Coordinates - February 2005, All Stages

Problems

Late Again

Stage: 1 Challenge Level:

Moira is late for school. What is the shortest route she can take from the school gates to the entrance?

Making Shapes

Stage: 1 Challenge Level:

Arrange any number of counters from these 18 on the grid to make a rectangle. What numbers of counters make rectangles? How many different rectangles can you make with each number of counters?

Arrangements

Stage: 2 Challenge Level:

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Fred the Class Robot

Stage: 2 Challenge Level:

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

Cartesian Isometric

Stage: 2 Challenge Level:

The graph below is an oblique coordinate system based on 60 degree angles. It was drawn on isometric paper. What kinds of triangles do these points form?

Ten Hidden Squares

Stage: 2 Challenge Level:

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

Eight Hidden Squares

Stage: 2 and 3 Challenge Level:

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Isosceles Triangles

Stage: 3 Challenge Level:

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Lost

Stage: 3 Challenge Level:

Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.

Square Coordinates

Stage: 3 Challenge Level:

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Lost on Alpha Prime

Stage: 4 Challenge Level:

On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?

Tracking Points

Stage: 4 Challenge Level:

Can you give the coordinates of the vertices of the fifth point in the patterm on this 3D grid?

Something in Common

Stage: 4 Challenge Level:

A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

Exponential Trend

Stage: 5 Challenge Level:

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

Pitchfork

Stage: 5 Challenge Level:

Plot the graph of x^y = y^x in the first quadrant and explain its properties.

Polar Flower

Stage: 5 Challenge Level:

This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.