Jim Hogan Mathematics

A self paced opportunity to improve your mathematics and numeracy.

How Numbers Work
Numbers were invented by Humans. In the early days there were simple bead counts. Counting is almost as fundamental to language as words. Some cultures had little use for numbers and their mathematics is quite primitive.

Australian Aborigines used "1,2,3,many". I explain this as... if a hunter sees 1 kangaroo it could be the last, so he leaves it. If he sees 2 kangaroo he can kill one and leave the other. If he sees 3 kangaroo he can have a feast. More than 3 is not an issue and there is simply more. He can report all this using language. Note there is no 0, zero. It makes no sense to return from a hunt and confess to see"NO KANGAROO". His gin (wife) would think he had been in the bush too long and probably knock him on the nut. Notice also there is no negative numbers. Like zero it makes no sense to come home and report "NOT SEEING 2 KANGAROO". His gin would definitely knock him on the nut becasue not only is he not seeing kangaroo he is now counting the things he can not see! Fractions make no sense either because things happen in KANGAROO units and are never seen as HALF a KANGAROO. There is no need to add or divide so maths for the Australian Aborigine was pretty primitive. They did draw a lot of pictures but it is archeologists who scrapped away thousands of years of habitation and tagged a time line. They did count hiigher than 3 by "subitising". [Subitising is a term that was coined by the theorist Piaget and defined the ability to instantaneously recognise the number of objects in a small group without the need to count them.]. Looking at a flock of birds or a herd of kanagroio teh aborigine can associate 5, 10 etc to the group. We all do this.

Well, it is a good story anyway. See "First Footprints" a book,and a documentary,  about the 80,000year old journey of the Australian Aborigine. https://www.imdb.com/title/tt3286844/

The Romans used a peculiar system that prevented operations like adding and multiplying. They used symbols I, V, X, L, C, D and M for 1,5,10, 50, 100, 500 and 1000. The by combining these they made 4 = IV or 1 less than 5. Centuries are written on most big screen films using Roman Numerals so 1978 = MCMLXXVIII. It is more clear if we expand as M CM LXX VIII. The Romans had a few other things on their minds than doing maths but a lot of our language comes from them. A Centurion was one of 100 soldiers who made up a fighting unit as a 10x10 grid.

Who invented the Place-Value system? A simple Google Search reveals most things.

"Only three cultures, as far as we know, invented a place value numeration system: the Mayans, the Babylonians, and the Hindu people of India. The current, almost planetary wide, place value numeration system is derived from the Hindu system. It was transmitted to western Europe by the Moslem world."

"The most commonly used system of numerals is the Hindu–Arabic numeral system. Two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero."
PV    PV Visual

Place value means just that. The value depends on the place of the number. In the illustration of 369 we see 369 is actually 300 + 60 + 9 and we say three humndred and sixty-nine but we could say 3 hundreds, 6 tens and 9 ones.

Bilbo in the book The HOBBIT has an interesting and better interpretation of placevalue numbers. He counts one, two, three, ...eight, nine, ten, tenty-one, tenty-two, ...tenty-eight, tenty nine, two-ty, twoty - one,...ninety - nine, tenty-ten and so on. The Maori counting is similar with 11 being te ko ma tahi or 10 + 1. The Chinese system is similar. The words we use for teen-numbers has its roots in French, Italian and the Romans. Where on earth did eleven and twelve thirteen come from. There was an ancient Anglo Saxon word called fife or 5. Hence fifteen = fife and ten.

Problem Based Learning with Place-Value.
Problem-based learning is fun. There are no long boring exercises that you have to complete before you are allowed to go and play! in PBL you get to collaborate and share and talk. It is fun. As a teacher I am never convinced that what I am teaching is being learned and what is being learned has anything to do with what I am teaching! So you explore and ask questions and learn whatever it is you need to learn. That is how it works. I can still test you for knowledge, skills and problem solving. So, have a play with these problems and do not forget you can share pictures and ideas and questions with me at jimhogan2@icloud.com

1. Take a look at The Original Mind-Reader. Link = http://www.flashlightcreative.net/swf/mindreader/

Try and figure out how all this works! It caused havoc on the web when first released with people thinking it was infecting computers. It is a simple, well fairly simple, explanation based around place-value and designed in a brilliant way by Andy with some random effects added. Very entertaining and I use this a lot. By struggling to make sense of this problem you grow your brain mathematically.

2. Think up a three digit number, for example, 234. Now write twice to make the 6 figure number 234234. Divide this 6 figure number by 7, and then answer by 13 and then that answer by 11. The number you get should be 234 or the same as you started with? How does this work? This problem is a very cool problem and has connections to many other place-value problems. (Use a calculator!)

3. Here is a simple one. In the number 123,456,789, 000, or 123 billion, 456 million, 789 thousand, is there a "3". I could ask is there a 1, a 2, a 3, a 4 or any digit? What is your answer?

4. Write down a 4 digit number and best at this stage if all the digits are different. (For example 3761.) Now scramble the same digits to make another number. (For example 7136.) Now find the difference between the two numbers. (Hint large - small). (7136 - 3761 = 3375).
There are a couple of curious things about this number 3375.
- It is divisible by 9 even though neither 7136 nor 3761 were divisible by 9. I was fascinated by this as a young lad and spent a long time trying to figure out why.
- If you add the individual digits of 3375 we get 3+3+7+5 = 18 which is always a multiple fo 9 and if we keep adding we get 1+8=9 and 9 is what we will always get. Adding the digits gives a term called the "digital root" . Here is alink on n-Rich for Digital Roots - https://nrich.maths.org/5524and https://www.hitbullseye.com/Quant/Digital-Root-or-Seed-Number.php and another  https://banking.currentaffairsonly.in/digital-root-or-seed-number-trick-for-bank-exams/ All just for fun.

5. Here is a heap of place value problems. https://nrich.maths.org/public/leg.php?group_id=1&code=6

6. Rules of Division. Best here I think! https://www.mathsisfun.com/divisibility-rules.html

7. Here is test of what you know about place-value. Write down the numeral (use digits) for "eleven thousand, eleven hundred and eleven".

All of the above works because we use a Base 10 Place-Value Counting system. We can use any whole number greater than 1 for a base. Computers use Base 2 so their number look like 110101001 and only use 2 digits 0 and 1. Base 10 uses 10 digits 0,1,2,3,4,5,6,7,8,9. Base 16 uses 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F and is also used in computer work. https://www.mathsisfun.com/numbers/bases.html

Reflection is always good. So what did you learn? I liked the Mind Reader the best.

Remember always that anyone can learn anything if you put your mind to it.