# Definition

GeoGebra is free dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. It has received several educational software awards in Europe and the USA. It has many applications in mathematics but is particularly useful at modeling complex mathematical problems for Higher Level calculus students.

See the Geogebra website here for downloads - http://www.geogebra.org/cms/en/download

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# Practical examples to support learning

There are lots of examples of students using Geogebra on youTube. I have seen the program being used in an Higher Level Maths class to solve this problem.

In this task you are to model such a building and produce a report to the contractor with all necessary specifications. You are to design an office block inside a structure with the following specifications: The building has a rectangular base 150 m long and 72 m wide. The maximum height of the structure should not exceed 75 % of its width for stability or be less than half the width for aesthetic purposes. The minimum height of a room in a public building is 2.5 m.

- Create a model for the curved roof structure when the height is 36 m.
- Find the dimensions of the cuboid with maximum volume which would fit inside this roof structure.
- Use technology to investigate how changes to the height of the structure affect the dimensions of the largest possible cuboid.
- For each height, calculate the ratio of the volume of the wasted space to the volume of the office block.
- Determine the total maximum office floor area in the block for different values of height within the given specifications.
- Given that the base remains the same (72 m × 150 m), investigate what would happen if the facade is placed on the longer side of the base.
- You now decide to maximise office space even further by not having the block in the shape of a single cuboid.
- Review your model and calculate the increase in floor area and the new volume ratio of wasted space to office block.

*Image Credit: Arthur Wei - Grade 12 HL Maths 2010*

# Exercises to try

## NOVICE

- Watch the Geogebra video above.
- Download the program to you laptop. You may need to update your version of Java to run this smoothly.
- Check out the

## APPRENTICE

- Complete all of the novice tasks above.

## PRACTITIONER

- Complete both the basic and apprentice sections above

## EXPERT

- Complete all of the sections above

# Further reading/exploration

The Geogebra people have set up to resources to help teachers:

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