Problem-solving Pack

Mathematics

Year 8 · 8.5 Directed Number and Algebra Basics

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12

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.

Working space

2Problem 2 of 12

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.

Working space

3Problem 3 of 12

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}

Working space

4Problem 4 of 12

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)

Working space

5Problem 5 of 12

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

Working space

6Problem 6 of 12

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

Working space

7Problem 7 of 12

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

Working space

8Problem 8 of 12

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?

Working space

9Problem 9 of 12

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)

Working space

10Problem 10 of 12

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.

Working space

11Problem 11 of 12

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

Working space

12Problem 12 of 12

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?

Working space