**Section 2 (Three Credits)**

**Visual Mathematics – Seeing is Believing **

We all “see” patterns and shapes in slightly
different ways.
We all lead different paths through experiences and these
experiences in turn
alter how the world is perceived.

How many dots are there in the following
diagram?

*Task 1*

*The task is different however. Think about
how you
grouped and counted the dots. *

* *

*Using a template draw circles around the
groups to
represent how you “see” the dots. If you have different ways
draw these as
well. *

Mathematics is very visual. Being able to see
patterns is
the key driver to be good at algebra. Patterns in numbers are
represented in
tables and these are likewise represented as graphs. Generalising
patterns is
something animals and humans are very good at. This is algebra.

Example of an animal pattern. My dog expects
dinner at about
4pm every day and shows this by waiting at the door.

Example of a human pattern. I wake up at about
6:30am every
morning.

Task 2

*Record a few more examples of patterns by
animals and
humans. *

There are many patterns in nature as well. The
way the moon
changes, Spring and Autumn, the way plants grow and flower are
just a couple of
suggestions. These patterns can be described and represented
mathematically.

*Task 3*

*Here are a few pictures of patterns in
nature. Look at
each and describe what mathematics you see in each one. *

*Task 4*

*Go for a walk and photograph patterns.
Record and
describe these in a mathematical way. *

Life is a pattern and so is mathematics.
Mathematics is all
about patterns and making sense of the world we live in.

There are patterns in random events as well.
There are many
random or chaotic events in nature. Clouds are a very good
example. Even though
they are generated by a random condensation effect there are types
or patterns
of clouds.

The next task is an experiment to demonstrate
how patterns
form as a result of a completely random event called a coin toss.

*Imagine
you are a
coin and you flip yourself 100 times. Record ten rows of ten
outcomes you
create. One the other grid toss a coin 100 times and record
the outcomes. Give
the coin a good toss [or use a die and record odds (H) and
evens (T).] *

** **

*Task 6*

*Look at
both
results. One generated by you, pretending to be random, and
the other recording
a randomly generated result. What do you notice? Write about
any curiosities or
patterns you notice. *

Being random is actually quite difficult. For
hundreds of students
who have endured the above experiment not a single one noticed
anything. I
would ask them to put the two grids side by side and take great
enjoyment
walking around the room quite quickly telling each student which
one they
generated. My success rate was 100%. I would tell them I was a
mind reader
or a magician. In fact all I was doing was looking for the grid
that had a long
run of 5 or even 6 H’s or T’s.

Humans convince themselves that after 2 or 3
heads it is
almost certain that a tail will happen and never write HHHHH or
TTTTT. Did you
notice? Noticing is very important in mathematics. Learning how to
“see” from
different perspectives too.

**Square Numbers**

This is a group of numbers with particular
importance in
mathematics. A square is the number multiplied by itself. For
example 7 x 7 =
49. 49 is a square number.

The diagrams show the first 100 numbers listed
in 10 rows of
10 consecutive numbers, and the basic facts for multiplication for
all the
Natural numbers 1 to 10.

*Task 7*

*On both charts circle or colour all the
square numbers
you can “see”. *

*Task 8*

*Write about any patterns you notice. *

The obvious one on the multiplication tables
chart is the
sloping diagonal 1x1 to 10x10.

[Did you forget about 1x1?]

*Task 9*

*Explain why the diagonal of square numbers
appears. *

Now on the Hundreds Chart colour or circle all
the square
numbers. They seem to be chaotic or random in the way they bump
around from one
line to the next but all the time getting larger and larger.

*Task 10*

*Describe any pattern you see on the square
numbers on the
Hundreds Chart?*

Another fun and cryptic statement I would make
to my
students was…

*“The
difference
between the square numbers is very odd!”*

*Task 11*

*Can you explain the meaning behind this
statement?*

There are many sets of numbers and all of them
have
interesting patterns. In fact all numbers are “interesting”. If
you can name
one that isn’t then it will come “interesting”. In this way “All
numbers are
interesting!”

N
= the set of
Natural or Counting numbers

= {1, 2, 3,
4, 5, 6, …}

W
= the set of
Whole numbers

= {0, 1, 2,
3, 4, 5, …}

Another cryptic statement arises!

**“There is zero difference between the N,
Natural numbers
and W, Whole numbers.”**

P_{2}
=
the set of Powers of 2

= {1, 2, 4,
8, 16, 32, …} and these were mentioned in Section 1.

T
= the set of
Triangular Numbers

= {1, 3, 6,
10, 15, …}

**Time to Review Progress, Again**

Once again, there are many connections, new
words, and ideas
presented in this section.

*Task 12*

*Take 5 minutes and review recording as many
new words and
ideas, connections and questions you may have. *

Noticing is a very human thing. Seeing
connections and
learning a vocabulary of words that helps describe the connections
and patterns
is being mathematical or “thinking like a mathematician”.

In Task 1 many people see 2 + 3 + 2 or 3 + 1 +
3. There is
no “right” way to “see” the dots just “different” ways. The
important part is
being able to explain what you see.

Here is another Axiom or fundamental truth. **“Always
explain
your answer or show how you solved the problem presented.” **It
is
the thinking and how all the connections happen that is the most
important. The
answer becomes less and less important and likewise the time taken
to get
there.

My guide to students is S, R, W, L and D.
Speak, Read,
Write, Listen and Do your solution or concept you are trying to
learn. After
that comes A. Apply the knowledge. When you can apply knowledge
you understand
and know. That is learning.

That leads to yet another Axiom. **“Make sure
you have fun
in whatever you do”**. Mathematics like all other human
endeavours should be
fun and enjoyable. That is what makes us human.

I am having is having fun trying to create a
guiding pathway
through the labyrinth of mathematical concepts and language which
is not a
textbook but more of an activity book.

Section 2 bumped into Number, Algebra, Geometry
and
Probability. Look over the tasks and see if you can identify the
“bumps”.

**Assessment
**

**Section Two
(Three Credits)**

**Visual Mathematics – Seeing is Believing **

1.
Name the number sets
represented by these
patterns.

(a)

(b)

2.
Draw a dot pattern for
these number sets

(a)
P_{2 }= {1, 2, 4,
8, 16, 32, …}

(b) N
= = {1, 2, 3, 4, 5, 6, …}

3.
What do the three dots “…”
mean in the above
sets?

4.
Five people meet and shake
hands with each
other.

(a)
How many handshakes in
total are there?

(b) Draw
a
picture showing the handshakes

(c)
Which Number Set connects
to the answer? How?

5.
Illustrate how you “see”
the total number of
this dots in this pattern.

6.
Describe 5 different random
number generator
tools. (eg “flipping as coin”)

7.
How can a ten sided die be
used to emulate or
behave like…

(a)
A coin

(b) A
six sided die

*Note…dice is
the plural of
die. Two dice, one die. *

To gain 3 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 3 credits a fee of $5 will be requested for this
section.

Well done, two sections over! On to Section 3!