Operations with Rational Numbers

Warm-up

The all-terrain gauntlet, five minutes, whiteboards: $-\frac{3}{4} + \frac{5}{6}$, $(-0.4)(2.5)$, $-\frac{2}{3} \div \left(-\frac{8}{9}\right)$, $-1\frac{1}{2} - \left(-2\frac{1}{4}\right)$. All the machinery exists from Grades 7–8; today welds it into one system.

Rational numbers = every fraction, decimal, and integer, positive and negative — ONE number family, all four operations, no seams.

Explore

Integration circuit mixing signs with fractions/decimals deliberately: (1) number-line placement of mixed forms ($-\frac{7}{4}$, $-1.6$, $\frac{5}{3}$, $-0.\overline{3}$) — ordering negatives still flips intuition; (2) the operations relay where each answer feeds the next, forms alternating fraction↔decimal by whichever is cleaner; (3) the properties station: does commutativity survive negatives? ($-\frac{1}{2} + \frac{3}{4} = \frac{3}{4} - \frac{1}{2}$ ✓) Does division? ($\frac{1}{2} \div 2 \ne 2 \div \frac{1}{2}$ ✗ — order matters, as always).

Close with form-choice strategy talk: $-\frac{3}{4} \times 0.8$ — convert which way? ($0.8 = \frac{4}{5}$ cancels beautifully: $-\frac{3}{5}$. The fraction lane had the shortcut.)

Formalize

Formalize the rational numbers as a closed system:

$\mathbb{Q} = \left\{\tfrac{a}{b} : a, b \in \mathbb{Z},\, b \ne 0\right\} \qquad \text{closed under } +, -, \times, \div \text{ (except } \div 0)$

Closure said plainly: rationals in, rationals out — no operation on fractions ever needs a number outside the family (division by zero excepted, forever). Sign rules and fraction rules COMPOSE without interference: signs decide the sign, sizes decide the size, separately.

Practice

Practice: twelve mixed computations (forms and signs interleaved); two order-of-operations towers; one form-choice defence ("I converted to ___ because…"); one temperature/elevation story crossing zero with fractional values.

Exit ticket: $\left(-\frac{2}{5}\right) \div 0.25 + \left(-\frac{3}{10}\right)$. ($-\frac{2}{5} \times 4 = -\frac{8}{5}$; $-\frac{8}{5} - \frac{3}{10} = -\frac{16}{10} - \frac{3}{10} = -\frac{19}{10}$.)

Exit ticket

Practice: twelve mixed computations (forms and signs interleaved); two order-of-operations towers; one form-choice defence ("I converted to ___ because…"); one temperature/elevation story crossing zero with fractional values.

Exit ticket: $\left(-\frac{2}{5}\right) \div 0.25 + \left(-\frac{3}{10}\right)$. ($-\frac{2}{5} \times 4 = -\frac{8}{5}$; $-\frac{8}{5} - \frac{3}{10} = -\frac{16}{10} - \frac{3}{10} = -\frac{19}{10}$.)