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.

## 8 thoughts on “The Truth About Equivalent Fractions and Multiplication”

Mia MerrickI do not agree with your model for equivalent fractions. If you use an array model for multiplication of fractions you can arrive at 9/12. Multiplication, as in the number of copies, will support this model. 3/4 will be multiplied by 1 because in essence there is only I copy of that unit. Using the array model to multiply 1 in the form 3/3 will result in 9/12.

Michelle WilliamsThank you for your response, however this model correlates with generating equivalent fractions using a multiplication chart. The denominator of equivalent fractions are multiples so therefore the model has nothing to do with multiplying by a whole and if the fraction is 3/4 and you multiply by 3/3 then it is not a copy of itself. The whole would be 4/4. Anytime multiplication is used it refers to repeated addition. The numerator and denominator increase at different rates and the definition of the denominator is applied as the total amount in all. Using fraction wheels and the area model are essentially the same because they both refer the whole as 4ths. It’s a different way of processing that works for my students.

Mia MerrickPartitions are not being considered. I respectfully disagree.

Michelle WilliamsPartitions were not apart of the lesson because the students understood that the denominator gets smaller. Your initial argument was that the fraction 3/4 copied itself (3/3) which was an incorrect assumption. If you want to add partitioning the model that I presented is still correct. We can use your example 3/4 and 9/12 the 3/4 repeats 3 times the second time it repeats each 1/4 is multiplied by 2 which is 6/8 and be shown by partitioning each fourth. When 3/4 repeats a 3rd time each 1/4 is partitioned by 3 thus creating the equivalent fraction 9/12 which debunks your initial argument of multiplying by 1. Thanks again for the response it made me realize that I need to add the partitioning to my lesson to make it more explict!

Ronald Lohse“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 is NOT an assumption, multiplying the numerator and denominator by the same number is the equivalent of multiplying the fraction by n/n and n/n equals 1. We then use the identity property of multiplication that says multiplying a number by one does not change the value of the number. Or m X 1 = m. Therefore, multiplying a fraction by multiplying the numerator and the denominator by the same number (equivalent to multiplying by 1) does not change the value of the number, but it might change the way that value is expressed. 3/4 = (3 x 3)/(4 x 3) = 9/12 or equivalently 3/4 x 3/3 = 9/12. This concept that it is actually just multiplying by 1 is absolutely huge and used in various forms throughout the rest of your students mathematical lives. It is the key to unit conversions and simplifying many different algebraic expressions as well.

Michelle WilliamsThe rule states that fractions are equivalent when a/b = c/d. What this Rule says is that two fractions are equivalent (equal) only if the product of the numerator (a) of the first fraction and the denominator (d) of the other fraction is equal to the product of the denominator (b) of the first fraction and the numerator (c) of the other fraction. I understand that the identity rule of 1 states that any number multiplied by 1 is that number, however in elementary were teach multiplicative patters and equivalent fractions follow a multiplicative pattern. I was taught this rule in high school and I understand it in the sense that the fractions are equivalent therefore it creates a copy of itself. The title of my blog post is The truth about multiplication and equivalent fractions.

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Lauren SchweitzerEquivalent fractions are fractions whose figures are different but their value is the same. But computing them is not a simple task as what this post is showing. That is why I used a tool to confirm that the computation is correct. For checking if the calculation of fractions is correct, I used an app which shows the correct answer and display the step by step solution. It is actually a fraction calculator from http://www.fractioncalc.com. It is very useful in solving fraction and a very good tool to be used both by teacher and student for reference if their answer is correct and if their solution is right.