math · Math and Paper Folding

Piecing Together a Project, Over Land and Sea

Pieces of Paper

This story begins in a teachers’ lunchroom, a couple of years ago, in Upstate NY. I was sitting with some teachers when another member of the staff started talking to a first grade teacher, Mrs. K, about a new math mandate. It was something about using manipulatives to create a variety of shapes. I’m a bit foggy on this part but it seems to me that they were required to use rhombuses (or rhombi, both are correct) for their shape building.

A Rhombus
A Rhombus

Upon being told that she would have to incorporate these manipulatives into her math unit Mrs. K asked if there was any money in the budget for manipulatives. The answer was no.

After school I sought out Mrs. K and showed her some paper-folding and shape transformations that referenced rhombuses. This teacher seemed delighted with what I was showing her. I volunteered to send her something that I thought she might find useful, then went home and created these images for her, which are equilateral triangles that become a rhombus.

screen shot rhombi

I never asked Mrs.K if she used what I sent her. I recognize that what I sent was, unfortunately, not a project. Instead, it was just the bones, the beginning of a project that needed to be developed.  Every so often I’ve revisited these images, wondering what I could do with them. Then a few nights ago Malke, from Indiana, asked me about projects for a family night.

It was late, and we decided to resume the conversation the next day. The next morning, before Malke and I reconnected, I saw this post from Simon Gregg, in France:

I had an Aha! moment. It suddenly came together. I sent off this note to Simon:

Malke, who I included in the conversation, responded with a reference to a beautiful manipulative that I wasn’t familiar with, but which also showed that she immediately recognized what I was getting at with my DIY paper version of manipulatives.

Since Malke seemed to know exactly what I was thinking about I got to work creating the pieces for this activity. I’m pretty happy with how this has developed. It requires triangle paper, and matching paper shapes that can be printed on colorful papers. My thought is that simple, bold shapes can be created in sort of a free form way…

Big Shapes
Big Shapes

…or more challenging shapes can be drawn on to the paper…

Drawn Shapes
Drawn Shapes

…and filled in, while trying to make as few cuts as possible and being mindful about cutting along the lines defined by the triangles.

Filling in drawn shapes
Filling in drawn shapes

So, where can you get these papers to do a do-it-yourself shape building set? Right here. I’ve created a couple of PDF’s to get you started:

Triangle paper and Rhombuses

Make beautiful shapes. Send photos. Thank you.

Addendum: Take a look at Malke’s post on hands-on math: she collected and organized many interesting perspectives. It’s a fabulous piece of writing.

Addendum #2 (April 2016)  Malke liked working with smaller rhombuses so I made her this  PDF rhombi with spaces So far she is planning on using them without the triangle grid paper. Here’s a link to some images she’s created as samples for an upcoming project rhombusphotos

13 thoughts on “Piecing Together a Project, Over Land and Sea

  1. Kids will really like these! Printing the triangle or rhombus pdf’s on different colored paper will make it so easy, too.


  2. Paula, these are wonderful. Thank you so much for sharing and giving me plenty ideas to work with. I have one artistically inclined child and one who likes manipulating for the fun of it. I will post what we make. Are you interested in sharing a Pinterest board with these?


    1. Hi Sonya! so glad you like these. Are you going to be using lightweight or heavier weight paper with these? I’m not sure which is better. If you try both and have preference, let me know.
      I’m not sure what you mean about sharing a pinterest board…it’s something I haven’t done before….or maybe I have !? A pinterest board of my daughter’s shows up on with my boards, and neither of know how it got there. No matter what though, if you create a pinterest board I will include a link to it on this page.


      1. Paula, I will create a board when I have something to show for myself, and you will figure it out. It’s easy. Since I’m not really a social media person, I get it. I feel lost on Twitter.


      2. Seems to me that you’ve been getting the hang of twitter. To me, the best part of it is finding a way of engaging with kindred spirits, exactly exactly like what’s you’ve been part of. We can engage with each other, encourage each other, and share in the evolution of our thoughts and ideas.

        Liked by 1 person

  3. Twitter is one of the great finds of 2016. I am a little late to the party. I have found so much on there and I’ve been blessed by all the encouragement. A bunch of math nerds. I have to keep all this somewhere or I lose it. Pinterest has been the best place. I use it for inspiration. Thanks again and can’t wait to show you what we come up with.


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