Writing Fractions and Finding Equivalences

Introduction

Working with the Fraction Skittles and Fraction Circles has given the students opportunity to explore fractions using both their visual and tactile senses. This exploration provides the students with a concrete way (the method) to use the material as often as required to come to an understanding of the concept. The next step is to associate the quantity with the symbol. The symbol, or the written form of the fraction, is the more abstract learning concept, which has been supported by the experiential learning that has taken place with the skittles and insets.

Equivalences (or equivalents) of fractions are fractions that represent the same amount. However, they are written differently because the whole is not divided into the same number of parts. For example, 2/4 is equivalent to 4/8, but in the first case the whole is divided into fourths while in the second case it is divided into eighths.

Students in Years 1-3 find equivalences by manipulating the Fraction Circles. This limits the number of equivalences they can find. The sets of equivalences that can be represented using Fraction Circles are shown in the chart below.

Equivalences that can be represented in Fraction Circles
Fractions Equivalences
 Halves  1/2 = 2/4 = 3/6 = 4/8 = 5/10
 Thirds  1/3 = 2/6 = 3/9
2/3 = 4/6 = 6/9
 Fourths  1/4 = 2/8
2/4 = 4/8
3/4 = 6/8
 Fifths  1/5 = 2/10
2/5 = 4/10
3/5 = 6/10
4/5 = 8/10
 All  1/1 = 2/2 = 3/3 = 4/4 = 5/5 = 6/6 = 7/7 = 8/8 = 9/9 = 10/10

 


Understanding Written Fractions

Purpose:
To further understand the concept of numerator and denominator and see how fractions are written using symbols.

Material:
Fraction Circles
Small Slips of Paper
Pen or pencil
Math journals and pencils

Presentation:
Most Montessori teachers present this concept in Year 1 and review it in Year 2 and as needed in Year 3.
– Invite a student to continue to learn about numerators and denominators and how fractions are written.
– Ask the student to get the Fraction Circles and position them at the top of the work area.
– Say, “When we write a fraction, we draw a line.” Draw a line on a slip of paper and point to it. Remind the student that this line is called the fractus.
– Say, We write one number on the top and one number on the bottom.” Write a number on the top and a number on the bottom. Use a simple fraction that the student has heard before, such as 1/2.
– Say, “We have learned that the number on the bottom tells us the fraction family and how many pieces the whole is divided into. It is called the denominator.”
– Encourage the student to point to the Fraction Circle represented by the denominator. For the fraction 1/2, this will be the halves family and the Fraction Circle for halves.
– Say, “The number on the top tells us how many pieces are being considered. This is called the numerator.”
– Ask the student to take out the fraction inset(s) that represents the numerator. For the fraction 1/2, the student will take out one of the halves.
– Ask the student student to place the fraction piece(s) on the table or mat to make the fraction. For the fraction 1/2, the student will simply place one fraction piece on the mat. For a fraction such as 2/5, the student will place the corresponding number of fraction insets side by side.
– Repeat with other fractions until the student has a solid understanding of how fractions are written.
– Ask the student to write a fraction in her/his journal. Ask him/her to label the top number “numerator” and the bottom number “denominator.” Younger students may simply label them “N” and “D”.
– In Years 2 and 3, ask the student to write the definition of numerator, fractus, and denominator in his/her journal. Denominator: How many pieces the whole is broken into. Numerator: How many pieces are being considered. Fractus: The line between the numerator and denominator.

Extension:
In the classroom, find three whole items or groups and name the fractions they contain.
Example: The whole set of chairs contains ten chairs, three green and seven black. In this case, 3/10 of the chairs are green and 7/10 are black.
Example: The whole classroom has three doors, two closed and one open. In this case, 2/3 of the doors are closed and 1/3 are open.


Matching Fraction Tickets to Fraction Circles

Purpose

To learn to associate the quantity to the written form for fractions 1/1 to 10/10

Material

Fraction Circles
Fraction Slips
Math journals and pencils

Presentation:
Most Montessori teachers present this concept in Year 1 and review it in Year 2 and as needed in Year 3.
– Invite a student to learn more about fractions in written form. Ask the student to retrieve the Fraction Circles and position them in a single row at the top of the work area.

Note:
 It may be necessary to teach the symbols in more than one session.

Part 1: Associating Quantity to Symbol, 1/1 to 1/10

– Lay out the fraction tickets from 1/1 to 1/10 on the table or mat in no particular order.
– Take out the whole Fraction Circle inset from the frame and say, “This is one whole.”
– Ask the student if he/she knows which fraction ticket represents one whole (1/1). If he/she does not know, place the fraction ticket 1/1 beside the whole inset.
– Explain how the written form of the fraction 1/1 equals 1 (one whole). Further explain how the fraction 1/1 really means that the numerator of 1 is divided by the denominator of 1.
– Ask him/her what 1 divided by 1 equals (1). Confirm that he/she is correct.
– Invite him/her to move the fraction ticket for 1/1 below the inset.
– Ask the student to copy the symbol for one whole in his/her math journal.
– Ask the student to remove the halves inset from the frame and place it below the board and to the right of the whole inset.
– Encourage the student to find the fraction ticket for the fraction piece.
– If necessary, assist the student by reminding him/her about the denominator (the fraction family) and the numerator, and how many pieces (or family members) you have.
– Ask him/her to place the fraction ticket below the fraction piece.
– Conform that he/she is correct.
– Ask the student to copy the symbol for one half in his/her math journal.
– Continue in the same manner with the fractions 1/3 to 1/10.

Part 2: Associating Quantity to Symbol, 1/1 to 1/10

Lay out the fraction tickets for 1/1 to 10/10 on the table or mat so the student can see each one. The teacher may want to lay out selected tickets (e.g., ten fraction tickets) rather than all of them to make it easier for the student to find the right ticket.
– Using Fraction Circle pieces, construct a fraction such as 4/7.
– Invite the student to find the corresponding ticket and place it below the fraction pieces.
– If necessary, assit the student by reminding him/her about the denominator and fraction family, and the numerator and how many pieces (or family members) you have.
– Ask the student to say the name of the fraction, for example four sevenths, and then write the fraction in his/her math journal.
– Encourage the student to place the fraction pieces back in their frame and return the fraction ticket to the basket.
– Continue as above with other fractions (e.g., 5/5), encouraging the student to find the corresponding fraction tickets.

Note: When the student is competent at matching tickets to fractions, put away the tickets and ask the student to write the fractions instead.

Extensions:
With a classmate, construct fractions for each other to label.


Constructing Fraction Circles to Match Fraction Tickets

Purpose
To learn to associate the written form to the quantity for fractions 1/1 to 10/10

Material:
Fraction Circles
Fraction Tickets
Math journals and pencils

Presentation:
Most Montessori teachers present this concept in Year 1 and review it in Year 2 and as needed in Year 3.

– Invite a student to continue learning about fractions in written form.

Note:
 It may be necessary to spread this presentation over two or more sessions depending on how quickly the student grasps the concept.

Part 1: Associating Symbol to Quantity, 1/1 to 1/10

– Ask the student to choose a ticket from the basket (e.g., 1/4).
– Ask the student to read the fraction out loud. Coach him if needed.
– Invite the student to make the fraction that matches the ticket by taking the appropriate frame off the board and placing it in the mat. If necessary, discuss with the student how the denominator tells us the fraction family, and which frame to choose (fourths). Then, the student will remove the correct number of pieces (1) and place it on the mat above the fraction ticket.
– When the student has correctly made the fraction, ask him to trace the fraction pieces in his math journal, and label the fraction below the drawing.
– Ask the student to return the fraction pieces and frame to the board.
– Invite the student to choose another ticket (e.g., 1/7). Encourage the student to continue in the same manner until he is competent at matching fractions.

Part 2: Associating Symbol to Quantity, 1/1 to 1/10

– Invite the student to choose a ticket from the basket (e.g., 7/10).
– Ask the student to read the fraction out loud.
– Encourage the student to make the fraction that matches the ticket by removing the appropriate frame off the board to represent the denominator, and placing it on the mat (tenths).
– Ask the student to remove the correct number of pieces to represent the numerator (7), and place them together above the fraction ticket.
– Confirm that he is correct.
– Ask the student to trace the fraction pieces in his math journal, and label the fraction below the drawing.
– Encourage the student to return the fraction pieces and frame to the board.

Extensions:
Find a classmate to practice writing fractions with. Take turns choosing fraction tickets from the basket and reading them to each other. The person not reading writes the fraction correctly in their journal.

Use a ruler to divide a sheet of paper into three columns. In the first column, write a fraction in words, for example, “one half.” In the middle column, draw a picture of the fraction, for example, by shading in one half of a circle or square. In the right hand column, write the symbol for the fraction, for example, 1/2. Use a ruler to draw a horizontal line under the work and continue with another fraction. Repeat this exercise for these fractions; one whole, one half, three quarters, and fixe eighths.

Ask a student to bring you from the shel both the Fraction Circle inset(s) and fraction ticket for a given fraction (e.g., 1/8).


Finding Equivalences of Fractions

Purpose:
To learn how to find equivalences of fractions

Material:
Fraction Circles
Math journals and pencils

Presentation:
Most Montessori teachers present this concept in Year 1 and review it in Year 2 and as needed in Year 3.

Invite a student to learn about finding equivalences of fractions. Ask the student to retrieve the Fraction Circles and place them at the top of the work area.

Finding Equivalences of 1/1

– Move the Fraction Circle frames and insets for a whole and halves to the middle of the work area.
– Point to each frame and name it.
– Remove the whole inset from the frame and position it below the frames.
– Demonstrate to the student how the two halves fit perfectly in the place of the whole inset.
– Return the two halves insets to the halves frame.
– Bring down the thirds frame and insets and position the frame to the right of the halves frame.
– Ask the student to find out how many thirds pieces fit in the whole frame (3). Conform that she/he is correct.
– Ask the student to continue in the same manner for all the fraction Circles insets up to 10/10.
– Explain to her/him that although the frame is made up of various parts, the parts cover the same space so we call this equivalent fractions (or equivalences of fractions).
– Ask the student to write these equivalent fractions in her/his math journal (e.g., 1/1 = 2/2 – 3/3 = and so on).
– Encourage the student to return the insets to the frames and the frames to the board.

Finding Equivalences of 1/2

– Move the Fraction Circle frame and insets for halves to the middle of the work area.
– Point to the frame and name the insets (halves).
– Remove one halves inset from the frame and place it back on the board. The left side of the frame remains empty so that it can be filled to make the equivalences.
– Encourage the student to find equivalences of 1/2, using the fraction pieces only from one frame at a time, each time recording the equivalent fraction in her/his journal.
– When the student finds an equivalence such as 2/4 that fits in the frame with the halves inset, encourage the student to also notice that the halves inset fits into the space left in the fourths frame on the board.
– Encourage the student to find all the equivalences of 1/2. (They are 2/4, 3/6, 4/8, and 5/10.)
– Once the student has found all the equivalences of 1/2, encourage the student to return the insets to the frames and the frame(s) to the board.

Finding Equivalences of 1/3, 1/4, and 1/5

– Take the thirds frame off the board and place it on the mat. Invite the student to remove one third from the Fraction Circle and place it on the board, leaving one third of the Fraction Circle open to receive equivalences.
– Encourage the student to look for equivalencies of 1/3, each time recording the equivalent fractions in her/his journal. They are 2/6 and 3/9.)
– Once the student has found both the equivalences for 1/3, encourage the student to return the insets to the frames and the frame to the board.
– Repeat the exercise for the fourths Fraction Circle (1/4 = 2/8).
– Repeat the exercise for the fifths Fraction Circle (1/5 = 2/10).

Extensions:
Invite a student to use the Fraction Circles to find one equivalence for each of the following fractions: 2/3, 3/4, 3/5.

Invite a student to write all the equivalences of 1/2 on a sheet of paper with the numerators in one column and the denominators in the adjacent column, as shown below. Study the pattern in each column to figure out the next two equivalences.

Numerator Denominator
1 2
2 4
3 6
4 8
5 10
? ?
? ?

The pattern shows the next two equivalences are 6/12 and 7/14.

Demonstrate the concept of equivalence with the Constructive Triangles (e.g., Triangular Box). This will also connect ideas of geometry with fractions.
Emphasize that the smaller triangles can be sued to make the larger triangles.
Reinforce the names of the triangles as you work with them.

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