8.5 Directed Number and Algebra Basics

**Directed-number temperature.** A weather station records temperatures every 4 hours on a January day: 6 am: $-7°C$; 10 am: $-2°C$; 2 pm: $5°C$; 6 pm: $-1°C$; 10 pm: $-6°C$. (a) Find the temperature change from 6 am to 10 am. (b) Find the largest temperature drop between two consecutive measurements. (c) Find the mean temperature.

**Bank account.** A bank account starts at £100. The following transactions occur in order: - Deposit of £80 - Withdrawal of £200 - Deposit of £45 - Withdrawal of £60 - Bank fee of £15 (a) Find the balance after each transaction. (b) State whether the account is overdrawn at any time and by how much. (c) The bank charges a £5 fee per day overdrawn (whole-day basis). Find the maximum fee accrued if the account spends 2 days overdrawn.

**Substitution puzzle.** Let $a = -2, b = 3, c = -4$. Calculate each: (a) $a + b - c$ (b) $abc$ (c) $a^2 + b^2 - c^2$ (d) $\dfrac{ab - c}{a + b}$

**Distributive practice.** Expand and simplify. (a) $3(2x - 5) + 4(x + 2)$ (b) $-2(x - 3) + 5(2x - 1)$ (c) $-(3x + 2y) - 2(x - y)$ (d) $4(2a - 3b) - 2(a - 3b) + (a + b)$

**Like-terms puzzle.** Simplify each expression and identify the coefficient of $x$. (a) $5x - 3 + 2x - 4x + 1$ (b) $7x^2 - 3x + 2x^2 + x - x^2$ (c) $a + 2b - 3a + b + 5$

**Order of operations.** Evaluate each carefully, showing your steps: (a) $-3 + 4 \times (-2)^2$ (b) $(-3 + 4) \times (-2)^2$ (c) $\dfrac{-12 + 4 \times 3}{(-2)^2 - 4}$ — note the denominator (d) $-2^4$ versus $(-2)^4$

**Algebraic notation review.** Rewrite each in correct algebraic form (using $\times$ only when necessary, and exponent notation): (a) $5 \times a$ (b) $a \times a \times a \times b$ (c) $2 \times x \times x$ (d) $3 \times a \times b \times a$ (e) $(2a)^2$

**Two-step rectangle problem.** A rectangle has length $(2x + 3)$ cm and width $(x - 1)$ cm. (a) Write a simplified expression for its perimeter. (b) Calculate the perimeter when $x = 5$. (c) For what value of $x$ does the perimeter equal 28 cm?

**Distributive with brackets and minus signs.** Expand and simplify. (a) $-(2x - 3y + 4)$ (b) $4 - 2(x + 5)$ (c) $-3(2x - 1) - 2(x + 4)$ (d) $5(x - 2y) - 3(2x - y) + (x + y)$

**Magic square with directed numbers.** A 3 × 3 magic square (every row, column and diagonal sums to the same total) contains the values $-4, -3, -2, -1, 0, 1, 2, 3, 4$. (a) Find the magic sum. (b) Show that the centre must be 0.

**Coefficients and exponents.** Simplify each expression: (a) $3a^2 \times 4a$ (b) $\dfrac{12 a^3}{4 a}$ (c) $5x \times -2x^3$ (d) $(2ab^2)^2$

**Modelling with directed numbers.** A lift in a building starts at the ground floor (level 0). It performs the following moves: - Up 5 floors - Down 3 floors - Up 7 floors - Down 12 floors - Up 1 floor (a) Find the lift's position after all moves (relative to ground, where below ground is negative). (b) What is the largest height above ground reached at any point? (c) What is the lowest level visited?