5th grade-Decimal understanding and comparing.
I went into the class to model a lesson where students use
models to understand and compare decimals. My opening
was an empty number line with 11 hash marks – zero on the
far left and 1 on the far right.
I asked students if they knew what any of the hash marks on
the number line should be labeled. Only a few students raised
their hands, so I asked the class to talk about this at their
tables for a minute.
After a quick discussion, a boy was chosen to come to the front.
I asked him to point to the hash mark on the number line that
he thought he knew the label for. He pointed to the middle line.
What I would’ve done 15 years ago, is ask him what it should be
labeled and move on with the lesson. Instead, I asked him to
whisper what the label for the hash mark should be.
I thanked him and asked all of the groups to focus on the middle
hash mark on the empty number line and see if they could agree
on what it should be labeled.
This teaching strategy never ceases to amaze me – and neither do the
students. The conversations were incredible. Just allowing students
to share their ideas with each other and try to make sense of numbers
(fractions and decimals) on a number line.
In the beginning of their discussions, most students thought 1/5 (the
same thing the boy whispered in my ear). Their reasoning was that
there were 11 hash marks and the middle one was the fifth one over.
It made “perfect incorrect sense.” But I learned what misconceptions
were prevalent in the class.
As I talked with each group, students began to question their own
reasoning. One group, while defending the idea of 1/5 said, “Yeah,
the fifth one over is in the middle and . . . well, it is in the middle, so
it could be 1/2.” This was my time to ask, “Can it be both 1/5 and
1/2? You have 90 seconds to discuss this and I’ll be right back.”
By the time I got back, they had decided it had to be 1/2, because
they “knew” that 1/2 and 1/5 weren’t the same.
When we came back as a whole group, many of the students had
shared that they had thought it was 1/5 at first, but many changed
their minds because of the idea of the hash mark being in the middle.
Many changed their minds to 1/2, but not all. Some had decided that
since our standard was about decimals, the hash mark should be
labeled 5/10. The next discussion lead to proving that 1/2 = 5/10.
Once students were comfortable with the decision that 1/2 = 5/10, I
asked them to label the hash mark to the left of 5/10. The discussion
was quick and efficient. They knew it was 4/10 because there were
ten “sections” on the number line (no longer 11 lines), and that hash
mark was the end of the 4th of the 10 sections. They were thinking of
the number line as an equally divided line (fractions).
Finally, the students were asked to draw a number line in their journals,
like the one at the front of the room, and label all of the hash marks
with fraction and decimal notation.
It’s important to note that this opening to the lesson (that ended up
becoming the whole lesson) would not have been possible if the teacher
hadn’t developed group norms with the students at the beginning of
the year. This class knows, after 9 weeks, how to talk to each other,
discuss their thinking, and work together toward a common goal.