Maria Montessori created this very simple material to teach young children what fractions are.
1° What is a fraction?
Show the circle and explain that it is « 1 » because it is « 1 circle » and therefore, it is the number « 1 ».
Go on explaining that one day, smaller numbers than « 1 », were needed and so it was decided to cut « 1 » in two parts. Take the red circle “1” and cut it in two parts with a horizontal separation.
At the same time, draw a horizontal line on a piece of paper: « ______ ».
Ask the child how many parts we have made when cutting “1” into “two”.
He will answer: « 2 », so write down “2” under the line:
Then, take the green base with three red parts and ask: “how many red parts do I have in the family of thirds?”. The child will say “3”, so write down this number under the horizontal line, this way:
Explain that the number under the line is called the denominator.
Then, what is the denominator in the family of fourth showing the base with 4 parts. The child will answer “4” and so write down:
And so on until 10.
Then, take the green base where there were two red parts. Take one of the red parts and explain that when talking about one only member of the family, the family of seconds, we write down “1” above the horizontal line, as shown here:
If I want to talk about one member of the family of thirds, I write down 1 above the line, like this:
The number above the horizontal line is called the numerator.
If I want to talk about more than one member, for example two thirds, I write down:
When the child understands this writing, prepare flash cards with the different names of fractions and ask him to pick up the corresponding green bases and to place them under the right flash cards.
2° The equivalences to fractions
Tell the child: “Today, we are going to see if some fractions are equal to others. We are going to check if they occupy the same space in the circles”.
Start with 1/2. Take it out of the base and ask: “does 1/3 go inside? Do they have the same surface?” No… « Lets’ try with 1/4… Yes, two fourth covers the same surface of one half. Write down:
Continue with the sixth and yes 3/6 = 1/2 . Write down:
Go on to the eighth and yes 4/8 = 1/2 so write down:
Go on to the tenth and yes 5/10 = 1/2 so write down:
Conclusion: they occupy the same space, they have the same surface, they have the same value, the same fractional value.
The aim of this exercise is to familiarize the child to the concept of equivalence between two fractions. You are preparing him to operations done with fractions where he will be asked to express the results in the simplest way possible.
3° Operations involving fractions:
- Additions :
– Fractions with the same denominator.
Example :3/8 + 1/8.
The child first places 3/8 and then 1/8 in the empty space of the « 1 ». He knows that when adding, he needs to put everything together, so he gathers the eighth and counts 4/8 ; then he looks for the equivalence and notices immediately that it is equal to 1/2.
Wrote down the answer:
Conclude the exercise by saying: « When adding two fractions with the same denominator, we add the numerators, but the denominator stays the same”.
– Fractions with different denominators.
Example: 3/4 + 1/8
Have the child work with the empty circle. The child places the different elements of the fraction in the circle and sees that, since the elements are different, he can place them together but cannot answer. He tries a few different bases and realizes that with the bases of the eighth, he can find elements that are equal to the fourth. He finds that 3/4 equals to 6/8. He then places the eighth all together and writes:
Try other examples where fractions need to be reduced like: 1/2 + 2/5.
Conclude the exercise with the rule: «When adding fractions with different denominators, we first must reduce one or the two fractions so that the denominators are the same. ».
- Substractions :
– Fractions with the same denominator.
Example: 5/8 – 3/8
LThe child places 5/8 in the empty circle. Then subtracts 3/8, he ends up with 2/8 that can be simplified in 1/4. Write down:
End the exercise with the rule: « To subtract fractions with the same denominator, subtract the smaller numerator to the bigger, the denominator stays the same. Simplify or reduce if possible ».
– Fractions with different denominators.
Example: 5/8 –1/4
The child usually knows that he has to simplify because he has done it before. It is clear that 1/4 cannot be taken from 5/8.
Therefore, he will try to exchange 1/4 with the eighth and will find that 1/4 = 2/8, he can then take 2/8 from 5/8 and is left with 3/8. Write down:
State the rule: « When subtracting fractions with different denominators, reduce one or the two so that the denominators are the same ».
- Multiplication :
– Multiplication of a fraction with one whole number:
Example: 1/9 x 6
The child places 1/9 six times in the empty space, he puts them together and counts « 6/9 » that can be reduced by the equivalence « 2/3 ».
State the rule: « To multiply a fraction with a whole number, multiply the numerator, the denominator stays the same. Reduce if possible. »
– Multiplication of a fraction with a fraction:
Example: 1/2 x 4/7. The four seventh are placed in the empty space of the circle. Equally 1/2 x 4/7 can be written 1/2 or half of 4/7.
Now, when taking half of anything, what do we do? We break it in two equal halves. So when breaking 4/7 in two equal parts, each part is 2/7 and that is our answer.
Using the same method, it is possible to work on other examples, like 3/4 of 8/9. First break 8/9 into 4 equal parts (which correspond to the fourth of 3/4), these equal parts are therefore composed of 2/9th each. Since we are using 3/4 of the total quantity, we place three parts of 2/9th (corresponding to 3 of 3/4). The result is 6/9th can be reduced to 2/3.
Use other examples and state the rule: “When multiplying two fractions together, first multiply the numerators. Then multiply the denominators. In most cases, it is necessary to reduce.” . »
- Division :
– Division of a fraction by a whole number
Example: 4/7 : 2
The child places 2 green skittles as divisors, and places the four seventh in the empty space of the circle. Then, he splits the seventh equally into two skittles. The result of the division is as follows:
State the rule: « To divide a fraction by a whole number, simply divide the numerator by the divisor; the denominator stays the same. Reduce if possible. »
Second example: 1/2 : 4
The child places 1/2 (numerator) in the empty space of the circle, and takes out 4 skittles for the denominator. 1/2 cannot be divided with the skittles, so a change must be done. The childs knows or tries to exchange the 1/2 in parts that can be separated into 4. He finds that 1/2 equals to 4/8. Now, he can distribute the 4/8 to each skittle. The result is what is attributed to the green skittles. The answer is: 1/8
The rule is: « When the numerator of a fraction can be divided by the denominator, we immediately obtain the result. When the numerator cannot be divided by the denominator, multiply the denominator of the fraction by that divisor. Reduce if possible. »
– Division of a fraction by a fraction
Example: 1/7 : 1/3
In a division, the result is always what “one unit” has,
(what ONE has).
Since we cannot make “one unit” with only one third, we have to add two other thirds to obtain ONE with the three thirds. For this division of fraction: 1/7 : 1/3 into 3/7 : 3/3. We then unite the three thirds of the denominator in ONE and the 3/7th of the denominator corresponding to the result of the division of fractions:
Other example: 4/9 : 1/2
The same way, with this division of fractions, I add one half to ONE and attribute four ninth to eachof my halves. I unite the two halves denominators to make ONE et I unite my eight ninth which become the result.
State the rule: « To divide a fraction by a fraction, multiply the numerator of the dividend by the denominator of the divisor and the denominator of the dividend by the numerator of the divisor. »
Before asking the child to learn the rule by heart, have him practice with the material so that he obtains the proof that the rule is confirmed by the sensorial manipulation. The best is to let him find the rule on his own.
The photos of this article show you how the operations are done with Montessori material (green bases with red fractioned circles and fractioned wooden skittles). Parents can easily build their own material as shown above (or with ten squares, ten rectangles or ten bars fractioned in one whole…)