Return to Mathematics IndexA Fictional History of Place Value

Bette B. Jarrow Goudy Elementary

5120 N. Winthrop Ave.

Chicago, Il. 60645

312-989-5080Objective:

When presented with a two or three digit number the child will be able to

rename it in several different ways including expanded form. The child will

also be able to perform addition and subtraction of two and three digit numbers

with the application of regrouping and the use of zero as a place holder.Materials Needed:

Disks cut from construction paper with correct markings, done in marker,

and backed with small flexible sticky backed magnets (necessary ONLY if your

black board is magnetic). Small disks with markings can be used as hand outs so

that the children can work in pairs after the initial explanation has been made.

A sense of the ridiculous and the ability to present it with a straight face.Strategy:

Once upon a time ... in the days when cave men huddled about their fires. It

occurred that a piece of meat fell into the fire. By the time he was able to

stab it out of the coals he had discovered cooked meat and shish-ka-bobs. As

his enjoyment of cooked meat grew he realized that by keeping 'cave-cows' he was

able to always have meat and demonstrate wealth by being able to provide that

meat to his family when ever desired.

Each day the cows would be allowed out of the cave to graze. Each evening

they would be herded back into the cave so that they were not stolen or attacked

and killed.

A system of checking every cow as she entered or left was necessary to be

sure that they all returned. A 'one-to-one' correspondence was developed and

soon our brave rancher was counting his cows by matching one cow with one

pebble. If, at the end of a day when the cows returned, there was an extra

pebble the rancher knew that he was missing a cow. If there was a cow with no

pebble to match, then he knew he had an extra cow.

Time has passed and we see our hero at the local cave bar with a pouch of

small pebbles. He brags to his friends about how many 'cows' he has. As he

opens his bag and shows a series of pebbles, each representing a single cow,

there occurs the typical comparison of my "ugh" pebbles more than your "ughh"

pebbles. Spoken numeral names have not yet been developed.

As evolution occurs we see that our cave man has tired of carrying about

all those little pebbles. Because of his great powers of observation he would

use that which he saw every day, his fingers, as a basis orbasefor his

numbering. And soon a "group of fingers" of pebbles (we'll call it ten) became

one stone.

In our terms: one group of fingers or one stone

and three extra pebbles became one group of the

base and three extra or 1 ten and 3 extra = 13

But I modernize too quickly. While at the bar our young genius has found

that it is not always convenient to carry that bag with him at all times. He

must have a method of notation. His notation would be representative of his

pebbles and stones.

And so: each finger ---> a little pebble ---> o

a group of fingers --> a stone ---> (*)

Two stones ---> (*) (*) and a hand of pebbles ---> ooooo

A cave cow had a cave calf ---> o extra.

Now the addition problem: (*) (*) ooooo 2 groups of 10 and 5 extra

+ (*) oo 1 group of 10 and 2 extra(*) (*) (*) ooooo 3 groups of 10 and 7 extra oo (or) 37^{Every time a "group of fingers" of stones appeared, it was traded in for a }^{larger rock or: ooooor}_{= (*)}^{ And so we experience the first}^{ ooooot recorded history of carrying.}_{ (*) (*) (*) ooooo}_{ + oooo 39 3 tens and 9 extra }_{ (*) (*) oooo + 24 2 tens and 4 extra}^{ }^{ (*) (*) (*) (*) (*) ooooor}_{=(*)}^{ 5 tens and 13 extra (or)}^{ ooooot 63 6 tens and 3 extra }^{ ooo}-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- - And the subtraction problem: (*) (*) (*) oo 3 groups of 10 and 2 extra - (*) ooooooo 1 group of 10 and 7 extraTranslates to: (*) (*) ooooo ooooo oo 2 groups of 10 and 12 extra - (*) ooooooo 1 group of 10 and 7 extra(*) ooooo 1 group of 10 and 5 extra (or) 15 A one-to-one-correspondence for cancelling becomes obvious and so: (*) (*) (*) (*) ooooo o 46 4 tens and 6 extra - (*) (*) oooooooo-282 tens and 8 extraTranslates to: 18 (*) (*) (*) ooooo 3 tens and 16 extra ooooo ooooo o - (*) (*) ooooo 2 tens and 8 extraooo(*) ooooo 1 ten and 8 extra ooo (or) 18