#### Material

A board of paper with columns for categories from 1 to 10,000. Each column has a rectangular heading with the category written in decimal colors: 10,000 in blue, 1000 in green, 100 in red, 10 in blue and 1 in green. The columns have many rows each with 10 squares. Below the columns are two rectangles: the upper is for carrying the dots, the lower is for the final answer. A blank column on the right side next to the units column is for writing the problem.

A lead pencil and a colored pencil other than the colors of the Decimal System, e.g. orange.

For the Board, water soluble markers are needed (black and another color), and a damp sponge.

#### Presentation

##### Recording Dots by Whole Numbers

- Invite the child to the lesson.
- Show the child the paper with columns for categories 1 to 10,000.
- Bring the paper and underlay.
- Ask the child to bring a pencil, an orange colored pencil, and a ruler, on a tray.
- Sit down with the child at the table, talk about the columns and how there are columns for units, tens, hundreds, and thousands.
- Emphasize how there is a column for ten-thousand.
- Emphasize the black lines which separate categories.
- Point to the numbers 1, 10 and 100 at the top and say “This is one family.”
- Cover up the 1, 10 and 100 with your hand, showing only 1000 and 10000.
- “This is a new family.”
- Point at the boxes at the bottom.
- “This is where we write our answers.”
- On the right side of the paper, write a few numbers that can be up to four digits, but include a variety of numbers that consist of one, two, three, and four digits.
- Emphasize how you line the numbers up in columns.
- Invite the child to write his own numbers, remind him to write carefully in columns.
- Under the numbers, draw a straight horizontal line with the assistance of the ruler.
- “We’ve seen this line before, in the stamp game. What does it mean? Right, it means equal!”
- Write a plus sign “I’m going to put a plus sign here so that it says we are doing addition, and so that we remember and know we’re doing addition.”
- Start with the first number, such as 7586.
- Start with the units.
- “We have six units.”
- “We’re going to start here, next to this dark line, and make dots for the number.”
- From left to right, make a dot in six boxes, counting as you go.
- Continue this process for other categories of that number, such as eight dots in the 10 column, 5 dots in the 100 column, and 7 dots in the 1000 column.
- After every category of the number has been written on the paper in the form of dots, write a check on the right side of the number to show that you’ve finished that whole number.
- Repeat this process for a couple numbers/addends.
- Hand it off to the child.
- Watch the child, remind him to count as he writes the dots.
- Once the child finishes putting all the dots of all the numbers on the paper, let him admire his work.
- “I’m going to show you how to count them.”
- Begin with the units category, count aloud with the child, in a row, left to right.
- If a row has ten dots, draw a line through it.
- Pick up the orange pencil and make a dot in the top left corner of the first box in the category’s column.
- Repeat this process for each set of ten in a category.
- Whatever doesn’t make a set of ten, such as three dots, is the answer for that category.
- Write that number, for example, 3, in the bottom most box of that column.
- In the next category, before counting the dots, count the number of orange dots from the previous category, such as four, and write the number 4 in the top right corner of the first box of the current category’s column.
- Using an orange pencil, draw that number of orange dots, such as 4, in the boxes of the present category.
- Continue the process as before, counting dots and striking through sets of ten and exchanging them into orange dots in the top left corner of the first box in the category’s column.
- Hand it off to the child, but guide the child through it.
- Once all the categories are counted and the numbers are written at the bottom of the categories, place a comma in its place to the right of the number in the thousand’s place; emphasize the comma.
- Say how we read the answer, such as “twenty-nine-thousand, six-hundred, thirteen.”
- Write that same number – 29,613 – under the equal line on the right.
- “When we add all these numbers, we get this answer! 29, 613!”

##### Recording Dots by Hierarchical Categories

Once the child has mastered the dot game, demonstrate that you can record the dots in a different sequence.

- Invite the child to the lesson.
- “We’re going to do it a little differently this time. We’re going to count all the units first!”
- Start the process as before, ask the child to write his own numbers.
- Once all the numbers are written, record dots for all numbers in the unit category.
- Count the units the same as before, making color dots for exchanging.
- Record the final number of units.
- Move on to the ten’s category and column.
- Count the colored dots from the previous column, such as 2, and write the number in the same spot as before.
- But this time, mark the paper with those colored dots before marking the paper with pencil dots.
- Continue the same process, completing the 10 column by recording all dots for all numbers in the ten category.
- Hand it off to the child.

#### Exercises

Child works independently.

#### Direct Purpose

To show the decimal system in a more abstract form, that is the relationship of the categories.

To focus the child’s attention on the process of exchanging.

#### Indirect Purpose

Preparation for abstract addition.

#### Control of Error

Understanding the process of addition and exchanging are more important than the correct answer.

#### Age

5 1/2

#### Notes

This kind of work is appealing to children who are nearing the 2^{nd} plane; which means bigger, complex problems are of great interest to them. Children here will often be very independent.