# Understanding What Fractions Are

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The fractions curriculum is divided into four parts:

1. Understanding What Fractions Are
2. Writing Fractions and Finding Equivalencies
4. Multiplying and Dividing Fractions

These parts are arranged in order from introductory material to more advanced material. The activities within each part are also arranged from simple to more advanced.

Teaching Materials Used For Fractions

Lessons within this section rely on the following teaching materials:

• Fraction Skittles
• Fraction Circles
• Fraction Circle Box
• Green Skittles
• Mute Cards
• Fraction Tickets
• Prepared Equation Slips

The following paragraphs describe each of the teaching materials.

Fraction Skittles

The Fraction Skittles are a large set of wooden skittles. They are sometimes referred to as The Large Fraction Skittles. The Fraction Skittles are used to sensorially introduce the concept of fractions and to represent the divisors of whole, one half, one third, and one fourth in division equations.There are four wooden skittles, and each skittle is 6 in. (15 cm.) high. The skittles are usually contained on a small wooden tray for storing and carrying them. One skittle is whole and undivided. the next is divided into two equal parts, with the internal faces painted red. The third is divided into three equal parts, with the internal faces painted orange. And the last is divided into four equal parts, with the internal faces painted green. Students quickly learn to associate the color of the internal faces with the fractional quantities and names.

Fraction Circles

The Fraction Circles are a set of ten green metal square frames with red circular insets. This material is sometimes referred to as the Fraction Metal Insets. The Fraction Circles are one of the main Montessori fraction mateirals. They are used to introduce quantity and names of fraction parts, equivalencies, and the four operations, as well as measurements of angles in geometry. The frames are arranged in two rows of five, and each row is supported on a long wooden board for storing and carrying. Each Fraction Circle is 10cm (approx. 4in.)in diameter, and has a different number of equal size insets or parts. The first circle is a whole, and the remaining circles are divided into 2, 3, 4, 5, 6, 7, 8, 9, and 10 parts. Each part has a small knob which allows the student to easily remove and replace the parts. The Fraction Circles provide a sensorial foundation and a sense of continuity for all the fraction work of the elementary student.

Fraction Circle Box

The Fraction Circle Box is a set of plastic fraction circles contained in a compartmentalized wooden box. This material is sometimes referred to as the Cut-Out Labeled Fraction Circles. The Fraction Circle Box is used when more than one of each type of fraction is needed, The box contains ten whole circles and five sets of labeled fraction pieces (plastic cut outs) for each of the fractions from halves to tenths.

Green Skittles

The Green Skittles form part of the Long Division Material. In the fractions curriculum, skittles are used to represent the number of portions into which an amount must be divided (the divisor).

The Mute Cards

Mute Cards are a set of ten cards, one corresponding to each of the Fraction Circles from a whole through tenths. Each mute card the outline of a whole Fraction Circle plus a colored section to represent a single fraction piece. For example, the mute card for the halves Fraction Circle shows the outline of a circle and a colored section of one half of the circle. Note that there is a small red dot in the top left hand corner of the mute cards to indicate how it should be oriented.

Fraction Tickets

Fraction tickets are small, uniform-size pieces of paper, each one with a fraction written on it. Students use them to label fractions.

One set of fraction tickets includes only the following ten fractions:

• 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10

The other set includes all 55 fractions that can be made using the Fraction Circles, as follows:

• 1/1
• 1/2, 2/2
• 1/3, 2/3, 3/3
• 1/4, 2/4, 3/4, 4/4
• 1/5, 2/5, 3/5, 4/5, 5/5

and so on up to 10/10

Prepared Equation Slips

Numerous fraction activities in the 6-9 curriculum rely on prepared equation slips. A student can choose a single equation at random from a container such as a basket.

Understanding What Fractions Are

In simple terms, a fraction is a part of something. In math, a fraction is a number used ot name part of a whole or a group. A fraction is a way to represent numbers that are not whole numbers.

Common fractions are expressed as a number over another number, for example 2/3 or 21/45. The shirt line separating the numbers indicates division. It is called the fractus, from the Latin term meaning to break, cut, or divide. The number below the line is called the denominator and represents the number of pieces bring considered.

Common fractions are the only kind of fractions covered in these lessons. Other kinds of fractions, such as improper fractions, decimals, and percents are introduced in the 9-12 curriculum. Simplifying fractions is also addressed in the 9-12 curriculum.

Fractions in Montessori Math

In the Montessori classroom, Fraction Circles are the foundation material for presenting fractions to Lower Elementary students. Students first work with Fraction Circles in a bais, sensorial manner ot gain a concrete understanding of the quantities and symbols of fractions. Like most foundation Montessori math activities, the next step is to associate the quantity with the symbol. Working consistently with this familiar material , students build on their basic knowledge and move gradually and confidently toward more complex concepts. Over time they use the Fraction Circles to explore equivalent fractions; addition and subtraction with like and different denominators; and multiplication and division equations. A common analogy in the Montessori classroom is ot think of fractions like families. The family of fractions is used ot explain the denominator to the students, and the Montessori material is used to make this connection is the Fraction Circles. The student learns that the bottom number of a fraction tells us the fraction family (e.g., fourth). This introduces the concept of the denominator in a very concrete way, as the student will learn to associate the family of fourths with the Fraction Circle frame and insets for fourths. Once the student understands which family is represented, then she/he will look to the top number, the numerator, to discover the quantity of pieces needed, or using the family analogy, the number of family members present. Them the student will select the correct number of fourths insets from the frame (e.g., one fourths inset for 1/4). and place it on the mat to represent the fraction.

Understanding the Concept of Fractions

Purpose

To introduce the concept of fractions.

Material

Apple
Knife
Cutting Board
Plate

Presentation

Most Montessori teachers present this concept in Year 1 as a review.
This Activity is designed for a group of students,

Cutting the Apple into Fractions

-Invite a group of students to learn about fractions at a table with the material already laid out.
-Hold up the apple and ask the students how many apples they see. (One.)
-Invite the students to suggest a way of dividing the apple into parts.
-Show the students the knife and cutting board and tell them the apple can be cut into parts.
-Tell the students that if two children planned to share the apple it would need cutting in half. Cut the apple in half. Point out the apple is no longer a whole.
– Now explain tjat if four children are hungry for apple, the apple can be cut in half again to make one piece for each child. Cut the apple in half again to demonstrate this concept, and place the apple pieces on a plate.
-Invite the students to say how much apple each of the four children would receive. (One fourth, also called one quarter.)
-Review with the students that a whole can be divided into parts. State that each part is called a fraction.
-Give a piece of apple to each student to eat, if necessary cutting the apple to make enough pieces for all.

Introducing Fractions Using the Fraction Skittles

Purpose:
To sensorially explore fraction quantities.

Material:
Fraction Skittles
Math Journals and Pencils

Presentation
Most Montessori teachers present this concept in Year 1 as a review.
-Invite students to continue to learn about fractions.
-Ask the students to go with you to the shelf to get the Fraction Skittles, and demonstrate how to carry them to the mat or table. Position the Fraction Skittles at the top of the work area.

Part 1: Sensorial Exploration

– Remove the Fraction Skittles from the stand and take them apart. Place the fraction pieces randomly on the table below the stand, with the colored parts toward you.
– Scan the pieces, and then pick up the whole skittle and say to the students, “This is a whole skittle.”
– Invite the students to look at and handle the whole skittle, and then place it on the left side of the work area.
– Scan the skittle pieces and select the red halves pieces. Pick up one half and say to the students, “This is one half.”
– Continue in this manner until all of the skittles have been assembled and are placed in order. – State that when something is divided into equal parts, each part is called a fraction.

Part 2: Three-Period Lesson

– Scatter the Fraction Skittle pieces on the table below the stand.
– Pick up and place the whole skittle in the center of the work area. Say to the student, “This is a whole. Whole.”
– Return the whole skittle to the top of the work area, and place the halves skittles together in the center of the work area.
– Explain to the student that this is also a whole.
– Separate the two halves and say to the student, “e have two equal pieces. We call each piece one half.”
– Point to the halves skittle piece and say to the student, “This is one half. Half.”
– Return the halves skittle piece to the top of the work area, and continue in the same manner with the thirds and fourths Fraction Skittle.

Note:
When introducing the fourth skittles, use the term “fourths”. The student will use the “quarter” in later activities.
– Proceed with the recognition phase of the second period, e.g., “Show me one half,” and then the recall phase of the third period, e.g., “What is this?”

Extensions
– Ask the students to trace around the base of one piece of each Fraction Skittle in their math journals. Invite them to color and label the fractions accordingly.
Note: Assist students will labeling in necessary.

– Supply the students with colored construction paper (beige, red, orange, and green) that has been precut into squares, 5×5 inches (12.5 x 12.5cm).
Ask the students to use a ruler to measure and then cut the squares into equal pieces to represent the fractions whole, halves, thirds, and fourths.

Introducing Fractions Using the Fraction Circles

Purpose:
To further understand fraction quantities and names.

Material:
Fraction Circles
Math Journals and Pencils

Presentation:
Most Montessori teachers present this concept in Year 1 as a review.
– Invite a student to learn about fraction quantities and names.
– Ask the student to go with you to retrieve the material.
– Show the student where the Fraction Circles are stored.
– Invite him/her to carry one board as you carry the other.
– Position the Fraction circles in a single row at the top of the mat.

Note:
The board representing the whole to fifths is positioned to the left and the board of sixths to tenths is on the right.
– Remove the fraction Circle whole frame and inset from the board and position it below the board.
– Place the golden Bead in the center of the work area. Remove the whole inset from the frame and position it to the right of the Golden Bead.
– Remind the student that the Golden Bead represents one or one unit. The whole Fraction Circle also represents one unit. To hlp the student visualize how the bead is equal to the whole inset, explain that the bead has been squashed and as a result has been flattened and turned red.
– Point to he whole inset and say, “This is a whole. Whole.”
– Tell the student that to share a whole Fraction Circle or bead, he/she would have to divide it, just as he/she divided an apple in an earlier activity.
– Return the whole inset to the board, and move the whole frame to the center of the work area. Move the Golden Bead to the right of the work area because it is no longer needed.
– Move the Fraction Circle halves insets from the board to the work area, and position them together to the right of the frame.
– Point to the halves insets and say to the student, “This is also a whole. If you want to share it with two students, we can divide it.”
– Move the two halves apart and say to the student, “Now we have two equal pieces. We call each piece a half. This is a half. Half.”
– Return the halves inset to the frame on the board.
– Next ask the student to take out the thirds inset from the board and place them in the mat. Ask the student, “How many equal pieces is the whole divided into this time?” (Three.)
– Confirm that he/she is correct, and say, “We call each piece a third. This is a third. Third.”
– Ask the student to return the insets to the board, and then to position the fourths insets in the work area.
– Ask the student to count how many equal pieces the whole is divided into. (Four.)
– Confirm that he/she is correct. Ask the student if he/she knows what each piece is called. (A fourth.)

Note:
You may introduce the term “quarter” at this time.
– Ask the student to return the insets to the frame and the frame to the board
– Continue in the same manner through to the tenths if the student is still interested.
– Point to the Fraction Circle insets on the board and explain to the student that each piece is called a fraction or a fraction of a whole.
– Continue with the second and third periods.

Extensions:
As a group, discuss how fractions are used when people think about and measure time. (Examples: Minutes are fractions of an hour and days are fractions of a week.)

As a group, name coins (e.g., penny, quarter) and say what fraction of one dollar each is worth.

As a group, demonstrate the Metal Triangles (or Metal Squares) to the students by showing them the whole triangle, and then its divisions. Point out that the triangle is a whole and it has been divided into two, three, and four parts. This will reinforce the important concept that the whole shape is not important, but its division into equal parts is what makes it a fraction.

Putting Fractions in Order from a Whole to a Tenthx

Purpose:
To learn to recognize fractions from a whole to tenths and put them in order using Fraction Circles.

Material:
Fraction Circles
Math Journals and Pencils

Presentation:
Most Montessori teachers present this concept in Year 1 and review it if needed in Years 2 and 3.
– Invite a student to learn to recognize fractions from a whole to tenths at a mat with the material already laid out.
– Remind the student that the Fraction Circles are made of equal pieces. The first circle is made up of one piece, the second circle is made of two pieces, and so on until the last circle is made of ten pieces.
– Encourage the student to notice that the Fraction Circles are arranged in order from the whole (one piece) and ending with the circle made of ten pieces.
– Take the Fraction Circles off the boards and place them in a row in random order. Do not remove the insets from their frames.
– Ask the student to place the Fraction Circles back on the boards in the correct order, starting with the circle made of one piece and ending with the circle made of ten pieces.

Extensions
Invite a student to imagine it is his birthday and three friends are joining him for cake. How many straight cuts right across the cake will he need to divide it into four pieces?

Invite a student to imagine it is his birthday and seven friends are joining him for cake. How many straight cuts right across the cake will he need to divide it into eight pieces? Suggest he try drawing the cake in his math journal as a circle to help find the answer.

Matching Fractions From Whole to a Tenth

Purpose:
To learn to recognize and match fractions from one whole to one tenth.

Material:
Fraction Circles
Mute Cards
Math Journals and Pencils

Presentation:
Most Montessori teachers present this concept in Year 1 and review it if needed in Years 2 and 3.
– Invite a student to practice recognizing and matching fractions at a mat.

Part 1: Matching

– Ask a student to go to the shelf, retrieve the Fraction Circles, and place them in a single row at the top of the mat.
– Choose a mute card, for example one half, and place it in the center of the work area.
– Ask the student to position the mute card below the corresponding Fraction Circle. For example, the student will place the mute card for one half below the Fraction Circle that is divided into two halves.
– Next, ask the student to take out the Fraction Circle inset that matches the mute card and place it on the nute card.

Note:
The fraction inset is placed directly on the colored portion of the mute card.
– Have the student continue as above until all the mute cards are correctly lined up and each mute card has a fraction inset correctly placed on it.
Ask the student to draw ten circles in their journal using the whole fraction inset. The first circle is a whole circle. In the second circle, ask the student to place the halves inset in position and trace around it with a different color to make a thin outline mute card. Ask the student to do the same for the eight remaining circles. Ask the student to write the title of the drawings as Fraction Mute Cards.

Part 2: Memory Exercise

– Place the mute cards face down on the table or mat and invite a student to select one.
– Invite the student to look at the mute card she has selected and then place it face down in front of her in the work area.
– Ask the student to go to the shelf where the Fraction Circles are kept. Encourage her to select the fraction inset that matches her mute card and bring it back to the work area.
– Ask the student to turn her mute card face up and place the selected fraction inset to see if it matches. If not, she can return the inset to the shelf and try another one.

Extension:
Reverse the memory exercise activity. Give a student a fraction inset to examine and then leave it on the desk. Ask the student to go to the shelf where the mute cards are stored and retrieve the mute card.

Understanding the Parts of Fractions

Purpose:
To understand the parts of fractions.

Material:
Fraction Circles
Ten black fraction lines
Word Labels
Small slips of paper
pen or pencil
Math Journals and Pencils

Presentation:
Most Montessori teachers present this concept in Year 1 and review it if needed in Years 2 and 3.
– Invite a student to learn about the parts of a fraction at a mat with the material already laid out.

Note: Each part of this activity can be presented over a number of days.

Part 1: Denominator – Family of Fractions

– Position the halves frame and inset in the center of the work area.
– Remove the halves insets from the frame, and position them to the right of the frame on the mat.
– Move the fraction pieces apart, and remind the student that she/he has already started to learn about fraction names.
– Point to the halves fraction pieces and say that these are halves. A whole that has been divided into two parts os called the family of halves.
– Place the world label “halves” below the black line and say, “Halves.”
– Write the number 2 on a small slip of paper, position it below the halves label, and say, “Two pieces.”
– Return the halves fraction insets to the frame, and the frame to the board.
– Move the fraction line and word label for halves to the far left of the work area.
– Place the thirds frame and insets in the center of the work area, and remove the insets. Position a black fraction line to the right of the thirds fraction insets.
– Point to the thirds fraction pieces and ask the student if he/she knows the name of this fraction. (Thirds.) Confirm that she/he is correct and say, “Yes, these are thirds. A whole that has been divided into three parts is called the family of thirds.”
– Place the word label “thirds” below the black line, and say, “Thirds.”
– Ask the student to write the number 3 on a small slip of paper and position it below the thirds label. Ask the student how many pieces there are. (Three.)
– Ask the student to return the thirds fraction pieces to the frame, and the frame to the board.
– Move the fraction line and word label for thirds beside the one for halves.
– Point to the halves and thirds examples on the mat, and explain to the student that these family names and numbers are called the denominator. The denominator tells us how many pieces the whole has been divided into.
– Continue in the same manner through to tenths.

Part 2: Numerators of 1

– Position the halves frame and insets in the center of the work area.
– Remove one halves inset from the frame, and place it to the right of the frame.
– Place a black fraction line in the work area to the right of the halves of the inset.
– Remind the student that this black line is called a fractus and it divides a whole number into parts or fractions.
– Point to the halves fraction piece and say that this is one half.
– Ask the student to move the halves inset above the black line.
– Tell her/him this is one.
– Ask the student the name of the family of fractions she/he is working with (Halves.) Conform that she/he is correct. A whole that has been divided into two parts is called the family of halves.
– Write the number 2 on a small slip of paper and position it below the fractus.
– Point to the halves inset above the fractus, and explain to the student that the number above the fractus, and explain to the student that the number above the fraction line is called the numerator. The numerator tells us how many pieces of the denominator (or fraction family) we have.

Note: Using the “family” analogy, the numerator also tells us how many family members there are. In this case, there is one piece (one member of the halves family). Thus, the fraction is one half.
– Continue in the same manner through to tenths.

Part 3: Numerators Greater Than 1

If the student is still interested, or on another day, introduce numerators greater than one, starting with two.
– Position the frames and insets in the center of the work area.
– Remove two thirds insets from the frame and place them above the fractus. Then, write the number 3 on a small slip of paper and position it below the fractus.
– Say, “There are two pieces (family members) of the thirds fraction family.
– Read the fraction as two thirds.
– Return the thirds pieces to the frame and the frame to the board.
– Position the fourths frame and insets in the center of the work area.
– Ask the student to place two fourths insets above the fractus.
– Ask her/him to write the denominator on a small slip of paper (4), and place it in its appropriate spot.
– Confirm that the student is correct, and ask her/him to read the fraction aloud and then return the insets to the frame and the frame to the board.
– Continue with 2/5 and so on.
– On another day introduce fractions with a numerator of 3.
– Ask the student to trace the whole fraction circle in her/his journal. Ask her/him to trace a thirds inset in the circle and then move the inset over and repeat. Invite the student to color the fraction and label it “two thirds.”

Extensions:

Working with a classmate, challenge each other to make different fractions using the Fraction Circles.

Cut a sheet of paper into two equal pieces. What fraction of a sheet is each piece? (One half.) Cut each of the halves in half again. What fraction of the sheet is each piece now? (One quarter.) Proceed with one more cut to into equal pieces. What fraction of a sheet is each piece? (One eighth.)

Count out enough pennies equal to a dime. What fraction of a dime is each penny worth? (One tenth.)