Tuesday, March 24, 2015

No Tricks! Comparing Fractions with Manipulatives


Teach students to compare fractions using hands on manipulatives (such as fraction bars and pattern blocks) instead of math tricks that students forget.

When learning new math concepts, it is crucial that students get a lot of time using hands on manipulatives to help them gain a deep understanding of the topic. Some teachers don't want their students to rely on manipulatives because they will not be available for students during testing. I understand this concern but skipping this step is counter productive. Students won't build an understanding but will try to rely on memory and tricks. This leads to damaging misconceptions about math. I know because I have been guilty of trying to rush the learning process due to the demands of testing and pacing guides.

When introducing fractions, I have pattern blocks, tiles, snap cubes and fraction bars easily accessible.  I'll also raid the resource room and see what else I can find. I have been lucky enough to borrow fraction cards, 3-d fraction models, and plastic fraction circles.

Teach students to compare fractions using hands on manipulatives (such as fraction bars and pattern blocks) instead of math tricks that students forget. Teach students to compare fractions using hands on manipulatives (such as fraction bars and pattern blocks) instead of math tricks that students forget.


The manipulatives I have are available to students whenever they need them to solve problems. First we work on identifying fractions using these manipulatives. For example, students explore the pattern blocks to determine that a triangle is 1/6 of a hexagon.

When we have a good grasp on identifying fractions, students continue to use manipulatives to compare fractions.  I find the fraction bars especially helpful for this concept. We even make paper ones that I send home for students to use during homework (or when the constant tinkling of plastic pieces gets to be too much for me). If you would like a set scroll to the bottom for a link to the ones I use. They're free!

I pose questions to my students such as which is more: 1/2 or 1/4. I have them look at models and explain their answers. I don't tell them, at least not at first. They have to discuss it. This is a good time to use small groups. Each group decides on an answer and writes and explanation. During this time students are free to use any of the manipulatives they need. Then we share. We do this a lot! Most students need way more repetition than we typically give them. It is necessary for them to have this problem solving time and experience to make discoveries.

I have learned not to start comparing fractions with an explanation that if you have a larger denominator your pieces are smaller therefore 1/4 is less than 1/2. I used to do this, thinking that it would save time. It didn't. Most of my students do eventually figure out how the denominator affects the size of the pieces through discussion and practice with manipulatives. Trying to teach them this fact before they have enough experience to understand why, doesn't work. They try to remember a rule or trick and don't build an understanding of fractions. Students develop misconceptions such as:
  • 1/3 is less than 2/6 because thirds are bigger pieces: This student remembers that larger denominators make bigger pieces but doesn't take the numerator into account. 
  • 2/4 is greater than 3/4 because the fraction with 2 has bigger pieces: This student is relying on a trick they don't understand. They remember the teacher told them a bigger number actually makes smaller fractions. They don't understand that the number has to be in the denominator because the denominator determines the number of pieces and therefore the size of them.
The student's above likely did not try to use manipulatives, visualize the fraction or draw a picture because they were focused on the trick. I have noticed that when introduced to "tricks" even my students with higher math understanding will make mistakes because they are trying to use a shortcut. They view the tricks as more advanced and faster and don't rely on their problem solving skills which is what we should be focusing on.

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