Purpose

Direct:
The exercises of numeration recapitulate the function of the decimal system by making the child realize once again the following:
1) That ten of one category make one of the next higher category and then in each category there can be no more than 9.
2) The value of the numerals is determined by the place they hold.
3) The function of zero is that of a place holder.
The whole set of exercises brings the child to the realization that when he writes down addition and subtraction problems, he must place all the figures belonging to the same category in the same vertical column.
The exercises also provide the children with opportunities to apply all they have learned before and thus prepare them further for abstraction.

Indirect:
Both the exercises on numeration and the multiplication exercises prepare the child for the distributive law of multiplication where it becomes necessary to analyze the numbers into their hierarchical values.

Materials:
A frame with support to enable it to stand.
The frame has four wires across, each strung with ten beads.
Top wire: 10 green beads for units; second wire: 10 blue beads for tens; third wire: 10 red beads for hundreds; fourth wire: 10 green beads for units of thousands.
On the left side of the frame the numeral category of each wire is written.
Background for hierarchy of units is white; background for thousands is grey.
Notation paper for small frame is lined in the same colors as the numerical categories.
Pencil and ruler.

Control:
The child may be shown how to check his work.

Age:
5 – 6 years

Presentation:
Introduce the Small Bead Frame Individual exercise.
At a table, examine the frame.
Ask the child to identify the numerals along the left-hand side of the frame.
Note the new background color at 1000, it denotes where we write a comma.
Note the beads correspond in color to the stamp game.
Introduce the notation paper to the child. The child reads the headings at the top of the paper.
Explain that as we count each bead we can write its number directly on the color coded line under the appropriate column.
Start with the units counting the beads by sliding them individually across the wire from left to right while recording the number down in its appropriate place on the notation paper.
Upon counting 10 units remind the child that we must exchange.
Slide one ten bead to the right and slide the 10 units back over to the left.
To record – write a 1 on the blue line and note there are no units therefore we write a zero in the units category.
Continue counting beads and noting the numbers on paper up to 1000, stressing exchanges.
Make large numbers.
The teacher writes a four digit number, the child reads and makes the quantity on the frame.
Note: We read the number on the right-hand side on the frame.
Repeat for a few examples – reverse process and use zero.

Exercise: On the same or on another day.

Write an equation on the notation paper which does not require exchanging.
The child sets out the first addend on the frame.
Remind the child to begin adding at the units.
The child reads the number of units in the second addend and counts the corresponding number of unit beads to the right.
The child counts and records the sum of the unit beads which are at the right of the frame.
Repeat the process for the remaining categories, tens to thousands respectively.

Write an equation which will require exchanging.
The child proceeds as above, reads the equation and sets out the first addend.
Beginning with the units, the child reads the number and counts the corresponding number of unit beads to the left.
He runs out of unit beads – 10 unit beads lie on the right-hand side of the frame – therefore he must exchange the 10 units for a ten.
Slide one ten to the right and the 10 units back to the left.
The child continues to count the necessary unit beads to the right.
The child records the number of beads that sit on the right.
Repeat the process for the remaining categories, exchanging as necessary.

Proceed as above except this time add all the units first, then the tens, hundreds and thousands.
The child reads the numbers in the units column – sets out the first, then adds on the second, exchanging as necessary and records the sum.
Repeat for the remaining categories.

EXERCISE 3: Static Subtraction –
Write a problem on the notation paper that does not require exchanging.
The child reads and sets out the minuend.
Remind the child to begin with the units.
The child reads the units in the subtrahend and takes them away by counting the appropriate number of beads from right to left.
The child reads the number of beads remaining at the right of the frame, which is the difference and records.
Repeat for the remaining categories, tens to thousands respectively.

DYNAMIC SUBTRACTION –
Write an equation that necessitates exchanging.
The child reads and sets out the minuend.
Beginning at the units, the child notes the number of beads which must be taken away.
The child counts the corresponding number beads from right to left, however the child will run out of beads.
Remind the child of how to get more units.
Slide one ten bead to the left and slide the 10 unit beads to the right.
The child continues to count the appropriate number of beads to the left.
The child reads the number of beads remaining at the right of the frame which is the difference and records it. Repeat for the remaining categories.