alternating signs run through mathematics — alternating sums, checkerboard colourings, the $(-1)^n$ that flips series terms in Calculus II. Today's flip-flip table is the first meeting with a pattern that never stops being useful.

MATERIALS

•Whiteboards

•Ladder circuit cards

•Powers-of-negatives tables

•Practice set (PDF)

WATCH FOR

-2^4

read as

(-2)^4 = 16

. Exponent binds first:

-16

. The warm-up pair exists for this.

!Sign rules over-generalized to addition ("minus and minus is plus, so

-3 + (-9) = 12

"). Rules are operation-specific.

!Substitution without parentheses:

x^2

x=-4

becoming

-4^2 = -16

. The poster rule.