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Number Systems & Counting

Animated YouTube Clip on the Development of Numbers - Click here

Babylonian Maths

 

Motivate has created a free online multimedia resource pack for the Key Stage 2/3 transition, based around short video clips introducing key concepts in Babylonian mathematics. Each pack includes related investigative activities and worksheets for students, and teacher support notes.

Find the free multimedia resource pack at http://motivate.maths.org/BabylonianMaths

Babylonian Numbers 1 to 9 YouTube Clip - Click Here 
Babylonian Numbers 10 to 50 YouTube Clip - Click Here 
Babylonian Numbers 60 and beyond YouTube Clip  - Click Here

Roman Numerals

Introduction to Roman Number System - Click here

References and Resources

Silly YouTube Clip - Click here
Selection of Worksheets - Click here
Site with a range of related resources - Click here
Roman Numerals Information
Roman Numerals PowerPoint - Quiz
Roman Numerals Online Quiz 1 - Click here
Converting Roman Numerals Online Quiz 2 - Click here

Roman Numerals Jigsaw Puzzle

The Roman Numeric System  Click here

Converting online - Click here
Conversion Worksheets  Click here

Investigation – The Secret Code  Click here

 Tasks  (mostly suited to lower ability sets)

  • Discussion:  Where do we see Roman numerals?  Why do you think the numbers 5 and 10 are so significant?
  • Simple understanding of the system – converting to and from Roman Numerals
  • Investigation (using ‘The Secret Code’)
  • Finding longest numbers
  • Encoding numbers

Number Bases

A lesson on Binary - Lesson Outline

                             Binary Birthdays

                             Binary Message

                             Binary Quiz


Binary: YouTube Counting in Binary   Click here
Binary Lesson Plan            Counting in Binary on Your Fingers

Online Binary Game

Octal and Hexadecimal  Click here

Spreadsheet for doing base conversions.

Binary Clock

Tasks (mostly suited to middle and high ability sets)

  • Examining decimal (base 10)
    • What do we call the column headings? (1, 10, 100 etc)
    • Why do we use base 10?  What is significant about the number 10? (fingers – digits!)
    • Which digits/symbols do we use? (0, 1, …, 8, 9)
    • We don’t have a symbol that means 10, we need to use two.
    • How do they relate to each other?
    • What about the columns to the right of the decimal point? (0.1, 0.01 etc).
    • How do we multiply by 10, 100 1000 etc?  (shifting left)
  • Examining octal (base 8).  This is a good first other base to look at.  I usually introduce it by considering an alien species who have 8 fingers.
    • What are the column headings? (1, 8, 64, …)
    • Which digits can we use? (0, …7.  no symbol for 8)
    • The ‘point’ in now called the ‘Octal Point’
    • Explain subscript notation to indicate which base we are using? (e.g. 738)
    • Try converting to and from octal and decimal.
    • Addition and subtraction using the column method?  Is it any different?
    • What happens if we shift the digits to the left or right?
  • Examining binary (base 2). – Follow same points as for octal.
    • ‘Binary digits’ are called ‘bits’ for short
    • Counting activity.  Have a number of students on chairs in a line – one student per column (say 5).  Standing up represents 1, sitting represents 0.  Start with them all sitting (i.e. zero) then try to count by standing and sitting as necessary.
    • A similar activity to the above can by shading squares on graph paper.  Each student will need just one strip no more than 8 small square wide.
    • Note that computers use binary as the memory is made up of a number or on/off (i.e. 1/0) switches.  That is why we see number like 512, 1024 etc showing the size of out USB memory etc, 32-bit games systems)
  • Examining hexadecimal (base 16)
    • Need for more digit symbols
    • Conversion to and from binary is easy (bits grouped into 4-bit ‘nybbles’

Egyptian Maths

Ancient Egyptian Maths Problems - Click here

Researching Egyptian Maths - Click here

Maths Related to the Egyptian Pyramids - Click here

Pyramid Challenge Game - Click here

Tower of Hanoi 1 - Click here                Tower of Hanoi 2 - Click here

Egyptian Fractions

The Ancient Egyptians used unit fractions (i.e. numerator of 1).  To make other fractions they added these unit fractions.

Example:  3/4 = 1/2 + 1/4

Unit fractions could not be repeated, so 2/5 = 1/5 + 1/5  is not allowed.   

Egyptian Maths - PowerPoint Presentation             NRICH Task

References and Resources

Lots of other references (complex)   Click here

Math Cats – interactive creation of Egyptian Fractions  Click here

Tasks

  • Show 3/5 = 1/2 + 1/10,   3/7 = 1/3 + 1/11 + 1/231
  • Write fractions with 2 as the numerator (e.g. 2/5, 2/7, 2/9, …) – describe any patterns you notice.  Can you find a rule for writing fractions of the form 2/n?
  • Investigate writing other fractions as Egyptian Fractions (when the numerator is 3, 4, 5, …)

Counting Systems

The calculator has its history in mechanical devices such as the abacus and slide rule. In the past, mechanical clerical aids such as abaci, comptometers, Napier's bones, books of mathematical tables, slide rules, or mechanical adding machines were used for numeric work. This semi-manual process of calculation was tedious and error-prone. The first digital mechanical calculator was invented in 1623 and the first commercially successful device was produced in 1820. The 19th and early 20th centuries saw improvements to the mechanical design, in parallel with analog computers; the first digital electronic calculators were created in the 1960s, with pocket-sized devices becoming available in the 1970s.

References and Resources

Early Numeration   Click here

Methods of Counting and their Uses  Click here

Making a Quipu  Click here

Counting Board Click here

Abacus

‘Abaci’ – Powerpoint presentation giving an overview of the history

Different Types    Click here

History    Click here

Chinese and Japanese related materials - Click here
 

Japanese Abacus (Soroban)

 

YouTube Clips

Introduction    Click here   
Addition         Click here    
Subtraction     Click here   
Abacus being used by experts      Click here   

 

 

How to do addition and subtraction (and other operations) on a Japanese abacus  Instructions

 

Superb set of Activities using the Japanese Abacus - Click here

 

Software/online abacuses

School type   Click here

Chinese type  Click here          Chinese Abacus to Colour - Click here

Japanese type Click here

 

Lower ability student might find it easier to look at 'school' abacuses which have horizontal wires with 10 beads on each.  The top wire represents units, then tens, etc.

Higher ability students may prefer to look at 'Japanese' abacuses.  These are essentially base 10, but each column is broken into two parts.  The top most bead means 'add 5' to the bottom beads.  (see Click here).  Template for use with counters - Click here

 

Note about addition and subtraction.  If you are adding with a Japanese abacus (and several other types) you have to be aware of number bonds to 5 and 10.  For example, if a column already contains the digit 8 and you wish to add 3, you do not count on 3 and do the carrying, instead you add 10 (i.e. one to the next column) and subtract 2 (8 = 10 – 2).  Similarly, it the column contains 4 and you wish to add 3, this is the same as adding 5 and then subtracting 2 (i.e. set the ‘heaven’ bead and subtract 2 from the Earth beads).  Subtracting happens in the opposite way – essentially implementing ‘borrowing’.

 

Multiplication is very similar to traditional long multiplication except you add the intermediate results as you go.

 

Research Tasks

  • Look at types of abacuses

  • Which number base (if any) do they use?

  • How do they deal with fractions?

 

Other Tasks

  • Learn how to use an abacus to add and subtract

  • Can you redesign it to work in a different number base?

  • Can you work out how you might use the abacus to multiply?

Slide Rules

Slide Rule: How does it Work                Introduction      

Make Your Own Slide Rule - Template 1            Template 2

Online Slide Rules

Click here
On this original version, the cursor and the slide may be moved with your mouse - just click and drag. The number overlay shows the reading and identification of the scale directly under the mouse pointer (or the hairline reading if over the cursor).

Click here
This applet is designed to be a real-time, configurable, slide rule emulator. It currently supports zooming and on-the-fly scale reconfiguration. The emulator can handle multiple slides, and multiple cursors. Cursors can contain multiple gauge marks and hairlines. Scales can be placed anywhere on the stators, slides, or cursors. The size and texture of the rule body/surface can be changed. In short, it's totally reconfigurable!

Click here

Instructions

The following page gives numeric examples of the basic calculations that a slide rule can do. Just follow the step-by-step instructions and you will be amazed by the power and versatility of the venerable slipstick.         Click here

Instruction Leaflet:  Front Side     Reverse Side

From Wikipedia, the free encyclopedia: Basic concepts     
Multiplication ,   Division,   Roots and powers,   Trigonometry,   Logarithms and exponentials History     Click here

Various downloadable Slide rule Manuals     Click here

Making a Slide Rule

The Slide Rule Museum have pulled together a large set of templates to make your own slide rules, including circular and cylindrical versions.  Click here 

Thanks to some very public-spirited slide rule experts (one in the UK, one in Canada), we now can provide some computerized tools that will allow you to build several different types of slide rules.     Click here

Students will learn about the history of slide rules, how they work, and then make their own. Click here

Logarithms Explained: Click here

 

Alternative Forms of the Slide Rule

Otis King Calculator - Click here

Circular Slide Rule - Click here 

Speed Calculations

Speed Calculations - Ideas

 

Napier's Bones

Brief Outline of Napier's achievements for display - Click here

Abacus, Napiers Bones, Slide Rule and Logarithms - Interactive Features: Click here

Napier's Bones printout of each rod and exemplar calculation for display - Click here

 

Napiers Chessboard Calculator - Click here

Other Calculating Machines

Curta Mechanical Calculator - Youtube Demo