Arts in Education · Math and Book Arts · Math and Paper Folding · Math with Art Supplies

Peek-a-Boo Skip Counting for First-Graders

Peek-a-boo skip counting
What number is under the heart?

For weeks I’ve been burning through piles of papers and ideas trying to work out an engaging skip-counting project to make as part of a math-activities folder for first graders. Having just done a math activities folder with kindergarteners, which went really well, I’ve been wanting to do something similar for first graders. As I’ve also been doing math-with-art-supplies bookmaking projects with second graders, I’ve been keen to design something for the next grade up.

What I’ve  needed to get me going on this is a school to want me to create a project for them. A couple of weeks ago, late in the season, a school called me, asked if I had any time for them, and we struck a deal. We’re doing the project that I’ve been wanting to create.

There will be four hands-on projects in a folder that the students will be making. This post is about just one of the projects, one that supports skip counting, reasoning, and attention to numerical patterns.

Show me what under the butterfly, Oh, it's a 14.
Show me what’s under the butterfly, Oh, it’s a 14. Still not sure what’s under the heart.

Skip counting is a big deal in first grade. Not only does it set the stage to understand multiplication, it also is helps with learning to count money.

My work with second graders has piqued my intereste in skip counting. The projects we’ve been doing, which is making designs with “coins” that add up to $1.00, has been interesting in that I’ve noticed that even though a student can count by fives, you know, 5, 10, 15, 20, 25, 30, 35, 40….,, they have a really hard time doing this same counting by 5’s when you ask them to start at any number other than zero. So, if they have 25 cents plus two nickels they are at a loss as to how to proceed.

Show me what's under the flower. Oh, it a 6, Now can you tell me what's under the heart?
Show me what’s under the flower. Oh, it a 6, Now can you tell me what’s under the heart?

 

Maybe by now you’ve guess what is under the heart in the photo above. Maybe not. If you need more hints, I can reveal that there is an 8 under the star. This will likely finally be enough for you know know that there’s a 10 under the heart.

We’re not just counting by twos here. I’ve made a paper that slides under the windows that helps with counting by 2’s, 5’s, and 10’s.

I consider this to be an elegant design. One piece of folded paper for the holder, with a one piece of paper for four different number series. The little designs on the peek-a-boo doors are cut with paper punches, which I’ve collected over the years. The rhombus shaped window are made by folding the paper and cutting triangles on the fold.

One of my thoughts with this project is that  it can support students in practicing with going both forward and backwards with their skip counting. For instance, if they see two numbers, say 80 and 85, can they tell me the number that is before the 80 and after the 85? This takes some practice, some thinking, and reasoning, but if they can figure out what number is behind the hidden door, I anticipate the pleasure at solving this puzzle will delight them when the peek-a-boo door reveals the answer.

I do plan to share the template for this after I try it out with some real live first graders. To be continued.

Addendum 6/11/2019

Two classes of first graders made this with me. It went really well! To teach them to use it, I do the demonstration on the board, drawing doors that they could “look” behind for clues.

Here’s a video of what playing with this looked like:

Here’s a template so you can make these yourselves: skip counting first grade

Art and Math · folding · geometry and paperfolding · Math and Paper Folding

Constructing a Shape then Admiring It

Paper folding in the morning
Paper folding in the morning

Yesterday I watched a video that showed the Lawler family looking at shapes.

https://mikesmathpage.wordpress.com/2017/01/13/exploring-3-intersecting-cylinders-with-3d-printing/
https://mikesmathpage.wordpress.com/2017/01/13/exploring-3-intersecting-cylinders-with-3d-printing/

One of Mike’s sons said he liked the top shape in the image above. You can’t see from the photo, but it’s a full sphere. The image above is only half of the sphere, the other half looks no different than the half that is showing. I’ve played around with constructing foldable versions of shapes that look something like the one above, and I thought I’d be able to make a foldable version of what was on Mike’s screen.

Template for folding and cutting
Template for folding and cutting

I’m not showing all the steps that led to this map of folding and cutting because what I’m most interested in showing here are the wonderful visuals I got to experience along the way of creating the final structure.

Folding and Cuttin
Folding and Cutting

Silly as it may seem, one of my first realizations was that the indoor, nighttime lighting in my workspace was just all wrong for photographing what I was about to fold. Morning light would be best. So I went to bed.

Assembling in the morning light
Assembling in the morning light

Of course I forgot to recharge the battery of my phone camera before I went to bed, so I didn’t get the earliest light.

wp-1484487824204.jpg

Hmm, I don’t really want to say much more about this process. I’m just going to post pictures now. (Haven’t had my coffee yet.)

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wp-1484487640608.jpg

img_20170115_082242.jpg

img_20170115_082605.jpg

zzzz

Ok. Time for coffee. Am heading to Rochester today to bring my daughter back to college. Will be thinking about all shapes that this structure made. (Which reminds me of a question someone once asked me, “What, do you just sit around thinking about folding paper?” Well, yeah.)

math · Math and Paper Folding

Piecing Together a Project, Over Land and Sea

Pieces of Paper
Pieces

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.  http://mathinyourfeet.blogspot.com/2015/04/some-thoughts-on-hands-on-math-learning.html

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 https://www.facebook.com/MathInYourFeet/ rhombusphotos

Math and Book Arts · Math and Paper Folding · Uncategorized

A Fractions Number Line Project for Fourth Graders

A Fractions Book

I’ve finished up this fractions/number-line project that I’ve been thinking about. I worked with a class of fourth graders who were just starting their fractions unit. My plan from the start was to try to present a project that was dynamic enough to capture their interest. The center piece of the project was to make a “magic wallet,” which is a shortened variation of a Jacob’s Ladder. I’ve been using the name “Li’l Jacob” instead of “magic wallet” because, originally, I couldn’t remember the magic-wallet name, and Li’l Jacob seems properly descriptive.  This structure opens in two different ways, to reveal two different visuals. It’s tricky, and seems magical. I am happy to report that these students were over-the-moon happy to learn how make this.

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The students told me that the first step of arrangement of papers reminded them of a person

After showing the students what the finished book project would look like we dove right into making the L’il Jacobs. Making this requires a completely non-intuitive sequence of precise folding and gluing. The students have to keep track of where they are in the sequence in order to get the folding to work. I was nervous about how I could get them to see for themselves what was going on. A great surprise was that they offered me the best description I could hope for: they saw the arrangement of papers and immediately recognized it a human figure, legs and torso. Perfect! Now I was completely convinced that I would continue calling this a Little Jacob.

???????????????????????????????The students decided that this next step was best described by saying that they were covering Jacob’s innards.

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As the sequence of folding continues, the Jacob becomes smaller. (Notice the paper that student is using  to protect the desk from getting mucked up with glue.)

Jacob squared
Li”l Jacob squared

The last fold reduces the paper into a square.

Each student made four Li’l Jacobs. Each of these had a set of equivalent fractions written on and in it. But we didn’t even start with the fraction labeling until the third class. Our second class was about making the book that was going to hold our fraction cards.

We folded a 33″ x 4.5″ paper int halves, then quarters, then eighths, to make an eight page accordion. I know that most people don’t have access to paper this size, but with a some thought this can be created by combining smaller sheets of paper.

Pockets
Origami Pockets on every other page. Student chose their own colors.

 

Students then made origami pockets out of 5.5″ squares of paper. Starting with the second page, these were glued on to every other page of the book.

zero page
Creating a cover with a fold-in, to extend the number-line to inside

Next came the cover. The book needed an extension  so that the number line could start at zero. To accomplish this we attached an extra long cover piece then folded it over. I know I’ve explained that badly, so I hope the pictures above are adequate explanation.

fractions-Zero-to-OneFinally we were ready to label the Little Jacobs with equivalent fractions. I talked to students about how fractions could be a way of counting to one: one-fourth, two-fourths, three-fourths, four-fourths(one). I showed them my animated zero-to-one gif and some static images of equivalent fractions  but they seems to like the image above the best. We circled the columns of fractions equivalent to eighths. The really seemed to get the concept, and kept referring to it as they did their labeling.

four

The picture above is my sample that shows the labeling, with different ways to write equivalent fractions, as well as a simple addition problem, using the fractions.

labels

Here are some images of the students finished books.

https://bookzoompa.files.wordpress.com/2015/02/fractions-zero-to-one1.pdf

The even pages hold the fractions in the pockets.

odd pages

On the odd pages, students wrote out the fractions. These fractions had no equivalents on our chart. Forth graders don’t do fractions beyond the twelfths, so 1/8, 3/8. 5/8, and 7/8 stand alone.

K's book Some students were more inclined than other to do decorations on the books.

k2

One thing that was wonderful about this class was that the students were incredibly helpful to each other. I could have never gotten this far with this project if I had to problem shoot with each child individually. The students who grasped each step were enthusiastic about working with a classmate that didn’t quite get a step.

big line

 

We lined up the fractions so the eighths showed, thus showing the 1, 2, 3, 4, 5, 6, 7 and 8 in the numerator. By the end of my third meeting with these students, just about everyone had finished with their books. This is one class that I can say is excited about equivalent fractions.