How hard is it to make your own ruler?  

PROVOCATION:   We   estimated lengths – things that are about an inch, a foot, a yard, a mile. These are personal references, quick and relatively easy ways to estimate the lengths of objects by comparing them to other objects whose length is known (a paper clip is about an inch , an adult shoe is about a foot long).   In our math workshop conversations, the children said that a sheet of paper is about a foot long, so we asked them to try making their own twelve-inch rulers with a simple sheet of paper, using what they already know about inches ,   without looking at a “real” classroom ruler for guidance. They then set themselves to this task with thinking pens and blank sheets of copy paper (which were only 11 inches long, unbeknownst to them). We asked them to be sure to put their names on their rulers and number their tries.

DAY ONE: FREE-HAND ATTEMPTS AT MAKING A RULER (with 11 inch long paper)

Hailey kept making rulers that ended at 12 near the edge of the paper, but not all the way to the edge of the paper. Knowing that a ruler has to be 12 inches, she decided to simply cut off the bit of paper at the end to make it 12 inches [literally making a short cut]. Many children saw that their concept of inch kept shifting as they noticed the variety of inch lengths they were producing.  Even if they got all the way to twelve at the edge of their paper, they could see that some of their inches were bigger (or smaller) than others.

Some children tried using their digits – which they think are close to an inch – to try making their rulers, laying their fingers carefully across the page.  After several attempts to make a ruler, Gabriella and Juliet expressed ideas that related back to an understanding of standardization:

This initial process required a lot of patience and persistence. Several tries were made by each student as they strove to create a “more exact” ruler.  While other students seemed frustrated, Alexander kept making ruler after ruler without giving up.  “I don’t back down from a task that easily.  I’m on my fourth ruler now,” he calmly stated.  “You need to use one of your digits and line them up to make an inch.  For rulers #1 and #2, I used millimeters and inches.  Now I’m just doing inches.”  Alexander wasn’t the only child who was using and confusing metric and customary systems of measurement in these first tries, but as they kept working, these children came to this realization, on their own and in conversation with their classmates, and reverted to working solely in inches.

Is there flexibility in an inch?

In looking at the students’ ruler making, we noticed that many of them weren’t iterating units – perhaps they don’t quite believe the whole notion of iteration.  In teaching measurement, they have heard (and have remembered) over the years that there should be no gaps, no overlaps, but it appears that they haven’t been convinced of the standardization of unit length and iteration.  We assume they get it, but this experience is showing us otherwise – and that some of them are a bit confused about the two systems of measurement.  Metric and customary systems are both used for measuring the same things, so it’s not completely illogical for some children to try a “mash up” ruler as a way to understand equivalencies.

DAY TWO:  FREE-HAND ATTEMPTS AT MAKING A RULER (with 12 inch long paper)

On the second day, we revealed to the students that the paper they had used the day before was not a foot long – it was merely eleven inches long.  There were several jubilant cries of “I knew it!”  We then supplied them with 12 inch long paper to try again.

The paper we used today was exactly twelve inches long.  Did knowing that make creating your ruler any easier?

Reese: Yes, because I knew it would be [12 inches] all the way across

  Can you make your own inch? 

If we all understand something about “inchiness” and rulers,  why then is it so hard to draw an inch?

Why did I ask you to make rulers?

What happened each time you made another ruler?  Did it get better? 

Why do inches need to be the same size?

DAY THREE:  FREE-HAND ATTEMPTS AT MAKING A RULER:  FOLDING PAPER METHOD

Why might folding be a good strategy?

DAY FOUR:   FREE-HAND ATTEMPTS AT MAKING A RULER:  USING INCH TILES

The folding method the day before was getting the children closer to an understanding of iteration.  If each fold was an inch – or close to an inch – wide, then you could make twelve folds of the same size to develop a twelve inch ruler.  Anyone who has ever made an accordion fold with paper knows that this exercise involves a good deal of careful repetition.  On the fourth day of ruler making, we decided it was time to give the children inch tiles to help them move forward – with manipulatives that gave them a better semblance of “inchiness.”

Using the inch tiles, do you think creating a ruler will be easier?

Would it work if you just traced it?

With the inch tiles, several children “got it” on their first try – and we asked them to go back to their rulers to add in 1/2 and 1/4 marks.  Others still had difficulties.   Below:  even with the inch tiles, there was the challenge of figuring out how to keep them aligned to trace and record the measurements exactly.

Later, some intrepid souls tried their hand at making meters, even trickier as the units are smaller (centimeters and millimeters).

Robbie, thinking back to our study of Hindu myths and stories, remembered the tale of Ganesha, who was asked by the poet Vyasa to write the Mahabharata with his tusk, a story “so long that no man could ever write the whole thing – all the pens in the world would break when before it was done.”  Any attempt to make your own meter stick, Robbie said, “would take so long that all the pens in the world would break…even Ganesha’s tusk would break.”

_________________________________________________________

In processing this lesson with Sabot’s math coach Cat Henney, she directed me to a quote that is particularly apt:

“Motion is a resource for coordinating measure and generation of attributes of length, area, volume, and angle.”  [p. 38, Developing Essential Understanding of Geometry and Measurement, Grades 3-8 (2014)]

Regardless of the measurement tool, measurement implies motion – the physical act of taking something and measuring it quantitatively – and so this exploration speaks to the necessity of real physical exploration.   An inch is an inch is an inch…which seems irrefutable…but is it?  The social learning involved in making your own ruler reveals the subjective nature of such a task.  We agree that a foot is a foot, an inch is an inch, but everyone has very flexible ideas based on perceptual judgement.  Giving children the opportunity to use what they know in a tangible sense affords them the ability to grapple with measurement conventions in a deeper way.  A ruler assumes that you know the transitive property and is loaded with cultural conventions and assumptions.   A twelve-inch ruler is a a mutually agreed upon standard, but when you ask someone to draw a ruler, there is no real agreement about actual length (again, the flexibility of an inch).  The inch that I draw may not be the inch that you draw, or that someone else draws. The relativity embedded in measurement is so revealing.  It is all about perspective of length relative to individual understandings and experiences of length, which then shape our unique personal benchmarks.

That these children could use this experience to think not only about concepts of measurement but exactitude – the need for it and the inherent problems in achieving it – is quite astounding.   When you think and work spatially, there is a high level of discourse and engagement.  It’s all in the talking-it-out (and figuring-it-out) that these concepts come together.  Most children construct unit iteration born of transitive reasoning by fourth grade, but hopefully this purposeful exercise afforded them a “task rich” opportunity to progress to their next stage of understanding.

Striving for accuracy is one of the sixteen habits of mind that Sabot educators work to inculcate in our students, in all grade levels.  I asked the children to think about what this habit means to them:

So what does striving for accuracy mean?

The striving is in trying to improve your skills, inch by inch…and in so doing, finding new understandings…and questions.  

The post Inch By Inch appeared first on Sabot at Stony Point.