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    Home > Developing Mathematics for Modeling > Measurement |
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   Measurement |
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    Personal
Unit Tape Measure:
Part A |
    Goal |
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Students
are each
provided
a thin
rectangular
strip
of paper,
named
after
them.
Each
personal
unit
is at
least
one foot
long
to help
create
the need
to partition
the unit
to measure
objects.
Alternatively,
each
student
is given
a foot-long
length
and informed
that
it is
one foot.
This
choice
may allow
students
to capitalize
on what
they
know
about
foot
measure
(Or any
other
standard
measure).
The choice
is up
to the
teacher:
Will
the convention
foreclose
student
thinking
or support
it? The
assessments
conducted
in the
previous
lesson
might
be the
best
guide.
If students
are firmly
convinced
that
only
conventional
measures
are possible,
it is
likely
better
that
unconventional
measures
be introduced.
In part
A of
the lesson,
various
splits
of units
are explored
and symbolized.
The lesson
employs
paper
folding
as a
model
of partitioning.
In part
B of
the lesson,
the splits
are put
to use
measuring
objects
and deciding
when
measures
are equivalent.
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   Teacher
Notes |
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If
students
say that rounding
is good enough,
introduce a four
foot long strip,
called a “giant
foot,” and
ask students
to measure
something about ¼ as
long as the
strip. In this
case, the visual
perception
of the
distance
makes the need
for a part-unit
clear.
In
this
activity, we
will use the
language, “of,” and
model fractions
as “a
measured
in
b.”
For
example, if
we have a length
measured in
feet, then “1/2
of 1 foot” means
split or
divide the
foot into
2 congruent
parts. ½ is
1 copy
of the 2
parts.
What has
been done
is that
the new
unit of
measure
is 2, each
of which
is half
as long
as the
original
measure,
the foot.
2/2 means
2 copies
of the
new unit,
3/2 means
3 copies
of the
new unit,
or “3
halves
(giving
the new
unit a
name that
relates
it to
the
old unit,
the foot).
It is
also
possible
to give
the ½ unit
a new
name,
so that
if the
length
of the
personal
unit
is “Bob,” then ½ Bob
could
be called
something
else
too,
such
as “click.”
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   Activity
Structure |
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Whole
Group Demonstration/Discussion,
Individual or
Pair Problem Solving
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   Teacher
Demonstration |
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Let's
try to create
parts
of our units
by
folding our strip
paper. Try to
create
2 parts. Demonstrate
splitting the
unit
by folding into
2 congruent pieces.
Ask: What do
I call the length
from here (start)
to here (midpoint),
moving your fingers
along the paper-strip?
[Students will
likely say “one
half”]
Emphasize
that each partition
is congruent
(by folding),
that there are
2 partitions,
and that the
new
measure of
the length is
2: 2 halves. ½ means
1 copy of
the new unit-the
half. [It could
be called anything
but we usually
call it a half
to remind ourselves
of its relation
to the original
unit – ½ foot
or ½ <personal
unit name>.
Be sure to
emphasize
that the
part-of-the
unit traveled
is half as
long as the
unit, and
that the
unit is twice
as long as
the part.
Be sure that
students
replicate
your motion
and not just
point to
the part of
the unit.
1
foot
is 2 times
as long as ½ foot.
½ foot
is ½ times
as long as
1 foot.
Ask:
What might 2/2
mean? How many
copies? Of what?
What about 3/2?
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    Problem
One |
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How
could I make ¼?
Let student
work in pairs
to answer this
question, then
come back to
the whole group.
Compare solution
strategies.
One common strategy
to fold the
paper in half
twice. Be sure
to symbolize
this as ½ of ½ of
1 <unit-name> to
emphasize that
the symbolism
captures what
is essential
about the activity—even
though the
result is a
different length
for differing
personal units,
the result
is the same:
1 is now measured
in 4. Emphasize
again that
the part-of-the-unit
traveled is
one-quarter
of the unit
and that the
unit is four
times as long.
1 <unit-name> is
4 times as long
as 4 <unit-name>.
1/4 <unit-name> is
4 times as long
as 1 <unit-name>.
Ask
students
to
draw a length
that is 1 ¼ <unit-name>,
1 2/4 <unit-name>,
and 5/4 <unit-name>.
Which ones
are the
same
length?
Why?
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    Problem
Two |
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Students
work individually
or in pairs.
Draw a length
on the board
or give students
a length
drawn on a
piece of paper.
Say: this
length is ¼ times
as long as
1-unit-name
(Give the
unit a name,
perhaps the
name of a
student.)
Draw the unit-name.
Have students
compare solution
strategies.
Teacher
note: Another
way to
emphasize
the reciprocal
relationship
between
the
part and
the unit-length
is to ask
students
to consider
how they
might “undo” the
part to recreate
the unit-length.
For example, ½ can
be undone
by iterating
the ½ unit
twice, ¼ can
be undone
by iterating
the ¼ unit
4 times.
This
reciprocal
relationship
provides
another
way
of creating
a unit—Make
a fractional
length,
and replicate
it n times.
So, for
example,
1/5 is
made
with the
tape by
creating
a length,
then folding
that the
tape to
reproduce
that length
again,
then
again,
then
again,
and
finally,
one more
time (5
lengths
including
the original).
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    Fraction
Strips I |
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Teacher
note: The intention
of Fraction
Strips
I is to provide
students with
experience repeatedly
splitting a unit
and then observing
the resulting
partitions of
that unit. For
example, ½ means
that the unit
is split into
2 congruent
partitions and
we move from
the beginning
to the end of
the first partition
(visible as
a fold line
on the paper).
We use an informal
language “of” to
signify copies
of splits of
a unit, as in ½ of
1 unit-name.
We explore
splits
of 2, 3,
and 5, and
combinations
of 2 and
3.
We suggest
an implicit
model of
addition
when we
ask
about continuing
a motion
from,
for example,
1/8 of
a strip
to the
next
fold (another
1/8), and
symbolize
that distance
traveled
as
2/8.
There
are many problems
in Fraction
Strips I. We
suggest letting
students solve
at most 2 problems
at a time, then
sharing their
solution strategies,
then continuing
with the lesson
in this manner.
Alternatively,
you might want
all students
to solve selected
problems and
then assign
different problems
to different
students or
groups of students.
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    Making
Thinking Visible |
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Mal-ruler . A teacher, Deb
Lucas, constructed
a mal footprint
ruler that was
intended to probe
students' thinking
about iteration
and identical
units. The mal-ruler
was constructed
by mixing different
length foot-prints,
by leaving gaps
between units,
and by overlapping
units. She measured
a few objects
with the ruler,
reported results,
and asked students
what they thought
(and why) of the
resulting measurement.
Was it accurate?
What did a simple
count of the number
of feet tell us?
Summarizing,
Representing
Units in Measure
Walks. Deb
Lucas asked
students
to summarize
what
they knew
about
rulers. During
the course
of one such
summary-discussion,
a controversy
arose
about “space,” with
some students
saying that
the marks denoting
the “end
of units” and
others denoting
more densely
notated marks
as meaning, “less
space.” To
follow
up on this
emerging
idea of “space,” Rich
asked students
to represent,
with drawings
and with
a drawing
of a ruler,
three different
types of
measure
walks. One
walk traversed
a distance
on the
floor heel-to-toe,
step-by-step.
The second
mixed the
units,
heel-to-toe,
and an
exaggerated
stride.
For Rich,
this corresponded
to a situation
in which
the units
were mixed.
The third
walk included
a jump,
along with
heel-to-toe.
Rich intended
this as
corresponding
to gaps
between
units. Student
solutions
were quite
variable
and overall,
a good way
of provoking
representational
discussion—what
about
the rulers
corresponded
to
what about
the walks?
Students
did some
surprising
things:
what Rich
took as
gaps,
students
saw as
another
way of
traversing
the length.
But during
the course
of discussion,
Rich asked
what would
happen
if we
measured
again
using each
method?
Most students
suggested
that the
heel-to-toe
was likely
most repeatable.
Some students
suggested
that for
longer
distances,
use of
strides
was better,
as long
as they
could
be calibrated
to feet.
A few
students
suggested
the jumps.
They acknowledged
that they
were least
replicable
(Rich
jumped
several
times
from the
same starting
point),
but the
jumps
had
the advantage
of quick
measure.
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Last Updated:
March 17, 2005
All Rights reserved. |
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