Section 3 (Four Credits)
Geometry, Number and Patterns – The Shape of
Things
Shape drives this world as much as random does.
We are not often
aware of this truth. The natural world evolved and has perfected
shape and size
to optimise survival and adaption.
Dimension
In the beginning was a point. The current
theory of the
universe can work it’s way back to an tiny splinter of a second
after the Big
Bang. The suggestion is that there was a point at which the
universe began in
both Space and Time. It was about 14 Billion years ago. The
religions on Earth
got to that point centuries ago!
In geometry the statement “There exists at
least one point”
is the primary Axiom about which all other deductions happen.
There are a
couple of others that also represent what we “see” in this world
and are
essential for a consistent and true representation.
Having zero dimensions is about being
a point. A
point has no size. We represent a point as a dot but that is
nonsense because a
dot of graphite pencil has length, breadth and thickness if you
look closely
and it also has mass. Our mathematical imagination “sees” a point
as
dimensionless and the point I think of is exactly and precisely
the same as the
point you think of. This is very different from language where we
often create
our own meaning. A metaphoric language like Te Reo struggles with
this
perception.
Having 1D or One Dimension
There is a book called Lineland
creates
a consistent geometry in a story located on a line. The line could
be bent but
any two points or characters can only ever know and met firsthand
two points,
their neighbours. Our roads can be considered to be one
dimensional but most do
run in both directions. There are several types of lines. Can you
name them?
Two Dimensions or Flatland
is
a much more interesting place. The surface of the Earth is a
Flatland and life
moves on the surface pretty much. The 3^{rd} Dimension of
length or
height is occupied by birds and rockets.
Euclid
is the father of two dimensional geometry and wrote about shapes.
Not the
complex shapes that animals, plants and rocks form but a much more
simplified geometry
of triangles, squares and circles. These in turn are then applied
to the wider
world.
The Triangle
A triangle is a polygon with three sides. It is
the simplest
of all two dimensional shapes.
Task 1
Sketch the triangles named scalene,
isosceles,
equilateral and right angled.
Task 2
Label the triangles you drew with capitals
for vertices
and lowercase for the opposite sides.
Angles
A triangle has 3 internal angles. There are
different ways
to label angles.
The Quadrilateral, or literally “four sides”,
is the next complexity
of shape and there are several types all with different names.
Task 3
Draw and name as many different
quadrilaterals as you
can.
Diagonals
A curious feature of the quadrilateral is that
each has two
diagonals.
This diagram is a check sheet for Task 3 and
also shows the
two diagonals in each type.
Polygons are shapes with different numbers of
straight sides
or line segments. A pentagon has 5 sides, a hexagon has 6, a
heptagon has 7
sides and the octagon has 8.
Now for some connections and problem solving.
Polygon 
# of Diagonals 
3 Triangle 
0 
4 Quad 
2 
5 Pentagon 

6 

7 

8 

9 

10 

n 
? 
Task 4
Complete the table showing the number of
diagonals in
polygons.
Carefully count up
the number of
diagonals in each ngon. The diagram shows the first 4 with the
diagonals
already drawn.
Problem to Solve
There is a
formula that
connects the number of sides n to the total number of diagonals.
Can you find
the formula? Write the answer you create where the “?” is in the
table. The “n”
stands for “any” sided polygon.
Constructions
A great source of
fun in geometry
is constructions. There are just a few basic skills to master and
fascinating
colourful creations appear. Here is one of them.
Skill 1  To draw a line on a point put the
pencil on the point first
and slide the ruler gently to the pencil, adjust for the other
end and lightly
draw the construction line past the points. Construction lines
are always quite
faint.
Skill 2  To draw a circle, set
the compass to the desired
radius, place the point on the circle centre and leaning the
compass a little
bit in the direction of rotation lightly draw a complete circle.
This is a bit
tricky at first.Persevere!
Skill
3  To bisect a line
segment
Skill
4  To bisect an angle
And
pretty much the
rest is your imagination!
During the COVID
Lockdown in NZ in
early 2020 and the uncertain times generated I created a series of
files for
students and teachers to send to students to work on at home and
continue to
learn mathematics.
Here is the link to
all of them. http://schools.reap.org.nz/advisor/COPVID19.html
Task 5
Locate #12 in the
Covid File
list. Read and complete the tasks.
Isometry 
Reflections and
Rotations
Shapes have
properties.
An isosceles
triangle for example
has an axis of reflection.
Task 6
Using the Covid
File link
above locate the lessons on Reflection, Rotation, Translation
and Enlargement.
Choose one that looks like fun and complete the tasks.
Theorems of
Geometry
There are many and
this is where
the Greeks spent most of their time developing mathematics.
An example of a
“proof” is to
show there are exactly 180 degrees in a triangle, every triangle.
Prove the three angles of a
triangle add to 180˚.
[Hint – Draw a line
parallel to one of the sides through the vertex]
Construct a line parallel to the base and
through the vertex at c. This
creates two pairs of rotated angles that are the same a and b.
The three angles
inside the triangle are the same as the three angles on the
straight line. We
know that a straight angle is 180˚ by definition so the three
angles in a
triangle sum to 180˚ as well.
There is
nice connection of transformation knowledge here and the logic
of “If A = B,
and B = C then A = C”.
Each theorem is a
building block
used to create another theorem. In this way a huge knowledge is
developed and
recorded. See COVID
FILE
#15.
Time to Review Progress, Again, Again
Once again, there are many connections, new
words, and ideas
presented in this section.
Task 7
Take 5 minutes and review recording as many
new words and
ideas, connections and questions you may have.
Section 3 introduced the language of shape or
geometry.
Having a robust knowledge of the language of geometry is important
because much
of our world is described using this language.
Task 8
Look around the house and/or go for a walk
and identify
and name all the shapes you find.
The circle was used in constructions but also
can be a deep
study with connections to astronomy, architecture, engines,
physics and nature
in general.
Geometry makes strong connections to Number and
the
generalisations called Proofs (Algebra). The next Section deals
with
quantifying or measuring the dimensions of the world we live in.
Assessment
Section 3 (Four Credits)
Geometry, Number and Patterns – The Shape of
Things
1.
Construct this
pattern using a compass
and a ruler.
2.
Sketch angles that
are acute, straight,
obtuse and reflex.
3.
Construct an
isosceles triangle and prove
that the base angles are equal.
4.
We say there are 360
degrees in a full circle. Why
is this?
To gain 4 Credits for this section screenshot
or photograph
your answers or otherwise email a copy to jimhogan2@icloud.com
for registering(first time), checking and getting feedback. This
is intended to
be a painless process and all questions are accepted. Once you
have been
awarded the 4 credits a fee of $5 will be requested for this
section.
Well done, three sections over! On to Section
4!