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.

 

 

 

 

 

Task 5

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.”

 

P2        = 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)   P2 = {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!