American Mathematics Competitions (AMC 8 & AMC 10)
On Tuesday, November 17th, several of our 7th and 8th grade students will participate in the American Mathematics Competition 8 (AMC 8), a worldwide math competition. The competition will consist of a 40-minute, 25-question test to be given during the participating students’ math class.
In February, my Euclidean Geometry class will take the AMC 10. This contest is a 75 minute test.
Check back later for practice problems for both competitions.
Politically Incorrect Math Problem
A man and his wife are expecting their first child when he finds out that he has a terminal disease and will not likely live to see the birth of his child. He writes a will with the following conditions: 1) If the child is a boy, the man’s assets will be divided such that the boy will receive 2/3 and the mother will receive 1/3. 2) If the child is a girl, the mother will receive 2/3 of his assets and the daughter will receive 1/3.
The man dies, and the mother gives birth to…twins–one boy, and one girl.
You are the judge who will decide how the man’s assets will be distributed. What fraction of the man’s assets would you give each individual, remembering that your job is to honor the deceased man’s intent?
I believe I originally saw this problem in Twenty Years Before the Blackboard by Michael Stueben or in Dr. Wilson’s EMAT 6600 course at UGA (I forget which came first.)
Where’s the missing dollar?
This is an old problem…but here goes.
Three businessmen decide to save money on a business trip by sharing a hotel room. The men are charged $30 for the room. They each pay with a $10 bill and retire for the evening. The hotel manager then notices that she should have only charged the men $25 for the room. She gives the bellhop 5 one-dollar bills and gives him instructions to take the men the $5. On the way to the room, the bellhop decides that he will help himself to a $2 tip. (Five isn’t divisible by three anyway.) He gives the remaining money to the men, and they each take $1.
At this point, the three men have each paid $9 for the room. 3x$9 = $27. The bellhop has $2 in his pocket. $27 +$2 = $29.
Where is the missing dollar?
Monty Hall Problem
Read the following problem and think of your answer before reading the rest of this post.
“Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?”
Well, what do you think? Should you switch, stay, or does it matter?
I first saw this question when I was in college. It appeared in a 1990 Q&A feature of PARADE magazine. Marilyn Vos Savant, the author of the feature, correctly advised that contestants should switch their choice to double their odds of winning. She was subsequently deluged with faxes and letters (no e-mail at that time) telling her she was wrong. Some of these letters were from Ph.D.’s in mathematics. She wrote a couple of follow-up articles, and most readers eventually seemed to be convinced that she was correct.
Link with the full original article, responses, and follow-up articles The responses from Ph.D’s and others (all men, BTW) who rather rudely tell her she’s wrong are hilarious. Especially since she’s correct!
Simulation Click here to try out a simulation of the problem.
Article on Robotics Team
Click here to read an Athens Banner Herald article on the robotics team. Mr. Herring, Andrew Herring, and Brandon McFarlin are quoted.
Calculus TV
Follow this link to some excellent Calculus videos that Kyle found today. An example is shown below…
Looks like Kyle put a sick day to good use. Thanks for sharing! Let me know if anyone else finds similar videos or helpful sites.
Congratulations, ACS Robotics Team!
The ACS Robotics team captured first place at the state competition this weekend. They were also honored as the best rookie team (obviously). Congratulations!
Should football teams punt on fourth down?
You are the coach of a high school football team. Your team has the ball on its own five-yard line, and it’s fourth and eight. Do you punt?
Not if your name is Kevin Kelly, the head coach at Pulaski Academy in Arkansas. His team won the 5A championship in 2008. They didn’t punt a single time. You can read his story here. To add to the drama, they also went for an onside kick 75% of the time!
His reasoning was as follows. He figured that his team would convert on fourth down at least 50% of the time. Statistics show that opposing teams that get the ball inside the opponent’s ten-yard line score 90% of the time. By going for it on fourth down in that particular situation, he statistically gives up a touchdown 45% of the time. If his team punts instead and the opponent starts on the 38 yard line (assuming an average punt with return), statistically the opposing team will score a touchdown 77% of the time. Thus, he is better off to go for it on fourth down, even on his team’s side of the field.
Similar reasoning led him to the conclusion that if his team failed to recover the onside kick, on average they were only giving up 14 yards. Pulaski ran seven different versions of on onside kick throughout the season.
If you read the article, you’ll notice that Pulaski based his strategy on the work of several mathematicians/economists, as well as taking into account the abilities of his current players. Apparently, the application of mathematics to football decisions is becoming more commonplace. Remember, if you take care of the math, the math takes care of you!
What do you think? Do you think you would trust the math, or stick with conventional wisdom?
Just leave every question on the SAT blank!
Juniors and sophomores at ACS recently took part in an annual tradition known as the PSAT. The PSAT, of course, is simply a warm-up to the SAT which these students will be taking in preparation for applying to the college(s) of their choice.
One concern I have is that so many students talk about leaving large numbers of questions blank if they didn’t know the correct answers. It appears that some test preparation books/services are issuing the edict, “don’t guess if you don’t know the answer.”
Problem: too many students made the comment, “If you leave a question blank, you aren’t penalized.” The perception seems to be, leaving questions blank is a good thing.
Let me ask the question: If you aren’t “penalized” for leaving questions blank, why not leave EVERY answer blank? Seems to me, you’d make a perfect score.
It is my contention that students ARE penalized for leaving questions blank.
Let me explain. The mathematics portion of the SAT has 44 multiple-choice questions. You receive one point for each question that you answer correctly. If you answer every question correctly (which is possible, by the way), you would receive 44 points onto your raw score. (The raw score is later converted into the score between 200-800.) For each question that you do not answer you receive no points (which sounds an awful lot like having a one point penalty per blank answer. Because it is.) For each question that you attempt, but are incorrect, you have 1/4 of a point deducted from your raw score.
You could look at the SAT multiple-choice math section this way. You start with 44 raw points. For each question that you leave blank, you have one point subtracted. For each incorrectly attempted answer, you have 1.25 points deducted from 44.
Bottom line: leaving answers blank is not ideal. You should try your best to finish, and you should try your best to answer every question.
Anticipated Followup Questions
- Why is an additional 1/4 of a point deducted for a question that is completed incorrectly?
Answer: To adjust for the fact that you have a 1/5 probability of answering the question correctly if you simply guess. Example: Suppose you have a 40-question multiple-choice test, and each question has five possible solutions. Suppose you filled in the bubbles at random without even reading the questions. You could expect to answer 8 (that’s 1/5 of 40) of the questions correctly! However, you would have answered 32 of the questions incorrectly. 1/4 of 32 is also 8, so that number would be subtracted from your total answered correctly, leaving you with zero. Thus, you would rightfully receive the same number of points as someone who left every answer blank.
- What if I don’t know the answer to a question, should I guess?
If you absolutely don’t know the answer to a question, then no. You’ll waste time coloring in the little bubble. However, before totally giving up on a question, look at the answers and try to eliminate at least one of the solutions. You may not know the answer to the question, but can you identify one of the answers as being obviously incorrect? If so, you should guess at the remaining choices. You just switched the odds ever so slightly in your favor.
- What if I’m having trouble finishing the test?
Don’t feel too badly about this. There are some careers where slow methodical thinkers are preferred. If I’m having brain surgery, for example, I do not want a surgeon who’s racing to finish before the guy operating next door. However, if you are a slow test taker, there are some things you can do:
- Take TIMED practice tests. Several practice tests are available online and at bookstores. Practice one or two of these while someone else times you. I know that online/computer tests are popular, but studies show that students who use pencil/paper practice tests score higher. (The idea is, you’re practicing using the methods that you’ll use on the actual test.)
- Complete the easier problems first. If you’re struggling with a question, skip it until you’ve answered other easier questions. In general, the easier questions appear at the beginning of each section.
- Be aware of the time. I realize that no one wears watches anymore, but you might consider wearing a watch on test day, preferably one with a stopwatch. At the very least, find out where the clock is before the test begins.
- Don’t waste time coloring. Seriously. Some students spend too much time filling in the little bubbles. In fact, some test preparation services have clients practice filling in sheets of bubbles. Be aware of the time that you’re coloring. Saving three seconds a question gives you over two minutes just on the mathematics MC questions. One thing that helps is using pencils that are a little dull. Also, the bubbles don’t have to be colored perfectly. (I know this bothers some of you who are a little OCD, but it’s true.)
- What is the best strategy for success on the PSAT/SAT?
Answer: Take challenging classes. Study hard. Read a lot. Practice, practice, practice. Sorry, but there really is no substitute for hard work.
- Mr. Mealor, why the diatribe?
First of all, “diatribe” is an excellent vocabulary word that might appear on the SAT Verbal. Secondly, your goal should be to make a perfect score! (Why not?) Ok, maybe that’s not realistic for every student, but hopefully you have a goal of a relatively high score. If you’re leaving a lot of answers blank, it ain’t gonna happen. (“Ain’t” and “gonna” will most likely NOT appear on the SAT verbal.)
Related topics:
SAT MATH question of the day (Yahoo!)
More EC, but there’s a catch…
First student to e-mail the correct answer gets 5 extra-credit points, second one gets 4, and so on…
Deadline: E-mails must be time stamped before 8 A.M. tomorrow.
Problem: Find the slope of the line perpendicular to the tangent line of g(x) = sinx at x = 0.234. No work required.
