# 1.17. Analyse and manipulate representations to solve problems

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In this section you are required to achieve results through efficient and correct analysis and manipulation of representations, while presenting problem-solving methods clearly, logically and in mathematical terms.

In order to do so, you must ensure that your solutions are correct and are interpreted and validated in terms of the context of the problem.

The range of problems you will need to solve includes:

- Using nets of prisms and cylinders
- Using and interpreting scale drawings of plans (e.g., plans of houses or factories; technical diagrams of simple mechanical household or work related devices such as jacks
- Using road maps relevant to the local community
- Using the Cartesian coordinate system in determining location and describing relationships in at least two dimensions

Using nets of prisms and cylinders

A net is a diagram showing how the plane faces of a solid are joined to each other. Only 3 dimensional shapes have nets (cut open shapes).

Cube | |

Prism | |

Cylinder |

**Using and interpreting scale drawings of plans**

Before we look at scale drawings we first need to understand the relationship between ratios and scales.

A **ratio** is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to write the ratio of 8 and 12. We can write this as 8:12 or as a fraction 8/12, and we say the ratio is *eight to twelve*.

**Examples: **

Jeanine has a bag with 3 DVDs, 4 marbles, 7 books, and 1 orange.

1) What is the ratio of books to marbles?

Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer would be 7/4. Two other ways of writing the ratio are 7 to 4, and 7:4.

2) What is the ratio of DVDs to the total number of items in the bag?

There are 3 DVDs, and 3 + 4 + 7 + 1 = 15 items total.

The answer can be expressed as 3/15, 3 to 15, or 3:15.

Normally, when we refer to a scale, we’re usually talking about a ratio. For example, take model building: most models are NOT the same size as the real thing and the scale is a ratio, an expression of the relationship, between the size of the real thing and the size of the model.

So if something is referred to as ‘quarter scale’, it means that the model is one quarter of the size of the original. In other words, something that was twelve metres tall in real life would be 3 metres tall as a model. Similarly if the real thing is 200mm long, the model would be 50mm long. It doesn’t matter what unit you measure it in, just as long as the measurement on the model is a quarter of the measurement on the real thing.

Instead of saying ‘quarter scale’, we can say ‘1/4 scale’. But we can also say ‘1:4 scale’, which means the same thing; it’s just a different way of writing it. In fact when you’re expressing a ratio, it’s the correct way to write it.

So, now that we understand quarter scale (or 1/4 or 1:4 scale), we can take a leap forward and discuss 100^{th} scale (or 1/100 or 1: 100). 1:100 scale means that something that’s 100 units long in the real world will be 1 unit long as a model.

For example, this drawing has a scale of “1:10”, so anything drawn with the size of “1” would have a size of “10” in the real world, so a measurement of 150mm on the drawing would be 1500mm on the real horse.

**Scale Drawings of Plans**

Scale drawings of plans would include:

- plans of houses or factories
- technical diagrams of simple mechanical household or work related devices such as the mechanical jack (scissor jack) described below:

A mechanical jack is a device which lifts heavy equipment. The most common form is a car jack, floor jack or garage jack which lifts vehicles so that maintenance can be performed. Car jacks usually use mechanical advantage to allow a human to lift a vehicle by manual force alone. More powerful jacks use hydraulic power to provide more lift over greater distances. Mechanical jacks are usually rated for a maximum lifting capacity (for example, 1.5 tons or 3 tons). The jack shown at the right is made for a modern vehicle and the notch fits into a hard point on a specific spot under the car. Earlier versions have a platform to lift on the vehicles’ frame or axle.Jackscrews are integral to the Scissor Jack, one of the simplest kinds of car jacks still used. |

Below is an example of a scale drawing of flagpole and a stick, to show the relationship that the lengths of their respective shadows have:

If you were representing a two-bed roomed house with rooms of the following sizes,

Room A = 6m² and Room B = 8 m² respectively, your scale could be: 1cm = 1m (depending on the size of the paper that you would use to create your scale drawing)

Therefore you will have your representations as follows:

- Room A = 6m²
- Room B = 8m²