MATH AND THE OLYMPICS:
AN INQUIRY UNIT
Unit designed and taught by
Denise Anthony and Cynthia Callard
Documentation prepared by Raffaella Borasi
SELECTED DOCUMENTATION
List of attached documents:
Note: The code following the title of each document indicates the
section of the unit (first letter) and then the subsection (or day) within
that section when the document was generated/used.
- List of winter sports generated by the students and recorded on newsprint.
(a1)
- Newspaper article briefly describing the sports played at the Winter
Olympics. (a1)
- Student-generated list of how math is used in the Olympics. (a1)
- Olympic unit initial survey. (a1; f2)
- Two examples of students' responses to the Olympics survey. (a1)
- Handout guiding students' initial estimation of what sport is "fastest"
(b1-3)
- Excerpt of a reading about speed records in various sports (b1-3)
- Homework sheet asking students to create a double bar graph to compare
estimated speeds and actual speeds in various sports (b1-3)
- Homework asking students to read and discuss an article on using the
metric system (b1-3)
- "Warm-up" work sheet asking for some conversions in the context
of speed skating (b1-3)
- Hand-out to guide the comparison of average speeds in 2 different speedskating
events (b1-3)
- Report of a class discussion on what learned about speed (including
a conceptual map created by the teacher based on students' input) (b4-5)
- Record of teacher's modeling how the proposed framework for the project
could be used to summarize the results of their inquiry around "Who's
fastest" (b4-5)
- Reading on scoring criteria in figure skating (c1-2)
- Handout created to help students organize the tabulation of the scores
assigned by each student (c1-2)
- "Warm-up" asking students to order and find various averages
using data from the Men's Mogul race (c3-4)
- Handout created to structure a first comparison between two alternative
medal counts (d1-2)
- Report of a 10-minute class discussion on alternative medal rankings
(d1-2)
- Excerpt from an article explaining key elements of the biathlon (e1)
- Handout articulating a measuring task designed to help students better
understand what is involved in the biathlon (e1)
- "Warm-up" asking students to compare the size of various
kinds of skis (e1)
- Ideas about measurement within the Olympics (e2)
- First handout describing project requirements (f1)
- Further clarifications provided about project requirements and evaluation
(f1)
- Project checklist (f1)
- Form for recording the question informing each pair's project and the
data they would gather to address it (f2)
- List of questions generated by the students for their individual projects
(f2)
DOCUMENT 1.
List of winter sports generated by the students and recorded on newsprint.
[Introduction to the Olympics -- Day 1]
Winter Sports
- mogul skiing
- bobsledding - 4 or 2 men teams
- luge - 1 or 2 men (laying down)
- downhill skiing - fastest (time)
- freestyle skiing - jumping
- speed skiing
- speed skating
- ski jumping - longest (furthest)
- hockey
- figure skating - singles/doubles
- cross-country
- biathlon
|
DOCUMENT 2.
Newspaper article briefly describing the sports played at the Winter Olympics.
[Introduction to the Olympics -- Day 1]
DOCUMENT 3.
Student-generated list of how math is used in the Olympics.
[Introduction to the Olympics -- Day 1]
How is math used in the Olympics
- scoring
- points for medals
- averaging scores
- time - more than 1 run
- distance
- speed
- "hang time"
- height of jump
- height/weight/ age of athletes
- integers - hockey - goals/penalties
- statistics
|
DOCUMENT 4.
Olympic unit initial survey.
[Introduction to the Olympics -- Day 1; Project development]
Olympics Unit Survey:
Answer each of the questions below. Please take the time to answer every
question carefully as we will be using your responses to help
us develop the unit to include your interests!
1. What Winter Olympic sports do you enjoy watching:
1.
2.
3.
2. Tell us about a Winter Olympic sport that you know alot about (the
athletes, the sport itself, etc.)
3. If you were a sport reporter assigned to the Winter Olympics, what
questions would you be interested in investigating and reporting about?
1.
2.
3.
4. For a final project for this unit, you will be working with a partner
to investigate a question that you have about the Winter Olympics, List
TWO people below that you could work with on this project. (NOTE: This project
will require you and your partner to spend time together out-of-class to
work on it.)
|
DOCUMENT 5.
Two examples of students' responses to the Olympics survey.
[Introduction to the Olympics -- Day 1]
Olympics Unit Survey:
Answer each of the questions below. Please take the time to answer every
question carefully as we will be using your responses to help
us develop the unit to include your interests!
1. What Winter Olympic sports do you enjoy watching:
| 1. |
bobsledding |
1. |
figure skating/ ice dancing |
| 2. |
luge |
2. |
luge/ bobsledding |
| 3. |
ice skating |
3. |
trick jumping |
2. Tell us about a Winter Olympic sport that you know alot about (the
athletes, the sport itself, etc.)
|
I know that for bobsledding there are groups of two or four, and they
go across an ice covered track. They reach very high speeds, and race to
beat the highest time. |
|
None |
3. If you were a sport reporter assigned to the Winter Olympics, what
questions would you be interested in investigating and reporting about?
| 1. |
If the team members get dizzy going around the track (bobsledding/luge) |
1. |
What is it about the sport that you nejoy? |
| 2. |
How many years of practice goes into the sport |
2. |
When did you first start --- ? |
| 3. |
A day in the life of an athlete |
3. |
What went through your mind when you knew you were going to the Olympics? |
4. For a final project for this unit, you will be working with a partner
to investigate a question that you have about the Winter Olympics, List
TWO people below that you could work with on this project. (NOTE: This project
will require you and your partner to spend time together out-of-class to
work on it.)
DOCUMENT 6.
Handout guiding students' initial estimation of what sport is "fastest"
[Who's fastest? -- Days 1-3]
DOCUMENT 7.
Excerpt of a reading about speed records in various sports
[Who's fastest? -- Days 1-3]
DOCUMENT 8.
Homework sheet asking students to create a double bar graph
to compare estimated speeds and actual speeds in various sports
[Who's fastest? -- Days 1-3]
DOCUMENT 9.
Homework asking students to read and discuss an article on using the metric
system
[Who's fastest? -- Days 1-3]
DOCUMENT 10.
"Warm-up" work sheet asking for some conversions in the context
of speed skating
[Who's fastest? -- Days 1-3]
Math 8
Warm-up
When Cathy Turncr qualified for the Olympic team in speed skating, she
broke two American records in the 1000 meter and the 500 meter races.
1. Her time in the 1000 meter was 1 minute 35.7 seconds. What was her
speed in meters per second?
2. Cathy skated the 500 meter race in 46.53 seconds. Her old record was
46.86 seconds. By how much did she break the record in this race?
3. How many miles per hour would you estimate Cathy is traveling?
(A photo of Cathy Turner skating was also reproduced at the end of
this worksheet) |
DOCUMENT 11.
Hand-out to guide the comparison of average speeds
in 2 different speedskating events
[Who's fastest? -- Days 1-3]
DOCUMENT 12.
Report of a class discussion on what learned about speed
(including a conceptual map created by the teacher based on students' input)
[Who's fastest? -- Days 4-5]
| (SCAN IN THE CONCEPTUAL MAP)
(From an observer's fieldnotes)
To tie together their discussion of speed and finish looking at their
question of who is the fastest, the teacher asks the students to think about
the following:
-- what have we talked about
-- what have we read about speed
-- what have we done in class
-- what is some of the math we have been working with
The students do not seem quite sure how to enter this discussion so the
teacher asks the students to think about where they started and what were
the first things they did when trying to ascertain who the fastest is. This
was a good way to get the class starting to discuss the topic. They slowly
began generating what they had done. While the class was discussing the
topic, the special education teacher assigned to this blended class was
recording everything they said on the board in a diagram form (which ha
been reproduced below).
After they discuss some of the activities they have done, the teacher
reminds them that they also discussed instantaneous and average speed. She
then asks the students what things they read about for today or what things
they know about that affect speed. As they are discussing this topic, the
teacher has a couple of overheads prepared of articles dealing with some
of the topics that students are raising. This is a good discussion and the
special education teacher also connects it back to the work they did on
their warm-ups. |
DOCUMENT 13.
Record of teacher's modeling how the proposed framework for the project
could be used to summarize the results of their inquiry around "Who's
fastest"
[Who's fastest? -- Days 4-5]
DOCUMENT 14.
Reading on scoring criteria in figure skating
[Who wins? -- Days 1-2]
DOCUMENT 15.
Handout created to help students organize the tabulation
of the scores assigned by each student
[Who wins? -- Days 1-2]
DOCUMENT 16.
"Warm-up" asking students to order and find various averages
using data from the Men's Mogul race
[Who wins? -- Days 3-4]
DOCUMENT 17.
Handout created to structure a first comparison
between two alternative medal counts
[Medal counts -- Days 1-2]
DOCUMENT 18.
Report of a 10-minute class discussion on alternative medal rankings
[Medal counts -- Days 1-2]
(From an observer's fieldnotes)
[When asked how they could rank countries with respect to their performance
in the Olympics Games,] a student suggests listing the countries alphabetically,
including all of the countries that participated. Other students who had
thought of the same thing suggest listing only the countries who had received
at least one medal.
A couple of other students suggest assigning point values for the
medals and then ranking according to the total point value of each country:
Gold: 3 points, Silver: 2 points, Bronze: 1 point.
The teacher asks why the students think this method might be more fair.
She then suggest that they try this ranking method, looking at Norway,
Germany, and Russia, since those three changed between the US and Italian
ranking. Then the teacher, writing this down on the board as they go, leads
the class in determining the points those three countries would have been
awarded based on the point system just suggested. The class discovers that
Norway would have come in first, Russia second, and Germany third. This
results in a different ranking than found in either newspaper.
Another student comments that she still likes the US way of ranking because
the difference between the medal you receive is so minimal that just receiving
any medal is great.
The teacher asks "Do you think there is an answer to this?"
The students respond no and that it is based on opinion.
A few other students briefly share their ideas:
- If the ranking is close, use the results of the competition to determine
ranking.
- group the countries by size or number of athletes competing
- Use percentages.
|
DOCUMENT 19.
Excerpt from an article explaining key elements of the biathlon
[Issues about measurement and its history -- Getting a sense of common
measurements]
DOCUMENT 20.
Handout articulating a measuring task designed to help students
better understand what is involved in the biathlon
[Issues about measurement and its history -- Getting a sense of common
measurements]
Warm-up
Note: Each group of students received only one of the following three
handouts:
"BULLET"
Draw and cut out a model of the bullet. Refer to article for dimensions.
|
"TARGET"
Construct (Draw and cut out) the two biathlon targets. Compasses are
available. |
"COURSE"
Go to the hallway and stand at the classroom door and from this starting
point measure the length of the course. |
|
DOCUMENT 21.
"Warm-up" asking students to compare the size of various kinds
of skis
[Issues about measurement and its history -- Getting a sense of common
measurements]
DOCUMENT 22.
Ideas about measurement within the Olympics
[Issues about measurement and its history -- What, how and why we
measure]
From: Document shared as a stimulus for discussion
WHY THINGS ARE MEASURED:
- mostly, it is a (usually preliminary) step to determine who wins an
event and to rank the other participants -- (NOTE: in most sports, the
ranking ends up being "complete", but not always -- ex: hockey;
but this is a point may be to consider more when I think about the issue
of order)
- measurements necessary to prepare the right conditions to play the
game (ex: length the speed skaters have to cover in each specific event;
position of poles in slalom; dimensions of hockey field and exact position
of "goal door" -- (NOTE: I realize I am lacking a lot of precise
sport terminology! the need to use precise terminology here could provide
an interesting analogy with the use of precise terminology in math -- something
worth thinking about...); etc.)
- in a sense, we could also talk of the use a person can make by knowing
his/her competitors "measures" in an event (what do they need
to do in order to "beat him/her" or win, how it compares with
their "average" or "best" performance (NOTE: this could
actually lead to some statistics and probability considerations... worth
thinking about)
WHAT IS MEASURED:
- time taken to complete an event (ex: most skiing, speed skating, bob,
sled) (not relevant in: ski jumping, figure skating)
- speed (average? istant?) (ex: speed skiing)
- accuracy of performance -- this could be more or less objectively determined;
objective ex: number of targets made/ missed in decathlon(?) -- subjective
ex: artistic score in figure skating
- distance -- ex: how far one jumps with ski jumping; what a speed skater
has to run;
- medals won by a country (notice here how gold, silver and bronze are
essentially not comparable, unless we explicitly create "ways"
to do so)
NOT SURE IF "WHAT" OR "HOW":
- "scoring" used to combine target shooting and speed in decathlon(?)
- measuring of time, etc. when participants go in "batches"
(ex: 2 speed skaters a time)
HOW IS MEASURED -- i.e., what UNITS are used:
- TIME: hours, minutes, seconds, 1/100th of second (NOTE: scores are
"mixed numbers")
- DISTANCE: miles/ km / m etc. (when would you choose one versus the
other? why? how do you convert?)
- SPEED: miles/hour; kilometers/hour; meter/sec. (why these "compounded"
units? when would you choose one versus the other? why? how do you convert?)
- many things are measured in whole numbers, by just counting
HOW IS MEASURED -- i.e., what TOOLS and APPROACHES are used:
- speed -- requires actually measuring distance and time; average vs.
instant
- how can a skier time really be measured? (technology used, etc.; it
may be quite different for a cross country skier or a slalom; also, you
probably have two different people and instruments measuring at "beginning
and end" -- how can those be coordinated?)
- "elimination rounds" (ex: hockey) (how do they really work?
when are they necessary/ useful in a sport)
- competitors going one at the time (ex: slalom), in groups of 2-n (ex:
speed skating), altogether (some cross country skiing, perhaps?) -- why?
how different is this from having elimination rounds?
- how are borderline cases decided (ex: bullet hitting the target on
line)
- averaging several runs (ex: bob, sled)
WHAT IS COMMON IN PREVIOUS EXAMPLES:
- essentially all measurements end up being expressed in some kind of
number (is it really true all the time? is it really necessary and if so
why?) (NOTE: it might be interesting to consider what kind of numbers are
or not represented -- ex: no irrationals!)
- in all the "objective" measurements, you use the same process
(choose a unit, etc.)
WHAT IS DIFFERENT IN PREVIOUS EXAMPLES:
- some measurements use whole units (ex: goals in hockey; targets in
shooting; scoring in figure scating), others a continuum (distance)
- competitors going one at the time (ex: slalom), in groups of 2-n (ex:
speed skating), altogether (some cross country skiing, perhaps?) -- why?
how different is this from having elimination rounds?
- some events use "elimination rounds" and others don't (why?)
OTHER QUESTIONS/ ISSUES THAT COULD BE RAISED:
- how do different events deal with ties? are there events when ties
are just NOT possible?
- was any of the events measured differently in the past? if so, why/
why was it changed?
- how did people come up with criteria to measure more subjective things
(ex: figure skating)?
- how "precise" the measurements taken really are? how can
these imprecisions affect the events? what can be done to make those measurements
more "precise"?
- what is already done to make measuring more "fair"? what
else could be done? (ex: figure skating -- dropping highest and lowest
mark)
- what is the "fastest" winter sport? (NOTE: one should also
consider the speed of sportman who are not measured on speed -- ex: skating
speed of hockey players)
- how would you judge who is the "best" skier? or even the
"fastest" skier?
- how would you judge whether the winner won by a large margin or not?
(difference? ratio? why? how do you compute each of those?)
DOCUMENT 23.
First handout describing project requirements
[Project development -- Project requirements and expectations]
DOCUMENT 24.
Further clarifications provided about project requirements and evaluation
[Project development -- Project requirements and expectations]
MATH 8
OLYMPIC UNIT PROJECT
You are a math reporter covering the Winter Olympic Games in Norway.
Your assignment is to write an article about an aspect of the winter games
that involves mathematics.
Your project will have 5 parts worth the following points:
QUEST - A question in your area of interest (5 points)
DATA - From magazines, newspapers, T.V., etc. (10
points)
GRAPHS AND TABLES - Frequency table and 2 graphs
(choose from bar, line, pictograph, or circle graph) (25 points)
BEHIND THE SCENES: What factors effect your data?
Identify and provide information about the story behind the data. (10 points)
CONCLUSIONS - Written in newspaper article form
with headline and any supporting graphs or pictures. This article should
include mean, median, mode and range, if appropriate to your data, in order
to help you draw your conclusions. (30 points)
PROJECT DUE MONDAY, MARCH 7, 1994
PROJECT TOTAL: 80 POINTS
This project must be planned ahead! Do not leave it until
the last
minute, and if you need help, get it now!! |
DOCUMENT 25.
Project checklist
[Project development -- Project requirements and expectations]
DOCUMENT 26.
Form for recording the question informing each pair's project
and the data they would gather to address it
[Project development -- Supporting students in developing questions
for their project]
DOCUMENT 27.
List of questions generated by the students for their individual projects
[Project development -- Supporting students in developing questions
for their project]
(From an observer's fieldnotes)
The following is a list of what the different groups' topics are and
what they are trying to find out in that domain.
- medal count: surveyed students as to who they thought would get the
most medals and are comparing that with the actual medal count.
- Ski jumping: comparing past medals with present medals.
- Ski jumping: focusing on this year's distances.
- Mogul skiing: looking at all of the different jumps, different speeds,
turns, and who won.
- Figure skating: looking at the scores for individual competitors and
judges - do the judges favor their own country?
- Luge: Comparing mens' and womens' scores.
- Ski jumping: Focusing on distances.
- Hockey: Surveyed people about who they thought would win and comparing
that with the actual results. (Surveyed about 40 people.)
- Bobsled: 2 men vs. 4 men times.
- Men's cross-country skiing: examining places and times.
- Figure skating: Looking at the scores.
- Giant slalom
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