Captain Ahab had Moby Dick. Wile E. Coyote's got Road Runner. I have
3.MD.B.4. Teaching kids how to measure to the nearest quarter inch bedeviled me as a grade 3 teacher, and has continued to do so in my role as math specialist.
Recently I described our attempts to tame this unruly standard, and while I was pleased with the results, I felt we needed to look for more ways to attack this beast. My latest inspiration came from an unlikely source:

I found this book buried in a closet. 

page 12 
The game was familiar. This past summer, at the
Middlesex MathScience Partnership program at Middlesex County Community College,
Dr. Milou had challenged us, playing from 120. He took great pleasure in trouncing all combatants, one after another, until someone in the group figured out the winning strategy.

This caught my eye. Might this game be employed in the struggle against 3.MD.B.4? 
I asked Jen if I could use her class to experiment, and she agreed. First, I taught the kids how to play a basic game, with the first player saying either the number 1 or the numbers 1 and 2, and the players then alternating the count by one or two numbers in sequence from where the other player left off, with the player saying 20 the winner. I channeled Dr. Milou, beat everyone in class, and left the defeated third graders with the assignment to play the game with each other whenever they could.
When I came back the following week, they were excited to tell me that they had figured out the strategy: the player who can say 17 is assured of winning. They hadn't gotten any further, but it was a start, and I urged them to continue playing. In the meantime, I explained that I had a different game to teach them. I explained the "fraction variation": same rules, but instead of using whole numbers and counting from 1 to 20, we were going to count by 1/4s from 1 to 5.

I played a demo game on the board. I recorded my count in red and the student's in blue. I wanted them to record their counts, in the hopes that we could connect this number line to the quarter inch ruler.


They picked it up quickly. 

We had to tape two pieces of paper together to get the right length. 

This sample, placed under the document camera, provided the teaching points at the lesson's close: I love how these kids wrote a 1 on top of 4/4 and a 2 on top of the 1 4/4. We also talked about how to write a mixed number, so that 3 and 3/4 would not be confused with 33/4.

Jen reported that the kids liked this game, and when I did an item analysis on the unit assessment I was pleased:

This is the item I was interested in. 17 out of 18 kids had gotten all three parts correct. 
But 3.MD.B.4 never sleeps, and I knew we couldn't rest on our laurels. The conceptual groundwork we were developing here would only help the kids as they moved into the more complex measurement and fraction standards awaiting them in the upper grades.

I took our standard grade 3 ruler and photocopied and enlarged it. 

I whited out the numbers and put two on an 11" by 18" page. 

Would you look at that! It's a ruler! And marked at quarter inch intervals, too. 
My hope is that, if they play enough, the kids can start to notice certain patterns: that the whole numbers live on the longest lines, the 1/4 and 3/4 on the shortest, the 2/4 right in between. (And I would like to transition the kids to writing 1/2 instead of 2/4, although this is a good way to reinforce these equivalent fractions.)
My hope is to avoid Captain Ahab's fate, and one day vanquish 3.MD.B.4 once and for all. So... what's your white whale?
Awesome, Joe. Real evidence of student understanding. I wonder if any of the students knew about the 1/2, but wanted to keep it fourths to maintain the beauty of the pattern.
ReplyDeleteThanks! Knowing about the 1/2 but trying to maintain the pattern by writing 2/4 is something I hadn't thought of. Now I need to revisit this class and see if, as they play, they are continuing with 2/4, or if some have switched to 1/2. That will generate an interesting conversation.
ReplyDeleteLove it! Thanks Joe.
ReplyDeleteYou're welcome. Thanks for the comment!
DeleteLove how your aggrandized the ruler. IT's like you are wordlessly making an editorial comment about fractions.
ReplyDeleteBe sure to take a look at my equivalent fractions PDF https://bookzoompa.files.wordpress.com/2015/02/fractionszerotoone1.pdf
Paula
Thanks Paula. Looking at your "Counting with fractions on the number line" makes me think to use colored dot stickers for the same effect. The visual is striking.
DeleteYep, 3.MD.4 is a beast. Maybe there's something wrong with the standard, not the teaching. Or maybe continuing to require the use of rulers that have inches is the problem, as the rest of the world wisely uses the metric system, in which subdivisions are invariably powers of 10, and where fractions such as 5/10 more easily transition into...decimals.
ReplyDeleteA more pointed grammatical ambiguity (read: weakness) in 3.MD.4 is why it states "halves AND fourths of an inch". If a ruler is marked off in fourths, then its halves are already marked, and the wording is redundant. If a ruler is only marked off in halves, then there are no fourths. Do the CC authors implicitly suggest that you start with halves rulers and then move on to fourths rulers? That would require keeping two sets of rulers, and if so, why don't they come out and say it? Or is the standard just sloppily written? That's the more likely option.
The line plot portion of the standard, "whole numbers, halves, OR quarters" implies a teaching sequence, not a single version of a line plot. Changing from AND to OR suggests doing line plots first with just whole numbers, then doing line plots with halves, then finally doing line plots with fourths.
How can Common Core suggest doing only one type of measurement but more than one type of line plot? Answer: you can't, unless you're standards authors with weak English skills unwittingly confusing both teachers and students.
A class could easily spend weeks on this one standard by taking its wording literally, and not even reach closure, because, as you accurately point out, there are "more complex measurement and fraction standards awaiting them in the upper grades".
This particular observation exposes the precise problem in American K12 mathematics: the habit of revisiting and rehashing concepts that should have been done with.
Thanks for the comment. I think you've articulated what's troublesome about this standard, and why it has presented us with so many challenges. I'm intrigued with your observation about, "revisiting and rehashing concepts that should have been done with." Could you say more about that?
Delete