Section 3 (Four Credits)
Geometry, Number and Patterns – The Shape of Things
Shape drives this world as much as random
We are not often aware of this truth. The natural world
evolved and has perfected shape and size to optimise survival and
adaption. Think about teh shape of different teeth and their use.
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 of which you think. This is very different from language where we often create our own meaning. A metaphoric language like Te Reo struggles with this "point" perception.
Having 1-D 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 point or character can only ever know and met first-hand two points, the neighbours on each side. Thank goodness we live in 3-space and can know more than 2 people.
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 3rd 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 simplifications are then in turn applied to the wider world. Simplification is a problem solving approach.
A triangle is a polygon with three sides. It is the simplest of all two dimensional shapes.
Sketch the triangles named scalene, isosceles, equilateral and right angled.
Label the triangles you drew with capitals for vertices (A) and lowercase for the opposite sides (a). Equilateral means 3 equal sides and is also equiangular. Isosceles has two sides the same length and a scalene has no sides of equal length. A right angled triangle has a right angle in one corner.
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.
Draw and name as many different quadrilaterals as you can.
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.
Complete the table showing the number of diagonals in polygons.
Carefully count up the number of diagonals in each n-gon. The diagram shows the first 4 with the diagonals already drawn. Complete the table.
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.
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
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.
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.
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.
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.
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 firstname.lastname@example.org 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! Use the Navigator