A Division Mystery: The Meaning of Remainders

We encountered a puzzling situation with remainders this week.  Until now, the students expressed remainders as R4, for example, rather than as a fraction or decimal.  The problem that arose this week helped build the conceptual understanding of what remainders are all about, and now we know why you express remainders as a fraction or decimal, not just that you express remainders as a fraction or decimal.

One strategy the students use to solve a division problem is to make an easier, equivalent problem by dividing the dividend and the divisor by the same number, which won't effect the quotient (12 / 2 = 6 / 1).  When they used that strategy to solve 376 / 6, however, they encountered a problem--halving and halving didn't produce the same remainder as other strategies, but no one could find an error in their work.  We revisited the problem today, and everyone tried to figure out why the remainders were different.  We ended with a meeting to discuss our findings.  Here's the original question as well a poster that shows our conclusions:

A Division Mystery

Students solved 376 / 6 in different ways on Wednesday.  Two different answers came up, but there doesn’t seem to be a calculation error in either strategy.  What’s happening here?

376 / 6

Strategy 1

60 x 6 = 360

2 x 6 = 12

376 - 372 = 4

376 / 6 = 62 R4

Strategy 2

376 / 6 = 188 / 3

60 x 3 = 180

2 x 3 = 6

188 – 186 = 2

188 / 3 = 62 R2

376 / 6 = 62 R2

The students worked together to come to the following conclusion, and several students volunteered to make this poster:

ELA Monday and Tuesday

Hi Families,

Just a reminder that the ELA is Monday and Tuesday.  Day 1 will be 45 minutes, and the students will answer approximately 25 multiple choice and some short answer questions.  Day 2 will be 50 minutes.  I'll read an article to the class twice, they'll take notes, and then they'll answer some multiple choice and short answer questions about the passage.  The test ends with a brief editing passage.

I administered a practice run of Day 1 of last year's test so the class knows how it will  feel.  They  did really well!  I'm sending it home today or tomorrow in case  you want to see what Monday will be like.

You've heard it before, but remember that the best thing for your kids to do on testing days is to sleep tight and breathe easy!

Take care,

Lauren

The Meaning of "Democracy" and Fate vs. Free Will

We wrapped up our conversation about revolution last week and are not talking about democracy.  Today in social studies we looked at several famous quotes (below) about democracy.  Each student chose one, and wrote about what he or she thinks the quote means.  Tonight's homework is to draw a sketch that shows the quote in action.  This would be a lovely project to check out if you're so inclined.  If you're looking for something exciting to discuss at dinner, it might be an interesting thing to talk about, as would the philosophical question that came up...


One of the quotes we looked at was credited to Aristotle.  We quickly discussed what a philosopher is, and that lead to an interesting conversation about free will vs. determinism, or fate.  While this was a bit tangential, it was an interesting conversation, and there are clearly some philosophers in the room.  Students, if you'd like to read more about free will vs. fate, page 86 of this book might be of interest (Click "contents" for links to the pages in the book.  You might have to play with the zoom to get it to look right.  Click and drag to scroll.):

Social Action: Day 1--What's the Recipe for Revolution?

We started our Social Action Study today!  My colleagues and I are happy to be writing this brand new study, and the first day was lovely.  The focus question of our study is, "How do people make change?"  We began by looking at the American Revolution (talk about change!) and thinking about what the "Recipe for Revolution" is--What do all revolutions have?  What are the constants in revolutions?  The variables?  We'll linger here for a few days, then it's on to the Constitution.  Later, we'll see how our definition of revolution stacks up against the other "revolutions"--women's suffrage and civil rights--that we'll be studying.  These are the notes from today's conversation: