The Montessori Fraction Circles are red plastic or wooden circles representing fraction pieces from 1-10. Children can easily understand the concepts of fraction families, equivalence, improper fractions, mixed numbers, and operations with fractions using the circles. The Fraction Circles are one of my Top Ten Montessori Math materials.
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Preceding Material
The child should have mastered whole number work (including operations) with the Bead Bars, Golden Beads, Stamp Game, and Bead Frames.
Introduction to the Concept
We always want to start with what the child knows, her background knowledge, and connect it to the unknown new concept. The introductory lesson aims to ignite the child’s imagination and impress the concept upon the mind.
- Show the child a Golden Bead representing one unit. One is the beginning of all numbers, but we can divide the bead by breaking it with a hammer. Demonstrate.
- Show the child an apple. Show her that we can divide the whole apple into smaller parts. Divide it into halves, then fourths. If we were to share the apple with four children what would be their share? Each child’s share would be one-fourth.
True fractions are, of course, equal in size and share, which will not be the case with a fractured bead or knife cuts on an apple. This is an impressionistic lesson meant to strike the child’s imagination and impress a concept on her.
Introducing and Naming Fraction Families
Whenever we introduce a new concept or material, we connect the known to the unknown, background knowledge to the new concept.
- Compare the red bead from the Bead Bar material with the whole red fraction circle from the Montessori Fraction Circles box.
- We can look at fractions like families. We call the whole circle the whole family, equal to one whole unit.
- Show each “fraction family” from halves to tenths by counting each part that makes the whole and labeling them. Example: The Halves Family, The Fifths Family, etc.
The key concept is that a unit can be divided into equal parts and we can name those parts. Children make booklets or posters of all the fraction families as a follow-up activity.
Introducing Written Fraction Symbols
- Choose one fraction family to work with at a time. For example, the fifth family. Take out five of the fifth fraction pieces from the Montessori Fraction Circles box and arrange them together to make a whole circle.
- Demonstrate that if we want two of the members from the fifth’s family, we can show it by placing a strip of paper above the fifth’s family to represent the fraction bar and create the denominator. Next, move two-fifths above the strip of paper to create the numerator. Show the written form 2/5 on paper.
- Continue exploring all the fractions for the fifth family 1/5, 3/5, 4/5, and 5/5.
Allow the child to work independently with the other fraction families. Find all the numerator and denominator combinations for each family. Make the quantity with the fraction pieces, write the numeral symbols, and read the fractions aloud.
Equivalence
Once a child is familiar with different fraction quantities using numerators and denominators with all the fraction families from whole to tenths and has had sufficient practice naming, reading, and writing fractions, she is ready to find equivalent fractions.
Equal to One Whole
- Begin with the whole family.
- Take out one whole circle.
- Take out 2 halves and place them on top of the whole circle. Two halves take up the same amount of space as one whole.
- Take out three of the thirds and place them on top of the whole circle. Three-thirds take up the same amount of space as one whole.
- Continue on with this experimentation through the tenth’s family using the Montessori Fraction Circles.
- The child should record 1/1= 1, 2/2= 1, 3/3= 1, 4/4= 1, etc.
Equal to One Half
Next, take out a one-half fraction piece from the Montessori Fraction Circles box and investigate what fractions are equal to one-half.
*It must be noted that we cannot mix families when finding equivalence!
- Start with the whole fraction and note that the whole is too big for the space of one half.
- Take out the thirds and place them on top of the one half trying to fill the same space as one half. She will discover it does not work.
- Take out the fourths and place them on top of the one-half trying to fill the same space as one half. She will discover two-fourths take up the same space as one-half. 2/4 = ½.
- Continue testing equivalence to one-half from the fifth through the tenth families.
Recognizing Equivalence to One-Half Patterns
Every other fraction family in the sequence is equivalent to one-half.
The numerators ascend: 1, 2, 3, 4, 5. The denominators are the multiples of two: 2, 4, 6, 8, 10.
Can she extrapolate the next equivalent fraction (s) even though we no longer have more fraction pieces to manipulate? (6/12, 7/14, 8/16, etc.) Some children will see these patterns naturally and others will need some explicit guidance.
The mathematical rule is that you get an equivalent fraction when you multiply the numerator and the denominator by the same number.
Continue Exploration and Discovery
Children independently find equivalence for the fractions 1/3- 1/10 using the Montessori Fraction Circles. Extend the exploration by finding equivalence for 2/3, 2/4, 3/4. 3/6, etc. Continue looking for numerical patterns in the numerators and denominators.
Adding Fractions with Common Denominators
Show children how to add fractions from the same family with the Montessori Fraction Circles. These are fractions with common denominators. For example:
- Take one-third piece out and place a plus sign to the right of it.
- Place another one-third piece out to the right of the plus sign followed by an equal sign creating the equation 1/3 + 1/3 =.
- Move the two third pieces to the right of the equal sign for a sum of 2/3.
- The child will write this on paper.
- Continue adding with 1/3 + 2/3 = 3/3 = 1 whole.
Children should continue working to add fractions with like denominators through the tenth’s family summing to no more than one whole (proper fractions).
They discover the mathematical rule when adding fractions with a like denominator is that we add the numerators together while keeping the denominator the same. You can point it out to her if she does not notice this.
This is a great addition fact practice also!
Simplifying Fractions
Show children how to reduce fractions to their simplest terms when adding fractions with the Montessori Fraction Circles. When we simplify a fraction, we use the least amount of fraction pieces that are equivalent. Examples: 3/3= 1 and 3/6=1/2.
Proper/Improper Fractions and Mixed Numbers
After the child has worked with adding fractions with like denominators that sum to one whole (proper fractions), we can introduce adding fractions that sum to more than one whole (improper fractions). For example, 2/3 +2/3 = 4/3. This is an improper fraction. The child will see before her one whole fraction and a 1/3 fraction piece. We write the fraction as 1 1/3. This is called a mixed number.
Subtracting Fractions with Common Denominators
Teach fraction subtraction by starting with the largest fraction family, the tenths. Show children how to subtract fractions from the same family with the Montessori Fraction Circles. These are fractions with common denominators.
- Take out ten-tenths and form them into a circle. Write a subtraction sign on a small piece of paper and place it to the right of the tenths.
- Take away five-tenths from the ten-tenths placing it to the right of the subtraction sign. This represents 10/10 – 5/10. Write an equal sign on a small piece of paper and place it to the right of the one-tenth piece.
- There are 5/10 left. Move these pieces to the right of the equal sign to show the difference left over after taking away five-tenths.
- Simplify: 5/10=1/2
Children continue subtracting fractions with like denominators with all the fraction families. They discover the mathematical rule when subtracting fractions with a like denominator is that we subtract the numerators while keeping the denominator the same. You can point it out to her if she does not notice this. Simplify fractions when needed.
This is a great subtraction fact practice also!
Multiplying Fractions by a Whole Number
Show the child how to multiply each fraction in a family by whole numbers with the Montessori Fraction Circles resulting in products less than one whole. For example:
- Begin with the tenth’s family.
- Start by multiplying 3/10 x 3.
- Take out 3-tenths and place them on the work surface.
- Take out 3-tenths a second and third time, placing them to the right of each previous set.
- Place an equal sign to the right of the three sets. Three-tenths taken 3 times is 9 tenths. Move the pieces to the right of the equal sign to show nine-tenths.
- Write on a paper 3/10 x 3 = 9/10.
- The child can continue with 3/10 x 2 and 3/10 x 1.
The mathematical rule is to multiply the numerator by the whole number while keeping the denominator the same.
After a child has practiced multiplying by whole numbers that have a product less than one whole, you can generate problems for them that have a product of more than one whole resulting in mixed number products. For example, 2/7 x 4 = 8/7 = 1 1/7.
Multiplying fractions is a great way to practice multiplication facts.
Dividing Fractions by a Whole Number
Begin with the tenth’s family. Use the green wooden peg “skittles” from the Stamp Game or the Division Bead Board materials.
- Take nine-tenths and divide them evenly among three “skittles” to represent 9/10 ÷ 3. This results in each skittle receiving 3/10. Write on paper 9/10 ÷ 3 = 3/10.
- The child can continue with 3/10 ÷ 3 and 6/10 ÷ 3.
The mathematical rule is when dividing a fraction by a whole number, divide the numerator by the whole number while keeping the denominator the same.
The child can independently continue dividing tenth fractions by a variety of whole numbers. Next, the child can move on to the ninth’s family, eighth’s family, etc. dividing by whole numbers.
The above picture shows another way to divide fractions with the Montessori Fraction Circles. To find 1/8 of 16, use a material to divide 16 equally among the eighths. The quotient is two. This is another excellent way to practice division facts!