Teaching the strange phrasing of technical mathematics

I’ve given my first geometry test this year (yes, I’m back on geometry, high five to all my geo-buds) and this is the first test I’ve given with the PARCC in mind.

Specifically, I made the phrasing match the technical language of PARCC questions, and I had my first experience with what happens when the students encounter something truly alien to them, like:

The point R is at (0, 3) and the point S is at (16, 7). Draw the line segment \overline{RS}.

Find a point L on the line \overline{RS} such that \overline{RS} is four times as long as \overline{RL}.

The first line didn’t go so bad. In a way students can plow through without reading it (draw a line segment? ok there must be points …. there they are!) but the second line had lots of bafflement.

It is a circumstance where vocabulary isn’t the issue, but phrasing is.

The issues seemed to be
a.) Not being clear where the question was; in this case it was directive to “find L”
b.) Students who got past the first hurdle were unclear in juggling the phrasing after; essentially the student brain seemed to go — first I find L, but to do that I need to worry about RS and RL, and somehow RS is — wait what?
c.) That rather than being told what to do (find the midpoint between X and Y) they had to hold a conditional in their head and fuss with a bit before they even understood they wanted a quarter-point to answer the problem.

This sort of conditional indirection seems to be common in PARCC questions: rather than being told what needs to be accomplished, be given some geometric object to assign THEN be told what will happen once that geometric object is in place THEN try to start unpiling what needs to be accomplished for those conditions to hold.

This sort of thing is routine in formal math texts but does not seem to be in the high school experience at all.


(From the PARCC sample Geometry test.)

Does anyone have experience teaching this sort of thing? How does one get students — both fluent and English language learners included — to read statements like the one above without blanking out?

Observations on the PARCC sample Geometry exam

Part 1: Observations on the PARCC sample Algebra I exam
Part 2: Observations on the PARCC sample Algebra II exam
Part 3: Observations on the PARCC sample Geometry exam

Calculator part: 18 of 25

Use the information provided in the animation to answer the questions about the geometric construction.

To pause the animation, select the animation window.

The students are supposed to watch a video of a construction and then say things about the proof enacted through the constructions. This is a very specific skill that needs to be practiced. Daniel Schneider kindly sent me a link to a website with a large number of construction animations (along with proofs) in case you need more to use in class.

However, there’s a serious interface problem. Here’s what the video looks like when paused, as well as a question to go with it:


Point “C” is completely covered. Whoops.

Non-calculator part: 6 of 7


This is one of those simple-looking questions which has enough of a trick to it I’m not sure how many students will get it right.

Part A requires students to work a double-completing-the-square manipulation, hopefully not getting sidetracked by the presence of b on the right hand side:

x^2 + y^2 - 4x + 2y = b
x^2 - 4x + y^2 + 2y = b
x^2 - 4x + 4 + y^2 + 2y + 1 = b + 4 + 1
(x-2)^2 + (y+1)^2 = b + 5

Part B requires noticing that a radius of 7 means the right hand side will be 49, so b + 5 = 49 and thus b = 44.

In principle this problem is solvable, but the lack of partial credit on a problem with a “trick” that I worry a student who can normally complete the square would still get no points due to the indirection.

Calculator part: 13 of 25


This problem’s rough for three reasons:

a.) Even with the phrase “the pipe is open at both ends” placed in there, this is something of a background knowledge problem; the students need to know the “outer surface” excludes the circles on the top and bottom.

b.) There are volume formulas on the formula sheet but not surface area formulas. Thus the students need to have memorized \pi d h or be able to extrapolate it, and know enough to exclude the circles.

c.) If \pi is set to be 3.14, there answer comes out to be 1356.48. If \pi is set to be 3.14159 or something with more digits (not unusual since graphing calculators have a “pi” button) the answer comes out to be roughly 1357.168. Rounding to the nearest integer thus can give either 1356 or 1357 as an answer.

Non-calculator part: 7 of 7


Out of all the problems on the PARCC final exam for geometry, 28% are related to transformations.

I can understand a transformational emphasis in general: it leads to a function transformation understanding of graphs (which is far more powerful and useful than looking at each kind of graph individually). However, why do so many of the dilation and rotation problem — 4 out of the 9 — involve centers not at the origin? This is not rhetorical; I really want to know where the utility is.

Non-calculator part: 20 of 25


This is one of the easier problems on the test, but assumes background the students don’t necessarily have. I can assume what a “collar” means here (even though I’ve never heard the word used in this context) but my ELL students are more likely interpret it as gibberish.

Calculator part: 10 of 25


This is very similar to the other problem in relying somewhat on background knowledge. Technically speaking one can ignore all the external stuff about merchant vessels and probes and focus on the math, but the brain of the ELL student doesn’t have an easy time removing the context.

Also note the weirdness of the rounding; in problem 20 the rounding needed to be done to the nearest tenth, while in this problem the rounding needs to be done to the nearest integer off the list.

Calculator part: 3 of 25


I’m noting this one because nothing in my current textbook (Carnegie Learning, written for Common Core) has anything resembling this kind of problem. Anyone have a source with problems that are similar?

Calculator part: 23 of 25


I don’t think I’ve ever give this much emphasis to the vocabulary of proof. Getting my students to keep the reflexive, symmetric, and transitive properties of congruence straight is going to be a nightmare and a half.

Ok, one last problem, from HS Sample Math Items, 7 of 10 (so not the final exam, but the open response part):


Here the angle bisector video returns (complete with unhelpful play button covering the diagram when paused) but the student is supposed to free-write a proof.

Here is how you type the first line as given:

1.) Pick “geometry” on the side and pick the short line in the upper right; that’s a “line segment” and will give you a blank box under a line segment so you can type letters.

2.) Type the letters you want under the line segment. If you accidentally type more than two letters any extra keypresses will be ignored.

3.) Go to “relations” above “geometry” and find the congruence symbol. Pick that. This will give the congruence symbol and a blank box.

4.) Pick the “line segment” and it will take the blank box that just appeared and put a line segment over it.

5.) Type the letters you need for the other line segment.

Now you have one step of the proof, now you just need to give a reason and then do four more steps.

(What would constitute a valid reason here, by the way? The mathopenref site I linked to early in this post just states “They were both drawn with the same compass width” — would this be considered valid by graders?)

Robot maze puzzle


The puzzle above has a robot (marked with an arrow pointing up, or “north”) that you can control with a set of command cards that either move the robot forwards a set number of spaces (the + numbers) or backwards (the – numbers). After using a command card, the card is used up and can’t be used again.

If the robot hits either the border of the grid or one of the black spaces, the robot stops moving and any remaining steps on the command card being used are ignored.

Landing on one of the spaces marked with circles causes your robot to turn 90 degrees. (That is, if the robot faces north it turns east, if east it turns south, if south it turns west, and if west it turns north.) The robot starts facing north. Can you get the robot to the star?

Observations on the PARCC sample Algebra I exam

Similar to my breakdown of the PARCC Algebra II, here’s some question-by-question comment on the sample PARCC Algebra I Final Exam. I’m not too discouraged by the actual items on this one, but the interface needs work.


This is the second problem of the test. You click points to set a line, then click “solution set” to shade in one side. Note (just like the Algebra II test) the points snap to half-grid points, not to grid points, a circumstance I find hazardous. Also, if you click on solution set to color a side, then realize your line was wrong, you have to click back on “line” again but the color goes away, so you have to add the color back again after the line is fixed. This is true even if switching the line from solid to dotted.

I should add this is one case where I see the superiority of open response to multiple choice. Here’s an inequality graphing problem from our old state test:


The lack of choices makes the problem a dotted-or-solid / above-or-below question where the actual shape of the graph is given away.


The question here is fine, but what if a student drags in the wrong number and wants to fix their answer? Removing a number only works if you drag the number back to the original number boxes, just “tossing” the number to a random position outside the answer box doesn’t work.


One common technique in the PARCC interface is for students to fill in sentences with a drag-down menu. By my eye, though, the interface doesn’t look much like a sentence, and I could imagine a student not understanding they are placing words between f(2) and g(2) and so forth to produce something that is meant to be read from left to right.


I guess 4 answer boxes — clear overkill — is better than the situation with 2 answer boxes where the suggestion seems strong to fill both of them even if one of the answers turns out to be extraneous.


I’m pretty sure logarithms aren’t supposed to be on the Algebra I test? Also, the graph is drawn automatically through the points, unless it can’t like in the example above. It took me a bit of deciphering to realize there’s an asymptote on there (right on the y-axis) and the asymptote can be slid around, so the reason the graph wasn’t showing up is the points were on opposite sides of the asymptote.


Do the blanks really have to be so large? I admit to getting confused because the symbols spread out in a single function looked to me like function-break-really small expression-break-random parenthesis and I had to do a double-take before I realized what was going on.


I hope students have their window large enough to realize (or least deduce from there being a “Part A”) that there is a “Part B” to the question.


There’s a truly weird option to change colors of things. Sometimes I can get it to trigger but I’m not sure how. The upper right inequality in pink shows what things look like after you’ve messed with the color.


There’s even an interface for systems of linear inequalities. Notice how there’s still a snap-to-half-grid feature even when the y-axis goes up by 5.


Why does one “find the zeros” question have a drag-and-drop interface, while this one gives a list?

Observations on the PARCC sample Algebra II exam

Fresh off the presses at


you can now find complete PARCC sample exams. Since it’s the one I most likely have to worry about next year, I tried out the Algebra II exam.

While I haven’t checked the other tests yet, the Algebra II has laid off the “show your work” type question. I’m going to put this in the positive column (I was deeply worried about open responses being graded by an army of interns; plus the turnaround needs to be very fast because this is technically the final for our Algebra II class. Imagine having to shave off an extra two months from the end to give time for the test to be graded). [ADD: According to this comment, this only applies to the final exam, and there are still “show your work” type questions on the other tests.]

Some question-by-question comments:


I admit I was baffled enough by this one to check Wolfram Alpha (I got m = -15/2 where m = 1 was extraneous), until I realized the key phrase “you may not need to use all of the answer boxes”. We’re going to have to specially train students for that trick, I think.


I tested this one on my students. The majority put the value for x which makes the statement true (1) not the extraneous solution (-3/4). I can understand the motivation here: forcing students to go through the factoring rather than just test numbers or even eyeball the thing, but putting the extraneous solution as an answer is rare and I can imagine a legion of students being confused by this.


I always considered the trick of rejiggering a number’s base (turning 4 into 2^2) to be something of a side trick; I’ve never had to use it outside a college algebra book.


Because saying “complete the square” would be too easy. In all seriousness, this is one of those problems I would have many students know how to do but get intimidated by the language of the question.


So the first 8 questions (which I was just quoting from) are in the non-calculator portion of the test. The remaining 26 are from the calculator portion. The students will have access to a TI-84 emulator.


I’ve got mixed feelings on this one. In a way it’s a neat bit of factoring (yank out the (x-y) terms, and what remains will be a difference of squares, tossing an extra (x-y) on the pile) but it also looks nothing like the kind of factoring from any textbook I’ve used; I’m fairly sure our Carnegie Learning book (which was written from scratch specifically for Common Core) has nothing like it.


The number of fancy widgets to enter answers has dropped considerably, but here’s one; you drag and drop in order.

Also noteworthy: there are two problems involving average rate of change. I think in our textbook there was enough material to squeeze out a single day. The PARCC writers must be putting rather more emphasis on it.


Here’s another widget: the dots are dragged around to form the graph. I was somewhat thrown by the dots not just snapping to the grid but also the halfway points. I believe this is a bad idea. A student could easily think they have a correct answer but leave the dot halfway rather than right on the appropriate point.


My students would be thrown off by the wording here. I think the intermediate step is actually helpful, but the phrase “Product of greatest common factor and binomial” looks so technical many of them would shut down.


I’m used to tests that make the statistics so easy students can answer them with no training whatsoever, but this is what I’d call a real statistics question.


This is a widget that lets you select an interval on the number line. In the process of trying to enter an answer the number line disappeared. Then going back a question and returning I was unable to get anything to appear at all.

Mathematics classes do not teach mathematics literacy

Both the phrases “mathematics classes” and “mathematics literacy” can be ambiguous, so I am meaning–

“Mathematics classes”: The high school curriculum delivered to a typical student in the United States. This usually excludes classes like “Statistics” or “Financial Math”.

“Mathematics literacy”: The kind of “good citizen” math that people refer to in articles like Headlines from a Mathematically Literate World. The word can also mean “ability to problem solve”, but that’s not what I mean here.

Taking the Headlines article and the classes of a typical high school mathematics student, how many of the headlines would a a student understand?

At the very least, understanding the entire list requires knowing about: correlation vs. causation, inflation, experimental replication, estimation of large numbers, incompatibility of comparisons with different conditions, understanding how tax brackets work, meaninglessness of predictions within a margin of error, statistically unlikely events, and reversion to the mean.

None of these will ever occur in an traditional math class. In other words, in the list of supposed math literacies, the typical math student in the US receives zero of them. (Some might possibly show up in a class labelled “Economics” or “Free Enterprise”, but those don’t get called Math Classes).

It’d be fair to argue I’m being highly specific in my starting definitions, but I often see the “good citizen” argument used during a general “why are we teaching math” type discussion which assumes a traditional math class track. That sort of argument only works if people are prepared to also overhaul the curriculum (by putting, for example, statistics before calculus as Arthur Benjamin discusses at TED).

Why is a negative divided by a negative a positive?

So there’s a whole lot of posts, including one from this very blog, which give intuitive explanations of why a negative times a negative is a positive.


I haven’t seen nearly as much material for a negative divided by a negative. One can certainly appeal to the inverse — since 1 \times -1 = -1, \frac{-1}{-1} = 1. Google searching leads to answers like that, but I’ve found nothing like the multiplication picture above.

Can anyone explain directly, at an intuitive level, why a negative divided by a negative is a positive? Or is the only way to do it to refer to multiplication?


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