Adding and Subtracting Fractions

Introduction

Fractions to be added are called addends. For example, in the equation 1/3 + 1/3 = 2/3, the fractions 1/3 and 1/3 are addends. Adding fractions that have the same denominator is much like adding whole numbers. The numerators are simply added.

Montessori students learn to add fractions by building addends with Fraction Circle insets and then counting the number of fraction Circle insets to find the total. In the following lessons, students start with the easiest fractions: those that have the same denominator and add up to one or less.

Students will learn how to add fractions with different denominators in Year 4. However, in year 3, those who are ready can be introduced to the concept with either the Fraction Circles or the Fraction Circle Box. The important thing for students to learn in Year 3 is that they must express fractions with the same denominator before they add them. The same denominator is called the common denominator. A common denominator is a denominator that two or more fractions have in common. For example, if a student wishes to add 1/3 + 1/2 = , the two fractions can be expressed using equivalent fractions with a common denominator of 6. The equation then becomes 2/6 + 3/6 = .

Year 3 students find common denominators by manipulating Fraction Circles using trial and error. This is a concrete, hands on way for students to understand the process of finding common denominators. They need easy examples that they can solve using Fraction Circles. That is, the fractions must add up to less than one. After students are successful in this way, they can be reminded of their work with equivalent fractions and lowest common multiples, and how these methods can be used to quickly determine common denominators. At this point, the students will be using more abstract means to find common denominators.

Students who can add fractions with the Fraction Circles are ready to learn how to subtract fractions with the same denominator using the Fraction Circles. Subtracting fractions with the same denominator is much like subtracting whole numbers. One numerator is simply subtracted from the other to arrive at the answer. For example, in the equation 3/5 – 2/5 = , the numerator 2 is subtracted from the numerator 3 to arrive at the answer 1/5.

It is easiest to start with fractions that will fit in a single Fraction Circle (i.e., that equal one or less). The minuend is set up in the Fraction Circle frame, and the subtrahend is subtracting by removing the corresponding number of fraction insets from the frame. The remaining insets in the frame represent the difference.

Students will learn to subtract fractions with different denominators in Year 4, but some more advanced students may be ready in Year 3. The emphasis in Year 3 is on manipulating Fraction Circles (insets or plastic cut-outs) using trial and error. For this reason, the fractions (minuends and subtrahends) should be less or equal to one.


Adding Fractions with the Same Denominator That Add Up to One or Less

Purpose:
To learn how to add fractions with the same denominator.

Material:
Fraction Circles
Prepared Equation Slips
Math journals and pencils

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

Note: It may be necessary to present this activity over two or more sessions.

Preparing the Materials:

– Invite a student to learn to add fractions that have the same denominator at a mat with the material already laid out.
– Invite the student to choose an equation slip and read it, for example 1/3 + 1/3 = .
– Encourage the student to record the equation in her journal. Help the student if necessary.
– Explain to the student that the best way to do this activity is to start with two empty frames. Invite the student to remove the whole inset rom the frame and return the inset to the board. Ask the student to place the empty frame in the center of the work area.
– Invite the student to place the thirds insets on the board, and position the empty thirds frame beside the first one.

Building and Adding Fractions

– Invite the student to build the first addend in the empty frame on the left by placing the a single thirds fraction piece in it.
– Encourage the student to build the second addend in the empty frame on the right by placing a single thirds fraction piece in it.
– Remind the student that in addition we are putting items/numbers together. Therefore, we will place both addends together in the right hand frame to find the find the sum.
– Ask the student to move the thirds fraction piece from the left frame into the right frame to find each sum.
– Ask the student to count how many thirds there are in the right frame. (Two.)
– State, “There are two thirds.” Ask the student to write 2/3 to complete the equation in her journal.
– Ask the student to replace the thirds frame and its fraction pieces.
– Encourage the student to continue in the same manner with other equations until she can easily add fractions with the same denominator that add up to one or less.

Extensions:
Use a set of measuring cups and a glass measuring cup to add fractions. For example, to add 1/3 + 2/3, use the one- third cup to measure the correct quantities of water into the glass measuring cup. Read the total off the glass measuring cup when you are done. Do the same with 1/4 + 1/4 =, 1/2 + 1/2 = , and 1/4 + 3/4 = .


Adding Fractions with the Same Denominator That Add Up to More Than One

Purpose:
To learn how to add fractions with the same denominator.

Material:
Fraction Circles
Prepared Equation Slips
Math journals and pencils

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

Setting Up The Equation

– Invite a student to learn to add fractions that add up to more than one whole.
– Ask the student to get the Fraction Circles and Fraction Circle Box and bring them to the mat.
– Invite the student to choose an equation slip and read it, for example 2/3 + 2/3 = .
– Encourage the student to write the equation in her journal.
– For 2/3 + 2/3 = , the student will need the thirds insets and a one-third fraction piece from the fraction Circle Box.
– Demonstrate to the student that the metal inset and plastic cut-out for one third are the same.

Adding the Fractions

-Encourage the student to build the first addend, 2/3, with the insets on the mat.
-Encourage the student to build the second addend, 2/3, on the mat.

Note:
The second addend will use one metal thirds inset and one plastic thirds cut-out from the Fraction Circle Box.

– Invite the student to push the addends together and count all of the fraction pieces. (Four.)
– Encourage and coach the student to say the equation, “Two thirds plus two thirds = four thirds.” Ask the student to record the sum in her journal.
– Show the student that the 4/3 is equivalent to 1 1/3.
– Encourage the student to return the Fraction Circle insets and plastic cut-outs to their respective places.
– Invite the student to continue in the same manner with more equations until the student has mastered adding fractions with the same denominator.

Extensions:
Invite the student to use the Fraction Circle Box to practice adding fractions with more than two addends, such as 1/3 + 1/3 + 1/3 = .
Some more sums ot try:
1/2 + 1/2 + 1/2 = ;
1/3 + 1/3 + 2/3 = ;
3/4 + 3/4 + 1/4 = ;
1/5 + 2/5 + 3/5 + 3/5 = .

Invite a student to use a set of measuring cups and a two cup glass measuring cup to add fractions. For example, to add 2/3 + 2/3 = , use the one-third measuring cup to measure the correct quantities of water into the glass cup. Read the total amount off the glass measuring cup when you are done. Repeat this exercise with 3/4 + 3/4 = .


Introduction to Adding Fractions With Different Denominators

Purpose:
To learn how to add fractions with different denominators by manipulating Fraction Circles.

Material:
Fraction Circle Box
Prepared Equation Slips
Math journals and pencils

Presentation:
Most Montessori teachers present this concept to students who show readiness in Year 3.

Setting Up The Equation

– Invite a student to learn more about adding fractions with denominators at a table or mat with the material already laid out.
– Invite the student to choose an equation slip and read it, for example, 1/2 + 2/5 = .
– Ask the student to write the equation in her/his math journal.
– Ask the student what he/she notices about the two denominators in the two fractions. (They are different).
– Explain that before the two fractions can be added, their denominators must be the same.

Finding a Common Denominator and Adding the Fractions

– Invite the student to build the two addends on the mat.
– Encourage the student to make equivalent fractions that have a common denominator using trial and error.
– Ask the student to rewrite the equation in his/her journal using the two equivalent fractions she/he found, for example 5/10 + 4/10 = .
– Encourage the student to add the two fractions by placing 5/10 and 4/10 together.
– Ask the student to record the sum in his/her journal, 9/10.
– Invite the student to return the fraction pieces to the box and continue in the same manner with other prepared equations.

Extension:
Look closely at a ruler with inches divided into halves, quarters, and eights. It does not matter whether or not the ruler is divided into sixteenths. Use the markings on the ruler to help you add 1/2 inch + 1/4 inch = . What is the total length? Now use the markings on the ruler to help you add 1/2 inch + 3/4 inch = .What is the total length?


Subtracting Fractions with the Same Denominator

Purpose:
To learn how to subtract fractions with the same denominator.

Material:
Fraction Circles
Prepared Equation Slips
Math journals and pencils

Presentation:
Most Montessori teachers present this concept in Year 1 and review ir in Year 2.

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

Setting Up The Equation

– Invite a student to learn to subtract fractions with the same denominator at a mat with the material already laid out.
– Invite the student to place the Fraction Circles in order at the top of the mat.
– Encourage the student to choose an equation slip and read it, for example 3/4 – 1/4 = .
– Ask the student to write the equation in her math journal.
– Invite the student t take the frame she will need and place it on the mat. For this example, it will be the fourths frame.
– Encourage the student to represent the minuend 3/4 in the frame by removing one fourths inset and returning it to the board.

Subtracting the Fraction

– Invite the student to subtract the subtrahend 1/4 by removing one fourths inset from the frame and placing it on the board.
– Ask the student to count the remaining fraction pieces and write the answer to the equation in her math journal, 2/4.
– Invite the student to return the material to the board and choose another equation slip.
– Continue until the student is proficient at subtracting fractions with the same denominator.

Extensions:
Invite a student to use a set of measuring cups and a glass measuring cup to subtract fractions. For example, to calculate 2/3 – 1/3 = , fill the glass measuring cup with water to the 2/3-cup mark. To subtract 1/3, pour water from the glass measuring cup into the 1/3-cup measure until the 1/3- cup measure id full. Read the marking on the glass measuring cup to see how much water remains. Repeat for 3/4 – 1/4.


Introduction to Subtracting Fractions with a Different Denominator

Purpose:
To learn how to subtract fractions with the same denominator.

Material:
Fraction Circle Box
Prepared Equation Slips
Math journals and pencils

Presentation:
Most Montessori teachers present this concept to students showing readiness in Year 3.

Setting Up the Equation

Invite the student to learn to subtract fractions with different denominators at a table or mat with the material already laid out.
– Invite the student to choose an equation slip and read it, for example 5/6 – 1/3 = .
– Encourage the student to record the equation in his math journal.
– Ask the student what he notices about the denominators of the two fractions. (They are different.)
– Explain that to subtract two fractions their denominators must be the same.
– Invite the student to make the minuend (5/6) with the fraction pieces. The student will need five sixths fraction pieces.
– While the subtrahend is not usually represented with this material, this first exercise will demonstrate it this way for the purpose of showing the equivalent fraction.
– Encourage the student to make the subtrahend (1/3) with the material to the right of the minuend.

Finding Equivalent Fractions and subtracting Them

– Encourage the student to use trial and error to make equivalent fractions with a common denominator for the subtrahend. The student will find that 1/3is equivalent to 2/6.
– Ask the student to return the one thirds and two sixths fraction to the box.
– Encourage him to subtract 2/6 by removing two sixths fraction pieces and placing them in the box.
– Invite the student to count the remaining Fraction Circle pieces and write the answer, 3/6, ni his journal.
– Ask the student to return the fraction pieces to the box and continue in the same manner with another prepared equation.

Extensions:
Invite a student to look closely at a ruler with inches divided into halves, quartes, and eights. It does not matter whether or not the ruler is divided into sixteenths. Use the markings on the ruler to help you subtract 1/4 inch from 1/2 inch = . What is the remaining length? Now use the markings on the ruler to help you add 1/2 inch from 3/4 inch = .What is the remaining length?

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