# Finding hidden fractions

Fractions have always been one of my favorite topics to teach. Yet the majority of students run away screaming when they hear the word fractions. I believe that this fear comes in part from not growing rich images in their minds for the idea, but are rather left at the mercy of their symbolic/numeric expression, such as 4/5. I feel that language especially early on is crucial for connecting student images to their numeric symbols – which is why I would prefer to initially speak about the fraction as four of five when first starting out with a student. I try to use this terminology and connect it to the more common usage by saying them both together, “ four of five or four-fifths”.

Regarding activities with fractions, I use many different manipulatives or models for fractions to help students Image Make (Pirie & Kieren, 1994) but the one in particular I like is Cuisenaire Rods (these can be purchased off of Amazon, I recommend the wooden ones). Georges Cuisenaire was a Belgium elementary school teacher. He invented these now famous rods, sometime in the 1930’s, as a means of helping his students understand and visualize the relationships within certain aspects of mathematics. He found that by engaging children’s natural inclination to play and giving them an appealing physical material which held meaning for the relationships on which mathematics is based, it was possible to provide a solid foundation of understanding for all his students.

The work started by Cuisenaire remained relatively unknown for more than twenty years until he was introduced to a visiting lecturer from the University of London, Dr. Caleb Gattegno. Gattegno was a mathematician and educator, who immediately recognized this tool’s power and educational value.

Through his work developing the application of the rods, Gattegno gained profound insight into the ability of children, which led him to the realization that they are far more capable than was being revealed in the typical classrooms at the time. This realization has been firmly established over the years by children from all over the world who have startled teachers with their remarkable grasp of mathematics concepts through the use of these rods.

If you have not worked with Cuisenaire rods before then spend a few days getting familiar with and playing with the rods. Each rod has a specific color and length relative to the other rods, these rods could be connected to the numbers 1-10 (see image above). Once familiar, begin the activity by discussing how often smaller numbers are hidden in bigger numbers for example 9 contains three 3’s, or 4 contains two 2’s. “Can you think up some more examples?”

“Is this also true of fractions?” Write the fraction 5/10. and have the student build a representation for this fraction.

“Now look for a single rod that will make the length of both rods.” For example using only the yellow rods I can create the same lengths:

The important piece of this activity is that you are building something that directly relates to the symbols. I’m assuming the student has a general familiarity with fractions and the idea of 5/5 is equal to 1 has been introduced.

Ask the student, “What do you notice about the numbers in the equation?” Make sure to write it the same way as below, this will help the student in their noticing.

Once they notice 5/5 and equate that to one, have them draw a big one around the 5/5.

Then rewrite the equation:

“What is any number

times one equal to?”

“Does this make sense? Let’s look at our blocks, how could five-tenths be equal to one half?” Listen to the students’ ideas and discuss why it makes sense.

“Hmmm. So, one half was hidden inside of five tenths. Is that the same as there being three 3’s inside of 9? Then does it make sense to say that 3 = 9? Is this something different then?” You can then introduce, if you haven’t already, the idea of proportions – how numbers can be used as comparisons. For example, 5 compared to 10 is one half. If they don’t totally understand that is okay, this is a topic to built upon throughout this activity and time spent with other activities.

A general idea of the concept is all we are looking for at this point.

“Let’s look at another one. Can you find the hidden fraction inside of six eighteenths? There may be more than one. Can you find them all?”

This question is somewhat different as there exist multiple equivalent fractions, which give us the opportunity to discuss equivalent fractions and the simplified form of a fraction.

Go through the same process but now you have multiple builds that could occur and should occur. Have the student find as many as possible. Once they’ve found a rod that creates both lengths, leave it built and write the symbol version beside it.

Find another one, leave it built, and write the symbol version beside it.

Let them keep trying until they have found the third one, leave it built, and write the symbol version beside it. The order doesn’t matter. Ask them, “Do you think there are anymore, and if not why? How do you know?”

Once again, have the students explain what they notice about the equation. Eventually, they will notice that each of these equations also contain a one. Have them draw a big one around the fraction, as below.

Since any number time one is just itself, the hidden fractions must be:

“Why are all these considered equal? Does it make sense? Let’s build each fraction and see what we notice.”

Encourage them to discover the relationship on their own, and then discuss proportionality and how each representation contains a 1 to 3 relationship. Show how they can all be reduced to one third, which would be called the simplified form of the fraction.

Once again, I can not emphasize enough the importance of the student doing the building and eventually the writing themselves. From here you can move on to many more examples finding the hidden fractions. Eventually leading to something like: