Download Zero-Sum Game - Mathematical Sciences

2010-2011 Workshop Series for
YISD Teachers of 6th Grade Math
Conceptual Understanding
& Mathematical Thinking
Workshop 3 (Jan 31)
Kien Lim
Dept. of Mathematical Sciences, UTEP
Goals:
• Strengthen our mathematical knowledge for
teaching signed numbers and operations
involving them
Question for Thought
How can we help our students understand deeply,
so much so that they can explain clearly,
why the product of two negative numbers is positive?
Goals:
• Strengthen our mathematical knowledge for
teaching signed numbers and operations
involving them
• Revisit the idea about the necessity principle
• Explore how to make learning interesting and
fun for students
Raise your hands if you use games in your classrooms
at least 3 times a year.
Activities for Today’s Workshop
• The Zero-Sum Game
• Discuss the underlying math concepts
• Discuss pedagogical issues
• Design a sequence of activities for use in
your classrooms
The Zero-Sum Game
 Number of Groups: 4
 Goal: Increase assets and decrease debts
 Materials:
o
o
o
o
o
15 debt cards and 15 asset cards (ranging from $1 to $15)
Recording sheets (one per group)
A small board to write and show your group’s answer
One pot (for collecting cards)
An elmo and projector to reveal the cards in the pot
 Number of Rounds: 5
o Round 0: Each group randomly picks 3 debt and 3 asset cards.
o Rounds 1-4: Put 1 debt and 1 asset into the pot; and then the
groups will take turn to pick 1 debt and 1 asset.
The Zero-Sum Game
 What happens in each round?
o A question will be posed. The question in each round is
different.
o Each group will write their answer on a board and all group
members stand-up to signify completion.
o The first group that has the correct or closest answer will get
to pick cards first, followed by the second group, and so forth.
o Speed and accuracy counts!
o At the end of each round, record the transactions, update the
net worth of each group, and get ready for the next round.
The Zero-Sum Game
 Instructions for Recording Transactions
o Each group will receive a sheet to record what cards are
received and what cards are removed
o Each group is advised to record their own transactions as well
as other groups’ transactions
The Zero-Sum Game
 Instructions for Recording Transactions
o Each group will receive a sheet to record what cards are
received and what cards are removed
o Each group is advised to record their own transactions as well
as other groups’ transactions
o It is advantageous to update your records of every group’s net
worth after each round
Are You Ready?
Round O
 Randomly pick your 3 asset and 3 debt cards
(no cheating)
 What cards each group received are public information
(each group must honestly report what cards your group received)
Round 1
 Each group puts a debt card and an asset card into the pot
(face down; you don’t need to tell others what you put into the pot)
 Get ready to write your answer
Are You Ready?
Information: Suppose the referee receives
all the remaining cards.
Question:
How much do you predict
the referee’s net worth is?
Results for Round 1
 Let’s see which group is the first to give the correct (or
closest answer).
 The first group with the correct/closest answer will get
to pick a debt card and an asset card from the pot.
Announce the cards you pick.
 The second, third, and fourth group will pick their cards
and announce the cards they pick.
 Each group will have 5 minutes to update their
recording sheet .
What Can We Learn from Round 1
Strategy-sharing:
How did you make your prediction?
Why?
Note the name of the game!
Let’s be more efficient in recording
 Did any group write the word “Debt” or “Asset” on the
recording sheet?
 What are some efficient ways to symbolize Debt and
Asset?
Are You Ready for Round 2?
Round 2
 Each group puts a debt card and an asset card into the pot,
and get ready to for the question
Are You Ready?
Action: The referee turns over all the asset
cards in the pot to reveal their value.
Question: What is the value of all the debt
cards in the pot?
Results for Round 2
 Each group will take turn to pick a debt card and an
asset card.
 Update your recording sheet .
What strategy did you use?
Did it work? Why, or why not?
Round 3
 Each group puts a debt card and an asset card into the pot,
and get ready to for the question
Are You Ready?
Action: The referee turns all cards in the pot.
Question: Excluding the debt card with the
greatest number, what is the total
value in the pot?
Results for Round 3
 Each group will take turn to pick a debt card and an
asset card.
 Update your recording sheet .
What strategy did you use?
Did it work? Why, or why not?
Round 4
 Each group puts a debt card and an asset card into the pot,
and get ready to for the question
Action: The referee will pose a math-like
question using 3 of debt-asset cards
and 3 operator cards. (Here is an
example)
Answer: A debt of $2
Asset
Add
$ 3.00
Asset
Subtract
$ 1.00
Debt
Add
Twice
$ 2.00
Action: The referee will pose a math-like
question using 3 of debt-asset cards
and 3 operator cards. (Here is an
example)
Question: What is the net result?
Are You Ready?
Debt
Add
Twice
$ 1.50
Asset
Add
$ 2.00
Debt
Subtract
$ 2.50
Action: The referee will pose a math-like
question using 3 of debt-asset cards
and 3 operator cards. (Here is an
example)
Question: What is the net result?
Results for Round 4
 Each group will take turn to pick a debt card and an
asset card.
 Update your recording sheet .
What strategy did you use?
Did it work? Why, or why not?
How can we represent these transactions
mathematically?
Debt
Add
$ 1.25
+ (-1.25)
Asset
Subtract
$ 0.25
– (+0.25)
Asset
Add
Twice
$ 0.75
+ (+0.75)
+ (+0.75)
+ (-1.25) – (+0.25) + 2 (0.75)
Follow-up Group Discussion
1. What mathematical concepts can students learn
from playing this game?
2. What challenges do you foresee if you were to
use Game 1 in your classroom?
3. How will you address those challenges?
(i.e., how will you change the same to suit your
classroom)
A Follow Question
Somebody spilled coffee
on Fantasia’s net worth
statement. She is trying to
figure out what transaction
took place to give her a
net worth of $12,000.
List two possible
transactions.
Fantasia’s Net Worth
Statement
Net Worth: $10,000
Transaction:
Net Worth: $12,000
Possible Activities from an Article
Stephan, M. L. (2009). What are you worth?
Mathematics Teaching in the Middle School, 15(1),
NCTM.
Who is worth more, Brad or Angelina? Justify your answer.
Which bachelor is financially most favorable?
Determine whether each transaction increases or decreases
one’s net worth.
The Necessity Principle
“For students to learn what we intend to
teach them, they must have a need for it,
where by ‘need’ is meant intellectual need,
not social or economic need.” (Harel, 2007)
Does this game provoke an intellectual need for
students to learn a certain concept? If so, how?
The Key Idea
 Does adding an asset a good thing?
 Does subtracting an asset a good thing?
 Does adding a debt a good thing?
 Does subtracting a debt a good thing?
Debt
Subtract
$2
– (-2)
Asset
=
=
Add
$2
+ (+2)
The Product of Two Negative Numbers is Positive.
Why? For example, why (-3) x (-4) = 12
Think in terms of multiplier and multiplicand.
• 3 x 4 means adding 3 times of 4 to 0
i.e., 3 x 4 = 0 + 4 + 4 + 4 = 12
• 3 x -4 means adding 3 times of -4 to 0
i.e., 3 x -4 = 0 + (-4) + (-4) + (-4) = -12
• -3 x 4 means subtracting 3 times of 4 from 0
i.e., -3 x 4 = 0 – 4 – 4 – 4 = -12
• -3 x -4 means subtracting 3 times of -4 from 0
i.e., -3 x -4 = 0 – (-4) – (-4) – (-4)
=0 + 4 + 4 + 4
Recapitulation
The context of debts and assets help students understand
• the difference between multiplier and multiplicand
• the difference between a negative operation
(subtraction) and a negative value (negative integer)
• why the value of subtracting a negative number
is essentially adding its absolute value
(i.e., removing a debt of $2 has the same effect on
net worth as adding an asset of $2)
• why -3 times -4 can be interpreted as repeatedlysubtracting negative 4 three times