For information:
Fred W. Duckworth, Jr.
c/o Jewels Educational Services
1560 East Vernon Avenue
Los Angeles, CA 90011-3839
E-mail: admin@trinitytutors.com
Website: www.trinitytutors.com
Copyright © 2012 by Fred Duckworth. All rights reserved. This publication is copyrighted and may not be linked to directly, reproduced, stored in a retrieval system, transmitted in any form by any means, electronic, mechanical, photocopying, recording, or otherwise without written permission from the publisher.
You may print this entire publication or portions thereof directly from this website, but may neither download to your computer nor store any text or images comprising this work. Moreover, you may not place on any other website for others to access, nor distribute to anyone else in any manner any of the pages comprising this work. This material is to be utilized only for your personal use in a homeschool or tutoring environment, and you may print out any or all of the pages herein only on an individual, one-time basis.
Furthermore, all copyright notifications must be included and you may not alter them in any way. Classroom use and/or use in a public or private school setting is expressly prohibited. Anyone wishing to use this material must come to this website to access it. Any use beyond these terms requires the written permission of the author/publisher. This publication is being provided at no cost and may not be sold under any circumstances.
TrinityTu tors
Math
TrinityTutors.com
The Origins of Mathematics
Math is the study of patterns and relationships that exist between numbers. But, what are numbers?
A number is something that tells us how many.
Numbers can be represented using different types of symbols. For example, a number can be represented using units.
A unit is a basic amount used to count or measure.
There are all kinds of things that can be used as units. For example, we will sometimes use something known as a cube—a solid shape that has six square sides (or faces), eight corners, and twelve edges.
This means we will often use cubes (similar to the leather cube pictured above) for counting.
A unit that is all by its self represents an amount known as “one.”
An amount of one can also be represented by the numeral one, pictured below.
l
Or, it can be represented by the word one, written here in blue.
one
Numbers can also be represented using something called tally marks, which we will learn more about later on.
The number line was invented when people began to see a need for keeping track of individual events in their lives...
...like the number of animals they killed on a hunt, how many villagers died in a plague, or the number of cycles of the moon.
Keeping records of these kinds of events helped ancient people to survive, and with such record keeping, counting was born.
Counting was originally done on number lines, and the very first number lines were actually tally marks chiseled into stones, or slashes made on pieces of bone.
The disadvantage of this kind of marking system is if, for example, you lose or damage your bone, your information is lost as well.
So then, a unit is something used to count an amount...
and a number is a symbol that tells us how many units.
A unit may be a single thing, such as a penny, or it may be a group formed by putting together an exact number of things, like a dime.
We can think of a penny as a unit because it is a basic amount that is used to count money.
But, if we put together a group of exactly 10 pennies, we would form a new, larger unit, known as a dime, which is also used to count money.
As already suggested, when we want to know how many, we use an activity with which you are probably already familiar—an activity called counting. And when we count, we normally start with the number one.
This all began long, long ago with what has become the cornerstone of mathematics: the number line. The number line gives us another way of representing different amounts.
LESSON 1
LESSON 1
 | ||||
1
To solve this problem, unique symbols called digits were invented that made it possible to know the number of marks without having to actually count them.
The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Please memorize them!
But, from where did we get the zero?
Well, the number system that we use today, which we call the base-ten number system is actually a Hindu-Arabic number system, and one of the things that made this system unique was the fact that it not only had all the counting numbers, but it had a symbol for zero as well.
The word system can mean an orderly arrangement. So by number system, we am referring to an organized arrangement of numbers.
You see, the Hindu-Arabic number system is a positional numeral system, meaning when there is more than one digit in a number, the value of each individual digit is determined in part by exactly where it is located.
Since the value of a given digit depends on its position in relation to the other digits in the number, we often speak of place value. Here is a list of facts about place value in the Hindu-Arabic number system:
- Place value is based on groupings of ten (which explains why we use the term: base ten).
- In order to identify the value of each place, we give them different names, such as the ones place, the tens place, and the hundreds place.
- Each name identifies a different kind of unit—but each type of unit is always a multiple of ten.
- Since the names and values of the different types of units depend on their relative locations, their positions never change.
- Each place has a value ten times greater than the place directly to its right.
- To determine the value of a digit located in a particular place, multiply that digit times the multiple of ten which names the place.
All of the above information, taken together, makes up the concept we know as “place value.” You will need to develop a solid understanding of base ten and place value to do well in math.
Another way to present all of the above information is to display it in what is known as a Place-Value Chart, which is pictured on the next page.
Your instructor will explain how each of the above facts relates to this chart, which displays a number written as 326 in standard form.
You can practice representing numbers using base-ten blocks on this web page:
On this next webpage, make sure you select Place Value when instructed to choose a lesson.
Here are a couple of worksheets you can use to practice applying what you have learned:
2
3
4
5
6
7
8
9
10
