Introduction:
The beginning exercises of the decimal fraction work should be introduced by the age of 7.5 years. The work should continue until the exercises have been presented to and internalized by the child. This process takes about 2 to 3 years. (My thought is that some of this work with the materials is better suited for some of the younger children around 6.5-7 years of age (late 1st grade, early 2nd grade)).
Before beginning this work, the child must understand the hierarchy of numbers (that 10 of these equal 1 of those, etc.), multiplication and division by powers of ten (the bank game), and that fractions are divisions of units (the circular inset work).
Materials: Fraction metal insets (whole and tenths), a dish with beads in the three hierarchical colors (green, red and blue), decimal cubes.
Presentation 1: Presentation of Quantity
1. Ask the child to identify the unit circular inset, then place it on the mat.
2. Ask the child what one green bead is (also a unit), then place it under the unit inset.
3. “What do I get if I divide a unit into 10 equal parts?”
4. Place the tenth piece to the right of the unit inset, introduce it as a tenth.
5. Introduce a pale blue cube as a tenth as well, and place it below the tenth inset piece.
6. Replace the unit and tenth in their frames, restating that 10 tenths make a unit.
7. Place a cube onto one tenth of the tenths frame. Have the child place cubes on the other tenths in the frame.
8. Put a green bead on the mat with a blue cube beside it. Ask the child to imagine how many pieces the unit would have if each tenth that made it was divided into ten equal parts.
9. Place a pink cube to the right of the blue cube, and introduce it as a hundredth.
10. Ask the child to imagine how many pieces the unit would have if each hundredth were divided into ten equal parts. Place a pale green cube to the right of the pink one, introducing it as a thousandth.
11. Continue as above with a lighter blue cube for ten thousandth and a light pink cube for a hundred thou-sandth.
12. Ask the child how many of each category it takes to make the next. Also ask her the names of the parts.
Put the cubes away.
13. Place a green bead on the mat, and ask the child what you get if you multiply it by ten.
14. Place a blue bead to the left of the green one. Ask the child what you get if you divide a unit by ten.
15. Place a blue cube to the right of the green bead.
16. Continue as above, alternating multiplication and division and placing the appropriate bead or cube in its place on the mat to a hundred thousand.
Child’s Work:
• Child may write labels using words and symbols for the beads and cubes.
• Child can use large newsprint paper to draw large circles section off the circle (or square) into 100 pieces. If you have 100 pale pink cubes, place one on each section of the 100 piece circle.
• They can make a circle or square sectioned off into 1,000 pieces but this is meticulous work and will only be accepted at the right moment.
Presentation 1: 1 Part
Presentation 2: Symbol Linked with Quantity
Materials: Hierarchical beads and cubes, the white (product) cards from the bank game to 100,000, the decimal cards, several small black dots, the decimal propeller and a pencil
1. Place a green bead on the mat. Ask the child what represents ten units. Have him lay out a blue bead.
2. Alternate between multiplication and division, laying out beads and cubes, as in the previous lesson.
3. Bring out the product cards and have the child label the beads, starting with the unit.
4. “Hmm, I wonder if there are any labels for these? (point to decimal cubes) Would you like to see a magic trick?”
5. State that the unit will stay, take the ten card, turn it over, and place one of the small black dots as a deci-mal point. Introduce as one tenth. Repeat with the 100 and 1000.
6. Ask the child if he would like to do some magic. Have him turn over the cards and place them above the cubes with small black dots appropriately placed. Place the decimal cards out, returning the product cards to their places.
7. Ask the child what happens when you multiply the unit by ten, divide by a hundred, etc. Use the three period lesson format (This is…, Show me ten times that…, What is this…, etc.).
8. Point out that the zero on the left of the tenth’s card stands for the unit, and we know the one to the right stands for a tenth because of the decimal point separating it from the zero.
9. Introduce the decimal propeller (or decimal balance), stating that the decimal system is balanced; let the child see how the zeros on the propeller correspond. Spin the propeller on your pencil.
Presentation 3: Formation and Reading of Quantity
Materials: The decimal board, the hierarchical beads and cubes, a small aluminum foil crown, a paper can-delabra that matches the spacing of the decimal board, and the decimal and whole number cards.
Passage One
1. Place a green bead in the unit space on the decimal board, announcing that the board is the kingdom of the unit.
2. Place the aluminum foil crown around the green bead. Put out beads and cubes representing tens and tenths, hundreds and hundredths, alternating between multiplication and division, and explaining the relations.
3. Point out the numbers across the top. Place the candelabra on the board, stating the unit supports the whole structure.
4. Remove the beads and cubes, and slide the candelabra up so that the dots representing place value appear
Presentation 2 Presentation 3
Passage Two
1. Place 3 blue cubes in the tenths’ place. Have the child find the card representing this amount (0.3) and put it next to the board.
2. If there is more than one child, have them place quantities and cards for each other. If not, alternate placing quantities and cards with the child. Use one category at a time.
3. After the child has had some practice, place more than ten cubes in a category, and ask the child to find a card for the amount.
4. When child says they can’t, acknowledge that, encourage her to exchange, and show her how to place the cards.
5. Ask the child how we read a number like this. Discuss different ways it might be read (see note below).
6. Using more than one category, lay out beads and cubes, practice reading the quantities.
7. Dictate numbers to the children.
Note: A number, such as 32.674 can be read in several ways:
1. Thirty-two and six hundred seventy-four thousandths, or
2. Thirty-two and six tenths, seven hundredths, and four thousandths, Extensions:
1. Have one child put out a quantity with the cards and have another put out a quantity with the beads and cubes. See who has created the greater quantity.
2. Take a quantity of beads and cubes, have the child take another quantity, see who has more.