A Fictional History of Place Value
Bette B. Jarrow Goudy Elementary
6634 N. Artesian Ave. 5120 N. Winthrop Ave.
Chicago, Il. 60645 Chicago, Il. 60645
312-764-5544 312-989-5080
Objective:
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 or base for 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
- (*) ooooo
oo 1 group of 10 and 7 extra
Translates to:
(*) (*) ooooo
ooooo
oo 2 groups of 10 and 12 extra
- (*) ooooo
oo 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
- (*) (*) ooooo
ooo -28 2 tens and 8 extra
Translates to: 18
(*) (*) (*) ooooo 3 tens and 16 extra
ooooo
ooooo
o
- (*) (*) ooooo 2 tens and 8 extra
ooo
(*) ooooo 1 ten and 8 extra
ooo (or) 18
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