Foundations of Fractions Online Course

Over the past 2 weeks, I’ve released a series of fraction videos in collaboration with my good friend Christina Tondevold (@BuildMathMinds). Each video addresses some of the ways I tackle fractions in an elementary classroom.

In the series, I shared some fraction subitizing cards, a couple new 3-act tasks you won’t find on this site, in addition to some student work. The site activity and participation in the comments only emphasize the need for quality fraction resources, as well as our desire to grow and learn together.

Over the past 3 years, a large part of my work has centered around math progressions and the way student thinking grows and develops over time. Building content knowledge is a huge undertaking in our K-5 world and many times, it’s more work to be away from class than it is to be present. It’s for this reason that I’ve created a Foundation of Fractions online course as a follow-up.

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The course will be broken down into 6 weeks:

  • Week 1: Meaning of a Fraction
  • Week 2: Equivalence and Comparison
  • Week 3: Addition and Subtraction
  • Week 4: Multiplication of Fractions
  • Week 5: Fraction Division
  • Week 6: Rate

Each week will feature 3 main components:

  • Launch the Concept: We’ll work through a 3-act task. We’ll also anticipate and analyze student work samples within the progression of learning. The analysis of student work will set up the learning that’s about to take place.
  • Connecting & Extending our Thinking: We’ll engage in meaningful tasks supported by an instructional video that models how I would teach the lesson. Through modeling the task, teachers will understand how to harness these tasks and activities to move students through the progression of learning.
  • Building Fluency: Once a conceptual understanding is developed, it’s critical that students build fluency. It’s here where we’ll engage in meaningful practice that builds and supports automaticity.

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    Blank Fraction Tiles from ETA

If you want to know more about the course be sure to check out our registration page. The 6-week course begins October 6th which leaves a small window to register.

Registration closes October 4th and there is an amazing early bird special courtesy of ETA Hand2Mind…6 full sets of unlabeled fractions tiles.

Looking forward to learning and growing alongside you and hope to see you soon.

 

Graham

 

Posted in Against the Norm | Leave a comment

Reasoning with Fractions Through the Lens of a 10 Yr Old

My daughter has laid down a marker when it comes to reasoning with fractions. Well, at least in my small world. She’s 10.

She won’t read this post for a long time because her mom and I know what she reads on the internet…at least we hope we do. Maybe she’ll never read it but I know she has great things in store and that makes us proud.  There might be a day or time when she feels defeated or maybe doesn’t feel like she has anything to contribute.  If that’s the case, I’m keeping this nugget in my back pocket.

In my 14 years of playing math with students AND teachers, she did something that teachers don’t teach, textbooks don’t show, and test prep academies might never understand. We were working on a task which led us to a parallel question.  I want to share the sequence of events that took place.

We started here…

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Solve the problem BEFORE pressing play and think about how you solved it.

What the hell just happened?!?  I get it… but what the hell just happened?

When she said “I don’t like multiplying fractions” I immediately thought she was going to jump to some trick or algorithm she’d been taught in 4th grade. I wanted to know more so I asked her to explain.

Partial products for multiplying fractions! I’ve never seen a student or teacher use this strategy and to be completely honest, I’ve never thought of it myself. But it’s beautiful and makes perfect sense.

Lesson 1: Don’t let anyone tell you that you have to multiply straight across when multiplying with fractions. Partial products work just as well.

With this being a student invented algorithm I wanted to try and break it. It’s what we do.  So I changed the 6 to a 7 because it isn’t so tidy when divided in half.

Another little gem of understanding was uncovered. Any predictions?

Silly me. It’s a student-invented algorithm so she owns the understanding.  Everything she’s doing here has me vigorously nodding my head in agreeance.

Lesson #2: Don’t let anyone tell you that you need a common denominator to add fractions. Partial sums work just as well.

Let’s change the other factor and see how you tackle it.

She didn’t do 60 questions in a minute. In fact, she never has because that’s not our jam. I don’t think anyone would argue that our daughter isn’t fluent.

Lesson #3: Don’t let anyone tell you fluency = speed.

If the definition of fluency asks students be efficient, flexible, and accurate she nailed it. So why is the definition of fluency different when we talk about addition and subtraction or multiplication and division of single digits? The same way our daughter decomposed numbers and used distributive property with fractions is the exact same way she learned her basic facts. Her understanding of number is scalable.

We started slow and took our time when she was in kindergarten because the turtle always wins the race. As our good friend Tracy Zager says, “The ROOTS of the work are in K. The FRUITS of the work are in 5th”.

Lesson #4: Memorization of facts and algorithms is a learning objective stopgap. It will never prevail in the long run.

Our work is paying off and I’m a #prouddad.

Posted in Against the Norm | 18 Comments

Behind the Scenes: The Creation of a Progression Video

In recent months, I’ve received a lot of questions asking how I create the progression videos. Here’s how it all goes down.

Step 1: Research

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There are so many great resources available for early number concepts but I continually found myself coming back to Clements and Sarama for this progression. Their research is highlighted throughout Teaching Student-Centered Mathematics (PreK-2).  

I also leaned on the work of Cathy Fosnot and Kathy Richardson.  Every time I revisit their work I’m reminded how much room for growth I have as an educator. They keep me hungry.

Step 2: Sketch a draft progression

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Step 3: Set the stage 

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Lay out the manipulatives in order.

 

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Adjust the high-tech overhead video capturing hardware system.

 

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Roll out the butcher paper.

Step 4: Go Time

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It never turns out right the first time…or the fifth time for that matter. The toughest part is that it all has to be done right in one shot.

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 I jam out to music while working on the videos.

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Step 5: Video Editing

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I mute all the sound and speed up the entire video. I’ll slow roll small pieces where I need to verbally flesh out more understanding. The turtle and rabbit show where I played with speed.

I use Apple Motion for most of my video editing but for the progression videos, iMovie does the trick.

Step 6: The voiceover

I’ll highlight one or two things on index cards and let the rest flow. The toughest part is always the first 15 seconds.  That usually gets a full card so I don’t sound like a blundering fool. Although some may say that I still do.

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Once the notes are scribbled, I open up Quicktime, start a new screen recording, play the video full screen, and do the voiceover.

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Step 7: Publish and share

Here’s the published version of the Progression of Early Number and Counting. If you or your students give this a try I’d love to check it out.  If you have any questions just let me know.  All of us are smarter than one of us.

Cheers!

Posted in Against the Norm | 10 Comments

The Progression of Early Number and Counting

If you’re not a pre-k, kindergarten, or 1st-grade teacher, you need to find one and give them a hug after watching this video.  They do the work of an army and many times their work goes unnoticed. There’s so much happening in the early years of school, that without this progression of early number and counting, we’d all be out of a job.

Here’s the 5th installment in the Making Sense Series. If you’re looking for other progression videos you can find them here.

Stay thirsty my friends!

Posted in Against the Norm, counting, K-2, Making Math Accessible, Making Sense Series, Math Progressions, Number Sense, Strategy Development, Teacher Content | 15 Comments

3-Act Task: A 5th-grade lesson captured

A while back I shared a kindergarten lesson and I was really happy with the way it turned out.  The 5th-grade lesson below, not so much.  The students did great but there are definitely some things I need to improve.

We recently finished up a district PL where we used The Apple and we decided it was a great place to launch our upcoming unit on fractions. Last year, we started using 3-acts at the beginning of our units because they help identify what our students know and don’t know.  As a formative assessment tool, they help unveil the misconceptions we’ll need to address in the upcoming weeks.

In the spirit of vulnerability and #ObserveMe, I’m sharing this 5th-grade lesson.  The lesson was taught in January, which means the majority of the students haven’t explored fractions in almost a year. Please share any feedback or questions you might have in the comment section below.

What went well? How can I improve?


Act 1 & 2

 

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Here’s how some students solved:Screen Shot 2017-01-23 at 1.54.48 PM.png

Some students drew models…

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….and we hit some bumps in the road.

 

Some students used repeated addition…FullSizeRender 2.jpg

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Lots of misconceptions began to surface…

The follow-up visit with table #4…


 

Act 3-Reveal

 

Posted in 3-5, 3-Act Tasks, 6-8, Estimation, Fractions, Intellectual Need, Math Progressions, Modeling, Teacher Content | 12 Comments

This Week a Webinar. Next Month a Workshop

On Monday night I had the pleasure of presenting the webinar 3 Act Math Tasks: What They Are & Why You Need Them in Your Class.  The webinar was hosted by my good friend Christina Tondevold and focused on the implementation of 3-Act Tasks in the elementary grades.

Christina is doing some pretty amazing things within her online community, so I was more than honored when asked me to present.



Next month I’ll be presenting a 2-day workshop in Anaheim, Califonia hosted by Grassroots Workshops. The workshop will take place January 25-26 and is open to all K-5 teachers, coaches, and administrators.

Over the course of 2 days, we’ll examine the progressions of learning through the lens of 3-act tasks and other meaningful activities. For more information check out the video below or the workshop landing page at Grassroots Workshops.

There’s one more day until the holiday break and maybe this could be a learning gift from your administrator. There will be lots of takeaways which will make our time together, the gift that keeps on giving.

 

 

 

Posted in Against the Norm | 4 Comments

The Progression of Fractions

I’m excited to share the 4th installment of the Making Sense Series which explores meaning, equivalence, and comparison of fractions.

Fractions are the gatekeeper of algebraic thinking and probably a big reason why we suffer from arithmophobia as a society.  I’m hoping this progression helps provide some relief and courage moving forward.  Let’s make sense of fractions together.

Happy viewing and stay thirsty.

Posted in Against the Norm | 24 Comments

I’ll Rip Your Face Off: The Art of Defacing Manipulatives

It’s our fault. We have no one to blame but ourselves.

We unknowing pigeonhole student thinking with the manipulatives we use. Take fraction tiles for example. Much to my disappointment, they come with labels and it kills me.

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Manipulatives that come pre-labelled ruin everything I want from a lesson. Sure you can flip them over but the label on the backside keeps rearing its ugly face and traps lots of student thinking.

Sure there’s Cuisneaire Rods but most teachers don’t have $200 to fork out for a class set. But I think it’s fair to say that most teachers would fork out $4 for some fine steel wool.

Presto! Fraction-Cuisen-Part-Whole-Tiles!

As I finish up planning for my Grassroots Workshop in Anaheim next month, I can’t help but think how faceless manipulatives help us guide students through the progression of learning because of how they can be flexibly used.

When we label items we avoid lots of opportunities to listen and build on student intuition. This was something I took away from Tracy’s most recent post. Tracy helped me see that I need to provide students with more opportunities to play and explore…WITHOUT INTERFERING.

I think this gives them a much better chance.

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What the value of the orange? It sure isn’t a third.

With that being said, even when we do get our hands on unlabelled manipulatives we usually assign the same value to each piece…every time.

Pattern blocks are a perfect example. Most of the time we assign the hexagon a value of a whole. This creates a false sense of understanding which is really hard to unmask.

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Where’s my head at right now?

I’m continually seeking ways to undo student learning and identify what understanding they truly own. In order to do that, I need to be sure I’m not “pigeon-wholing” student thinking.

Question: Where else in mathematics do we pigeonhole student thinking? This can be within our instruction OR through the use of manipulatives.  

Please share your thoughts below.

Posted in 3-5, 6-8, Against the Norm, Fractions, Making Math Accessible, Making Sense Series, Math Progressions, Math Tools, Strategy Development, Teacher Content | 24 Comments

Where’s Poly? An Exploration in Geo-Dotting

What’s geo-dotting?  I have no clue but that’s what I’m calling this lesson.

We started by asking, “What do you notice?”

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Our favorites:

  • Looks like Pac-man
  • I see dots and they make a “Y”
  • Looks like someone went crazy with a hole punch

We needed to wrangle in student thinking a bit so we gave them some information…

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Unanimous vote. “I see a square and a triangle.”

We asked students to explain their reasoning and one said:

I know there are 7 corners, I mean “vertexeses”, and 4 of them make up a square which leaves 3. I can’t make a shape with less than 3 dots because then it’s not a shape. So the only shape I can make with 3 dots is a triangle.

We have a winner…

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Now that students had the hang of it, we went here next…

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What do you notice?

We let them play, talk, and share for a couple minutes and triangles seemed to be the shape of choice.  Then we revealed the mystery polygons.

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By now we felt students were ready to tackle the opening slide again.

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On our second time around there was no Pac-man or letters, only shapes.  But this time instead of just talking about the dots, students were encouraged to put their thinking on paper.

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Students used only the top three boxes for about 5 minutes. This allowed them to flush out each other’s misconceptions.

This helped students construct their own understanding.

After about 5 minutes we slow-released the following criteria, giving them one new nugget every 3 minutes:

  • Total of 5 shapes
  • No dots left over and each dot can only serve as 1 vertex for 1 shape
  • Shapes can overlap
  • Only 2 triangles
  • One square and one rectangle
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Students compared work to ensure the criteria was met.  “Looks like you have 2 rectangles in the bottom corner. Try again.”

As we wrapped things up, students came to the board and shared their solutions.

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My takeaways:

  • Talking about the shapes and their properties before moving to paper really allowed for students to engage in SMP#3 once we made the leap.
  • The slow release of information allowed students the opportunity to build problem-solving stamina.

If you want to give the lesson a try here’s the slides in a pdf file and student work mat. Please report back and let us know how it goes.  I’m wondering what takeaways you can share.

Posted in Geometry, Making Math Accessible, Who Knows? | 14 Comments

GCTM 2016 – Ignite Talks

For the second year running, we tackled Ignite Talks at Georgia’s Math Conference.

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Mike Wiernicki


Katie Breedlove


Jenise Sexton


Karla Cwetna


Carla Bidwell


Brian Lack

 

 

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Posted in Against the Norm | 3 Comments