Yesterday, I was thinking about some new number routines for the 4th and 5th graders who will soon be in their fraction units. I wanted to find something that both grade levels could engage in so the teachers could try them out and have a common talking point when we met to discuss what we learned about student thinking. I tweeted it out and didn’t want to lose all of the great thoughts so I going to compile them here!

This was the image I created:

Before launching this with the students, I thought the teacher would establish that a full circle was 1 and ask the question we typically use with our dot images, which is

*How many dots do you see and how do you see them?*

I thought there could be a variety of responses, but I was most anticipating these two responses:

- I know two halves make 1 and so the first row is 3. There are 4 rows so 4 x 3 =12.
- I know two halves make 1, so the first column is 2. There are 6 columns so 6 x 2 =12.

I knew questioning would be important to move from those descriptions to what the equation would look like and imagined we would get to some like this:

- 6 x 1/2 x 4 = 12
- 4 x 1/2 x 6 = 12

**Other interesting things I am now thinking about thanks to others on Twitter:**

Why am I stuck on dots? Circles makes so much more sense!

Yes, why not leave it completely open,not tell them that circle is 1 and talk about the unit?

It is funny that the same image conjures up different student responses, I always love that! I had not anticipate half of an array. I also love the idea of messing around with quarters here!

Awesome to have them do some scaling of the dots to find how many are there? And then I can see them saying it is half of the array like Michael suggested once it is uncovered!

I like the idea of moving them around however the cutting and tearing thing never goes well in my room!! I do love changing the value of the whole a lot!

I debated this part a lot when I was making it! I thought for the 1st one, a whole number would be best and then move to non!

There is the running list so far! Add away in the comments!

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xiousgeonz… and ask if it matters which way the halves are facing…

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math on the edgeThis is a fantastic idea. I love the idea of trying to build toward the equations and showing equivalence. Will you ask the students to record their own thinking? I am thinking that the differences in how students see what they see will be really subtle. For instance, I can see a student adding the three groups of halves 3(1/2+1/2), but saying “I see three wholes in the top row and then 3,6,9,12” How would you record a statement like this with an equation? How will we know if they are thinking purely additively or multiplicatively? Can’t wait to hear how this goes.

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