My Top Eight Teachable Moments Dan Kennedy Baylor School “A teachable moment, in education, is the time at which learning a particular topic or idea becomes possible or easiest.” Moment #1: The Magic Numbers for Retail Wine Pricing Take the wholesale cost of a case of wine. Add the excise tax ($1.21/gallon today). Divide by 12 to get the bottle cost. Add 30% retail markup. Discount the price 10% (a store policy). Oh, how he yearned for the ability to accomplish all this bothersome math in a single step! Praying I would not make this look too simple, I did a little Algebra I on the back of a paper bag… (x + 1.21 * 2.37755) ÷ 12* (1 + .30)* (1 - .10) = 0.0975 x + 0.28049 So, I told him sagely, all you need to do is multiply your wholesale case cost by 0.0975 and add 0.28049. I thought the man was going to cry. He later mounted the paper bag on the wall of his office: Price * 0.0975 + 0.28049 I also walked away with a free bottle of champagne…and a new respect for Algebra I! Moment #2: “Magic” Math E-mails from the Clueless How many of us have received this e-mail from friends wondering what sorcery is behind this trick? 1. Pick the number of times a week that you would like to eat chocolate (try more than once and less than 10). 2. Multiply this number by 2. 3. Add 5. 4. Multiply this number by 50. (You might need a calculator). 5. If you have already celebrated your birthday in 2010, add 1760. Otherwise add 1759. 6. Now subtract the four digit year that you were born. You should have a three digit number left! The first digit is your original number of how many times you want to eat chocolate each week. The second two numbers are: YOUR AGE. (Oh Yes it is!!!!!). Amazingly, 2010 is the ONLY YEAR that this incredible trick will work! This is a wonderful Teachable Moment for algebra teachers! 1. Let d be the number of days a week I want to eat chocolate 2. Double it: 2d. 3. Add 5: 2d + 5. 4. Multiply by 50: 100d + 250. (Who needs a calculator?) 5. Add (for me) 1759: 100d + 2009. 6. Subtract (for me) 1946: 100d + 63. (Don’t try this if you’re older than 99!) Moment #3: My college French grade from Boom Boom Fortier* (*Not his real name) Grade on first test: 72 Grade on second test: 80 College policy: Exam grade counts 1/3 of semester grade I figured that I could make a B with an 88 on the exam. I studied hard and nailed the exam. I got a C for the semester. I went to see Mr. Fortier in his office. He congratulated me on my many trips to the language lab. “I decided not to count those, though. Not enough people went to the lab.” “What about the exam?” I asked. He looked in his grade book. “Let’s just say you got a 92 on the final exam. That still doesn’t get you a B, because the final only counts 1/3 of your grade.” Mr. Fortier showed me the math. There were two tests in the semester, each of which he counted 2/3. The final counted 1/3. 72 72 tw o th ird s 80 80 tw o th ird s 9 2 on e th ird 396 5 = 79.2 = C 3 9 6 sem ester to tal Convinced it would make no difference, he grumbled and agreed to try it my way. Semester test average: 72 + 80 2 = 76 76 + 76 + 92 = 244 244 3 = 81.33 = B He stared at the numbers for a few moments of apparent confusion before opening his grade book again. “Of course, there’s no way of saying for sure that you got a 92 on the exam. Maybe you got an 82.” I assured him that there was no need to crunch the numbers again. I thanked him for his time and left. Moment #4: Scaling Grades on the TI Calculators Something I learned about assessment from the AP program: It is perfectly OK, perhaps even necessary, to scale grades! AP Grade Conversion Chart Calculus AB Composite AP Grade Score Range* 75−108 5 58−74 4 40−57 3 25−39 2 0−24 1 *The candidates' scores are weighted according to formulas determined by the Development Committee to yield raw composite scores; the Chief Faculty Consultant is responsible for converting composite scores to the 5-point AP scale. 75% =5 At our school, 75% is not a good grade. In fact, 65% is a minimal pass. Is this reasonable? Think about it. •The all-time NBA record for field goal percentage in a season is 72.7%. •The all-time record batting average for major league baseball is .440 (44%). •A salesperson who makes a sale on 75% of first contacts is a genius. So how can we expect 75% success from someone who is just learning? 99 92 82 • • 71 • 30 • 20 75 • 93 An Important Disclaimer: Scaling grades is not about building self-esteem. Scaling grades is about teaching mathematics. Assessment should support your efforts to teach your students mathematics. It should not get in the way. ClrHome:FnOff PlotsOff :ClrTable:ExprOff 6 Xmin:100 Xmax 0 Ymin:124 Ymax 0 Xscl:0 Yscl Input "RAW SCORE: ",A Input "CURVED TO: ",B Input "RAW SCORE: ",C Input "CURVED TO: ",D (B−D)/(A−C) M "round(MX+B−AM,0)" Y1 IndpntAsk DispGraph Text(1,1,"TRACE OR USE TABLE") Text(7,1,"TO ENTER RAW") Text(13,1,"SCORES.") Scaling grades on the TI-84 Plus I showed this calculator program to my Baylor colleagues during an in-service training on assessment. The English teachers were excited about its potential for grading papers. Suddenly our students had a ready market for their old TI-82 calculators! Moment #5: Baylor School’s Graduated GPA The Challenge: Design a sliding scale that our school could use to convert our numerical (percentage) grades to grade-point averages on a 4-point scale. The assumptions I made: 1. Our lowest D (65) should get 1.0. 2. An average A (95) should get 4.0. 3. The GPA curve should be steeper at the low end than at the high end I decided to use a power function of the form x - 65 G(x) = 3 30 1/ p + 1. (65, 1.0) I gave the faculty a choice of curves for various p-values, and the runaway winner was p = 1.7. p-value 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 65 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 70 1.5 1.6 1.7 1.8 1.8 1.9 2.0 2.0 2.1 2.2 2.2 75 2.0 2.1 2.2 2.3 2.4 2.4 2.5 2.6 2.6 2.7 2.7 80 2.5 2.6 2.7 2.8 2.8 2.9 2.9 3.0 3.0 3.1 3.1 85 3.0 3.1 3.1 3.2 3.2 3.3 3.3 3.4 3.4 3.4 3.4 90 3.5 3.5 3.6 3.6 3.6 3.7 3.7 3.7 3.7 3.7 3.7 95 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 100 4.5 4.5 4.4 4.4 4.3 4.3 4.3 4.3 4.3 4.3 4.2 104 4.9 4.8 4.7 4.7 4.6 4.6 4.5 4.5 4.5 4.4 4.4 Grade Baylor continues to use this sliding conversion today, over 25 years later. It continues to provide teachable moments for educating students, faculty, administrators, and, of course, parents! x - 65 G(x) = 3 30 1 / 1.7 +1 Moment #6: My Year with NUMB3RS January 2005: CBS premiered a new show in which an FBI agent and his mathematician brother solved crimes using mathematics. It was called NUMB3RS. Chuck Biehl (Charter School of Wilmington, DE) Br. Patrick Carney (Depaul Catholic HS, Wayne, NJ) Pat Flynn (Turner HS, Kansas City, KS) David Bressoud (Macalester College, MN) Ron Lancaster (University of Toronto, ONT) Dan Kennedy (Baylor School, Chattanooga, TN) Tom Butts (University of Texas at Dallas, TX) Jonathan Farley (Stanford University, CA) Terry Wyberg (University of Minnesota) Johnny Lott (University of Montana) Terry Souhrada (University of Montana) Kathy Erickson (Monument Mountain Regional HS, MA) Ed Burger (Williams College, MA) Sue Eddins (Illinois Mathematics & Science Academy) Karen Longhart Lenda Hill Brett Morrow Heather Gunsallus The NUMB3RS Team at NCTM Why is this man smiling? In the episode “Double Down,” some college students beat a casino at Blackjack by cracking the algorithm used by the automatic shuffler to randomize the cards. Eventually, of course, there was also a murder. Which of the following sequences of black and red cards is most likely to result from a random shuffle? BRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBRBR RRRRRRRRRRRRRRRRRRRRRRRRRRBBBBBBBBBBBBBBBBBBBBBBBBBB BBBRRRBRRRRRRRRRBBBBRRRRRBRBBBBBRRRRBBBBBBBBBRBBBRRR Answer: All three are equally unlikely! Each has about one chance in 496 trillion of occurring. Which of the following lottery tickets is least likely to be a winner? 11 12 13 14 15 16 09 13 22 25 31 43 17 21 33 34 38 41 The odds against each: 7,059,051 to 1. The activity Now You See It, Now You Don’t introduced students to a method of randomly coding pictures, called steganography. Moment #7: Twinkle, Twinkle, Little Star One of the neatest math articles I ever read was a piece by Martin Gardner in the September 1998 issue of Math Horizons. He called it “Ten Amazing Mathematical Tricks.” Twinkle, Twinkle, little star; How I wonder what you are, Up above the world so high, Like a diamond in the sky; Twinkle, twinkle, little star; How I wonder what you are. 7 7 6 4 3 1 6 4 3 4 2 5 3 5 2 4 4 1 7 2 3 3 7 7 6 4 3 1 6 4 3 4 A few months later I attended an entertaining NCTM session on “Mathematical Magic” at which the speaker showed this trick. He admitted that he had no idea why it worked but that he would love to know the secret. After the talk, I shared my discovery. It was another Teachable Moment! Moment #8: Rebecca Flake’s Portfolio Entry Rebecca Flake In my classes, each student hands in a portfolio of items. The students choose the items they want me to see and grade. The main point of this assessment is that they are not responding to a stimulus from me (as in a test or a quiz). My primary directive for student portfolio entries is this: Give me evidence of your learning that I otherwise would not have! This was my first year to be a peer tutor, and I enjoyed helping the girls in the dorm a lot. Last night, though, I finally saw the importance of my peer tutoring. My roommate came in at 10:00 extremely upset over her Precalculus test that was the next day. I calmed her down and told her that I would help her if I could. Carrie, who had been in the play, had gotten behind in her work, so she didn’t understand what they were doing. She showed me the problem. I knew the answer, but I wasn’t sure how to explain it to her in a way that was not confusing. I thought about it for a while, and I ended up trying several approaches (with Clara’s help) that I had learned in Calculus, until I finally got through to her. Then I made her work a few problems for me, and she did them perfectly. She understood! I was so happy to be able to help her that I had forgotten I was supposed to be studying for my own Calculus test. She was so happy she understood that she began to cry. She really began to cry. It’s great to be able to use the things you have learned to help other people learn too. A happy footnote: Carrie really did understand. She scored 93 on the Precalculus test the following day – a personal best for her, and a full 9 points above the class average. Actress Carrie Rebecca had obviously had a wonderful Teachable Moment with Carrie. Thanks to her portfolio entry, she also had a wonderful Teachable Moment with me! I wish each of you a life filled with Teachable Moments!