
Steve Arnold

Areas and PaperFolds
Download Student Worksheet file (PDF)

Who would have thought that there could be so much mathematics in simply folding a piece of paper? This activity spans the years of secondary school, beginning with measurement, data collection and interpreting scatter plots in the early years, through linear functions, Pythagoras' Theorem and trigonometry, right through to calculus in the senior years.
The mathematical focus at each level is different  from finding the largest area to discovering functional relationships between the sides of a rightangled triangle, and on to optimisation. While algebra and calculus can be used to prove this result, it actually takes some geometry to understand why the final result is true.

M2: Applications of area and volume
M3: Similarity of twodimensional figures
M4: Rightangled triangles



Steve Arnold

The Falling Ladder
Download Student Worksheet file (PDF)

What does it feel like to be at the top of a ladder as the bottom begins to slide away? Do you fall at a steady rate? If not, then what is the nature of your motion  and when are you falling fastest?
This modelling problem is suitable for students across the secondary school, from consolidation of work on Pythagoras' Theorem in the early years, to optimization using differential calculus in the senior years. At all levels, it is a realistic and valuable task, which links a variety of mathematical skills and understandings with a practical realworld context.

M2: Applications of area and volume
M3: Similarity of twodimensional figures
M4: Rightangled triangles



Steve Arnold

The Diminishing Square
Download Student Worksheet file (PDF)

Study the diagram provided. A smaller square has been constructed inside a larger square, as shown.
A point x is located on the base of the larger square. (As shown) The smaller square is constructed using similar points on each of the remaining sides of the larger square. If x is the midpoint of the base, what is the ratio between the area of the larger square and the smaller square?
Explore the relationship between the position of this point and the area of the smaller square.

M2: Applications of area and volume
M3: Similarity of twodimensional figures
M4: Rightangled triangles



Steve Arnold

Cones and Witch's Hats

From a sheet of cardboard 40 cm square, I need to make a conical witches hat for my child's party. If we assume she has a circular head of diameter 14 cm, what is the tallest hat I can make?
Assume I will make the cone by cutting a sector from a circle: what angle must I make this sector? 
M2: Applications of area and volume
M3: Similarity of twodimensional figures
M4: Rightangled triangles

