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Some fractions even with different numerators and denominators have the same value.
For example, consider the fractions 
and 
. Though the denominators and numerators look different, the value of both the fractions is the same. They are called equivalent fractions.
Equivalent fractions have exactly the same value but expressed using different fractions.
Consider a pizza cut into half, another one into 4 slices and the third one cut into 8. Eating a slice from the first pizza is equivalent to eating two slices from the second and eating four slices from the third. That is, the pizza fractions calculated any of the three ways is the same.
That is, the unit fraction is equivalent to , and again equivalent to 
.
Equivalent fractions have the exact same area covered. Also, equivalent fractions occupy the exact same place on a number line.
Simple equivalents of fractions can be generated by doubling or tripling the given fraction. Equivalent fractions calculator multiplies the numerator (the number above the fraction bar) and the denominator (the number below the fraction bar) by the same number to generate equivalent fractions. Similarly by reducing fractions to halves or quarters also give equivalent fractions.
The below discussed math manipulatives would help to understand the concept of equivalent fractions for fourth graders.
1. Fraction Charts / Fraction strips: This is one of the most commonly used manipulative for learning fractions.
The definition, as well as the equivalence of fractions, can be explained using fractional strips. Children can make such equivalent fractions chart using multicolored chart papers. As they do the activity by themselves, they enjoy independent learning the as well as the concept gets reinforced.
2. Area models: These are another similar math manipulative for teaching fractions.
The area covered by the shaded portion of each circle is the same, though the number of parts of the whole is different. The equivalent fractions reduce to 
if you simplify fractions given in the area models.
Fractional strips can be used to show tenths and hundredths as shown:
A three-dimensional version of these fractional strips can be like base ten blocks. The smallest cube is called 1 unit or 1 hundredth. Each rod consists of 10 such hundredths can be called 1 tenth. The flat square consists of 10 such rods and is called 1 whole or 10 tenths or 100 hundredths.
Once the concept of fractions is clear, the equivalence of fractions should be introduced using activities and fraction games for kids. A mere robotic repetitional practice can give children computational fluency. But for a better understanding of the topic, it should be demonstrated using real-life activities and math games.
Matching Equivalent Fractions: Given a set of fractions, the child can pair the equivalent ones.
The difficulty level can be increased by giving the models without giving the fractions represented. Introducing timer in later levels would help children concentrate more and calculate faster.
Each time a fraction is given to the child, on finding an equivalent fraction from the grid, it opens up. The child who first identifies the character behind the grid wins the game. The difficulty level of the game can be controlled by giving more complex fractions.
Two fractions are equivalent if they have the exact same size.
Here, the fractions 
are equivalent fractions. The area shaded in all the three fractions are equal. Though they are expressed differently, the value of all the three is the same.
Now, consider the fractions 
.
When these are represented using area models, the area covered by the first two fractions as a part of the same whole are equal. But for the third fraction, the area shaded is more than the other two. Thus, 
is not equivalent to the other two.
That is, 
.
When the fractions 
are plotted on a number line, they occupy the exact same point.
Plenty of equivalent fraction calculators are available online. But this might not give a clear idea about the concept.
A child can practice solving fraction equations to find the missing number in equivalent fraction equations at equivalent fractions with models.
The 3rd grade fractions skills include generating equivalents for common fractions like 
and 
by simply doubling or tripling the numerators and denominators. At this level, the equivalent fractions worksheets available at equivalent fraction without models children can practice by solving fractions.
In 4th grade fractions, children can initially generate equivalent fractions using models.
Consider the fraction 
. This can be represented using area model as shown.
Here, the whole is divided into 4 equal parts. Now, to form a fraction equivalent to this, say with one with denominator 8, the whole should be divided into 8 equal parts. For that, one need to sub-divide each of the above parts into two.
The circle is divided into 8 equal parts and the shaded parts are 2. That is, the above area model represents the fraction 
and it is equivalent to the unit fraction .
Further, if the whole is divided into 16 parts, you get an equivalent fraction 
as shown:
In a similar way when the divisions of a whole are merged together, the resultant is an equivalent fraction too.
Consider the area model representing the fraction 
.
Here the whole is divided into 6 equal parts and 4 parts are shaded in blue. Now, merge the two rectangles in each row to get a bigger rectangle.
Now the whole is divided into 3 equal parts and 2 of them are shaded. So, the fraction represented is 
.
But the area shaded in both are equal and thus, and are equivalent.
The algebraic significance of this generation process will be discussed in the next section.
In the previous section, each subpart of the area model for the fraction was subdivided into two. That is, the number of divisions in the whole is doubled. As a result, the number of parts that are shaded also doubles. Precisely, the numerator and denominator get multiplied by 2.
That is, 
=
Similarly, if the numerator and the denominator get multiplied by 3, 
= 
.
Thus, the equivalent fractions can be generated by multiplying the numerator and the denominator with the same number.
In the second example, when the two rectangles in a row were merged to get a bigger rectangle, the division of the whole was reduced to half. That is, the numerator and the denominator were multiplied by .
=
So, and are also equivalent fractions.
Finding an equivalent fraction for any given fraction can be practiced at equivalent fractions.
One way to add mixed numbers is to separately add the integral and fractional parts.
In this method, children have to be really careful if there is an improper fractional part after addition.
For example, 
= 3 + 
Now, the fractional part is an improper fraction. Convert it to an equivalent mixed number. 
= 3 + 
= 3 + 1 + 
= 4 +
While subtracting mixed fractions, the chances of making mistakes are even higher.
Consider the subtraction problem, 
.
When the integral and fractional parts are operated separately: 
= 2 + 
To avoid this, write the mixed numbers as equivalent improper fractions. A mixed number say, 
, can be converted to an equivalent fraction as follows:
= 
= 
= 
= 
Consider the initial addition problem, 
.
First, convert the mixed numbers to equivalent improper fractions. Make sure to follow math order of operations for the computations.
= 
= 
Adding fractions: 
Now, convert the improper fraction back to the mixed number.
This method is more systematic and children tend to make fewer mistakes.
The subtraction problem can also be solved in a better way.
Convert the mixed fractions to improper ones.
= 
= 
Subtracting fractions:
Tenths are special fractions with denominator 10 and hundredths are the ones with denominator 100.
How to generate a fraction equivalent to 
with a denominator 100?
It can be generated by multiplying the numerator and denominator by 10. 
That is, one-tenth is equivalent to ten-hundredths.
This can be represented using fraction model as shown:
A three-dimensional version of these fractional strips can be like base ten blocks. The smallest cube is called 1 unit or 1 hundredth. Each rod consists of 10 such hundredths can be called 1 tenth. The flat square consists of 10 such rods and is called 1 whole or 10 tenths or 100 hundredths.
Further, children learn to convert non-unit tenths to hundredths.
For example, 3 tenths = 30 hundredth.
The converting fraction worksheets provided by SplashLearn has a handful of questions that models tenths and hundredths using fractional strips and their conversions.
Using the equivalent fractions tenths and hundredths, children learn how to add fractions with unlike denominators of 10 and 100.
For example, to find the sum of 
and 
, first rewrite
Now the child has two like fractions to add.
Another skill learned at the level is comparing decimals based on their sizes.
For example, to compare the decimals 0.5 and 0.25.
A child whose foundation of the topic is not clear might think that 5 < 25, so 0.5 < 0.25.
One needs to realize that comparisons are valid only when the two decimals refer to the same whole.
First, the child needs to learn how to convert decimal to fraction.
Like whole numbers, decimals also can be written as a fraction with a denominator of 1.
So, converting decimals to fraction, they can be written as 
.
Now, write the equivalent fractions of the two such that decimal points do not exist in the numerator.
For that, multiply the numerator and the denominator of the first fraction by 10.
Similarly, multiply the numerator and denominator of the second fraction by 100. 
When you convert the decimals to fractions, 
and 
the two fractions do not have the same whole. To compare them, rewrite the first fraction.
Now, the two fractions refer to the same whole. Comparing the numerators,
.
Thus, 0.5 > 0.25 .
In simpler words, 0.5 is 5 tenths and 0.25 is 25 hundredth. Five tenths are equivalent to 50 hundredths. It is obvious that 50 hundredths are greater than 25 hundredths.
Thus, 0.5 > 0.25.
Further, children use this principle to list decimals in ascending and descending orders.
Evaluation:
To evaluate the equivalent fraction skills one can use the below assessment questions:
9 tenths = ? hundredths
0.35 ? 0.53
Ans: <
. Ans: 28
