Multiplying and Dividing Fractions

Introduction

Lower Elementary Montessori students learn to multiply and divide fractions by whole numbers. These operations are similar to multiplying and dividing whole numbers.

Multiplying Fractions

To multiply a fraction by a whole number, the numerator is simply multiplied by the whole number. For example, to multiply 2/9 by 4, the numerator 2 is multiplied by the whole number 4, for a product of 8/9.

Montessori students first learn this concept by manipulating Fraction Circles. It is best to start with prepared equations with products less than or equal to one. This way, the products fit in a single Fraction Circle and are easier to visualize.

Dividing Fractions

To divide a fraction by a whole number, the numerator is simply divided by the whole number. For example, to divide 8/9 by 4, the numerator 8 is divided by the whole number 4, for a quotient of 2/9. Montessori students first learn this concept by manipulating Fraction Circles. For example, to divide 8/9 by 4, the student separates the eights fraction pieces into four equal groups. The student then sees that each of the four groups contains two pieces.

The amount to be divided is called the dividend, and the result is called the quotient. Then introducing the concept of dividing fractions by whole numbers, it is easiest to use prepared equations with dividends and quotients less than or equal to one. This way, each equation can be completed using a single Fraction Circle.

The number by which the dividend is being divided is called the divisor. For this introduction, it is necessary to choose equations in which the divisor divides evenly into the numerator.

Not all equations are as simple as those in the above examples. It is sometimes necessary to express a fraction in different terms so the divisor will divide evenly into the numerator. For example, the equation 3/5 ÷ 2 = can be expressed as 6/10 ÷ 2 = . Two divides evenly into 6, giving a quotient of 3/10 for the equation.

Montessori students learn this concept by working with Fraction Circles and experimenting until they find an equivalent fraction that can be divided evenly. To do this, the student needs prepared equations that can easily be worked out using Fraction Circles.


Multiplying Fractions by Whole Numbers

Purpose:
To learn how to multiply fractions by whole numbers.

Material:
Fraction Circles
Prepared Equation Slips
Math journals and pencils

Presentation:
Most Montessori teachers present this concept in Year 3.

Setting Up the Equation

– Invite a student to learn to multiply fractions by whole numbers at a mat where the material is already laid out.
– Encourage the student to choose an equation slip and read it, for example (3/10 x 3 = .
– Tell the student that the equation means taking three tenths three times.
– Ask the student to write the equation in his/her journal.
– Invite the student to take the tenths frame and place it on the mat.

Solving the Equation

– Encourage the student to take three tenths and place them on the mat to the right of the frame.
– Invite the student to repeat this twice more, leaving a space between each group of three. She/he will end up with three groups of three tenths.
– Invite the student to count the fraction pieces on the mat and say how many there are. (Nine.)
– Say to the student that three tenths taken three times equals nine tenths. In other words, three tenths times three equals nine tenths.
– Ask the student to write the answer in his/her journal.
– Ask the student to replace the tenths in their frame on the board.
– Invite the student to continue in the same manner with anther question.
– Encourage the student to do several equations until she/he is proficient at multiplying fractions by whole numbers.

Extension:
Invite a student to copy a prepared equation into his/her journal and find the answer without using the Fraction Circles. Repeat for other prepared equations.


Dividing Fractions by Whole Numbers

Purpose:
To learn how to divide fractions by whole numbers.

Material:
Fraction Circles
Green Skittles
Prepared Equation Slips
Math journals and pencils

Presentation

Most Montessori teachers present this concept in Year 3.

Setting Up the Equation

– Invite a student to learn to divide fractions by whole numbers at a mat where the material is already laid out.
– Invite the student to choose an equation slip and read it, for example 6/8 ÷ 2 = .
– Ask the student to copy the equation in his/her journal.
– Review the terms dividend and divisor with the student.
– Encourage the student to make the dividend 6/8 at the center of the work area by placing six eighths pieces together on the mat.
– Ask the student to select enough skittles to make the divisor and place them to the right of the dividend on the mat. (Two skittles for the example of 6/8 ÷ 2 = .)

Solving the Equation

– Remind the student that division means sharing equally.
– Encourage the student to begin sharing equally, placing one fraction piece under each skittle until they are all shared.
– Remind the student that the amount one skittle receives is the answer (quotient) to the division question.
– Ask the student what the quotient is for 6/8 ÷ 2 = . (It is 3/8.)
– Ask the student to write the answer in her/his journal.
– Encourage the student to place the eighths pieces back in their frame on the board
– Encourage the student to continue in the same manner with another equation.

Extension:
Invite a student to select a prepared equation slip and find the answer without using Fraction Circles. Repeat for some other prepared equations.


Dividing Fractions by Whole Numbers When Equivalent Fractions Must Be Made First

Purpose:
To learn how to divide fractions by whole numbers when making equivalences is necessary prior to solving equations.

Material:
Fraction Circles
Green Skittles
Prepared Equation Slips
Math journals and pencils

Presentation:
Most Montessori teachers present this concept in Year 3.

Setting Up The Equation

– Invite the student to learn more about dividing fractions as a mat with the material already laid out.
– Invite the student to choose an equation slip and read it, for example, 1/2 ÷ 3 = .
– Ask the student to copy the equation in her/his journal.
– Invite the student to make the dividend 1/2 in the center of the work area by placing a halves piece on the mat.
– Invite the student to select the right number of Green Skittles to make the divisor and place them to the right of the dividend on the mat. (Three skittles.)
– Ask the student what he/she notices about the dividend (One half can not be equally shared among three skittles.)

Finding the Equivalent Fraction and Solving the Equation

– Encourage the student to find a fraction that is equivalent to 1/2 that can be shared equally among three skittles. Coach the student as necessary until she/he finds an equivalence.
– Once the student has found the right equivalence (3/6), ask him/her to record the new equation in the math journal as follows: 3/6 ÷ 3 = . Encourage the student to return the halves inset to the board.
– Encourage the student to begin sharing fraction pieces equally among the three skittles, and stop when the fraction pieces have been used up.
– Remind the student that the amount one skittle receives is the answer to the equation.
– Ask the student what the quotient is. (One sixth.)
– Ask the student to complete the equation in his/her math journal as follows: 3/6 ÷ 3 = 1/6.
– Encourage the student to return the sixth pieces to their frame on the board.
– Invite the student to continue in the same manner with more equations until he/she is proficient at dividing fractions that require equivalences.

Extensions:
Invite a student to imagine he/she has one-third of a chocolate bar to share with one friend. How much chocolate will each person get? Invite the student to find the answer using Fraction Circles and Skittles.

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