CCSSI 7.G.2 states, ``Draw
(freehand, with ruler and protractor, and with technology) geometric
shapes with given conditions. Focus on constructing triangles from three
measures of angles or sides, noticing when the conditions determine a
unique triangle, more than one triangle, or no triangle.’’

Some preliminaries before we parse this standard.

Protractors

We
here at ccssimath.blogspot.com LOVE protractors. The way we see it, a
(non-toxic, teething) protractor should be given to every newborn in
their bassinet (thanks to Charles Schulz for the inspiration); perhaps
the first one can be hanging from a mobile. A protractor (like a banjo) is a happy
thing: it’s a smile, or a big letter D. A protractor is easy to hold,
it’s too big to choke on, and it’s covered with numbers. It even has
mystery and intrigue, for why would the numbers run in opposite
directions? Rulers, in contrast, though useful, are dangerous things;
they fit in the mouth and can be brandished as swords.

***

Angles

Formal
introduction of the concept of an angle and angle measurement belong in
the fourth grade. CCSSI has followed international benchmarks and we
find no fault with this placement. (We’ve previously
critiqued CCSSI, though, for talking about angles earlier than that, in 2.G.1,
before students have even learned the concept.)

4.MD.5 states, in part, ``Recognize
angles as geometric shapes that are formed wherever two rays share a
common endpoint, and understand concepts of angle measurement...’’
The definition of an angle and the concept of measurement are fairly
straightforward, although no subsequent reference is made of the need to
explain why a circle is divided into 360 equal degrees, and not some
other number. That 360 is an arbitrary number (with an
historical basis—cross discipline!) and would be unrecognized by Martians is an important idea, and should not
be glossed over. What fourth grade teacher knows why 360 was chosen?

4.MD.6 begins, ``Measure angles in whole-number degrees using a protractor.’’
Whoa! Isn’t this jumping the gun? What about 90º, 180º angles, etc.?
What about the 45º angle of a typical slice of pizza? To the
uninitiated, these numbers are downright strange, but the ``common’’
angles all meet the definition given in 4.MD.5 and can be determined
without direct measuring, i.e., without a protractor. Shouldn’t this
come first? Shouldn’t every student be given the task of calculating
the common angles with their newly learned division skills? This
recalls a central theme of this blog: that students should apply
concepts and skills in useful ways to abstract the concept. What would
CCSSI have students do? Take things on faith and let teachers throw
these angle numbers at them? CCSSI completely skips a critical
intermediate step in learning about angle measurement, but more
importantly, misses an opportunity for effective student-led discovery.
It belongs as a standard.

Right
angles are mentioned for the first time over in 4.G.1, but a right angle is
never defined. Adding the standard we suggest, and letting students figure
out the arcane numbering of angle measurement, would fill this gaping hole.

As
an aside, we know that classroom discovery has gotten a bad rap in the
return to ``fundamentals’’, but unlike the absurd notion of
``constructivists’’ that students should figure out every math principle
and algorithm on their own, calculating the common angles is the sort
of project that belongs.

Once fourth graders understand and are proficient at calculating the common angles, protractors are ready to make an appearance.

4.MD.6
should both be expanded and clarified. Protractors need to be
adequately explained, and the dual scales connected to the textbook
concepts acute and obtuse angles. Teachers, be prepared for the astute
student who will already understand why 90 is the only number at the
same place on both scales.

That
we have encountered countless high school students who cannot use this
basic tool, from how to position it, to how to read its dual scales,
tells us that they haven’t had enough practice using one, or worse,
can’t even connect the numbers with those textbook concepts. CCSSI
could be making an effort to address this problem, but hasn’t.

Finally,
does 4.MD.6 envision using a protractor to measure angles greater than
180º, which would require extra steps? Who knows, but it should, and it
should say so.

***

``Sketch angles of specified measure.’’
We also find fault with the second part of 4.MD.6; it should afford
more respect to the highly important skills of the professional
draftsperson.

With
all due respect to Pablo Picasso and his ilk, ``sketch’’ sounds like an
art class. Angles are a serious business, a far cry from Jackson Pollock's irreverent flung paint and cigarette ashes.

Actually, unbeknownst to many, world-class artists have tremendous mathematical skill, so good, in fact, that analysis of priceless artwork
often finds geometric meticulousness underlying the daubs of paint
forming lilies, still lifes, and naked ladies, perhaps explaining why
those particular works have commanded the attention (and value) that
they do. CCSSI's term ``sketch’’ belittles the manual dexterity, the
eye-hand coordination, and the precision that are necessary to
accurately draw an angle—a non-trivial task.

CAD
notwithstanding, beginning architects still learn to draw blueprints
manually, and creating an angle of out of nothing is not ``sketching’’,
it is drafting. Those of
us who have dexterity often fail to realize that drawing with precision
is a skill that not everyone gets right away, and many never adequately
learn to perform basic drafting tasks with even the simplest of tools, a
ruler and a protractor.

How can we make such a blanket assertion?

In
2005, the following open-ended task was given to students on the
National Assessment of Educational Progress (the NAEP, otherwise known
as ``The Nation’s Report Card’’)

This
problem combines several manual skills, plus presents the test taker
with a choice. The easiest skill (hopefully) is drawing two segments
each 8 cm
long that form an angle and then connecting their endpoints to make a
triangle. The second skill is getting the angle between those segments
to be correct (the rubric requires accuracy to within ±1°). The rubric
only requires accuracy for part b. to be within ±2°, which would be
understandable if a student’s best approach to finding those angles were
by measurement, but it ignores the obvious solution that those angles
can be found exactly by the triangle angle sum theorem. Now wait, you
say. The triangle angle sum theorem is taught later, in middle school
(now 8th grade, by CCSSI’s reckoning), and this problem requires only
fourth grade skills, so that’s why NAEP scored in that way.

Aha!,
but no. This question was actually given to 12th graders. And only
20% of graduating seniors performed these fourth grade level tasks
correctly.

We don’t see CCSSI having taken steps (i.e., fixing the curriculum) to address this problem.

***

We’ve
discussed the issue before, but when students are measuring and
drawing angles, it is a perfect time for them to be discovering (on
their own) that the angles of a triangle add to 180º. As we stated
before, classroom discovery has gotten a bad rap in the return to
``fundamentals’’, but this is another project that belongs. We’ve
already written that the triangle angle sum theorem can be proven as
early as the fifth grade, not the 8th grade as CCSSI would have students
do. To omit specific reference to this obvious classroom exercise in
4.MD.6 is to unnecessarily separate two closely connected ideas.
Disjointed standards positioned years apart lead to confusion and
unnecessary reteaching, rather than forward progress.

No mention of protractors appears in either fifth or sixth grade in CCSSI, so we know they
will be collecting dust. Again, unless we intervene. There are lots
of interesting things to do with protractors other than measuring and
drafting angles. That’s just the beginning. Put
this into the fifth and sixth grades: students can use a ruler and a
protractor to design things. Or they can create those polygons that
they’ve been hearing about because CCSSI has been foisting it upon them
since kindergarten (read our previous post about hexagons in
kindergarten). Using a protractor to draft a regular hexagon is a
challenging activity. (Edwin Abbott Abbott’s book Flatland comes to mind here.)

***

Finally we arrive in the seventh grade, and 7.G.2, whence we started.

We
first take the opportunity to point out an inexplicable blunder by
CCSSI’s authors in this standard. When you are given lengths of the
three sides of a triangle, it is impossible, in mathematical terms, to
draw the triangle with only a straight edge (ruler) and protractor.
You need another tool, an integral part of a comprehensive mathematics
education, to manage the task. It’s called a ``compass’’. Oh yes,
CCSSI does mention a compass once: in high school geometric
constructions. But along with protractors, students should be working
with compasses long before high school. (Of course, we don’t recommend a
compass in the crib.)

On
the topic of compasses, to skip back to the earlier paragraph about
drafting a hexagon with a protractor, it becomes an even more
interesting project to draft a regular hexagon with a compass (and
ruler). (And then to compare and contrast the techniques—a cross
discipline essay for an ELA class). Projects that include drafting fit into the elementary curriculum, to deepen, rather than
widen, mathematical exploration.

But
when to introduce the compass? Drawing a triangle using a compass given lengths of the
three sides also belongs in the fourth grade (why delay until seventh
grade, as 7.G.2 would have students do?)

And
as an added connection, learning to use a compass can easily transition
into the triangle inequality theorem, which states that if a, b, and c
are the lengths of the sides of a triangle, and a≤b≤c, then a+b>c.
Using a compass to try to draw a triangle with measurements that
violate the theorem is the easiest way to see it, and well within the
capacity of a fourth grader.

***

Teachers
are forever looking for ``hands-on’’ activities. Compasses and
protractors are REAL mathematical tools, not contrived, and should be
incorporated into the curriculum every year. from fourth grade onwards.
And teachers should make the argument (and students should understand)
that a compass is a precision tool, and is considered ``exact’’, whereas
a protractor is a measurement tool (but can be considered exact when drafting). The distinction is important. But
as compasses and protractors are often so closely associated, they
should both be introduced in the fourth grade.

Can’t
think of compass tasks in every year from the fourth grade onwards?
You kid us. At first, students can practice drawing circles with a
compass (it’s not so easy to do neatly as it sounds); then they can
practice copying (creating equal) line segments.

Elementary
educators: forget about those silly math activities you’re finding on
Pinterest. Teach your students real skills using real math tools
instead.

As for the remaining problems inherent in 7.G.2, we will delve into that more in Mathematical Tools – Part 2.