Thursday, November 20, 2014

Math problems of the week: Common Core-inspired geometry problem

The Common Core Standard in question:


Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

The source:

The problem:

The justification, solution and scoring:

Student reactions to this kind of problem:

An alternative, more tradition proof:

Extra Credit:

Which proof do you like better?

Use transformations to prove that the two proofs are (or aren't) similar.

Tuesday, November 18, 2014

Conversations on the Rifle Range16: Parallelograms, the Mercy of the Court, and Kit Kat Bars

Barry Garelick, who wrote various letters under the name Huck Finn, published here, is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number 16:

A “prep period” –a period in which teachers have no class—is one of education’s most sacred and cherished traditions. Mine was first period and involved making copies and putting finishing touches on lesson plans, as well as pacing nervously in anticipation of the day ahead. The stage fright dissipated when class began but would return if I didn’t pace things right and had slack time left over at the end of class. The resulting restlessness of students spelled disaster and invited clock-watching, students getting out of seats, general disruption, and lining up at the door (though I put an end to that practice quickly).

My fourth period pre-algebra class was the one class I dreaded most, though in the end it turned out to be my favorite. It was the most crowded, and also had six students who were “English Learners”. A Spanish-speaking aide was in class to help them. Mrs. Halloran had explained to me that because they were “low” in ability, they weren’t part of the regular class, and were relegated to the back of the class where they worked on an online course on computer tablets. They were given a special “pass/no grade” credit for the class. With the aide working with six students, the conversations at the back of the room served as a stimulus for other students, many of whom had self-control problems to begin with, to start talking. When the din got to a certain volume I had to raise my voice to quiet them. If that failed, Allysandra, a rebellious Mexican girl would yell “SHUT UP” at the top of her voice. This would generally do the trick.

Then there was Trevor, a disruptive boy, not well-liked by the other teachers. He got into fights and was even suspended for a week for one of them. He had a gift for arguing convincingly even when wrong. I first noticed this when he put some make-up work in the wrong place. When I told him where the correct bin was, he said “Chill.” My reaction was so swift it caught both me and the class by surprise: “Don’t you EVER tell me to chill!” I said. The class became a tomb.

“No, I didn’t mean it like that,” he said. “I meant ‘chill’ in the same way you say ‘cool’ or ‘OK’. That’s how I meant it.” He was convincing so I backed off and the class din resumed after about a minute. When I related this tale the next day to a teacher, she told me “Oh, that’s Trevor! He always has some excuse about how he didn’t mean this or that. He’s good at that.”

I mentioned to her that he was on the school’s mock trial team. This team competed with other schools in a mock trial, judged by a real court judge. “Given his gift for arguing and being on the mock trial team, I would guess he’ll end up being a lawyer,” I said.

“I hate to think who his clients will be,” she said.

I put that thought aside and tried to reach out to him. He would sometimes read a book quietly if he either 1) finished his homework early or 2) was avoiding doing the homework. Rather than try to find out which it was, I would ask him about the book he was reading and he would tell me. I sensed a more cooperative side to him, but wasn’t sure whether it was because I was showing an interest in him, or because of the impending championship mock trial coming up that was helping him focus his energies. Since I had members of the mock trial team in all of my three pre-algebra classes, I decided to tap into these students’ argumentative gifts during the unit on geometry.

I was finishing up a chapter on parallelograms. I decided to put a figure up on the screen in each of my pre-algebra classes and asked if anyone could tell me if two particular line segments in the figure were congruent and why. Some parallel lines were marked as such. The two line segments of interest were marked as being perpendicular to one of the two parallel lines and were, in fact, opposite sides of a parallelogram and therefore congruent.

I offered a Kit Kat bar to any student who could answer the question. Students immediately rose to the challenge. In all classes, some student would inevitably say “Can’t you just measure the two segments?” to which I would reply “Inadmissible evidence! The court will not allow rulers or any type of measurement devices in this trial. Demonstrations must be made using definitions and theorems only.”

As expected, the members of the mock trial team rose to the challenge. Some tried to get around my restriction of no measurement devices by saying “They look equal” but I easily put that to rest. “Not adequate. Visual comparisons are not allowed.”

In fourth period, upon hearing the “Inadmissible evidence” warning, Trevor rose to the challenge. He stood up and said “I got this! I got this!” and then proceeded to make spirited, breathless demonstrations that didn’t quite make the case.

I gave him some hints. “Do you think these two line segments are parallel?” I asked.

“Yes, definitely,” he said.


“Because they’re both at right angles, at right angles!” Trevor said as if pleading to a jury that his client did not deserve the death sentence. “What about the right angles?” I asked.

“It proves it,” he said.

“Proves what?” I asked.

“Proves that the lines are congruent.”


“Because they’re right angles, they’re right angles!”

I paused as if giving the matter great thought and the class quieted.

“Do you mean to say if two lines are perpendicular to the same line they are parallel?”


“So why are they congruent?”

Someone shouted “Because it’s a parallelogram!”

In the best spirit of courtroom drama, Trevor protested: “Unfair! I was going to say that!”

“There will be silence in the courtroom,” I ordered to no avail. “Counsel will be seated, please,” I said and continued: “The court will show mercy and recognize that counsel’s observations and arguments have merit and has provided indications that he knows that opposite sides of a parallelogram are…what?”

“Congruent!” Trevor shouted.

“One Kit Kat bar is awarded.” The class applauded, though the person who identified the figure as a parallelogram wanted one also.

“Come up and claim your Kit Kats,” I said and presented the awards. To Trevor’s credit and my satisfaction, after he took his Kit Kat, he shook my hand.

Sunday, November 16, 2014

Autism Diaries: reversing heart rates

J spent at least the least the first 15 years of his life relentlessly raising the heart rates of everyone around him. In years 1-5, he'd constantly throw things around and break them; turn appliances off (the lights in the evening; the refrigerator and/or freezer) or on (the heat in summer; burners); run into other people's yards or ahead of us into the street (face turned towards us, grinning); push and grab people and poke them in the eye; and vanish in pursuit of ceilings fans. In years 5-10 he'd disrupt his classes and alienate his teachers; charge through crowded hallways and thoroughfares, force random people to sign two; bother the heck out of his siblings; and vanish in pursuit of ceilings fans. In years 10-15 he continued to disrupt his classes and alienate his teachers and charge through crowds, as well as engaging in increasingly sophisticated mischief and vanishing in pursuit of ceiling fans.

Fast-forward a few years. The mother of a dear friend who works with J has been in the ICU all week. And so she, too, has been in the ICU all week, at her mother's side. But she misses J tremendously--as she often does when separated from him for more than a few days. She misses, in particular, the fresh air and levity he provides when times are tough: his innocent questions ("How is your mom's heart?") and hopes ("When she gets better, do you think we can go to the restaurant with fans?"). So, two days ago, she asked if I could drop him off near the hospital so that she and her mother (another fan of J's) could "get some J time."

Afterwards she wrote me a text message commemorating what has to be one of the biggest milestones we've seen in his 18 years:

I am literally in tears over how sweet J is. Thanks for sharing him with us. He was a calming presence and brought my mom's heart rate down to the lowest it's been since she got here.
Who could ever have predicted that, 18 years on, J could not only provide comfort when times are tough, but enter an ICU and bring someone's heart rate down?

Friday, November 14, 2014

Math problems of the week: Common Core-inspired geometry test questions

Here is the breakdown of the Common Core standard that has inspired this problem, CCSS.Math.Content.HSG.GMD:

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Extra Credit:

(Follow-up to last week's Extra Credit question)

1. Will a student who has never heard the phrase "Cavalieri's principle" know how to proceed on this problem?

2. Should a student who has never heard the phrase "Cavalieri's principle" end up with fewer points on this problem than one who has?

3. Should a student who explains without reference to "Cavalieri's principle" why the two volumes are equal get full credit for this problem?

4. Is it acceptable to argue that the volumes are equal because they contain the same number of equal-volume disks?

5. To what extent does knowledge of "object permanence," typically attained in infancy, suffice for grasping why the two stacks built from the same number of equal-sized building blocks have equal volume?

6. To what degree does this problem test knowledge of labels rather than mastery of concepts?

Tuesday, November 11, 2014

Conversations on the Rifle Range 15: Word Problems, No Guess and Check, and a Sound Bite for an Interview

Barry Garelick, who wrote various letters under the name Huck Finn, published here, is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number 15:

I had come to the point in the chapter on systems of linear equations in my algebra 1 class where the book presented mixture, rate and current, and number problems. To prep them for the onslaught, I included a word problem into one of the warm-up problems I had them do as I checked in their homework.

The problem was: “The length of a rectangle is 3 units more than the width. The perimeter is 58 units. Find the length and width."

Students asked me “How do you do this problem?” as I came around to check their homework. I offered one hint: “You can solve it using the substitution method”.

"What does this problem have to do with the substitution method?" a boy named Lonnie asked.

I answered his question when I went over the warm-up questions. “If you solve the problem to find length and width you will have two equations in two unknowns which you can solve by substitution."

Many students shouted at once.

“I can’t answer you when you all shout at once. Raise your hands.”

I called on Lonnie. "How do you get the two equations?" he asked.

I drew a rectangle on the board. “How do I express length (L) in terms of width (W) if length is 3 more than width?”

One person ventured an answer, a very smart and quiet girl named Anna. “L = 3 + W” she said.

“Now give me an equation for the perimeter.” After some struggling to remember the formula for it, they came up with 2L + 2W = 38. From there it was obvious how substitution played a role. With this framework now in place, I segued to rate and current problems (which I discussed in Chapter 7), some of which leant themselves to solving by substitution, and others solving by elimination.

After we worked through a few problems I said: “These problems seem hard now, but the problems in the book are really all the same. If you learn how to solve one, you’ve solved them all.”

Which is one of the many complaints that math reformers have against traditional math, though I hasten to say here that such an approach is traditional math done poorly. And Holt Algebra 1 is guilty of providing problems that are exactly alike in structure and vary only in the given values in the problem. I would also like to see problems that up the ante in difficulty, so that subsequent problems force students to extend their thinking from the worked examples.

“I expect that you will find these very easy after a while,” I continued. “So easy, you’ll find them boring. And when that happens, I’ll give you some that are a bit more interesting.”

They didn’t like this idea, and Naomi, a seventh grader, asked "When will we stop doing word problems?"

"We won't,” I said.

“So the answer is ‘never’?”

“Correct,” I said.

And with that I launched into the other types of word problems. Over the next two days we covered the various word problems. For mixture problems, they learned to put the information into a particular format, whether working with nickels and dimes summing to a certain amount, coffee beans selling for various prices per pound, or mixtures of acids of varying strengths. For example:

“A grocery store sells a mixture of peanuts and raisins for $1.75 per pound. Peanuts cost $1.25 per pound and raisins cost $2.75 per pound. Find the amount of peanuts and raisins that go into a 6 pound mixture.”

The information is placed into a table and information arranged in such a way that the associated equations become obvious.

1.75*6= 10.50

What I wanted to do was to introduce harder problems of my own eventually, once they caught on and found them easy. I wanted to progress to variations such as: “A grocer blends teas worth $0.66 and $0.48 a pound. If he interchanges the amounts, he saves $6 in a blend of 100 pounds. Find the ratio of the weight of the two teas in the original blend.” This is varied enough that they cannot simply plug the values in to a handy table.

But I never got to do this. Looking back, this would have involved giving them a bunch of word problems interspersed between some of the lessons. As it was, the class had started a little behind where other classes were, so I was playing catch-up. Not to mention writing lesson plans only about a week ahead of where I was. And as it was, not as many students achieved the facility with these problems that I wanted.

And then there was the eye opening revelation that came when doing number problems. These are the type such as: “The sum of the digits of a two-digit number is 3. When reversed, it forms a number that is 9 more than the original number. Find the number.” When going through these in class the day before the quiz, I asked a girl who got the answer correct (the original number is 12) how she did it.

“I guessed,” she said.

“What do you mean you guessed?”

“I just can’t do it the way you showed us. But I can get the answer really quick if I try certain numbers out till I get the right answer.”

An argumentative girl named Sandra who did very little work and had a knack for hijacking conversations (e.g., “What’s your favorite color, Mr. G?”) came to the other girl’s defense. Her teacher last year (Mrs. Perren, the math department chair) embraced some of the math reform philosophy and Sandra apparently absorbed what her teacher told her. "Guess and check is a perfectly legitimate mathematical procedure that requires deductive reasoning to narrow down the choices," she told me.

My response: "You need to show the equations for the problems on tomorrow's quiz or it will be marked wrong. Guess and check will not be allowed."

“Even if we have the right answer?” she asked.

“Correct,” I said.

This experience would serve me well, I thought. If I ever got to interview for a teaching job and I was asked to describe how I would work within the Common Core standards, I could say “Getting the right answer isn’t enough; students have to show their reasoning” or some such language.

In the meantime, with my position firmly stated, I went over once more how to solve the number problems using algebra.

Monday, November 10, 2014

Autism Diaries: An unexpected wildcard on the SAT math test

J has just taken the SATs—hopefully for the last time. To anyone in the know, his scores—with the several hundred point gap between verbal and math sections--cry out “autism” (or, possibly, “gifted math student from a non-English speaking country who only had a few years of English instruction”). Given that J’s autism-related communication difficulties have him reading at a 7th grade level, there’s only so much his verbal scores might budge upwards. But an 800 in math is theoretically possible, if only J would avoid the sorts of careless mistakes—there are always 2-3 of these--that he invariably makes on practice tests.

The wildcard here are the 10 “student produced response questions.” Here, instead of selecting among 5 multiple choice options, the student enters a numerical answer by filling in bubbles in a number grid. Everything else being equal, these SPR questions are much easier to get wrong than the multiple choice questions: they allow, after all, a much broader range of possible answers. But they’re also easier to get wrong for stupid reasons in particular. If you misread a multiple choice question, your misreading often becomes obvious to you when you look at the 5 choices and see that none of them fits. If you misread a student produced response question, there’s only the number grid to clue you in, and it will do so only if your response is, literally, off the charts.

Mathematically speaking, the SPR questions strike me as generally easier than many of the multiple choice questions. Despite this, they’re the only sort of question J gets wrong on practice tests. He misreads the question, and then has no clue that he’s misread it.

You can sort of see why SPR questions are theoretically appealing. Multiple choice questions always get a bad rap, and SPRs, by comparison, look refreshingly open-ended. But this open-endedness goes only so far—and remains a far cry from the open-endedness one finds in college entrance exams in many other countries (for example, Finland). The end point of America’s SPR questions is still a single, right-or-wrong response, with no opportunity to show your work (not to be confused with “explaining your answer”) and get partial credit. In a way, therefore, SPR questions combine the worst of both worlds: they don’t rule out stupid mistakes, as multiple choice questions do, and they don’t allow partial credit, as truly open-ended responses do.

So I can only hope that J managed somehow to show an unprecedented level of vigilance vis a vis the 10 SPR questions that confronted him this past weekend.

Saturday, November 8, 2014

My daughter can now date Barry's daughter!

Barry Garelick once told me that anyone who wishes to date his daughter must first successfully derive the Quadratic Formula. A few days ago my daughter proved up to the task.

In fact, she was able to derive it on her own the first time around, with minimal assistance: without having first had it demonstrated to her. In Constructivist parlance, she "discovered" it! But only after lots of cumulative, guided practice placed it squarely within her Zone of Proximal Development. (Cumulative, guided practice of the sort that, incidentally, is entirely missing from Reform Math Algebra, which, if it asks the students to learn the formula at all, has them do so not via conceptual understanding, but via rote memorization).

My daughter's cumulative, guided practice included solving dozens of quadratic equations of varying complexity (including so-called "literal equations" in which the coefficients themselves are variables): first by factoring, and then by completing the square. I credit in particular her honing the technique of multiplying the equation by four times the co-efficient of the squared term before completing the square. Not only does this simplify the process by eliminating the need for fractions; it also makes the Quadratic Formula derivation a tad more elegant.

It's the difference between this:

And this:

Or, in her own hand:

Of course, either way works--whether for handling quadratics, or for dating Barry's daughter.

Having to resort to rote memorization, on the other hand, substantially limits your future prospects: both mathematical and romantic.