Mathematical tools – Part 1

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.


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.



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.