Problem-solving Pack

Mathematics

Year 7 · 7.7 Shape and Measure

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

Fencing a field. A farmer wants to fence a rectangular field with area 72 m². She has exactly 34 m of fencing to use as the perimeter.

Find the length and width of the field. Show all your working.

Working space

2Problem 2 of 12

Optimal pen design. A farmer has exactly 40 m of fencing. She wants to make a rectangular enclosure divided into three equal pens by two internal fences parallel to one pair of sides (as shown below):

\underbrace{\Big[\;\big|\;\big|\;\Big]}_{\text{3 pens}}

Let

W

be the width of the whole enclosure (perpendicular to the dividers) and

L

be the length.

(a) Explain why the total fencing used is

2L + 4W = 40

.
(b) Express

L

in terms of

W

.
(c) Write the total area

A

as a function of

W

alone, and complete the table:

|

W

(m) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|----------|---|---|---|---|---|---|---|---|---|
|

A

(m²) | | | | | | | | | |

(d) What value of

W

gives the maximum area? What is that area?
(e) At the maximum, what is the ratio

L : W

? Does this surprise you?

Working space

3Problem 3 of 12

Trapezoidal swimming pool. A swimming pool has a trapezoidal cross-section: it is 1 m deep at the shallow end and 3 m deep at the deep end. The pool is 25 m long and 12 m wide.

(a) Sketch the cross-section and label all dimensions.
(b) Find the area of the trapezoidal cross-section.
(c) Find the volume of the pool in m³.
(d) Convert the volume to litres and find how long (in hours) it takes to fill at 1 000 litres per minute.
(e) A second pool is rectangular with the same length, width, and the same volume of water. How deep is the rectangular pool?

Working space

4Problem 4 of 12

Tiling a floor. A rectangular kitchen floor is 3.6 m long and 2.4 m wide. Square tiles with side length 30 cm are to be laid with no gaps.

How many tiles are needed? Show how you convert units consistently.

Working space

5Problem 5 of 12

Shape investigation. A shape is made from a rectangle and a right-angled triangle.

The rectangle is 12 cm long and 5 cm wide. The triangle is attached to one of the shorter ends (5 cm wide), and has a perpendicular height of 8 cm.

(a) Find the total area of the compound shape.
(b) Find the perimeter of the compound shape. The slant side of the triangle has length 8.5 cm (given).

Working space

6Problem 6 of 12

Nets. A cube has side length 4 cm.

(a) Draw a sketch of a valid net for this cube (describe it in words if you cannot draw).
(b) Find the total area of the net.
(c) Explain why a cross-shaped net with five squares in a column and one square to the right of the second square from the top is NOT a valid net for a cube.

Working space

7Problem 7 of 12

Algebra + area. A rectangle has length

(2x + 4)

cm and width

(x + 1)

cm. Its area is 40 cm².

(a) Show that

x^2 + 3x - 18 = 0

and solve it to find

x

.
(b) Write down the dimensions of the rectangle and find its perimeter.

Working space

8Problem 8 of 12

Perimeter puzzle. The perimeter of an equilateral triangle equals the perimeter of a square. The square has side length 9 cm.

(a) Find the side length of the triangle.
(b) Find the area of the triangle. (Use the formula: area =

\frac{\sqrt{3}}{4} \times \text{side}^2

, or split into two right-angled triangles.)
(c) Which shape has the larger area?

Working space

9Problem 9 of 12

Thinking about 3D shapes. A toy factory uses cuboid boxes with dimensions 6 cm × 4 cm × 3 cm.

(a) Find the volume and surface area of one box.
(b) The boxes are packed into a larger cuboid crate. The crate is 24 cm × 20 cm × 12 cm. How many boxes fit in the crate?
(c) What fraction of the crate's volume is taken up by the boxes? Simplify your answer.

Working space

10Problem 10 of 12

Area reasoning. Two shapes have the same area.

- Shape A is a triangle with base 16 cm and height

h

cm.
- Shape B is a trapezium with parallel sides 5 cm and 11 cm, and perpendicular height 8 cm.

Find

h

Working space

11Problem 11 of 12

Staircase border investigation. A "staircase" pattern is built from unit squares (each 1 cm × 1 cm). The

n

-step staircase has

n

columns: column 1 has 1 square, column 2 has 2 squares, …, column

n

has

n

squares (like a rising staircase from left to right).

A border of width 1 cm is painted around the outside of each staircase.

(a) Draw (or describe) the 1-step, 2-step, and 3-step staircases.
(b) For each of

n = 1, 2, 3

:
- Count the number of unit squares in the staircase.
- Count the perimeter of the staircase (in cm).
- Calculate the area of the 1 cm border painted around it.
(c) Complete the table:

|

n

| Squares in staircase | Perimeter (cm) | Border area (cm²) |
|-----|----------------------|----------------|-------------------|
| 1 | | | |
| 2 | | | |
| 3 | | | |
| 4 | | | |

(d) Find a formula for the number of unit squares in the

n

-step staircase.
(e) Find a formula for the border area around the

n

-step staircase.
(f) The border area of one staircase equals the number of squares in another staircase. Which two values of

n

satisfy this? (There may be more than one answer.)

Working space

12Problem 12 of 12

Open-ended investigation. A rectangle has a fixed perimeter of 24 cm.

Complete the table of possible integer dimensions, calculate each area, and identify which dimensions give the maximum area.

| Length (cm) | Width (cm) | Area (cm²) |
|-------------|-----------|------------|
| 11 | 1 | |
| 10 | 2 | |
| 9 | 3 | |
| 8 | 4 | |
| 7 | 5 | |
| 6 | 6 | |

What do you notice? What happens if the rectangle becomes a square?

Working space