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 
 two thirds
72 
80 
 two thirds
80 
92 one third
396 semester total
396  5 = 79.2 = C
Convinced it would make no difference, he
grumbled and agreed to try it my way.
72
+
80
Semester test average:
= 76
2
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
1/ p
 x - 65 
G( x ) = 3 

 30 
+ 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!
1/1.7
 x - 65 
G( x ) = 3 

 30 
+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!