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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Saturday, March 14, 2015

New Design

I am so excited to have my new blog design. Christi at Design by Christi did a great job incorporating everything I asked for. I wanted to use the logo I already have from Surfer Kids Clip Art. Christi added the words around it for my blog button. I love that addition. I needed a blog with primary colors to go with my logo. I never even told her that all my anchor charts are done in rainbow colors from red to purple. So the social media buttons are absolutely perfect for me!  The number background she found for me is just great since this blog is all about math. I know that this blog design is a lot different from the candy color trends right now but what I can I say, I've never been much of a conformist.

I had thought about having a blog designed for a couple years. I decided I better start blogging to see if I liked and would keep doing it. I definitely think I will. I thought I wouldn't have enough to say but I actually have a bunch started. I have more ideas then I thought. I like to start a post when I have an idea and go back and finish it when I have time.

While your here on my blog, make sure you check out the links to my pages at the top. You can read about me, download some freebies, and shop at my stores. 

Wednesday, March 4, 2015

Introducing Equivalent Fractions with Concrete Manipulatives


My class is just starting to work on fractions. They should have be able to name simple fractions from second grade but I got a lot of blank looks when we started. They could identify fractions with a numerator of one but it seemed that any other fractions were new to them. Once we started working on those fractions, some students started misnaming the denominator instead of 2/6 they would say 2/4 as in 2 pieces are shaded and  4 are not. I had 5 students that were ready to move on though. Even though they don't seem to have much of a background in fractions either, I thought they could start looking at equivalencies. I didn't want to move too fast though so we started with halves. I started by giving them this question.

equivalent fractions
Questions for introducing equivalent fractions.

When I pose a challenge question, it is either something that I haven't taught yet or an open ended complex problem. In this case, we are exploring a topic I haven't taught yet. They need to think about it first. Then we may do a turn and talk, write about it before sharing or if we are in a small group (like this instance) just have a discussion.

During our discussion, students came to the conclusion that while each robot's pizza had a different number of pieces the amount was the same. They did point out that Robbie's pizza would be more difficult to eat. We talked about how these were equivalent fractions, then I showed them the booklets that we were going to use. In your classroom, you could do this activity without the booklets by making a chart for students to record their work on.

To complete the task, students used a specified number of tiles to make a fraction that had the same number of red tiles and blue tiles. On page 4 (pictured below), students make a rectangle with 8 tiles, half of the tiles were red and half were blue. Some students wanted to sketch their picture without using tiles. For this activity though, I insisted that they use manipulatives. I knew that they did not have a strong enough understanding of equivalent fractions to forgo using concrete manipulatives.
Sample pages from Equivalent Fractions Booklet
Pages from Equivalent Fractions Booklet
I also included writing in math pages (like page 9 shown above) in student booklets to give students practice explaining the concepts in their own words. They did a great job on this activity. If you would like to purchase the booklet you can get it by clicking the picture below. Right now it is priced at $1.00.



http://bit.ly/1BRYSnL
Buy now by clicking the cover.

Students Working on Equivalent Fraction Booklets
Students working on booklets.