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

two angles sum to 45 degrees

a ppt is here

sum to 90 degrees: complementary
sum to 180 degrees : supplementary
sum to 360 degrees: explementary

sum to 45 degrees: halfrightlementary (or anything else...)

this is part of a series of tasks involving a square dot grid and expressing an angle as arc tan (fraction)
in this case, it is exploring what two angles sum to 45 degrees

these were used in several tasks, part of a session at Anne Watson and John Mason's house looking at the topic of 'angle'

angle bisectors on a grid

a ppt is here

this is part of a series of tasks involving a square dot grid and, in this case, expressing an angle as arc tan (fraction)
exploring when angle bisectors passing through integer (dot) grid points

these were used in several tasks, part of a session at Anne Watson and John Mason's house looking at the topic of 'angle'

you could use a few of the initial tasks to practice bisecting an angle with ruler, pencil and compasses

but the work continues, to see how pythagorean triples are involved and how to calculate the radius of the incircle for such triangles

grid reflections

based on KS2 SATs in England

a ppt is here