A Way In

February 27, 2015

metal doll 2

Metal Doll from Mexico


I used to wonder why it was that students who traveled more and did more things seemed to learn better than students whose worlds were more limited. Intuitively this seemed correct to me, but for a long time I carried around the question about why this was. Finally, there was moment that I realized that knowledge needs an entry point, and that children whose had more experiences have more ways to relate to, and thus integrate, new information.

 Rasberry Leaf, inked and pressed by hand

raspberry Leaf, inked and pressed by hand

I not only do hands-on work with students, but I also believe in the value of it. Lately one of my projects has been to develop flip books to animate math concepts to enhance student understanding. There are numerous absolutely outstanding on-line technologies that can do what I am laboring over, but what they are doing happens only in the virtual world. I am using the tools available to me through technology to design interactive books that are completely off-line and non-electronic. One question has popped up repeatedly: is there an advantage of creating a manipulative when such excellent 0n-line resources exist?

pencil drawing by Paula Beardell Krieg

Unripe Tomato

The short answer is that on-line resources are richer in information than any book I can make. Each book I make illustrates just one concept, whereas any on-line resource can illustrate what looks like an infinite number of concepts. An on-line resource doesn’t require extra purchases, it doesn’t get lost, it disappears once you click off of it so won’t clutter up your house. It won’t get stolen. Actually, books about math generally don’t get stolen either, so maybe I should leave it out of this list.

Hand and Friend, pencil drawing by Paula Beardell Krieg

Hand and Friend, pencil drawing

We’re always looking to relate what’s inside of us to what is outside of us. So, when I make a book that illustrates an idea in math that can be experienced visually, but that can also be picked up and flipped through, then stopped. then made to go forwards and back, when I hear the click of it in my hands, feel it’s weight, well, then, somehow, it begins to feel like mine.  Now, here’s a GIF of my most recent set of pages for a flip book. If it’s not animating, click on it.

equations of a Line, changes in B

equations of a Line, changes in B

I really like the image that happens with the changes in this equation, especially when I let them stack on top of each other. But if I want to share this with someone who has less than a positive history with numbers and equations, I have to deliver it in a way that they will accept it. Opening my computer and opening a graphing program is just not going to do it.

At this point, at gatherings of friends, I’ve pulled out some of these little books. People become engaged immediately, they ask questions and they want to share them with their children. One friend of mine said he never understood the equation of the line until he looked at my little books. Graphing equations always seemed so random to him. This understanding gives us one more entry point into information and deciphering relationships between things, which can feel really good. Then, maybe after this world of numbers feels better, it’s more possible, then, to explore things beyond the few little books that I create. But to get there, it’s good to have a friendly way in.

Quarry Road

An Invitation to Enter


Li’l Jacob

February 21, 2015


How to make a Li'l Jacob

Li’l Jacob

 In a previous post I had written about a structure that I am planning to teach to a group of fourth graders next month.  There’s a fun folding trick to it that feels like magic: its two panels flex in such a way that allows the front and back covers to reverse positions, and it offers two different configurations of the inside spread.  I’ve put together the instruction sheet above for anyone who feels up for something completely different. I think that the instruction page above will print just fine, but, just in case, here’s the Li’l Jacob PDF 

Now for my dire warnings.

If you attempt to make this or teach this you will be unsuccessful unless you release any attachment to thinking for yourself. Follow the directions precisely. Chances are you will not heed this warning until you have botched up the first few tries of making this. Oh well. I tried to warn you.little-3-6

Any students you work with should also be encouraged to work along with you lest their attempts fail. Precocious student will anticipate the next step, thinking that they see a pattern in the steps. These students will likely find themselves with a structure that is glued shut. I have a special name for this kind of mistake. I call it a bad Christmas present: it looks interesting from the outside, but it just won’t open. Can be funny, but ultimately it’s not something you want.


I did not invent this structure. I don’t know the history of it, or what it’s called. I have seen it used a novelty gag, something that will hide or reveal a dollar bill. If someone knows its official name, I would be grateful to hear it. Only recently, when I had been exploring the folk toy known as the Jacob’s Ladder, did I realize that this structure is a shortened version of the Jacob’s Ladder. It seems fitting, therefore, for me to call it Li’l Jacob. I really like this name.


Let me mention too that there is nothing special about the dimensions that I present in my instructions.  The squares can be rectangles. The strips of paper can be all the same width, or all different widths. The dimensions that I illustrated will be the ones that I will be using when teaching, so that’s what I based the tutorial on.

I’m looking forward to seeing how students react to this odd folding activity.  If you try it out please please please let me know how it goes.




Valentine Fig by Paula Beardell Krieg

Consider yourself hugged

Wishing you a good day.


I’m in my visiting-artist teaching season right now. This year I have some breaks between my classroom visits. This is unusual, but I am taking advantage of having more time to design some new projects. There’s a fourth grade teacher that I have worked with many times already, but I have yet to feel like I’ve developed just the right project to bring to his classroom. I’ve been playing with something completely different this year. I’m not sure that I can teach this to fourth graders, so, as is my way when I am nervous about something, I completely over-prepare.


The structure I am planning to show is tricky. Do you see what is going on here? Folded one way, the word Magnanimous in on the right, fold it the other way, it’s on the left. The back of the card looks like this:


But when opened in the other direction, the back of the card shows a completely different image:



It’s tricky, and interactive. It feels a bit magical, which I think will appeal to this group of fourth graders.  However, we’re not going to be using this for vocabulary words. My plan is that are going to work with  fractions, with an emphasis on equivalent fractions. I’m not sure how far along the class will be on their fractions unit so I’ve been making graphics like crazy, showing sets of fractions that I think the Common Core is looking for at this grade level.

fractions-Zero-to-OneIt’s tough to find the equivalents with the chart above, so I made three more graphics that pull out the patterns that I want the students to work with. Here’s one of the charts:

Halve, fourths, Eights and twelfths by Paula B Krieg

A chart to help visualize equivalent fractions


If you would like the complete set that I made, here’s the link to the Fractions PDF Fractions Zero to One. While working on this project it occurred to me that while studying fractions these students are doing something they have never done before, which is to think about numbers between zero and one. Up until this point the emphasis in school is on bigger and bigger numbers. This is their first U-turn into smaller numbers. Correct me if I’m wrong, but it seems to me that the studying of fractions could use a bit of levity. A what could be more flippant than a Fractions Gif? Here’s what I came up with (chances are you will have to click on the image to get it to animate. Please do.)

Fractions Gif by Paula Beardell Krieg

Fractions Gif…Click to animate

The plan is to draw the students in with playful, colorful graphic, then have them make a series of little oddly constructed cards that collect equivalent fraction groups, then make a long, pocketed accordion to store them in.

I let you know how it goes.

In the meantime, I will work on a tutorial page on how to make the cards.  See you then.


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