Preliminary Presentation for Non-Decimal Bases

Posted by Greg MacDonald on

A Note on Formatting

The following presentation utilizes a formatting system designed when Kay Baker, Peter Gebhardt-Seele and I worked together.  It requires the minimum number of keystrokes for the writing of a presentation.  I still use it for every lecture that I revise and for every workshop/conference presentation that I write – It’s super-efficient.  Additional formatting can be added (quotation marks for direct speech, colored font, etc.) according to individual taste.  

Formatting is as follows:

 Directions to Guide (12 point, left justified.)                                                               

 Guide’s verbalizations - What he/she says. (12 point, 0.5” indented.)

Possible (ideal) children’s responses. (12 point, 1.0” indented.)

Expressing the Same Quantity in Different Bases

 The work that we've done with numbers so far has used what we call Base 10, or the Decimal System ... Today we're going to see what happens when we use Non-Decimal Bases ...

Take out Chart of Numeration in Different Bases.

 We generally use Base Ten, the Decimal System, when we work with numbers. That’s why the left-hand reference column here is in base ten.

All of the other columns show how those different quantities that we usually express in base ten can be expressed if we use other bases.

Let’s just choose one of those base ten numerals – Say 22.

Point to 22 in Base 10 column.

When we express a number in any base, all that we are really doing is choosing to organize a certain number of units in a particular way.
In this case, this number of units…

Count 22 unit beads from the box of bead bars, collecting them together into a bowl..

 Obtain Number Base Board and place a ticket for 10, specifying that the base being used is 10.

How would these units be organized on the Number Base Board if we were working in Base 10?

 Children organize the 22 units into two bars of 10, and two unit beads, by placing all units in the units column, then changing base(ten) beads at a time for bars of 10.

            So we have two tens and two units.

             Let’s go back to where we began ...

 Convert the two bars of ten back into units, and move all 22 units to one side of the board again.

Suppose I wanted to express this quantity in Base 7!

Replace ticket for 10 with ticket for 7.

Children organize the same number of unit beads until three bars of 7 and 1 unit is on the board:

 So that same number of unit beads that was twenty-two (or two-two) base ten is three one in base seven!

Is that what our Numeration in Different Bases Chart says?

 Children check and confirm.

 Let’s try this for other bases, and for other numbers.

 Check with the Numeration in Different Bases Chart each time.

 Once the children have tried this with other values, suggest:

 Try this for another group of units – A group that is larger than 22ten,

Which is the largest value on our chart?  And try other bases!  Base six, or base eight or base three, for example.

© Greg MacDonald 2019

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