The Rookery

One of the challenges of writing a blog is that often, in a single day, there will be two or three things that I could write an entry about.  (This problem is compounded in January when all of my writing energy goes into narratives.)  By the time I peep my head back into the blogosphere, there are thirty to forty things that I wanted to write about.  So I'm going to start fresh today and share the three moments when I thought, "I should put this in a blog."

Uncanny Triangles

The best math lessons are those in which we are using new skills and ideas to solve a puzzle together.  Today, the Herons looked for Uncanny Triangles.  In the problem, "uncanny" triangles are right triangles whose area is equal to the sum of their two shortest legs.  How many uncanny triangles were there (if we limited it to sides whose sums were equal or less than 15).  We made sure everyone understood the task at hand and went at it.

I worked with a group at the carpet that wanted to explore the challenge in a more supported way.  We reviewed how to find the area of a triangle and went over "sum" again.  Then we started to draw triangles and test them out together.  Another group wanted to work more independently and headed off to the tables.

Soon someone exclaimed, "The bigger the numbers, the less likely that they'll work."  I stopped the class for a moment and shared that conjecture, labeling it the T Conjecture for the child whose idea it was.  A conjecture in math is like a hypothesis in science.  It's an idea that you look for evidence to prove or disprove it.  A lot of heads nodded and someone shared, "Well, the products get a lot bigger than the sum when you are multiplying big numbers so that makes sense.

The hunt continued and soon both groups had discovered the same uncanny triangle.  Were there more?

One student explained she didn't have a proof, she had something that was just true, "You have to have at least one even side because two odd numbers multiplied together always equal an odd number."

The rest of us took a second to catch up to her thinking and try a few out.  I considered going off on the tangent of how we can know this is true but decided not to (can you explain why?).  We wrote it down in our list of conjectures (and now proofs).

Another conjecture – "You can't have one as one of the sides."

Then some questions.  Can you have negative sides?  I was skeptical…how would you draw it?  They explained that if something went to the right and was 4 long then if you flipped it wouldn't it go to the left -4?  I asked them if they could use a ruler to measure it and what it would say.  "4 both ways."  We decided a triangle couldn't have a negative side.

Can you have a triangle with a zero side?  One of the students had drawn a line segment and labeled its lengths as 8 and 0.  "What makes a triangle a triangle?"  The class agreed that three sides were needed.  The triangle with a 0 side did not have three sides so didn't count.

I set up a table because I was running out of room to draw triangles.  For some students, that made sense; others preferred to stick with trying different triangles by chance.

We found a second uncanny triangle.  A new conjecture was made – since the sum/Area of one that we found was 9 and the sum/area of the other was 8, perhaps we should look at 10 and 7.  We wrote up a list of "tens buddies" from way back in kindergarten – the numbers that add to 10 and tried them out as triangles.  No luck.  We tried the "sevens buddies" still no luck.  We tried sums of 4, 5, 6, 8, 11.  No luck

Our time had elapsed but now we had a way to know for sure if we had all of the answers – we just had to try out the other sums.  (Editors note:  when linking the problem in the first sentence, I noticed that a single side was allowed to be 15 – not just the sum.  does this make a difference?)

Spoiler alert – the picture below shows both of our solutions…so far.  Can you keep another secret?  We did a lot of practice with the area of a triangle!

But Wait…There's More

The metric system was a topic of hot debate in our foundation math group this morning.  It wasn't supposed to be.  I made an off hand comment about how Newfoundland had gotten 75 cm of snow on Friday.  I through out the question, "How many meters is that?"  Crickets.  "How many centimeters in a meter?"  A few hesitant guesses.

Oops.  Mind the gap – time for a mini lesson.  The kids knew about meters (and centimeters) but didn't know how they were connected or why they make so much more sense than inches and yards.  We had 15 minutes.

I explained a (very brief) explanation of the adoption of the metric system – including its unfortunate timing with the english and french conflicts.  We ended up with the vastly inferior "customary" system.  Once students were aware of how our base 10 number system was the basis of the metric system (and we had done a few blissfully easy conversions between meters and centimeters and a few brutal conversions between fractions of yards and inches) they were up in arms.

As they left, they were planning for the student council to make a policy proposal to make metric the official measuring system of Prairie Creek.  Onward.

Many Returns – A Conversation to Honor Martin Luther King

The notion of having "The Talk" with your kids about "the birds and the bees" is outdated of course.  Ideally, we have many, many conversations about development and puberty and relationships.  The same is true for conversations about race.  A single conversation will never be enough to help our children construct an understanding of the history that has led to this point and the actions that are needed to begin to make the world a better place.  And so we return to conversations about race often, weaving in new information and broadening our understanding.

Today was another conversation in that work.  I had students read excerpts of "Oh Freedom!"  It's a book of interviews of people who experienced segregation and participated in the Civil Rights movement of the 1960s.  The interviews are done by 4th graders who interviewed people in their community.  The students could name Martin Luther King Jr., Ruby Bridges, and Rosa Parks but most are still building their understanding of the role of every day people in the movement and these interviews shared those stories in simple, approachable ways.  The students read and discussed in small groups the came back to the large group to share questions and thoughts that they had.

These conversations are necessary, but sometimes difficult and often unpredictable. I recently found a wonderful resource, Not Light, but Fire by Matthew Kay  (Download Author Interview With Matthew Kay), that emphasized the importance of teaching listening in conversations.  I prefaced our discussion today with those reminders.

The conversation was rich and, indeed, did go in directions I had not anticipated.  Kay talks about the necessity of taking these risks – the conversation must be genuine if it's going to shift students' understanding.  At one point, we talked about how people came to accept segregation (one of the interviews was with a white woman who remembers being a little girl first noticing segregation).  The Herons remembered a conversation we had earlier this year about fish and water – we don't always question what's around us.  Dr. King and the many people who marched along side him, had the ability to see what was wrong with the "normal" and speak up against it.  We talked about situations on the bus when we knew something wasn't O.K. but went along with it because everyone was doing it.  A child asked about George Washington…how could someone who was "nice" (his word) own slaves?  Another child pointed out that "the victors write the stories."  That took some unpacking but the students knew from our Silk Road theme and other conversations that there are a lot of stories missing from our history and that it's the job of historians to find them and tell them. 

Propaganda was brought up in the context of how the Nazis convinced people to follow them.  That led to talking about the "scientific" rationales (head size measurement) that people used to justify being against women from voting and the religious rationales that people used to justify enslavement.  One student shared that there were studies that showed that women were smarter than men.  That provided some more great conversation – how was the study done?  How do you define "smart"?  What tools did they use to test people?  Were the tests biased?  What IS bias? I interjected at this point and shared that I had never seen a valid study that indicated an intelligence difference between sexes and that even deciding how "intelligence" is defined and measured was controversial.

Eventually the conversation circled back to the interviews.  The students were interested in what happened when the Emancipation Proclamation was signed.  We talked briefly about Reconstruction and the Jim Crow laws that were a response to formerly enslaved people being given rights and property.  That cycle of freedom and unjust laws has continued and we talked (very) briefly about Black Lives Matter.

Whew.  Like any conversation of this type, there were a lot of loose ends.  Students come into the conversation with a lot of different experiences and a lot of different levels of historical understanding (one student shared a lot of detailed information about why southerners were against a central U.S. bank…) . But by starting the conversation, they know that this is something that we need to talk about and know about. 

Oh yeah – and we went skiing and worked on our opera!