Activity 1

Transcription

Activity 1

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Exploring Probability
Understanding and Expressing the Likelihood of an Event Occurring
Unit Standards for Grades 3-5
Pages
Formulate questions that can be addressed with data. Collect, organize and display data.
Lesson 1:
Probability
5– 6
Lesson 2:
Spinner Puzzles
7- 8
Lesson 3:
Graphing Probability
9-10
Lesson 4:
Calculating Probability
11-12
Lesson 5:
Sum of 2 Dice
13-14
Lesson 6:
Combinations
15-16

Lesson 7:
Fairness
17-18

Lesson 8:
Game Strategies
19-20
Lesson 9:
Data Collections
21-22
Lesson 10:
Review
23
Bibliography and
Websites
24

Collect data using observation, surveys, and experiments.

Represent data using tables and graphs.
d.
Inside this Unit
A Ten Lesson Unit
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Develop and evaluate inferences and predictions that are based on data.


Propose and justify conclusions and predictions that are based on data and design studies
that further investigate the conclusions or predictions.
Looking at median, mode and mean to make predictions and draw conclusions.
Understand and apply basic concepts of probability.

Describe events as likely or unlikely and discuss the degree of likelihood using such words
as certain, equally likely and impossible.
Predict the probable outcomes of simple experiments and test the predictions.
Understand that the measure of the likelihood of an event can be represented by a number
from 0 to 1.
Use the language of mathematics to precisely express mathematical ideas.
Communicate mathematical thinking coherently and clearly to peers, teachers, and others.
Apply and adapt a variety of appropriate strategies to solve problems.
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Teaching Tips
Evaluations:

Allow time for students to work on, and perhaps even struggle with, any given problem. It
is more meaningful for students to discover the solution on their own than for the teacher to
tell them the answer. Remember that once one student answers a question, thinking will
stop for all others. Slow down and let students apply the knowledge that they have learned
to the questions. (Some students will do this rapidly; others will need time to process.) One
way to achieve this is to say to students, “I will wait until I see six (or however many)
hands before I call on someone.”
At the first class, please have
the students fill out the PreMath Club Student
Evaluation. At the last class,
please have the students fill
out the Post-Math Club
 Feel free to experiment and alter activities as per the interest of the student group.
Student Evaluation. Doing so
 Students can record information in their journals as they go through the unit. Journaling
will enable the club teacher
helps students become aware of what they do or do not know and helps them connect prior
and Zeno to track student
knowledge. Journaling also helps students to organize their thoughts and to reinforce
progress.
appropriate mathematical language.
Also, teachers please fill out
the Curriculum Evaluation
 At the end of this unit are websites that can be used to enhance lessons or to further
form so that Zeno learns how
challenge students who complete their work early.
to better serve you and the
students in the future.
Supplies List for 24 students
Activity Sheets:
Supplies:
Student Contract
1
1
Teaching Manual
bag of bi-colored counters
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Spinners I
d.
Parent Letter
Spinners II
1
box paper clips
1
white board marker
6
decks playing cards
10
calculators
2—12 Number Lines
24
dice in color A
Combination of Two Dice
24
dice in color B
Fair Games
24
pencils
Fair Game Results
40
plastic opaque cups
Spinner Puzzler
Biased/Unbiased
Assigning Probabilities
Horse Race
Math Millionaire
ABCD cards
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Math Millionaire Money
Games:
1
Sequence Dice by Jax Ltd.
2
Pass the Pigs by Winning Moves
2
Rummikub by Pressman
2
Yahtzee score sheets
Replenishable Supplies:
1
pad sticky notes
12
paper bags
24
journals
Club Evaluations:
Return envelope for evaluations
Student Club Evaluation
Teacher Curriculum Evaluation
Page 5
Lesson 1 Focus: Probability
The likelihood that an event will occur.
Materials:
*Student Contract
*Pre-Math Club
Student Evaluation
*Parent Letter
Introductions, Student Contract, What is Probability?
Lesson Prep: Make copies of the Student Contract and Parent Letter.
Welcome students, go over the Student Contract and then discuss what they
currently know about probability and why it is important. Hand out the Pre-
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Remember to send home the Parent Letter at the end of class.
d.
Math Club Student Evaluation sheet (copies are in the manila envelope in the back of
the Teaching Manual), and ask students to answer the questions as best they can.
Remind students that this is not a test, but rather a way for you to understand their
thinking on number sense.
Activity 1: Is It likely?
On the board, write a probability line as follows:
Impossible
Not Probable
50/50
Probable
Certain
Vocabulary
CERTAIN
PROBABLE
LIKELY
IMPOSSIBLE
Ask the students to respond to the below statements/questions by using such
terms as “certain”, “impossible” or “50/50”. If students responses are none of
the above words, discuss where their results would fall on the scale, using
words such as possible, probable, likely, unlikely, chance, etc.
UNLIKELY
1. It will rain tomorrow.
PERCENT –comparison of
a number with 100.
2. There is a live dinosaur in the zoo.
3. Trees will talk to us.
4. The sun will rise tomorrow morning.
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5. Three students will be absent for school tomorrow.
6. Someone in this club will win the lottery this year.
7. When you grow up, you will be 10 feet tall.

If we were to give the word “impossible” a number value, what would
that value be?

What do you think the value of “certain” would be? Possible? Equally
likely?
Discuss the term probability—the mathematical process of determining the
possibility of an occurrence or result. The scale ranges from 0 to 1 or 0% to
100%.
CHANCE
POSSIBLE
EQUALLY LIKELY
RATIO—a comparison of
two numbers written as a
fraction
DECIMAL
Page 6
Activity 2: Cup Game
Set out the same number of cups as there are students in the class. Under one
cup (and out of student view) place a bi-colored counter. Ask the students
who would like to go first, then second and so on. Write the order of the
students on the board.
d.
Call upon the first student to pick a cup and turn it over. This student has 1
out of X (X equals however many cups) chance of selecting the cup with the
counter. For the purposes of this example, we will use 12. If there is no
counter under the cup, take that cup out, and then let the next student select
a cup. The second student in this example then has a 1:11 chance of getting
the right cup. Continue having students select cups.
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On the board create the ratio of each chance. Probability can be written as a
fraction/ratio (total number of ways an event can occur / total possible
occurrences), decimal or percentage.
1/12 1/11
1/10 1/9
8%
10% 11% 13% 14% 17% 20% 25% 33% 50%
9%
1/8
1/7
1/6
1/5
1/4
1/3
1/2
1/1
100%
Students continue selecting cups until the counter is found. Using a
calculator, demonstrate to students how to convert the ratios into percentages.
(One divided by the total number of cups remaining determines the percent of
each selection.)
If there is time, play the game again.
How likely is it that the first person to choose a cup is going to find the
counter?
NOTES:
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
Material:
*Calculators
*Plastic cups
*Bi-colored counters
Materials:
Material:
*Bi-colored
*Opaque
cups
counters
*Bi-colored
*Calculators
counters
*Cups
Page 7
Lesson 2 Focus: Spinner Puzzles
Creating and analyzing spinners to further explain probability concepts and
vocabulary.
Lesson Prep: Make copies of the Spinners I, Spinners II and Spinner Puzzler
sheets.
Materials:
*Spinner sheet
*Paperclips
Activity 1: Spinners
d.
Probability can be demonstrated with a spinner and can be expressed using
words such as: equally likely, twice as likely, certain, same chance as, etc.
A
B
C
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Hand out the Spinners I sheet and ask the students what they can tell you
about them. Can they describe these spinners with ratios or percentages?
Equal Chance, 1/3 chance, 33%
B
A
A is: twice as likely, 50%, 1/2
Now test the next two spinners. Once again spin fifteen times; record and
review the results.
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B
C
D
C
Give each student a paperclip; have the student create a spinner by
straightening the outer wire of a paperclip, then holding a pencil, pointed end
down, through the center of the clip in the middle of the circle. Spin the
spinner fifteen times; record and review the results. Note that only the tip of
the spinner counts when recording results.
C
A
B
A
C
B
A is certain to win; 100%, 1/1.
A
D
E
F
G
A and B are equally likely to win.
Page 8
Activity 2: Creating Spinner Puzzles
Material:
Material:
*Paperclips
*Circles sheet
*Spinner Puzzler
Materials:
*Circles sheet
*Paperclips
*Spinner Puzzler
sheet
Hand out a Circles sheet, a Spinner Puzzler sheet and a paperclip to each
student. Students will draw six spinners, each one matching one of the
descriptions of the Spinner Puzzler. Call on students to draw an example, or
show the students some examples.
B
A
D
C
B
C
D
E
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Examples:
A cannot possibly win.
A or B will probably win.
Students should not make all the spinners in order; once complete with their
work, instruct students to swap sheets with a classmate. The classmate will
then attempt to determine which spinner goes with which description.
Were you able to figure out which spinner went with what description?

Were some spinners harder to figure out? Why?

Were some spinners harder to create? Why?
NOTES:
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

a is certain to win

a cannot possibly win

a is likely to win

a, b, c, d, and e are
all equally likely to
win

a or b will probably
win

a, b, and c have the
same chance to win,
and d and e cannot
possibly win.
Page 9
Lesson 3 Focus: Graphing Probability Results
Learning to look at data, compare it to other data and make reasonable
estimates.
Lesson Prep: Make copies of the Biased/Unbiased sheet. Review vocabulary
terms with the class before starting any activities.
Activity 1: Biased or Unbiased Die
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Pair up students; give each pair a die and the Biased/Unbiased sheet. Each
student has an opportunity to roll the die ten times (total of twenty rolls per
sheet) and record the results on their graph. They are to first tally their
results, put that tally in numerical form, and then using that data, create a
bar graph. Once all the students have collected their data use the Biased/
Unbiased Club Results sheet to record class results. Then, ask students to
answer the following questions:
Materials:
*Dice
*Biased/Unbiased
sheet
*Biased/Unbiased
Club Results sheet
*Assigning Probability
sheet
*Journals

Are the bars all nearly the same height?

What would you think about the die if one bar was much higher than the
others?

Using the graph results, would you consider the die biased or unbiased?
Look at the results on the previously created chart at the bottom of the
Biased/Unbiased Club Results sheet; three columns show results that are made
up and one column is actual data.

Which column would you say represents the most accurate data? Why?
Students can make a bar graph of this data to better understand the
differences in the numbers.
How many different outcomes can you have when rolling a die? 6

How many ways can you roll a 1? 1 What is the ratio of rolling a 1? 1/6

How many ways can you roll a 4? 1 What is the ratio of rolling a 4? 1/6

What is the probability of rolling a 1 or a 4? 2 possible results out of 6
chances (2/6) or reduced to 1/3

What is the probability of rolling an odd number? An even number?
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
Display the Assigning Probabilities sheet; have students respond to the
questions in their journals and then discuss them as a class.
VOCABULARY
BIAS—one outcome is favored
over other outcomes.
MEAN– average or same
amount if distributed evenly.
MEDIAN– the number in the
middle of a set of numbers
arranged from greatest to least
or least to greatest. If it is an
even number of data, the
median is the average of the
two middle numbers.
MODE– most frequently
occurring number.
RANGE– from the least
number to the greatest.
RATIO—total number of ways
event occurred over total
possible occurrences.
Page 10
Activity 2: What’s the Record?
The object of this lesson is to see who can set a class record for rolling dice the
most times without getting doubles. Students will then graph all the data to
obtain its median, mode and mean.
First, have the students make estimates on how many rolls it will take before
getting doubles. (The probability of rolling doubles is 1 of 6.) Then give each
student 2 dice; students should keep a tally record in their journals of how
many times they rolled the dice before getting doubles. Once students have
rolled doubles, give them a sticky note to write on it the numeric total of rolls
(make sure they write it large enough for all to see).
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Once students have written out their numbers, have the student with the
lowest total come up to the front of the room — this is the beginning of the
RANGE. Then have the student who had the record number come to the
front of the room — this is the end of our RANGE. Have the remaining
students line up in numerical order between these two numbers.
Have students discuss options for figuring out the MEDIAN number, (i.e. a
student from each end of the line can sit down until only one student is
standing) and then figure out that number.
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Once they have figured out the median, discuss how they might figure out
MODE. The best way for figuring out mode is for the students to make a
human bar graph with students with the same number lining up together.
They can then tell which line has the most students.
Materials:
Material:
*Dice
*Dice
*Sticky notes
*Journals
*Journals
*Sticky
notes
Page 11
Lesson 4 Focus: Calculating Probability
Learning that the probability of an event occurring is between 0 and 1 or 0
and 100%.
Materials:
*Dice
Activity 1: Dados
Group students in pairs; give each pair two dice.
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One player rolls both dice. If the sum rolled is 2, 3, 4, 10, 11, or 12, Player A
gets one point, Player B gets zero. If the two dice total 5, 6, 7, 8, or 9, Player
B gets one point, Player A gets zero. The first player to 10 points wins. Ask
students to predict who is most likely to win (point out that one player has
six numbers, the other only five).
After students have played one round, ask them the following questions:
?
How many players with 5, 6, 7, 8, 9 won? (It should be a majority of the
class.)
?
Is the game fair? (Number 5, 6, 7, 8, 9 have 24 ways to score a point, the
other numbers have only 12 ways.)
?
Do the results (who won/who lost) support the answer to question 2?
(The answer should be yes.)
?
Ask students how they might change the game to make it fair.
(Suggestions could include alternating who is A and who is B; players
take turns picking the numbers that will give them points; one player takes
odds, the other, evens.)
Games:
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Students should play a second round, after which ask them to explain why 5,
6, 7, 8, and 9 will almost always win.
Included in this unit are three board games. These games should be introduced early on to allow students
ample opportunity to figure out strategies, and to practice mathematical skills and concepts.
Pass the Pigs by
Winning Moves
Sequence Dice by
Jax., Ltd.
Rummikub by Pressman
Page 12
Activity 2: PIG
Material:
*Dice
Group students in pairs; give each pair two dice.
Materials:
The goal of the game is to be the first to score 100 or more.
*Dice
d.
In their journals, students should draw two columns and write one of the
player’s names above each column. Players take turns rolling the dice. On a
player’s turn, she/he may roll the dice as many times as desired, keeping a
running total of the sum; the player rolling the dice should also say the dice
sums out loud so that his/her partner can check for accuracy. When the
player decides to stop rolling, both partners should record the total for the
turn in the journals, adding it to the total from previous turns.
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If a 1 comes up on either die, the player’s turn automatically ends and 0 is
recorded for that round. If a player rolls a double 1, their turn ends and the
total accumulated so far turns to 0.
JOURNAL QUESTIONS:
What strategies did you use while playing PIG?

Did those strategies work? What strategies did not work?

Did the probability of rolling a 1 or double 1 increase with the number of
times you rolled the dice? Why?
NOTES:
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
*Journals
Page 13
A Ten Lesson Unitfor Third, Fourth, and Fifth Graders
Lesson 5 Focus: Sum of Two Dice
Testing the probability of attaining a certain sum by rolling two dice, and
making predictions on how to win a game based on that probability.
Lesson Prep: Make copies of the 2-12 Number Lines and Horse Race sheets.
Activity 1: Number Line Game
2
3
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Give each student a 2-12 number line and 11 counters. The students are to
place their 11 counters on their number line in any arrangement. They may
put more than one counter on some numbers and none on others. The teacher
then rolls two dice and on each roll every player removes one counter that is
on the number that matches the sum of the dice. The winner is the first
player to remove all 11 counters. On the board, keep track of how many
times each number is rolled.
Materials:
*2-12 Number Line
*Dice
*Horse Race sheet
4
5
6
7
8
9
10
11
12
JOURNAL QUESTIONS:

What was your strategy for winning?

What is the best arrangement of counters on the number line?
Play another round as a class to see if the new strategies worked.
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Activity 2: Horse Race Game
This is another game in which students predict the most likely sum to be
rolled by a pair of dice.
Place students into groups of twos or threes, and then hand out the Horse
Race sheet, and a pair of dice to each group. Each student picks a numbered
horse and initials his/her selection.
The students are to take turns rolling the dice, then on their Horse Race sheet
they are to mark with an X the sum of the dice. Even if the sum is not a
number they selected, they need to place an X in that horse’s spot. Play
until one horse reaches the winner circle. The horse that wins may not be a
horse that the players selected.

Which horses received the most X’s?
Play the game again to determine if the results are similar.
Page 14
Activity 3: Combination of Two Dice
Group students in pairs; give each pair the Combination of Two Dice sheet
and two different colored dice. Students are to determine how many ways
there are to roll each sum, 2-12. Encourage students to look for patterns to
help them. Also, help students understand that a “red” 3 and a “blue” 2 is a
different roll than “blue” 3 and a “red” 2.
Materials:
*Combination of
Two Dice sheet
*Dice in two
different colors
d.
Have them start filling in the graph; those students who are capable of finding ratios and percentages can fill in that part of the graph. Students may
need help figuring out that there are 36 different combinations of rolling the
dice.
Possible
Totals
2
3
1,1
2,1
1,2
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Go over all results together.
4
5
6
7
8
9
10
11
12
3,1
4,1
5,1
6,1
6,2
6,3
6,4
6,5
6,6
2,2
3,2
4,2
5,2
1,3
2,3
3,3
4,3
5,3
5,4
5,5
5,6
4,4
4,5
4,6
1,4
2,4
3,4
3,5
3,6
1,5
2,5
2,6
1,6
1
2
Ratio
1/ 36
2/36
Percentages
3%
6%
NOTES:
4
5
6
5
4
3
2
1
3/36
4/36
5/36
6/36
5/36
4/36
3/36/
2/36
1/36
8%
11%
14%
17%
14%
11%
8%
6%
3%
3
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Total Ways
Page 15
Lesson 6 Focus: Combinations
Figuring out all possible combinations for rolling the dice and the probability
of getting a Yahtzee© on a first roll.
Materials:
Activity 1: Combinations

What is the ratio of rolling one die and getting a 4? 1/6

What is the ratio of rolling one die and getting a 2? 1/6

How many different combinations can you get from rolling two dice? 36
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To figure all combinations mathematically without counting and graphing all
possibilities, take the total number of sides on one die (which is 6), times the
total number of sides on another die (which is 6), so 6 x 6 = 36 (total number
of combinations from 2 dice).
Two bi-colored
counters
Give each student two bi-colored counters. If they were to flip these counters,
how many different ways can these counters land? Have the students flip
and record in their journals the different ways in which the counters landed.
Discuss results.

How many ways can one counter land? 2

How many ways can two counters land? 2 x 2, or 4
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Give each student another counter; have them figure out how many different
ways in which three counters can land.

How many ways can one counter land? 2

How many ways can three counters land? 2 x 2 x 2, or 8
NOTES:
Three bi-colored
counters
Page 16
Activity 2: Yahtzee by Hasbro
Materials:
*Cups
*Dice
*Yahtzee score
sheets
Ask students if they have ever played Yahtzee©; how does one roll a
“Yahtzee” ?
Using information learned in the previous activity, ask students to calculate
the total number of combinations five dice can roll (they can use calculators if
needed). 6 x 6 x 6 x 6 x 6, or 7,776 combinations
The probability of getting a Yahtzee (all dice with the same number showing)
on the first roll is a 6/7,776 chance.
Is it impossible, unlikely, probable, or certain to roll a Yahtzee on the first
roll?
d.

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Place students in groups of twos or threes; hand out Yahtzee© score sheets,
five dice and a cup to each group.
Instruct younger students to play only the top part of the score sheet. Older
students, or students who have played the game before, can play the entire
score sheet.
Control over game pieces can be a management problem. Tell students that
they may shake the dice in the cup only three times, at which point they
must turn it over.
It is best to have students sit on the floor while playing so dice do not go
flying off tables. If a group loses dice, the game is over for them because play
requires five dice to continue.
JOURNAL QUESTIONS:
Which outcomes were easier to roll, and which were harder?

Are there any strategies to utilize when building a large straight and a
small straight?

How does probability play a part in the dice you decide to re-roll?
NOTES:
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
Vocabulary
LARGE STRAIGHT—five
sequential numbers (i.e. 1-23-4-5 or 2-3-4-5-6)
SMALL STRAIGHT —four
sequential numbers (i.e. 1-23-4 or 2-3-4-5)
Page 17
Lesson 7 Focus: Fairness
Looking at the combination of outcomes to determine if a game is fair.
Materials:
*Fair Game I sheet
*Fair Game II sheet
*Dice
Lesson Prep: Make copies of the Fair Game I sheet.
NOTE: Place students in groups of three; give each group a Fair Game I
sheet. All groups should play all four games. After all game play is complete,
use group results to complete the Fair Game II sheet.
Activity 1: Fair Game #1
RED
YELLOW
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Place students in groups of three; give each group two bi-colored counters.
Each player decides if they are red, yellow or both (one red, one yellow).
Have the students take turns flipping the counter; if both counters are red,
red gets a point. If both counters are yellow, yellow gets a point. However, if
the counters land one red, one yellow, red/yellow gets a point. After fifteen
flips, the person with the most points wins. The teacher should tally total
group results on the board.
ONE RED & ONE YELLOW
Toss 1 Toss 2 Results
R
R
RR
R
Y
RY
Y
R
YR
Y
Y
YY
Review the results; is there any evidence that one player is more likely to
win?


Why do you think that player is more likely to win?
What are the possible results from tossing two counters? See the example
at right.

In probability, the likelihood of tossing either two reds or two yellows is 1 in 4
(25%). However, the likelihood of tossing one red and one yellow is 2 in 4
(50%).
Player A (same signs):
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Place students in groups of three to play the game “Rock, Paper, Scissors.”
All players make a fist, and on the count of three each player shows either:
Paper (flat hand)
Scissors (two fingers)
Rock (fist)
Decide who is player A, B, and C and play fifteen times with the following
rules:
Player A gets a point if all players show the same sign.
Player B gets a point if only two players show the same sign.
Player C gets a point if all players show different signs.
Tally the winning points:

Is this game fair?
SSS, RRR, PPP
Player B (two of the same sign):
SSR, SSP, PPS, PPR, RRP, RRS
Player C (all different signs):
RPS, SRP, PSR
Page 18
Place students in groups of three; two students are players and one student is
the recorder.
Give the two players nine bi-colored counters each and one die; players are to
decide which player takes odd and which player takes evens.
If Player A is even and Player B is odd, any time an even number is rolled,
Player B must give Player A counters to total the number shown on the die.
If an odd number is rolled, Player A must give that many counters to Player
B. Play continues until one player has all the counters.

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d.
The recorder shall fill in the graph documenting who won each roll, and how
many counters the player received.
Materials:
*Fair Game sheet
*Fair Game Results
sheet
*Dice
Why is the player who chose “evens” more likely to win the game?
Place students in groups of three; two students are players and one student is
the recorder.
The two players need a pair of dice and must decide who is even and who is
odd. The dice are rolled and the two numbers are then multiplied. The
recorder shall keep track of the points.
Roll the dice a total of ten times, scoring points under the following rules:
Player A scores a point if the product is even.
Player B scores a point if the product is odd.
Is the game fair? There are 27 even products and 9 odd products.

If not, how could you make the game fair?
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
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Lesson 8 Focus: Strategies in Games
Learn how probability affects the likelihood of winning a game of Rummy.
Materials:
*Decks of cards
Activity 1: Rummy
Start the lesson by talking with the students about a deck of cards.
How many cards are there in a deck? 52

How many Ace of Spades are in the deck? 1 What are the chances of
drawing that card? 1/52

How many cards in the deck are black? Half or 26

What are the chances of drawing a black card? 26/52 or ½

How many different suits are there? 4

What is the probability of drawing a heart? ¼

How many different cards are there in each suit? 13

What is the likelihood of drawing any club out of a deck of cards? 13/52 or
1/4
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d.

Place students in groups of two or four to learn how to play Rummy.
Game rules: The object of the game is to acquire four cards that are either of the same rank (a set) or the
same suit in a sequence (a run), and then three cards that are also a set or a run. The ace can be treated as
either a low (1) or a high (13) card.
Examples:
7,
7,
7,
7
set
4,
5,
6 or
8,
9,
10,
J
run
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

Each player is dealt seven cards, facedown. Each individual looks at his/her own hand, and decides which
cards they want to collect. The deck is placed facedown in the middle of the group; this pile is called the
stack, and the top card of the stack turned over for the discard pile. The player to the left of the dealer
goes first. He may either take a card off the top of the stack or take the top card from the discard pile. He
then must discard one of the cards in his hand onto the discard pile face up and it is the next player’s turn.
Play will continue until one player has a winning hand.
After playing a few hands of Rummy, ask the students what strategies they used when playing the game.
JOURNAL QUESTIONS:
 If you were dealt a 4, 5, 6 of clubs, what cards could be added to that sequence to make a run?
4,

6, you could add the
3 or the
7
If you were dealt a 4, 6, 7 of clubs, what cards could be added to that sequence to make a run?
4,

5,
6,
7 you could only add the
5
Which of the above scenarios is more probable to draw a winning hand?
Page 20
Activity 2: Games
Discuss strategies and probabilities with these games:
Rummikub by Pressman—same strategies as with the card game version of
Rummy
Pass the Pigs by Winning Moves– same strategies as rolling two dice. Some pig
landings are more difficult to get than others.
NOTES:
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d.
Sequence Dice by Jax Ltd.– same strategies apply as those with rolling two dice.
Materials:
*Pass the Pigs
*Rummikub
*Sequence Dice
Page 21
Lesson 9 Focus: Data Collection to Predict Outcome
Collecting data in order to make a predictions.
Lesson Prep: Prepare a paper bag as follows: eight dice in color A and four
dice in color B.
Activity 1: Best Guess
Materials:
*Cups
*Dice in two different
colors
*Paper bags
*Journals
d.
Show students the first paper bag prepared as described above; tell students
that there are a total of twelve dice in the bag (some in color A and some
color B), but that there is not all of one color in the bag. Students will collect
data to help them predict how many of each colored die are in the bag.
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Ask students what possible combinations are in the bag and record results on
the board. There are a total of eleven combinations. Write the combinations on
the board.
Each student, without looking, will draw one die from the bag, say aloud the
color and put it back in the bag. This set of actions is called sampling with
replacement. Have one student record the data on the board.
After everyone has drawn a sample from the bag, have the students write on
a piece of paper their predictions of how many “A” colored dice and how
many “B” colored dice. Predictions should be written as a fraction/ratio —
the number of “A” dice over the number of total dice.
Have all the students again do a sample and replacement; record the data and
once again ask students to make their predictions. In particular, students
should note if their prediction has changed.

Has your prediction changed? Why/why not?

On what are you basing your predictions?
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Proceed through one more round of sample with replacement. Have the
students record any changes in their prediction. Then empty the contents of
the bag for all of the students to see. Review their predictions.
“A”
DICE
“B”
DICE
1
11
2
10
3
9
4
8
5
7
6
6
7
5
8
4
9
3
10
2
11
1
Page 22
Activity 2: Let’s Make a Deal
Let’s Make a Deal is a game where the contestant selects a “door” and the
host then reveals what is behind a door that the contestant did not choose
(this door is never the big prize). Then the contestant is asked if they want to
stick with their choice, or switch to the other door.
Materials:
*Cups
*Dice
Group students into pairs: one student is the contestant, the other the host.
Give each pair three cups and one die. The contestant is to turn away while
the host hides the die under one of the cups.
d.
Next, the contestant is to point to one of the three cups, guessing which is
covering the die — note that the cup is not lifted up to reveal the hidden
object. If the contestant chooses the cup with the die, the host should
eliminate one of the two other cups (it does not matter which one). If the
contestant chooses a cup that is empty, the host should eliminate the other
empty cup.
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Next, the host gives the contestant the option of sticking with the first choice
or switching to the other remaining cup. Once the decision is made, turn over
the contestant’s choice to reveal whether or not the object is hidden
underneath. Let students take turns playing the game for about six rounds.

What is the better strategy — sticking or switching?
Have the students play fifteen rounds with the
contestant always sticking with his/her first choice.
Then, play another fifteen rounds with the contestant
always switching from his/her first choice.
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After recording all the results, ask students which is
the better strategy?
NOTES:
GAME RESULTS
1st pick
Revealed
cup
Remain- Results
ing cup
Dice
Empty
cup A
Empty
cup B
stick win
switch
lose
Empty
cup A
Empty
cup B
dice
Stick lose
switch
win
Empty
cup B
Empty
cup A
dice
stick lose
switch
win
Page 23
Lesson 10 Focus: Probability Review
Reviewing the information learned.
Lesson Prep: Make copies of the A B C D cards; cut apart. Optional — Make copies of the Funny Money sheet; cut apart.
Activity 1: Who Wants to Be a Math Millionaire?
Materials:
*A B C D cards
*Funny Money
* Math Millionaire
Questions sheet
*Math Millionaire
Answer Sheet
d.
Place students into groups of four; all groups play at the same time. Give each
group a set of the A B C D letters. Math Millionaire Questions are valued at
different dollar amounts — as the dollar amount gets higher, the questions
become more difficult.
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To play, the teacher reveals a question on the sheet and reads it and the four
possible answers aloud. The groups then privately confer as to which is the
correct answer. Allow one to two minutes for the groups to come up with a
solution. (A group can show that it is ready to answer by putting their hands
up. ) Once all the hands are up, the Math Millionaire Host counts to three; the
players reveal the letter that their group believes represents the correct answer.
Groups with the correct answer add the question value to their current total.
Example: Teacher reads the $100 question. The groups with the correct answer
can either collect that money from the teacher (if using Funny Money) or the
students can keep a tally on a piece of scratch paper.
Teacher reads the $200 question. Only three groups get the correct answer —
those three groups then add $200 to their previous total of $100. They now
have $300.
If a group needs help prior to answering the questions, they have three options,
but they can use each option only once during the entire game.
1. Ask the teacher (teacher can help that group solve the problem)
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2. 50/50 (teacher reduces that group’s answers from four possible answers to
two possible answers)
3. Survey the class (all students show with hands if they think the answer is A,
B, C or D)
Play continues until the million dollar question has been answered. The group
with the highest dollar amount wins.
HELPFUL HINTS:
 Each group should have one person keeping track of their dollar amounts.
 The student keeping score can also keep track of what help options the group may have used.
 All students in the group have to agree upon the answer.
 The teacher should reveal only one question at a time and cover up the other questions so there is no confusion

as to what question students are answering.
If a group can not come up with an answer within a pre-set time limit, they will need to guess.
Activity 2: Evaluations
Post-Math Club Evaluations:
Please ask the students to fill out a Post-Math Club Student Evaluation sheet. Remind students that this is not a test, but
rather a way for you to understand their thinking.
Also, teachers please fill out the Curriculum Evaluation form so that Zeno learns how to better serve you and the students in
the future.
Please mail all the Student Evaluations (those from the first day of class, and those from the last) and your Curriculum
Evaluation back to Zeno in the self-addressed envelope provided.
d.
Thank You
Bibliography
http://www.Magicofmath.com
Burns, Marilyn. About Teaching Mathematics 2nd Edition A K-8
Resource. Math Solutions Publications, 2000.
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Websites on Probability
http://bodo.com/applets/yahtzee
http://www.pagat.com/rummy
http://www.betweenwaters.com/probab
http://www.block.co.uk/education/mathfiles/gamewheel
http://www.matti.usu.edu
http://www.betweenwaters.com/probab/probab.html
Burns, Marilyn. Math By All Means, Probability Grades 3-4.
Math Solutions Publications, 1995.
Cushman, Jean. Do You Wanna Bet? Your Chance to Find Out
About Probability Clarion Books, 1991.
Jacobs, Harold R. Mathematics: A Human Endeavor, 3rd Edition.
W.H. Freeman and Company, 1994.
Johnson, Art. Classic Math: History Topics for the Classroom.
Dale Seymour Publications, 1994.
Moscovich, Ivan. Mindgames: Probability Games. Workman
Publishing, 2000.
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Van de Walle, John, Elementary and Middle School Mathematics,
4th Edition, Longman Publishing, 2001
Vorderman, Carol. How Math Works. Reader’s Digest Associa
tion, Inc.,1996.
. Math Yellow Pages for Students and Teachers. In
centive Publications, Inc, 2002
Dear Parents/Guardians and Students,
d.
Welcome to a Zeno math club. In this ten week unit, your child will be working on fun activities and
games to reinforce and expand upon their regular classroom math curriculum. This club is an overview
to the mathematics of probability for elementary students in third, fourth and fifth grades.
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During class, students will be playing a number of different games in order to figure out the likelihood
of an event occurring. Students will also be creating graphs and recording information so that they can
figuring out the ratio and percentage of different outcomes, such as all possible results of dice rolls.
They will be playing games to determine if the outcomes are fair, or if the game is weighted to one
particular player (and why). Students will also be encouraged to change rules so that games are fair
and unbiased.
A number of the games played in this unit — games such as Yahtzee, Pass the Pigs, Rummikub and
Sequence Dice — can be purchased at Target, Fred Meyer, or Toys ‘R Us. Students will also be
learning games like Horse Race, and Rummy; all they need to play these games is dice or a deck of
playing cards. Be sure to ask your child to teach you to play these games.
Sincerely,
Zeno
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We hope your child enjoys participating in math club. If you would like to learn more about our
programs please check out our website at www.zenomath.org or call our office at 206-325-0774.
Student Contract
Welcome to a Zeno math club. In order for all students to have a positive experience in
this club, we have developed the following guidelines. The guidelines explain the positive
behaviors we expect, as well as behaviors that are unacceptable.
Positive behaviors will allow us to learn, play, grow and have fun together. Unacceptable
behaviors will be handled by the club instructor, who may choose to contact the parents,
teacher, or principal of the misbehaving student. Consequences of unacceptable behavior
could include a warning or suspension from club activities.
d.
Positive Behaviors
Follow directions.
Wait quietly.
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Listen and cooperate with students and teachers in the program.
Be responsible and respectful with your words and actions.
Treat the materials carefully and use them as instructed.
Help with clean up.
Unacceptable Behaviors
Not following school rules.
Put downs, teasing, and swearing.
Roughhousing, pushing, tripping, hitting, kicking or play fighting.
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Damaging materials or taking them out of the room (without teacher permission).
I agree to follow these behavior guidelines and to do my best to help everyone have a positive experience.
___________________________________
__________
Signature
Date
Pre-Math Club Student Evaluation
Club Teacher: Please read aloud the questions below to all students and allow a few seconds for response
time. Please mail to Zeno pre and post student evaluations, along with the teacher evaluation at the completion of the math club. Thank you.
d.
Student Name: _______________________________________________ Date: __________________
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School: ______________________________________________________ Grade: _________________
Is this your first time in a Zeno math club? ____ Yes ____ No
————————————————————————————————————————————
I think math is fun.
No
Maybe
Yes
I am comfortable answering questions in math class.
Agree
Neutral
Disagree
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Mathematics helps me develop my mind and teaches me to think.
Agree
Neutral
Disagree
I believe I am good at solving math problems.
Agree
Neutral
Disagree
Math is _____________________________________________________________
____________________________________________________________________ .
Pre-Math Club Student Evaluation
1. Draw a line from the following words to their definitions.
the middle number in a set of numbers arranged in order
mode
the average or same amount if distributed evenly
d.
median
the most frequently occurring number
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mean
2. What is the probability of rolling a
3 when rolling a six-sided die? __________________________
3. What is the probability of rolling an even number when rolling a six-sided die? _______________
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4. If you have 3 red (R), 3 yellow (Y), and 3 blue (B) dice in a bag, what are 9 possible combinations of 3
colored dice that you could draw from the bag?
___RRR____
__________
__________
__________
__________
__________
__________
__________
__________
Spinners I
A
B
B
A
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Tally
d.
C
Number
A
B
C
Spin each spinner 15
times and record the
results under each
spinner.
C
Tally
Number
A
B
C
A
B
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B
C
C
F
A
Tally
D
E
Number
Tally
G
Number
Tally
A
A
D
B
B
E
C
C
F
Number
Zeno: Exploring Probability: Lesson 2 Activity 1
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d.
Spinners II
Spinner Puzzler
Create your spinner to fit one of the following descriptions:
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d.
Do not make these spinners in order because you will
switch worksheets with a classmate for him/her to
match spinners with descriptions.

a is certain to win

a cannot possibly win

a is likely to win
a, b, c, d and e are all equally likely to win

a or b will probably win

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
a, b, and c have the same chance to win, and d and e
cannot possibly win.
Biased or Unbiased
Roll the die 20 times; record the results on the chart (first in tally
marks and then in numbers.) Then, graph the results below.
Tally
Number of times
20
18
16
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14
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d.
Face
12
10
8
6
4
2
0
ASSIGNING PROBABILITIES
Use what you know about probability and ratios to
answer the following questions:
B. 3 or less?
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d.
Using one die, what is the probability/ratio of rolling….
A. a 2 or a 4?
C. more than 4?
D. Less than 6?
E. An even number?
F. A multiple of 3?
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G. A seven?
H. Six or less?
I. Are all the answers, from A to F, fractions?
J. What probability words would you use to describe G
and H?
Biased or Unbiased
Club Results
130
120
110
100
90
d.
80
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70
60
50
40
30
20
10
0
Frequency
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Score
(i)
(ii)
(iii)
(iv)
10
12
11
6
10
14
23
4
10
8
7
5
10
11
8
14
10
9
1
16
10
6
10
15
3
3
2
2
3
2
4
4
4
6
7
8
9
5
6
7
8
9
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.
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5
10
10
10
11
11
11
12
12
12
HORSE RACE
Horse Number
Winner Circle
2
3
6
7
8
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9
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5
d.
4
10
11
12
Combinations of Two Dice
Possible
2
3
4
5
6
7
8
9
10
Ratios
Percents
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Total
Ways
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d.
Totals
11
12
FAIR GAME I
Fair Game # 1
Fair Game # 2

Decide if you want to be all red, all yellow or both 
(red and yellow).

Decide which two players are going to flip counters and which player is going to tally.


Flip the counters 15 times and record the results.
RED
YELLOW
RED & YELLOW
Play the game Rock (fist), Paper (flat hand), Scissors (two fingers) but with the following rules:

Player A gets a point if all players show the same
sign.
Player B gets a point if only two players show the
same sign.

Player C gets a point if all players show different
signs.

Decide who is player A, B, and C and
play 15 rounds, recording the results:
PLAYER
TALLY
TOTAL
A
d.
B
C
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Is the game fair?
If not, who had a better chance of winning and why?
Is the game fair?
Which player would you rather be, and why?
Fair Game # 3
Fair Game # 4
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Two students are players and one student is the re- Two students are players and one student is the recorder. Both players get nine counters each and a corder. Students will need two dice and they will
die. Decide who is going to be even and who is going need to decide who is even and who is odd.
to be odd.
For this game students will roll two dice and then figA player rolls the die, if an odd number the even play- ure out the product. If the product is even, the player
er must give the odd player the number of counters that is “even” gets 1 point. If the product is odd, the
shown on the die. For example: if a 3 is rolled the player that is “odd” gets 1 point. The first player to
even player must give the odd player 3 counters.
reach 15 points wins.
The first player to collect all the counters wins.
Recorder records the number on die and the first initial of the person receiving the counters.
EVEN
ODD
The die
shows
Counters
gained by
Is the game fair?
Which player would you rather be, and why?
The die
shows
Counters
gained by
How might you change the rules to make this game
fair?
Zeno: Exploring Probability: Lesson 7 Activity 1-4
FAIR GAME II (CLASS RESULTS)
Fair Game # 1
Fair Game # 2
Possible outcomes
Possible outcomes
RED
YELLOW
RED & YELLOW

Player A gets a point if all players show the same sign.

Player B gets a point if only two players show the same
sign.

Player C gets a point if all players show different signs.
PLAYER
POSSIBLE
RATIO
PERCENT
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Percent:
A
d.
OUTCOMES
Ratio:
B
C
Fair Game # 4
Possible outcomes
Possible outcomes
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Fair Game # 3
Total points
Even
points
EVEN
ODD
Products
Products
Odd points
Can you change the rules to make the game
fair? _______________________________
___________________________________
Ratio
Percent
Zeno: Exploring Probability: Lesson 8 Activity 1-4
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d.
AB
CD
Math Millionaire Game
$100
What is probability?
$300
The likelihood that trees will talk to us in the afternoon.
A. Adding of numbers B. drawing of a graph C. likelihood of an event D. multiplication
B. 50%
C. 2%
D. 20%
If you have a 1 in 4 chance of drawing a heart out of a deck of cards, what percent would that be?
A. 25%
$2000
D. impossible
If you have a 1 in 2 chance of getting tails when flipping a coin, what percent would that be?
A. 25%
$1000
C. probable
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$600
B. possible
d.
A. certain
B. 40%
C. 80%
D. 20%
When rolling two dice, there are how many possible outcomes ?
B. 26
C. 36
D. 12
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A. 30
$4,000
How many outcomes are there when you roll three dice?
A. 36
$7,000
B. 9
C. 180
D. 216
If you had 2 shirts, 3 pants and 2 hats, how many different clothes combinations would you have?
A. 12
B. 7
C. 8
D. 24
Math Millionaire Game
$15,000
What is the median number in the following series? 3, 3, 4, 4, 5, 5, 5, 6, 8, 8, 8, 8, 10
$30,000
What is the median number in the following series? 24, 34, 28, 25, 25, 28, 32, 44, 50
A. 5
C. 50
D. 10
D. 28
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$65,000
B. 34
C. 8
d.
A. 25
B. 6
Which number represents the mode in the following series of numbers?
3, 10, 5, 11, 10, 13, 8, 11, 10, 9, 10, 8, 10, 11, 5, 12, 3
A. 10
$125,000
B. $4.50
C. $3.25
D. $4.30
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B. 2/9
C. 1/4
D. 1/25
One postcard will be drawn out of 1,000 received for free concert tickets. You mail in
30 postcards. What are your chances of winning the tickets?
A. 30%
$300,000
D. 9
Your company is giving away 5 free trips to Hawaii to the employees. There are 125 employees in your company. What is the probability that you will win the trip?
A. 1/10
$250,000
C. 8
Jack has $4, Sam has $3.50, Julia has $5.25 and Robin has $3.25. What is their average amount of money?
A. $4
$200,000
B. 3
B. 33%
C. 3%
D. 100%
Three boys and five girls are nominated for an honors band. One boy and one girl will be
selected at random. What are the changes of being selected? (2 answers are correct;
you need both correct answers to win)
A. 33%
B. 20%
C. 25%
D. 40%
C likelihood of an
event
$300
D impossible
$600
B 50%
$1000
A 25%
$2000
C 36
$4,000
D 216
$7,000
A 12
$15,000
A5
$30,000
D 28
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$65,000
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$100
A 10
$125,000
A $4.00
$200,000
D 1/25
$250,000
$300,000
d.
Math Millionaire Game Answers
C 3%
A 33% and B 20%
$
10000
$
1000
$
10000
$
5000
1000
$
$
10000
$
500
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1000
100
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$
100
$
500
d.
100
500
$
5000
5000
50000
$
50000
$
50000
Post-Math Club Student Evaluation
Club Teacher: Please read aloud the questions below to all students and allow a few seconds for response
time. Please mail to Zeno pre- and post-club student evaluations, along with the teacher evaluation at the
completion of the math club. Thank you.
d.
Student Name: _______________________________________________ Date: __________________
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School: ______________________________________________________ Grade: _________________
Would you like to attend another Zeno math club? ____ Yes ____ No
————————————————————————————————————————————
I think math is fun.
No
Maybe
Yes
I am comfortable answering questions in math class.
Agree
Neutral
Disagree
A
ll
Mathematics helps me develop my mind and teaches me to think.
Agree
Neutral
Disagree
I believe I am good at solving math problems.
Agree
Neutral
Disagree
Math is _____________________________________________________________
____________________________________________________________________ .
Post-Math Club Student Evaluation
1. Draw a line from the following words to their definitions.
the middle number in a set of numbers arranged in order
mode
the average or same amount if distributed evenly
d.
median
the most frequently occurring number
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mean
2. What is the probability of rolling a
3 when rolling a six-sided die? __________________________
3. What is the probability of rolling an even number when rolling a six-sided die? _______________
A
ll
4. If you have 3 red (R), 3 yellow (Y), and 3 blue (B) dice in a bag, what are 9 possible combinations of 3
colored dice that you could draw from the bag?
___RRR____
__________
__________
__________
__________
__________
__________
__________
__________
Curriculum Evaluation Form
Dear Club Teacher,
Thank you so much for making math fun for feedback from you regarding the
club lessons and games. Please include on the back of this form any additional
information you would like us to know. Thank you.
Ages/Grades of Students:
How many weeks long was the club?
How many lessons were completed:
How many students did you teach?
d.
__________________________________________________________________________________________________________
From the beginning to the end of math club, overall, did you observe any shift in student confidence? Please explain.
No change
More confidence
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Less confidence
_________________________________________________________________________________________________________
_________________________________________________________________________________________________________
What lessons and games did you find to be most helpful, and why?
__________________________________________________________________________________________________________
__________________________________________________________________________________________________________
What lessons and games did you find to be least effective, and why?
__________________________________________________________________________________________________________
__________________________________________________________________________________________________________
Is there anything that you feel needs to be changed or restructured?
__________________________________________________________________________________________________________
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__________________________________________________________________________________________________________
Do the daily lessons provide enough activities to fill an hour?
__________________________________________________________________________________________________________
__________________________________________________________________________________________________________
Were any supplies missing from the club kit?
__________________________________________________________________________________________________________
__________________________________________________________________________________________________________
Please return evaluation forms to:
Zeno
1404 East Yesler Way, Suite 204
Seattle, WA 98122
If you have any questions or concerns please feel free to contact:
Program Director, Jennifer Gaer at 206-325-0774 or Jenniferga@zenomath.org

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