Making Equilateral Triangles Outdoors

On Friday morning, our class bundled up and headed out to the outdoor classroom for math.  Cat was with us that morning as we continued our investigation of geometry and measurement.  The first challenge was to break into groups of three and then create an equilateral triangle using only yarn, scissors and their bodies.  A few students asked for tape measures, but the challenge was to construct the shape without a standard measurement tool.  

So why take this outdoors?

This is certainly an activity that could be done indoors, but this group has demonstrated that the level of ingenuity, problem-solving and cooperation is greater in our outdoor classroom.

So how did they solve the problem?

I noticed one group that started with a length of yarn.  The students positioned themselves in a triangular shape, estimating that they were an equal distance from each other.  I asked, “How do you know it’s an equilateral triangle?”  They responded, “We know because we’re standing the same distance from each other.  I pushed a little harder, “But how do you know for sure that the sides are of equal length?”  They then folded the yarn into thirds but quickly saw that their lengths were not quite equal.  A simple adjustment guaranteed that the sides were now congruent.

 Another group insisted that the only way to build an equilateral triangle was to measure using a standard tool.  Because the logs of the community space had already been measured at 15 feet each, the yarn triangle they created was an exact copy of the logs.  T said, “We already know this one is equilateral.”  They used their bodies as vertices, pulling the yarn behind their backs at the waistline.  Cat noticed that this distorted the shape somewhat, and C said that they could turn “sideways”, making it “thinner” and therefore closer to a point.  He also said that if they used their fingers, it would be “even better”.  What if we had not gone outdoors?  Here, they were able to build on prior experience to guide them in their cooperative problem-solving.

Cat asked them if it would be possible to make an even smaller equilateral triangle, and they struggled to come up with a strategy.  N came over and told them about how her group had accomplished it.  They had cut off a “random length” of yarn and then used that piece to measure off two identical pieces.  

By not allowing the students to use a standard measurement tool, the students got to the nature of congruence (equal measure), of measurement itself, and of proving something definitively.  Even the students said that they thought it would be “much easier” than it was.  After all, they could name and describe an equilateral triangle.  But building one was an entirely different matter. Years from now, in high school geometry, they will have to construct an equilateral triangle –again without using measurement tools– and prove how they know the sides are congruent.   

Thinking with a strength

At Sabot at Stony Point we feel it is important for children to draw on their strengths to help them take their ideas as far as possible.  We are writing fiction stories and the children have the option of drawing their ideas, or cartooning to write an illustrated story. One of the students had been having trouble writing in the details, so during his first draft he had chosen to illustrate instead.  I typed his story for him and left spaces for his pictures.  However, as he was revising and editing this final draft I noticed that he had chosen to take some of the pictures out, and instead include the detail in his writing.

This was an unexpected step.  But, it really made me think.  We do need to allow children to think and organize in a language that makes the most sense to them.  They may not choose to use it in their final presentation, but by drawing, this student allowed himself to add detail, and make his story more cohesive.

How tall is a tree?

Every Friday we go to the forest, this is in addition to our time in our outside classroom.  Sometimes the children have specific science activities to do and sometimes they are presented with a challenge.

This week's challenge was how tall is a tree, or indeed how can one measure the height of a tree.
I had a basket containing tape measures, yard and meter sticks, string and yarn, protractors and set squares,  and safety goggles for the children to use if they needed.

Well, as usual my class went at the task with gusto and I enjoyed watching the various ways they solved the challenge.

So how did they do it?

Find a small tree and measure it with a measuring tape or stick.

Find a dead tree and measure along its length with a tape measure.

Find a thin, bendy tree, pull it down from the top, attach a tape measure to it and let it spring back up.

Measure a tree that is leaning over.  - I added a question for this group - did you measure the height of the tree or the length?  They realized they had in fact measured the length and redid it measuring the height.

Get a group of students together, one person climb the tree, the other hand them a measuring stick, and yet another hold the tree trunk steady.

Find a really long tree branch, use it to measure a tall tree and then measure the branch.

Eyeball a tree, think it is about three times the height of a student - so estimating about 12 feet tall.

The idea of measuring shadows came up and caused a lot of discussion.  The conclusion grade was, that measuring with a shadow would not work because shadows change in length throughout the day. They also noticed that at the time their shadows were longer than them so it wouldn't be an accurate measurement.  One student actually got down on the ground to see if that would make his shadow match his height, but noticed it didn't.  - I did wonder that after this experiment whether they would take it further, noticing just how long their shadow was compared to them, then using that same ratio with a tree, but they decided shadows were not going to work, so moved on to a different idea.

What I liked the best about this challenge was the range of ideas the children came up with.  It worked whatever their ideas were, it worked for all kinds of thinkers. They stayed focused, they measured many trees, they worked in teams, they understood the importance of safety, shared ideas with each other, discussed tough questions and also had a lot of fun.