I often tell my co-workers that the students that I have this year are getting my best teaching! This school year has challenged my math content knowledge and pedagogy (instructional strategies) because my students have major deficits. On top of that I have many ELL students who are still trying to learn English.
Even with all of the obstacles that I face teaching these students, I find myself delving deeper into my content to find alternative ways to teach my students math content that should have been taught in previous grade levels while trying to teach the current 5th grade level standards. Last week I began my usual quest to teach my students how to create equivalent fractions. In previous years I have learned that students like mine always need explicit instruction because the smallest thing will thrw them off. Equivalent fractions are very confusing for most students because they are accustomed to looking at the numerator and the denominator for the answer. Now they have fractions that have different numerators and denominators and the fractions are equal.
About 2 years ago I was trying to find a model for creating equivalent fractions with multiplication and much to my surprise there was not one. I had already created a model for creating equivalent fractions through division or simplifying fractions, but I didn’t really understand why we could multiply the numerator and denominator by the same number and get an equivalent fraction.
Well this year I figured it out! Whoohoo!!! While creating a model for equivalent fractions and multiplication there were other ideas that I had to abandon. Like the assumption that when we multiply the numerator and denominator by the same number we are multiplying it by the whole number 1. This idea goes against the concept of multiplication although the multiplicative identity of 1 states that when a non zero is number is multiplied by 1 the value doesn’t change. In theory this is correct, however when creating the model for equivalent fractions it does not show a fraction being multiplied by 1. The factor of change is what creates a quantity that can be viewed as multiplying by 1. Multiplication is repeated addition and the concept of multiplication doesn’t change when it is applied to equivalent fractions. When we create equivalent fractions through multiplication the fraction repeats itself . For example the equivalent fractions 3/4 = 9/12 the fraction 3/4 repeats three times. Most teachers would assume that the fraction 3/4 would be added like we traditionally add fractions. This is a huge misconception because the role of the denominator is to tell how many parts or groups that the fractions has in all. So, as the fraction repeats the denominator or the total amount increases each time the fraction repeats, therefore creating multiples of the original denominator. When this repetition occurs each fourth is partitioned the same amount of times that it takes to create the equivalent fraction.
The numerator in the fraction also repeats the same amount of times as the denominator because it is a fraction and is part of the whole which is 4/4. I like to call these fractions repeating fractions. This is why 3 can be multiplied by the numerator and the denominator to create the equivalent fraction 9/12.
At first I thought that this idea of repeating fractions would only work up to a certain point but I had to keep reminding myself that multiplication means to repeat and that the role of the denominator stays the same in any situation that involve fractions. The only difference is that the denominator can be applied in a different way depending on the situation.
I took a huge risk teaching this to my students because I was afraid that it would confuse them. Honestly, they had to get use to referencing, connecting, and applying the concept of multiplication to equivalent fractions. After teaching my students how to create equivalent fractions using this model they really understood why behind the 3/4= 9/12. I’m really looking forward to seeing how this model for equivalent fractions will help my students with other fractional concepts that involve creating equivalent fractions.