A spider and a fly are in the same room. The fly is fast asleep and the spider is hungry. What is the shortest route the spider might take? Ah, I almost forgot! The spider can not swing itself like Tarzan, king of the jungle. It is limited to crawling on the walls, ceiling and floor.
The room is 15 meters long, 6 m wide and 6 m high. The spider is 0.5 m from the ceiling and 3 m from each wall. The fly is 0.5 m from the floor and 3 m from each wall.
I will give you a hint: The answer is not 21 m.
saylor.org offers a free course in beginning algebra.
In this course, you will study basic algebraic operations and concepts, as well as the structure and use of algebra. This includes the solutions to algebraic equations, factoring algebraic expressions, working with rational expressions, and graphing of linear equations. You will apply these skills to solve real world problems (word problems). Each unit will have its own application problems, depending on the concepts you have been exposed to. This course is also intended to provide you with a strong foundation for intermediate algebra and beyond.
The section called Solving Two Step Equations has a reading assignment, an assessment part (both pdf documents) and this video.
There are seven theorems for you to discover and prove.
- Go to this page to discover seven hypothesis.
- Go to this page to see the seven theorems summarised.
- Come up with seven proofs and put them, with your name, as comments below.
There are two kinds of shears.
1. Horisontal shear.
All points are moved horisontally, i.e. the y-value does not change. How much a point is moved horisontally is proportional to its distance from the x-axis.
In short: (x, y) -> (x + ky, y).
k is called the shear factor. The x-axis is called the invariant line. This means that any point on the x-axis does not move: (x, 0) -> (x + k*0, 0) = (x, 0).
2. Vertical shear
All points are moved vertically, i.e. the x-value does not change. How much a point is moved vertically is proportional to its distance from the y-axis.
In short: (x, y) -> (x, y + kx).
k is called the shear factor. The y-axis is called the invariant line. This means that any point on the y-axis does not move: (0, y) -> (0, y + k*0) = (0, y).
What kind of shear maps triangle X to triangle S? What is the invariant line? What is the shear factor?
By choosing a suitable unit you should be able to find how big each piece is.
Compare what you found with the image below.
Can you solve the mystery now?
James Tanton has made a video where he eats a piece of a cake. The white square below.
Carefully rearranging the pieces that are left he manages somehow to make the cake whole again!
Please explain how this is possible.
Khan interviewed by Charlie Rose last week.
On the usefulness of Khan’s videos.