Fitting It Together: Another Take on Number Visuals

A visual number pattern created by artist and scientist Stephen Von Worley forms the backbone of day two of youcubed.org's week of inspirational math.   

The lesson envisioned by the youcubed team is here and starts with a video of Jo Boaler talking about the importance of "brain crossing" - connecting visuals with symbols - followed by exploration of the number pattern, finishing with summarizing and sharing out.  The whole shebang in one day (possibly followed with an exploration of consecutive number patterns).  Whew!  

I had the opportunity to test this out with various classes ranging in age from 3rd grade to 9th grade, and some adventurous 3rd grade teacher colleagues of mine tried it out with their classes, too.  Low floor high ceiling problems like this have great potential to encourage creativity, build stamina, make and test conjectures, and apply and connect math concepts.  Although we found the youcubed lesson fun, and yes, inspiring, it felt a bit... unwieldy and unfocused.  

We found ourselves asking: How could we use it to hone our conceptual goals?  In this blog post, I am putting forth a possible answer to this question with this overall plan:

An overall instructional plan for Tier 1.  From beginning to end this could span about 2 weeks (more or less depending on how solid and flexible the students' grip on additive reasoning is). 

Essential Question  

We ground this low floor, high ceiling exploration with our first WCSU Non-negotiable Skill for 3rd grade which is understanding multiplication as repeated addition.  An essential question we can address is: What is multiplication and how is it related to addition?

This question can be addressed in an exit card or journal entry on day 1.  I would post it in the classroom and revisited it each day.

Skill Building

One of our prerequisites for this concept is fluency with addition and subtraction of numbers to 20.  We know that some of our students have gaps in number concept (including subitizing, making 10, additive fluency to 20, etc).  To address these gaps, we can spend ten minutes each day on building these skills.  

Spending extra time on day 1 (before launching Number Visuals) on an extended number talk on one of the number visual patterns would help to "prime the pump" making the loose objective of "exploring patterns" more concrete.  For example, take the visual of 9 pictured at right, flash it for 1-2 seconds.  Have students use Cuisenaire Rods to record what they saw.  Record all of the different ways that students saw this quantity.  Get them to talk about grouping patterns.

• For more information on how to run a number talk check out this blog post.  
• Here is a file containing larger versions of a few of the number visuals that could be used through this 

I would use this opportunity to address some language such as: cluster, group, sum, addend, multiple (in the intuitive rather than mathematical sense), repeated, double, triple.  I would also use this opportunity to develop the concept of equivalence.

For example, write 3+3+3 = 4+2+2+1 and ask: Are we saying that 3 plus 3 plus 3 is equivalent to 4 plus 2 plus 2 plus 1?  How do we know that?  How can we show that? 

A great formative assessment would be to present students with the symbol "=" and ask them to explain their understanding of it.  I did this with a second grade class recently, and we discovered all kinds of misconceptions (i.e. "it shows you that the answer is the number to the right") that made equivalence hard to pin down.

Launching Number Visuals

It's definitely worth having students watch Jo Boaler's "brain crossing" video first.  This will validate the number talk you had and emphasize the importance of multiple representations and modalities.

As you move into the launch, emphasize these norms in particular:

• Math is about connecting and communicating.
• Math is about creativity and making sense.
• Depth is much more important than speed.

Rather than simply providing students with this sheet containing all of the number visuals at once in "order,"  I would cut the visuals up beforehand and begin by presenting students with a shuffled stack of visuals for the numbers 1 through 21 and ask them:

• What patterns do you see?
• How could you sort them?
• How could you order them?

Here's a link to a sheet that can more easily be printed up on card stock and cut out.  Having the visuals separated encourages students to consider how to order and group them.  These are valuable discussions that don't happen as organically when we give them the patterns statically arranged in order.

A 3rd grade student sorts the visuals.

After students have a chance to get acquainted with the patterns, I would introduce using the number visuals as dominoes.  In order to put two visuals adjacent to one another, they must match either the overall number of groups or the number in each group.  This is an idea that my colleague Mary (who teaches third grade) found at this website.  Mary noticed that it was when students started looking at the number visuals this way that they started to really see the patterns.

I would withhold the visuals for numbers 22 through 28 and feed them to the students one by one.  As a high ceiling question, ask: What would the next number look like?  Students can:

• Make a conjecture.  Predict what it will look like and why.
• Compare their prediction to the actual.
• Think about why the pattern is what it is.

Using the Number Visuals to Address and Assess the Essential Question

I would thread work on this pattern throughout conceptual work on repeated addition.  There are many different places that students could take this pattern, but eventually, I would want them to be able to recognize that the patterns can be represented as repeated addition which is equivalent to multiplication and "counting by" a number and can be recorded using an open number line and both addition and multiplication expressions.

I would be sure to check my students' flexibility with the idea of equivalence.  So, for example, given the visual for 9:

• How can this represent repeated addition? 3+3+3
• How can this represent multiplication? 3x3
• How can it represent both operations at once? 3x2+3 or 3+3x2 or 3x1+3x2, etc...

The third bullet might feel like overkill.  However, without this flexibility, students will be missing the number sense that they need to develop fluency with multiplication which requires them to decompose and recompose products.

You may be asking yourself, Does that mean we need to teach third graders the order of operations?  To that I would say, if we teach our students about equivalence and the multiplicative hierarchy of our number system from the very beginning, we don't ever have to cram "order of operations" down their throats as some enigmatic new procedure that is useful for simplifying convoluted numerical expressions that are not grounded in reality.

Okay, I will step down from my soapbox now...  

Is there one right way to use this task?

The beauty of a low floor, high ceiling task is that it could be applied so many different ways to different grade levels.  I'm not implying in this post that the number visuals must be a third grade task nor that another task couldn't be used to ground repeated addition.  All I am presenting are ideas and thoughts based on reflection on experiences.  This task would work fine for introducing division, factors and multiples (GCF and LCM), prime factorization, fractions as a subset of a group, and so on.

I do want to thank my colleagues who are willing to share their experiences (both good and bad), collaborate and take chances.  I learn so much with you and from you.  Teaching is such an adventure!  I'm so glad we're in it together...