Equivalent Fractions

- Equivalent fractions can best be shown by
(rods lined up end-to-end) of the same length.*trains* - In comparing any set of trains showing equivalent fractions, the train
with the smallest number of rods
*represents*the fraction in its.*lowest terms* - There may be several
*groups*of equivalent fractions for each unit.

**When two fractions are equal in overall quantity or value they are
called equivalent fractions. We can say that two fractions are considered
equivalent when it can be demonstrated that each fraction can be
used to represent the same amount of a given object.**

**To Demonstrate an Equivalent Fraction:**

**For Example:**

Let's say the brown rod which represents 8 cm. is the **unit **(meaning
it is equal to 1). We can show the following 3 equivalent fraction groups.

**EQUIVALENT GROUP 1:**

In the above example, the two fractions shown are equivalent to each other. This is evidenced by their equal length. In the equivalent fraction group 1, we used the white and red rods because both can be evenly divided into 8. We know there are no other fractions that belong in this equivalent group because there are no other rods equal in length to 1 red rod and 2 white rods.

The numerator of each fraction is the number of rods used in the fraction.
The denominator of each fraction is the number of rods that would be used
if the train was equal in length to the unit. For example, in the first
fraction of the equivalent fraction group 1, the numerator of **2/8**
is the number of white rods used in the fraction (2) and the denominator
of **2/8** is the number of white rods that would be needed to equal
the unit in length (8). In the second equivalent fraction, the numerator
of** 1/4** is the number of red rods used in the fraction (1) and the
denominator is the number of red rods (4) that would be required to equal
the unit length.

**The following is another equivalent fraction group representing the
unit 8.**

**EQUIVALENT GROUP 2:**

In equivalent group 2 we use the purple rods in addition to the white and red rods because the purple rods can also be divided into 8 evenly. As you can see the numerators of all 3 fractions are equal in length and the denominators are also equal in length.

As stated above, the train with the smallest number of rods represents
the fraction in its lowest terms. The fraction **1/2** is the fraction
in its lowest terms.

**EQUIVALENT GROUP 3 :**

As in equivalent group 1, group 3 also only uses white and red rods
because no other rods which equal 6 in length *and *divide evenly
into 8.

The fraction **3/4** is the fraction in its lowest terms.

In the example above we see that there are **3 groups **of
equivalent fractions representing the unit 8. In order to determine these
equivalent fractions we created trains that must be equal in length and
their multiples must equal the unit which is 8 in this case.

**Let's try another example:**

In this example let's use the orange rod (which represents 10 cm.) as the unit. We can show the following 4 equivalent fraction groups.

**EQUIVALENT GROUP 1:**

In review we will use rods that are equal in length and that divide evenly into 10.

The fraction **1/5** is the fraction in its lowest terms.

**EQUIVALENT GROUP 2:**

The fraction **1/2** is the fraction in its lowest terms.

**EQUIVALENT GROUP 3:**

The fraction **3/5**_{ }is the fraction in its lowest terms.

**EQUIVALENT GROUP 4:**

The fraction **4/5 _{ }**is the fraction in its lowest terms.

In the example above we see that there are 4 groups of equivalent fractions. As in the first example, we created trains equal in length whose multiples equal the unit which is 10 in this case. Some students may want to insert a dark green rod in the equivalent group 3 or a brown rod in the equivlent group 4, however neither of these are possible because neither 6 nor 8 divide evenly into 10 so their is no fractional equivalent for this rod when the orange rod is the unit.

_{Now it's your turn.}

1.** **Which equivalent fractions (with values less than 1) can be
represented using a dark green rod?

2. Which equivalent fractions (with values less than 1) can be represented using a purple rod?

3. Assuming the blue rod equals 1 whole, what fractional value would be assigned to the light green and white rod?

Check your Answers.

Send comments to: silha@girltech.cs.rice.edu

**This page was developed through GirlTECH
'97, a teacher training and student technology council program sponsored
by the Center for Research on
Parallel Computation (CRPC), a National
Science Foundation-funded Science
and Technology Center. **

**© June 1997 Molly Silha **