grid geometry angles, using arc tan

using the tangent of the angle as a measure of the size of the angle
is better than a protractor!
convert an angle into a fraction (inv tan (rise / run), possibly by summing or subtracting) - so that it can be compared in size with other grid angles (i.e. as fractions)

a powerpoint is here

this presentation was part of a session in London, joint ATM/MA on march 16th 2019 and was intended to illustrate how a square grid might be helpful in geometry, an idea proposed by van Hiele in 1986 ('structure and insight') following the research of Dina and he from 1957 onwards

using arc tan in triangles

a ppt is here

continuing to explore the value of relating an angle drawn connecting (square) grid points to the tan of the angle, so that the angle could be worked out (using inv tan (fraction)) or properties of angle size related to the tan of the angle

grid geometry parallels

Pierre van Hiele maintained that geometry should be started off using a square grid

it might be advantageous to describe (and define) parallel lines as having the same gradient/slope/journey/vector on a grid

dwelling on the properties of shapes:

• parallel sides
• right angles (perpendiculars)
• equal sides
• convex/concave
• mirror symmetry and rotational symmetry

should probably precede the naming of shapes and classifying them

a powerpoint is here

angles on parallel lines

a ppt is here

justifying parallelism by using congruence and then inv tan to show that gradients are the same

opposite angles in a parallelogram are equal

grid geometry perpendiculars

it seems helpful to consider (and then define) perpendicularity in terms of a 'rotated' vector (through 90 degrees)

equal lengths can be established from the diagonals of rectangles (or, later, by pythagoras)

a powerpoint is here
 

show that the diagonals of kites cross at right angles