Hands-on Activities for Developing Conceptual Understanding of Fractions

So how important are fractions in math instruction? Well, a recent study out of Carnegie Mellon University shows that students' understanding of fractions and division directly predicts how well students will perform in algebra and more advanced math classes. Of course, fractions tend to be one of the most challenging math concepts for students to master, and so I've spent that last 6 years learning more about fractions and best practices in developing conceptual knowledge in students.

First, an short overview of the progression of fractions instruction in Common Core math: Students are first explicitly exposed to fractions in 3rd grade (indirectly, students are exposed to factional concepts as early as 1st grade in geometry units), where they are introduced to the concept of fractions as numbers and develop an understanding of fractions using part/whole models (including number bonds, bar models, etc.) and number lines. Students also explore the idea of equivalent fractions and comparing fractions in 3rd grade.

In 4th grade students extend on their fraction understanding by composing and decomposing fractions, ordering and comparing fractions with different denominators, and adding/subtracting fractions with unlike denominators and mixed numbers. And in 5th and 6th grade, students build on previous understanding of fraction operations as they begin to multiply and divide fractions and work with decimal fractions.

The hands-on activities below are activities that I have collected so far in the last 6 years of reading about, learning about, and attending trainings on how to teach fractions. There are tons of ideas out there, but these are my favorites. They are appropriate for all grade levels and easily adaptable for all skill levels.

Paper folding to create fractions

Build your own fraction strips
Many of us have given students fractions
strips to work with, or have had students cut fractions strips out of a piece of paper to maybe play with once and never again. But something I learned several years ago in a math institute provided in our district by San Jose State University is that it is much more powerful to have students create those fraction strips themselves through a process of guided folding and exploration of how one fraction can become another when folded in half or in thirds, etc.

After learning this instructional strategy, I immediately started using it with my 5th graders at the beginning of our fraction unit. Although 5th graders are already supposed to have a fairly strong understanding of what a fraction is, I found that most did not. Even if they could add and subtract fractions fluently, most had a fairly weak understanding of fractions as division, or of how the numerator and denominator are related (ie. adjective-noun theory and place value relationships) and creating the fraction strips helped strengthen their understanding of fractions and place value.

So how does it work? First, the teacher should prep strips of paper ahead of time, in different colors, to provide to students (I usually take 9"x12" construction paper of various colors and cut each sheet into 4 strips). You'll need a different color for each of the following fractions: 1 whole, halves, fourths, thirds, sixths, eighths, twelfths, sixteenths.

Here is a sample transcript of this lesson:
T:  (holds up 1 strip) This is our whole for this lesson. Let's label it '1 whole', '1', and '1/1'
Ss do this with teacher

T: (holds up another strip, different color) Now, take your strip of paper and fold it in half, ends touching, nice and neat and then open it back up. Each piece is worth....?

Ss: 1/2!

T: Each piece is worth 1/2. This is 1/2 (pointing to one side) and this is 1/2. Altogether I have 2 halves. How many halves do I have altogether?

Ss: 2 halves! 

T: And the name of the fraction is 'halves'. Say it with me.... 'halves'. This piece is worth how many halves?

And so on... labeling the piece as you go with the name of the fraction and the value of each piece in number form and word form. After repeating this with each strip, I finally have the students cut out their pieces and save them in a bag for future activities.

Fraction Strip Games

Cover Up game with fraction strips

-Cover Up: Students play in pairs and take turns rolling fraction dice and layering the fraction pieces on top of their 1 whole strip. The first student to cover up their entire 1 whole strip without going over "wins". Students write their winning equation on a recording sheet (ex: 1/2 + 1/4 + 1/8 + 1/8 = 1). The game introduces students to the idea of equivalent fractions and adding fractions.

-Fraction Four Corners: Students pick up a fraction piece at random and are then asked whether their fraction is closer to 0, 1/2, 1 or 2 (or similar) and move to corresponding corners of the room. Students compare their results with those standing in the same corner as them and determine whether they are correct in their thinking or not. Teacher then asks if anyone wants to move. In the next level, the teacher can ask students to randomly pick up two fraction pieces and then move to a corner of the room.

Human Number Line

-Human Number Line: It is important for students to work with fractions on a number line and
develop a strong understanding of how fractions are related to each other on a number line. Students that can tell whether a fraction is closer to 0, 1/2 or 1 are able to better determine reasonableness of an answer. In this game, students are grouped into teams of 4-6 and are asked to each pick up a different fraction piece. Then, each team is asked to create a human number line, making sure that they stand with their fraction piece in the correct order from 0-1.

Patty Paper Activities
Patty paper (yup, same paper they use to separate hamburger patties at the local diner) is great for comparing fractions and for modeling multiplication/division of fractions. Students can fold the paper (almost) squares to model fractions, use colored markers or pencils to label, and can layer each piece of patty paper to see the similarities and differences between fractions, determine equivalent fractions, and model operations.  Transparencies also work well for this!

Fraction Mind Maps
Mind Maps are great for building connections between ideas and concepts, and a great way to show the relationship between the different "types" of fractions. In the example to the right we created a mind map in our math journals as a review of fractions (unit fractions, decimal fractions, improper, mixed numbers, etc.) and included examples for each type of fraction.



Legos
Legos can be a fun way for students to model fractions, work with addition and subtraction of fractions, and explore equivalent fractions. Scholastic's Top Teaching Blog post on using legos to
model math concepts includes some guided activities and printables that you can use to help students explore fraction concepts. Or even better, you can have students model fractions and/or fraction equations with their legos then create a video where they explain their thinking verbally and visually.

Counting chips
Students can use counting chips, buttons, m&m's, plastic coins, or any other physical set of objects to explore fractions of a group/set. Have students set up their group and physically move pieces when determining fractions of a set. Having students work with fractions of a set is important in helping develop students' understanding of fractions as division (ie. 2/3 of  24 means we are dividing 24 by 3 then looking at the quotient 2 times).

Measuring cups
Measuring cups are another great way to help students understand fractions of a whole. Students can pour 1/2s or 1/4s of a cup, etc. into a 1 cup to explore parts to whole with fractions. And this is a great way to integrate measurement lessons with fractions lessons!

Pattern Block Fractions
Pattern blocks help students visualize geometric concepts while working with fractional parts, equivalent fractions, and addition and subtraction of fractions. Pattern blocks are easy to layer, too, so students can compare fractions by stacking and combining them in different ways. Using pattern blocks are a great way to build connections between fraction concepts and geometric concepts!

Fraction Quilt Squares
This is a fun, artistic activity for practicing fractions. On a hundreds grid, students use 4 colors to create some type of design. Then, students count up the total squares colored in with each color. Finally, students record what fraction of the hundreds grid is filled in with each color. Students are also then asked to convert those fractions into decimal fractions and percents, and since we're working with hundredths, this activity is a good introduction into the relationship between fractions, decimals and percents.

I like this activity a lot because students at all levels of understanding can explore fractions at their own levels. Some of my students would create a simple pattern, something that they knew would be easy for them to count. Others wanted to create more complex designs and color in just parts of squares (which then meant that later they were counting up halves and quarters of squares on the grid and having to add them together). Almost all students loved having a little art integrated into their math lesson.

Equivalent Fraction Dominoes/Fraction Path
There are quite a few variations of this game available online. The idea is that students are given a set of paper dominoes that they match end to end based on equivalence. The fractions are represented in number form, picture form and word form and equivalent values are matched up end to end. One version we played included only unit fractions and pictures, while the most complex version we played included mixed and improper fractions. I have my students glue these down on construction paper once they complete the path correctly, but you could also have students record their final path in a photograph or video instead (and submit via a class website, LMS, Seesaw, etc.) so that they can keep the game in a bag and play again at a future date.

3-D Modeling Fraction Bars
I was working on a 3-D modeling activity for a 3rd grade class and started playing around with the idea of 3-D printing our own fraction bars. Once I got going with the sample, I started to realize just how much math was involved in modeling fraction bars accurately-- more than just understanding how to model a 1 whole bar vs. a 1/4 bar!

In setting this up as an open-ended activity, students have to first determine what length they want their 1 whole bar to be, and they'll have to choose carefully if they want to be able to create not only a bar that is 1/2 of the whole but also 1/3 of the whole or 1/8, etc. For example, I first chose 100 mm and soon found that that number did not work well for me, because determining 1/3 of 100 would be tricky... Next, students will need to create a fraction bar that is 1/2 of the whole, then another that 1/4 of the whole, and so on. Students should use the measurement grid to ensure that they are created fraction bars that are precisely 1/2 or 1/4 or 1/3 of the original whole.

Additional readings on how to develop conceptual understanding in fractions:

• Fractions: Building a Strong Foundation Based on Conceptual Understanding (Yale Initiative)
• Fractions are the Key to Math Success
• Teaching Fractions According to the Common Core State Standards (UC Berkeley)

Always Carry Blue Tape or What I've Learned as a New TOSA

So I'm currently 8 months into my new journey as an elementary math/tech-integration TOSA in my district. It's been a roller coaster ride so far-- just as many ups as downs, and plenty of corkscrews in between. I never really know from day-to-day what my role will look like, and it's still hard to tell

what kind of impact I'm having in my district, but I am learning a lot as I go, and getting a better idea everyday of what my role is, and what it could be. And as a district level TOSA, constantly on the road, regularly traveling between 9 sites, working out of staff rooms more often than my own office behind the district office, I am definitely in a position much different than the classroom teacher role that I had become fairly accustomed to.

Here is what I've learned so far as a new district TOSA:

Carry your own stapler...and copy paper... and pencils... and most office supplies, really
Unfortunately, it is not just my former site's staff room that is depressingly bare of office supplies. It would seem that most staff rooms are more of a black hole than anything else. And I've also found that not every copy machine in my district has an unlimited supply of paper included. At the sites where teachers bring their own paper to the copy room, I have found myself swapping favors for copy paper. I have learned that if I need a stapler, or paper, or paper clips, or pens, or anything else, I should plan to bring my own.

Keep the secretaries happy
Especially since they are often my gateway to the paperless copy machines! That being said, this isn't much different than when I was a classroom teacher. All teachers and admin know that the secretaries are really the ones that run the school, and if you want anything, you need to go through them. Now, I just need to make sure that I stay on the good side of 9 secretaries (and my own at the D.O. for that matter) instead of just one. Make friends, bring treats, say 'hello' and 'thank you' often!

It's all about relationships
You will get nowhere as a TOSA until you have developed trusting relationships with the classroom teachers. With some teachers, this will take a long time. Sit in staff rooms, chat with people, get to know them, compliment their strengths, build them up as leaders... make sure they know that you are not there to be evaluative, you are not there as a "spy". And once you have developed trusting relationships with teachers, have them write your "Yelp reviews", so to speak. Have them work with you at PDs, or talk to their teammates about the work you're doing together. When teachers see other teachers working with you, they will be more open to working with you.

Make sure your bag includes chargers , lots and lots of chargers

Especially as a tech TOSA, I carry quite a few devices with me and there is no guarantee that there will be extra
chargers laying around at the school sites. Carry your own chargers, and for your own sanity, invest in a good organizer for those chargers. It is not ideal to have to run back to the D.O. for a charger while out on the road, or worse, to find yourself at a conference without your chargers. Getting ready to present at a conference and finding yourself with a dead device is a major bummer! Oh, and dongles, don't forget your dongles!

Always carry blue tape!
Yeah, this was my favorite tool as a classroom teacher and it is still my favorite tool as a district-wide TOSA. Blue tape fixes everything.

Always have a backup plan
Standard classroom teacher knowledge, and it doesn't end when you move into a "special assignment" role. So far, I've experienced everything from simple lesson malfunctions during a demo lesson to wifi going out during my presentation at a tech conference to arriving for a Saturday morning PD and being locked out of our meeting space. Always have a plan B!

The best gift is gift of time
I've offered class coverage during rainy day recesses so teachers can get a bathroom break, extra prep time for worn out teachers, and sub duty so teachers can observe other classrooms. A TOSA friend of mine shows up during yard duty at his sites and relieves teachers from yard duty so they can take a little break. Teaching for an hour here and there keeps my practices fresh and provides a little relief to stressed out teachers. And reminding classroom teachers that you are indeed still a teacher by actually doing some teaching helps build and/or strengthen those trusting relationships.

My PLN

Get connected
Being a TOSA can be a lonely job. Whether housed at a site or working at a district level, you aren't quite a classroom teacher anymore, and you aren't an administrator. It has helped me to find my "tribe" on Twitter via #TOSAchat, #educoach and #connectedTL. We swap ideas, share successes and failures, collaborate on projects, support each other, build each other up, remind each other why we love this job. Get connected!

The sticky from CUE16

This last week I attended CUE National in Palm Springs with an amazing group of teachers and teachers on special assignment (TOSAs) from my school district, along with our Director of Instructional Technology and our Technology Manager. It was my first time at National CUE, and it was epic. Big thinkers, innovators, game-changers, passionate educators (including those from my own district)-- surrounded by this group of people for four days was inspirational and reinvigorating, and it reminded me why I love my job. And, through all the conferencing, unconferencing, presenting, lunching, networking, hallway chats, and, of course, karaokeing, here is the stuff that stuck most strongly for me at 2016's CUE:

• It's time for a Joyful Rebellion. Part of my job as a teacher leader is to be a "joy rebel" (@thebradmontague). I am officially joining Kid President's Joyful Rebellion! I am in a lucky position this year, being a TOSA. I have the pleasure of spreading joy across the district, among students, teachers, admin, office staff... I need to focus on this more. Focus less on the negative and only on the positive, and all the ways that I can use my position to help spread joy!
  

• Listen more, talk less. I hear this often from more seasoned TOSAs and I'm working on it. I need to resist the urge to fix or suggest, and listen more. My tendency, when someone starts talking about something they want to do, is to jump in with some amazing tool (or two or three) that I love that I think will solve their problem. But really, I need to wait more, listen more, try to help those I work with come up with a solution. I just am so passionate about the power of technology in education that I sometimes get a little excited about sharing... but I need to work on resisting the urge to solve the problem all the time.

• Exploration before Explanation. Kevin Fairchild (@kfairchild6) reminded us that 21st century instruction means a complete shift to inquiry-based learning. Our students do not need us to teach them facts... they can Google the answers to facts. We need to teach them how to learn. Let students explore a concept before teachers explain further. Lisa Highfill (@lhighfill) echoed this idea and exemplified it with a sneak peak at her lesson design plan. The "explore" time for students falls before the "explain" time for teachers.

• We need creators & critical thinkers, not memorizers. A topic reiterated over and over again, in session after session, was the need to teach students how to be critical thinkers, not memorizers. I started reading Jo Boaler's latest book, Mathematical Mindsets, just before CUE and the message in Boaler's book aligned well with the message I kept hearing at CUE. It was just another reminder of the big work that we have to do to help teachers in all classrooms shift their instruction away from textbooks and fact dumps and towards inquiry-based instruction, research, and analysis and evaluation of information. Students don't need us in order to learn, they need us so that they learn how to learn. Nancy Minicozzi's (@coffeenancy) session on Imposter Sets gave me some great strategies for implementing critical thinking strategies in all subject areas. And Lisa Highfill's (@lhighfill) Hyperdocs session provided insight on how to develop our students into independent thinkers.

• We are indeed #bettertogether. The absolute best part of my time at CUE was being with my PLN in person! Face time with like-minded educators, with my tribe, passionate about changing the education system and innovating instruction, is priceless. It reminds me that I am not alone-- as Kid President (@iamkidpresident) said, we are a bunch of weirdos, and it was great to be with all those weirdos in real life! 

We are ready for #cue16 ! Question is, are they ready for us?! pic.twitter.com/sWfIoL5zDe

— Lindsey Blass (@LindseyBlass1) March 16, 2016

#tossachat in the house. #cue16 pic.twitter.com/Re5KiJ2oV5

— Jessica Vannasdall (@mrsvannasdall) March 17, 2016

#TOSAchat thank you @theedtechdoctor for the #irk pic.twitter.com/XCdhedJ7Wb

— Hans Tullmann  (@htullmann) March 19, 2016

Finally met my favorite Twitter math friend...yeeeesssss!!!! #cue16 success!! #elemmathchat @mathkaveli pic.twitter.com/B1XJRbpAYC

— Amanda Haughs (@MsHaughs) March 19, 2016

Breakfast with #TOSAChat at #Shermans before last day of #CUE16 #photobomb by @gochemonline! pic.twitter.com/E4Tg973TXs

— Hans Tullmann  (@htullmann) March 19, 2016