How far is the island?

A week or so ago I posed the following problem:  You are stuck on an island inhabited by man-eating crabs. Unless you swim to another island you will be eaten.  The problem is, you can't tell how far away the island is.  Is it two kilometers or five?  Knowing the distance with reasonable accuracy may be the difference between making it or not.

 How do you determine how far away an object is when you are unable to directly measure it?

Well this is where maths is so cool.  Maths allows you to reach beyond the mortal coil, to see into the future and stretch beyond your physical limitations.

I asked my children how they would work out how far the island was.  One said you could observe how fast the shadow of a cloud moved, and time it, as it traveled the distance.  Awesome.  My second eldest said you could use the sun traveling across the sky as a sort of timer and convert it to distance.  Very nice idea also.

There is another way.  It involves a right angled, 45 degree triangle.  That's a square folded in half for the uninitiated.  A VERY useful shape.  I cannot be bothered  writing down how I would do it - however I do want to amaze you with my brilliance so I'm going to have another go at a video.

A note:  I am a poor student.  I don't have ANY high-tech gadgetry.  This clip is raw.  But I have filmed in shaky cam style to provide some realism.  Think 'Blair Witch project' or 'The Borne Identity'.  In other words- sorry about the quality.  Also- I can only upload 2 minute clips (Errr), so it is in two two minutes blocks.

So there you have it - I hope that gives you an idea anyway.  If we know the length of one side of a right angled triangle and one of other angles we can determine the length of any side.  The 45 degree triangle rocks.  You can use it to work out how high cliffs are, trees etc.

This is essentially the system the navigators on ships would use.  They had Sextants that would work better than my bamboo square - but the same fundamental idea.

My question to you is this... What kind of scenarios might get your children/adult learners (or you) interested in exploring this concept of distance estimation further?

I may put up a learning plan in coming weeks as to how I begin to develop interest and knowledge of trigonometry with learners.