Grid Drawing Dilations: Algebraic Notation and Fractions

After studying coordinate point dilations using multiplication of whole numbers greater than one for enlargements and fractions between 0 and 1 for reductions, I wanted the children to explore the wonders of grid drawing dilations.

HISTORY: 

Artists have been using mechanical aids for centuries. In the 15^th and 16^th centuries, for example, a Master would create a small drawing and apply horizontal and vertical lines equally spaced. The artist would then enlarge the grid to the desired dilation. Some of the paintings were dilations of scale factors in the hundreds. They would then draw on paper his complete design for a fresco, then assistants would transfer an area of that design to damp plaster sufficient for the day's work. This they did by pricking holes along the lines of the drawing, holding the paper up to the plaster surface and then "pouncing" (pummeling) the lines with muslin bags of soot or carbon. The artist could then continue his work assisted by a representation of his drawing in the form of closely spaced black dots. 

I do not believe that this process diminishes the art. I have heard it said that the only test of an artist was the finished product, no matter what preparatory steps had gone into its making.

PROCESS: 

The Mathletes were given 20 drawings that were superimposed onto a grid of cells numbered A through J horizontally and 1-8 vertically, for example. Each cell can be identified by naming it by its corresponding letter and number like B5. The children are now very familiar with this algebraic notation. I then gave them another document that has blank grid paper that dilates to factors of 2 or greater (for the George Washington and Horse drawing, the dilation factor is slightly less than 1).

I recommended that the children choose one of the easier square cells in the picture and find the corresponding cell in the blank grid. Then find a point that is some fractions from one of the vertices (the corners). So if an intersection point is at the half way mark of the bottom of the cell, then make a small mark at the half way point in the corresponding blank cell. Then make another point using the same process and draw the line or curve while still looking at the picture. It is easy to see points at the half way point. It is more difficult to see points at 1/3, 1/4, 1/5 and so on, but the more accurate your estimations the better. 

If you want to work with your children to make exact dilations, use a centimeter ruler and measure the millimeters from a vertex to a point and then multiply it by the scale factor. The product is the exact location of the point on the blank grid.

MATERIALS FOR LARGE DILATIONS:

The easiest way to make large dilations is to use sticky notes as they are perfect squares (3x3 inches or 77x77 mm). Dilate each cell using a single sticky note and then paste them together with tape. These dilations can by up to 8 or 10 scale factors. Now for the real fun: I dilated to factors of 14-21 by using square pieces of 8.5x8.5 inch paper. This is created by taking printer paper (8.5 x 11 inches) and cutting out a square.  With your permission, the children can make huge murals on their bedroom or basement walls that are nearly identical to the originals. I showed them a dilation of 21 of the parrot in the attached pdf as well as a dilation of 14 of the Mona Lisa, also in the packet. 

I am exploring the creation of a dilation of a portion of The School of Athens with the detail of Plato and Aristotle.