Unit Title:
The Efficiency Expert
Day 1
You have been hired by the ACME Congruent Triangle Company to increase their productivity.
ACME Congruent Triangles are known to geometry students worldwide as the gold standard for
congruent triangle production. Each type of congruent triangle they produce has all of its sides
and angles carefully measured to make sure all of the triangles in a batch are indeed congruent.
However, the company has just been purchased by Bain Capital, and their boss, known to you
simply as “The Governor” doesn’t like what he sees.
“Your measurement costs are way out of line”, he says to you. “We are making six separate
measurements on each and every triangle we build. I need you to cut our measurement expenses
in half, or else I will have no choice but to fire everyone and close the factory down! But don’t
compromise the quality at all – we need to be sure the triangles are congruent. When you come
up with some ideas, you will need to demonstrate that they will work.”
“But the news isn’t all bad. If we can indeed get our costs down, I see some opportunities for
expansion. I have some clients interested in various quadrilaterals, and I think we can use our
congruent triangles to make them.”
“I will expect daily reports on your progress. Good luck!”
Being a good manager, you decide to consult with the employees in the measurement department. You give them the bad news that initially the department will be cut in half, but that
expansion into new markets may allow for some rehiring later on. You quickly find out that the
employees are divided into two camps – side advocates and angle advocates. Each camp wants
to retain all of their people and fire everyone else. As the efficiency expert, your task is to see
whether each method (or perhaps both) will work. Measuring only the sides will be referred to
as SSS, for Side-Side-Side, and measuring only the angles will be referred to as AAA, for AngleAngle-Angle. Also, if one of the methods does work, you will want to see if a similar procedure
will work for quadrilaterals.
Please follow the Task #1 worksheet to see if SSS or AAA is a feasible method for triangle
congruence.
Day 2
After sending your report for Day 1, The Governor meets with you via a teleconference. He
seems pleased. “You made a good start yesterday”, he says. “The SSS method that your team
devised will allow us to cut measurement time and expenses in half. But I have some
reservations. As you showed in your report, the method won’t work if we get into the
quadrilaterals market. I also think we need to retain at least some angle measurement capability
– I don’t want to fire all of my angle measurers. If we can develop some shortcuts involving
angle measurement that will increase our flexibility and keep the angle people engaged. Go back
to your team and develop some more techniques.”
You return to your measurement team and tell them the news. “The angle measurers and the side
measurers will need to work together to develop new shortcuts.” The angle measurement people
are relieved, as they were facing the elimination of all of their jobs. The side measurers are
upset, though, as they thought all of their jobs were safe. You finally get them to agree to work
with the angle people, but they still would like two side measurements, and only one angle
measurement.
After talking with the team some more, everyone agrees that there are 2 possibilities. The first
possibility is that the congruent angle will be between the congruent sides. The team decides to
call this SAS, for Side-Angle-Side. The other possibility is that the congruent angle is not
between the congruent sides. The team has trouble agreeing on a name for this. Some members
want to call this the ASS method, for Angle-Side-Side. This leads to so laughter and some
inappropriate comments. Others want to call this SSA, for Side-Side-Angle. You finally make
an executive decision on the matter. “The Governor is a very religious man. We can’t take the
chance that he will find ASS offensive. So, the name of the technique will be SSA. Now go, get
started, and see if these techniques even work!”
Please follow the task #2 worksheet to see if SAS or SSA is a feasible method for triangle
congruence.
Day 3
At the daily teleconference, The Governor is still cheerful. “Your team is doing good work,” he
says. “The SAS method gives us another way of proving that our triangles are congruent, which
increases our flexibility. “But there are a couple of things that concern me. First of all, why did
you call the other method, the one that didn’t work, as SSA? Wouldn’t ASS be a better acronym
for that procedure?”
“Well, er., um, the thing is that …”
“I know, it looks naughty, doesn’t it? But you need to realize that the first rule of acronyms is to
make them memorable. I doubt anyone would remember SSA, but I’m sure everyone would
remember ASS. As long as you say the letters, or say angle-side-side, I don’t see any problem
with it. But in the end, the point is moot because the method doesn’t work anyway.”
“The other thing that concerns me is far more important. We now have a little bit of angle
measurement, but between SSS and SAS we are still doing far too many side measurements.
This gives the side measuring employees too much power and I don’t like that. Go back to your
team and see if we can develop some techniques that involve 2 angle measurements and only 1
side measurement. Meanwhile, I’ll let you in on a little secret. My marketing research staff is
looking into new applications for our triangles. We should have the results soon.”
Once again, you report back to your measuring team. “We need to develop congruence
techniques involving 2 angle measurements and only 1 side measurement.” This makes the
angle measurers happy – more of their jobs can be retained. They are happy to help and quickly
come up with 2 possibilities. The first possibility is that the congruent side is between the
congruent angles. This will be referred to as ASA or angle-side-angle. The other possibility is
that the congruent side is not between the congruent angles. This will be called AAS for angleangle-side.
Please complete the task 3 worksheet to see if ASA and/or AAS is a feasible method for triangle
congruence
Day 4
“So, you seem to have covered all of the possibilities for measuring 3 parts of a triangle to show
congruence,” says The Governor at the daily teleconference. We’ve now got 4 techniques to
demonstrate triangle congruence, and the ASA and AAS techniques will allow more flexibility
and let us keep more of our angle measurement people. Say, is there any chance we can make
only 2 measurements on our triangles to prove they are congruent?”
“Well”, you reply, “we had a debate about this in the Right Triangle division. They were
claiming they could prove congruence with just 2 measurements – the hypotenuse and one of the
legs. They call it the HL method”
“Sound promising – what was the problem?”
“We still need to measure and make sure the triangles are right triangles. This angle
measurement would be a third measurement, and so it doesn’t save us any money.”
“But isn’t this HL method the same as the SSA (or ASS) method you said did not work?”
“Yes, sir. But it turns out that SSA (or ASS) works on right triangles, but no others. Internally
we are calling this right triangle method HL, so there is no confusion.”
“I see. So, it looks like we are as efficient as we are going to get with our congruent triangle
measurements. Now, in order to grow our business, we need to reduce waste. I see we are
throwing away approximately 2.87% of our triangle sides. Why is that happening?
“Our triangles need to meet stringent standards for congruence. When a side exceeds our
tolerance limits and is too short or too long, then we can’t use it.”
“OK, then. What would you say if we could bundle some of these scrap sides together to sell.
Our marketing people think that there is an opportunity to sell ’build it yourself’ triangles. Some
people want to make their own triangles and don’t care if they are congruent or not. Tell me, do
you think we can take any 3 sides and make a triangle out of them?”
You think for a minute, but are not sure of the answer. You tell the Governor that your research
team will need investigate. You end the meeting be promising to have an answer for him the
next day.
Please complete the Task #4 worksheet to see if we need to develop a criterion for being able to
make triangles.
Day 5
At today’s teleconference, the Governor’s expression is sterner than in the past few days. When
he explains, you understand why. “I was really hoping that we could take any 3 side lengths and
sell them as triangle kits. I could have then hired some minimum wage workers to package them
up and we wouldn’t have needed to worry about which sides went into each kit. I still think we
can make some money on this, but the workers will need to understand the triangle inequality
equation you developed, and that means I’ll need to pay more for skilled workers.”
His expression softened. “But don’t get me wrong – I’m glad that you discovered the inequality
criteria before we started selling our kits. I’d hate to have to deal with a massive triangle recall
and the bad publicity and expensive lawsuits that would have ensued.”
“But anyway, on to bigger and better things. Our marketing department says that there are
opportunities to sell special quadrilaterals, including parallelograms, rhombuses, squares,
trapezoids and kites. Could we make any or all of these by combining two congruent triangles?”
“Rhombi”, you reply. “The plural of rhombus is rhombi, not rhombuses.”
“I see”, he says. “But rhombuses is clear enough, isn’t it. In business, as well as in politics, its
often more important to be clear, than to be strictly grammatically correct. Most people won’t
get the meaning of rhombi, but everyone understands what rhombuses are.” And besides, you
haven’t answered my question. Are you stalling on me?”
You admit that you don’t know the answer, and that your research team will need to look into it.
You promise an answer by the next day.
“I see your political skills are improving. When a politician doesn’t know or doesn’t want to
give a direct answer to a question, they change the subject. One more thing – we need to let our
prospective customers know the properties of the quadrilaterals we are creating. Try to use what
you know about congruent triangles to determine the properties of these quadrilaterals. Now get
to work!”
Please complete the Task #5 worksheet to see what special quadrilaterals can be created by
combining two congruent triangles.
Day 6
At the daily video conference, you are surprised that The Governor is not there. Instead, his
executive assistant peers back at you from the screen. “The Governor appreciates your
continuing good work, and sends his regrets that he can’t meet with you personally. I just have
one question about your latest report. You were able to make all of the special quadrilaterals the
Governor requested except one. Why couldn’t you make a trapezoid for us?
“Actually we can,” you reply, “but it didn’t prove to be cost effective. We could make a
trapezoid by overlapping two congruent triangles sharing a side, and then adding a cross brace to
create a base parallel to the shared side.. But, we would have needed to cut the triangles up in
order to get them to fit on a single plane. The special equipment we need for that would be
expensive, and we didn’t think we could make a profit on it. Trapezoids are a small market
compared to rectangles, parallelograms and squares anyway.”
“Good thinking,” says the flunky, err, executive assistant. “I’ll pass that on. By the way, just
between the two of us, how do you feel about higher sided polygons? Could you see yourself
working in that area someday?
“Why do you ask? Is there something I need to know?”
“I’m not at liberty to say. But I think you will be very pleased when you do find out. But in the
meantime, be thinking about pentagons, hexagons, etc. and how your efficiency team might deal
with them.
Epilogue
The expansion into the quadrilaterals market proved to be a great success. Total sales
skyrocketed, and the ACME Congruent Triangle Company showed a huge increase in quarterly
profits. Total employment at the factory increased, although a few measurers still lost their jobs.
Then one day, you are called to Bain headquarters for a face to face meeting with The Governor.
As you enter the meeting room, you are surprised to see someone you recognize from lectures at
the Annual Shapes Expo: Wright Angle, the founder and CEO of Polly Gone, the largest
worldwide manufacturer of straight-sided shapes. Polly Gone is your chief rival, especially in
the low end scalene triangle markets. Their products are lower quality than ACME’s carefully
measured triangles (and now quadrilaterals), but they cost much less than yours.
“Great news!”, the Governor says, “Polly Gone has agreed to purchase ACME Congruent
Triangle Company. Mr. Angle has assured me that as long as you continue to meet performance
and profitibility goals, he intends to keep the factory open. You have demonstrated that good,
quality products can be produced without excessive measurement costs, and your high-end
triangles and quadrilaterals should mesh nicely with his er, um, more basic product line. As a
key employee, Mr. Wright is prepared to offer you a substantial salary and stock options
package. You will need to sign a non-compete clause, but that’s standard in these type of
situations. Of course, I’ve made another boatload of money on the deal as well. By the way I’m
thinking of starting a political career. Do you think I would make a good politician?”
You turn to Mr. Angle. “May we discuss the details of my new responsibilities and my
compensation?”
The Governor laughs. “See, our friend here is already executive material. Changes the subject
whenever there is an awkward question!”
So, what questions do you have for Mr. Angle? Do you think you will continue with the
company? Finally, do you think Bain Capital was a positive or a negative force in this scenario?