Fractions: Exploration and Operations

Materials:

10 green frames with red circular insets 10cm in diameter - one inset is a complete circle with a knob, other insets are divided into 2, 3, 4, 5, 6, 7, 8, 9 and 10 equal parts; each part has a knob by which to handle it; the background is white.
Two sloping green stands to hold the insets
A box of labels for each piece
Other containers with equations for simple operations with fractions
A box of green skittles
Papers and pencil

Purposes:

Age: 4.5 (Sensorial and language explorations)

6 (Mathematical operations)

Preparation:

For Sensorial Explorations: The child should be making designs with the Metal Insets.
For Mathematical Operations Work: The child should have a solid understanding of whole numbers and has already been introduced to the four operations at the Stamp Game level.

Parallel: Metal inset design work

Presentation 1: Designs

Select three fractions and have the child help bring them to the table.
Tell the child they are fraction circles and you will use the pieces to make a design on the table.
Make your design leaving one fraction piece in each frame and replace the pieces when you are done. Match the existing piece in the frame to the piece you are putting away, by placing them back-to-back.
Invite the child to make a design.
When the child is done, invite them to continue with another three fractions to make another design.

Note: On another day you can show the child how to trace their design onto the paper.

Presentation 2: Naming the Fractions

Show the child to bring all the fractions to the table.
Remove each fraction in order from the tray and place on the left edge and along the top and right edge of the table.
Bring the unit in a cup.
Explain that fractions are pieces of a whole like a pizza. We don't eat the whole pizza, we eat pieces of it.
Remove the whole place on the table and compare it to the unit bead-if we flattened the bead it might look a little like the whole circle. Give the name whole. Return the bead, move the whole to its frame.
Remove both pieces of ½ explain it is a whole broken into two pieces. Name each piece (one half) and replace one piece into the frame.
- Avoid the phrases: “divided into” or “there are __ equal pieces” because the language is familiar with the process of the operation of division. Right now, we want the focus to be on the broken whole.
- Use fourths not quarters because there is a correlation to 4 pieces.
Repeat for the thirds (one third).
Repeat for ¼ (one fourth).
Offer a Three-period lesson on ½ and 1/3 and ¼ .
Repeat introduction for 1/5-1/7 followed by a Three-period lesson.
If the child sees the pattern in the language there is no need for further introductions, if they do not offer a Three-period lesson for those later.

Note: On another day, invite the child to take out combinations of a fraction. “Take out 3/4ths.” “Take out 7/10ths.”

Presentation 3: Mathematical Concepts-Writing Fractions

Have the child set up the fractions around the edge of the table.
Bring the small slips of paper and a pencil.
Pull out the whole, set to the right of the frame and write a 1.
Return the whole to the frame and set the label on the top.
Remove the two halves and ask the child to name one.
Explain that you will show how to write the fraction.
Draw the line and explain that the total pieces that are in the fraction are written there.
- Explain that above the line, tells how many pieces have been removed from that whole fraction.
- Explain that fractions have two names, the name of their fraction and the name of their piece.
Isolate ½ and write a 1 over the 2 and explain that it is one of the two pieces.
Have the child write another label for the other ½ .
Replace them back into the frame with the labels on top.
Repeat up to ¼ and let the child finish independently as they understand the pattern.
Encourage the child to label them all.
Note: On another day, show the child the box of prepared labels.
Note: Show the child how to make labels for combinations of fractions. Ex: 4/9, 2/3, 3/4

Presentation 4: Equivalency

Invite the child to set up as before, bring the box of paper and a pencil.
Place the ½ frame in front of the child and remove one of the pieces.
Explain that you will fill the empty space but with pieces from the same frame.
Try with the thirds and then transfer it to the child.
The fourths will work. Set the 2/4 to the right of the ½ that was taken from the frame.
When other matches are found, set them to the right of the others.
At the end, show the child how to write them on the slip with an = in between each.
Explain that they are all equivalent, they fill the same amount of space.
Invite the child to try another.

Note: The child can find the equivalents of combinations.

Presentation 5: Addition

Note: In the 3-6 classroom, we only add fractions with like/common denominators.

Have the child bring the addition fraction box and tray of paper to the table.
Have the child pull one problem and the guide will write it onto a paper.
Guide the child to find the appropriate fraction circle.
Remove the numerator from the first fraction and place in front of child.
Remove the numerator from the second fraction and place to the right of the first.
Ask the child what addition is. Bring them together and count the pieces.
Show the child to write the answer.
Recite the whole equation with the child.
If the child comes to one that makes a whole, remove the whole and compare, write and equal sign next to the answer on the paper and write a large 1.

Presentation 6: Subtraction

Set up as before.
Invite the child to bring the prepared subtraction slips and the paper to the table.
The child will pull a slip and write the equation.
Guide the child to the correct fraction frame and to remove the numerator of the first fraction with the pieces placed in a circle.
Ask the child what subtraction is.
Pull the numerator from the second fraction down to the bottom of the table.
Have the child count what is left and write the answer.
Let the child read the entire equation.

Presentation 7: Multiplication (by a whole number)

Set up as before.
Invite the child to bring the prepared multiplication slips and the paper to the table.
The child pulls a slip and writes the equation.
Guide the child in locating the fraction and taking the pieces to equal the numerator.
Place the pieces to the right of the first to build the quantity indicated by the multiplier.
Ask the child what multiplication is.
Have the child combine the pieces into a circle, count, and write the answer.
Child reads the whole equation.

Presentation 8: Division (by a whole number no remainder)

Set up as before.
Invite the child to bring the prepared division slips, skittles, and the paper to the table.
Child pulls a slip and writes the equation.
Child removes the pieces needed from the fraction on the slip and builds them in the center of the table.
Ask the child what we do for division.
Child takes skittles to equal the divisor on the slip and lines them up vertically on the left side of the table.
Child shares to the skittles.
Ask what the skittle got.
Child writes answers and recites the equation.

Control of Error: None

Pedagogical Notes:

We do not:
- Provide problems with remainders
- Use improper fractions. Example 10/6
- Prepare problems that result in a mixed fraction 2 7/8
At some point you can casually introduce the terminology of numerator and denominator.
We want the child to understand that they are working with pieces of a whole. Pieces are smaller than a unit. They need experience working with quantities and time to understand this concept.