AAC strikes again! I found another great assessment for grade 6 called "The Penny Carnival"
Teacher Resources:
Smartboard Resources http://www.aac.ab.ca/resources/PA/EMMA6PennyCarnival.notebook
CONTEXT FOR
LEARNING
The school is organizing a Penny
Carnival to help raise money for a new community playground. A Carnival
Committee is in charge of choosing games for the event. Each student will
design a spinner game, complete with prize
recommendations, to present to the committee in hopes that it will be chosen
for the Penny Carnival. Students will build and test spinners to determine and
compare the experimental and theoretical probabilities of landing on each
section. They will choose prizes for their games, keeping in mind that all
carnival games will cost just 25¢ to play. Students will create presentations
for the committee, to convince them to choose their games.
This task provides
opportunities for students collect, display and interpret data, and to demonstrate
their understanding of experimental and theoretical probability. As they explore
the task, share their strategies and defend their plans, they refine their
ability to communicate and reason mathematically (see Program of Studies, 2007,
p. 4).
Teacher Resource Materials
include:
·
Learner Outcomes and Assessment Criteria (p.
2)
·
For Best Results (pp. 3-5)
·
Prize Page (p.6)
·
First Steps Student Support (pp. 7-8)
Student Materials
include:
·
Student Task (p. 1)
·
Checklist and Rubric (p. 2)
·
Optional Student Pages (pp. 3-5)
Assessment for Learning Tools:
•
Coaching Feedback
•
Student Self-reflection
ASSESSMENT AND EVALUATION OF STUDENT
LEARNING
This
performance task addresses the following learner outcomes (shown in Times New Roman font) from
the Mathematics Program of Studies:
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Grade Six Learner Outcomes
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Criteria for
Evaluation*
Students
provide evidence of their learning as they:
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Statistics and
Probability (Chance and Uncertainty)
General
Outcome: Use experimental or theoretical
probabilities to represent and solve problems involving uncertainly.
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4. Demonstrate an understanding of
probability by:
·
identifying
all possible outcomes of a probability experiment
·
differentiating
between experimental and theoretical probability
·
determining
the theoretical probability of outcomes in a probability experiment
·
determining
the experimental probability of outcomes in a probability experiment
·
comparing
experimental results with the theoretical probability for an experiment.
[C, ME, PS, T]
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· determine experimental
probability of outcomes
· determine theoretical
probability of outcomes
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Statistics and
Probability (Data Analysis)
General Outcome:
Collect, display and analyze data to
solve problems.
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3. Graph collected data, and analyze the
graph to solve problems.
[C, CN, PS, R, T]
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· create and analyze a
graph
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Number
General
Outcome: Develop number sense.
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2. Solve problems involving whole numbers and
decimal numbers.
[ME, PS, T]
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· determine expected cost
of prizes
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Mathematical Processes
Students are
expected to:
·
develop
mathematical reasoning
·
communicate in order to learn
and express their understanding.
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·
create
presentation for committee
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*
Criteria statements appear again in the first column of the evaluation tools
(checklists, rating scales and/or rubrics) and are the basis on which student
evaluation is made relative to the learner outcomes.
Mathematical
Processes
Mathematical
processes are skills that are addressed at all grade levels. They are not
taught as discrete skills, but are integrated into the specific outcomes. Links
to the processes are identified within square brackets after the specific
outcomes.
Throughout
this task, the following mathematical processes are specifically addressed:
- Communication:
communicate in order to clarify, reinforce and modify ideas.
- Connections: connect mathematical ideas to each other
or to the real world.
- Problem
Solving: develop and apply new mathematical knowledge through problem
solving.
- Reasoning: use
reasoning skills to analyze a problem, reach a conclusion and justify or
defend that conclusion.
FOR BEST RESULTS
This
section provides suggestions for additional instruction and assessment
for learning support. A student self-reflection tool and a peer-coaching
tool have been provided in this package. These tools are not intended to be used for grading purposes, but rather to
scaffold students along the way to successful completion of the performance
task. As not all students will require the same type and/or amount of
scaffolding, teachers make instructional and coaching decisions based on
student needs.
After
initial suggestions on preparing for the task, the information in this section
is organized around the criteria for evaluation. Thus, teachers can target the
areas where they feel students require additional support and guidance.
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To
help prepare students for the task…
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·
Share the context, using the notebook or PowerPoint
file if desired. What experiences have students had with carnival games? Spinners?
Would spinners always give you the results you expect? What are some variables
that could affect the fairness of a spinner? How could this impact a carnival
game? What would you have to consider as you chose prizes for your game?
·
You may choose to review the concepts of probability
in general, and theoretical and experimental probability in particular. What
could impact the reliability of your spinner? How does the number of trials in
your experiment affect your confidence in the experimental probability results?
·
Provide materials including the spinner page
and a paper clip for each student. Effective spinners can be made by
straightening one end of a small paper clip, and holding it in place with the
point of a pencil when spinning. Use the tip of the straightened end to accurately
mark the section, and be sure the spinner is on a flat, smooth surface.
Students may wish to work with a partner when testing spinners and recording
results. You may wish to remind students to count carefully, and stop after
exactly 100 spins. This makes comparisons and prize calculations more
manageable.
·
The probability of an event is represented
by a number between 0 (impossible) and 1 (certain). For this task, the spinner
provided in the Student Materials is a circle marked along the edge in
hundredths. Theoretical probability for each section can be represented as a
decimal number, fraction or percentage by considering the number of hundredth
segments included in that section. Suggest students check to ensure the
probabilities of all sections total 1.0 or 100%.
·
While students will be interested in
assigning prizes to each section right from the start, they will want to
revisit those choices once they have tested the spinners. Remind them that
players will only be paying 25¢ for a spin, and that they should consider the
amount they would collect for 100 spins at the carnival. The “Carnival
Committee” will not choose a game if it seems likely to lose money.
Differentiation
notes:
A
list of possible prizes, including prices, is included in this package (p. 6)
and could be supplied for students who need more structure for that part of the
task. The “First Steps” pages (pp. 7-8) also provide support. Students who need
more challenge could be given spinners with no markings on the edge, and asked
to use their knowledge of angles to calculate theoretical probabilities.
Students may also choose to test spinners more than 100 times, which will
increase the complexity of calculating experimental probabilities.
Combined Grade
Teachers:
This
task has many connections to curriculum outcomes at the Grade 5 level, and
would be very suitable for use in a combined grade classroom.
Assessment for Learning Support
ü
Share the assessment task, criteria and
rubric with students at the beginning of the activity to help focus their
learning during the unit of study.
ü
Exemplars are a powerful way to help
students understand the expected standard of performance by viewing work at a
variety of levels of proficiency. Check the
AAC website (www.aac.ab.ca),
where student exemplars will be posted as they are available.
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To
help students determine experimental
and theoretical probabilities (criteria #1 and #2)
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·
Help your students understand the difference
between theoretical probability, which is determined through mathematical
reasoning, and experimental probability, which is determined through
observation and the gathering of data. In some situations, we can quite easily
determine theoretical probabilities (flipping a coin, shaking a die, choosing a
card, spinning a spinner, choosing a marble from a bag, etc.) However, the
experimental probabilities for each of these situations vary depending on the
data you collect. Often, it isn’t possible for us to determine the theoretical
probability of an event (for example, the probability of having heart disease
if you are a smoker, or the probability of the school bus being late), so we
depend on experimental probability in these situations.
·
Typically we think of theoretical
probability as a ratio of the number of favorable outcomes to the total number
of equally likely outcomes. “Equally likely” is an important qualifier, and in
the case of the spinner game your students will want to consider each tiny
hundredth wedge of the circle as one equally likely outcome. This means that
the probability of landing on a section is determined by counting the hundredth
wedges in that section and dividing by the total number of wedges (100).
·
When determining the experimental probability
of an outcome, such as landing on a certain section of a spinner, it’s
important to gather data from a large number of trials. In this task, 100
trials is suggested in part because of the ease of comparing experimental and
theoretical probability. If students are prepared to test their spinners more
than 100 times let them, but be prepared to support them as they calculate and compare
probabilities.
·
Provide students with many opportunities to
calculate experimental probabilities for a variety of events, then determine
the theoretical probabilities and compare the two. This is particularly
interesting when the experimental results might be surprising and cause
students to question assumptions they might be making about theoretical
probabilities of certain results. For example:
o
Flip two coins to calculate an experimental
probability of two heads, two tails, or one head and one tail.
o
Shake two dice to calculate an experimental
probability for each possible sum.
o
Place two blue tiles and two red tiles in a
paper bag. Calculate an experimental probability of choosing two tiles of the
same colour or two tiles of different colours. Repeat using one blue tile and
three red tiles, or one blue tile and two red tiles. (Try this one yourself.
You will likely be surprised by the results!)
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To
help students create and analyze a
graph (criterion #3)
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·
With your students, create a list of
criteria for a quality double bar graph: informative title, clear and complete labels
for axes, suitable choice of scale, legend where applicable, accurate bars, etc.
·
Give your students frequent opportunities to
create and interpret graphs representing data collected during classroom
activities.
·
In cases where the experimental results are
significantly different from the theoretical probabilities, ask students to
speculate on the possible explanations. Homemade spinners are not always as
random as we may think. Encourage students to continue testing a spinner that
is not behaving as expected.
Assessment for Learning Support
You may choose to
use the Coaching Feedback tool or Student Self-Reflection tool to help your
students focus on the criteria for this task and to involve them in the
feedback process. In both cases, be prepared to change the questions if you
feel there are other areas for reflection that would be more helpful to your
students. These tools are not used
for marking purposes.
Note:
Allow
time for students to make adjustments to their work based on the feedback and
reflection before submitting it for
evaluation.
Coaching
Feedback Tool (Assessment for Learning Tools)
ü This is an opportunity for oral
conversation and feedback from either a peer or the teacher. Students
will discuss criteria with their coach to help clarify thinking.
ü Students will be asked to evaluate the
feedback they received, and use the feedback to improve their work in some way.
Student Self-reflection
(Assessment for Learning Tools)
ü
This
gives students an opportunity to reflect on the criteria and how successfully
they are meeting them.
Prizes for… (p. 6 of
this document)
ü This page provides prize choices and
prices to support those students who are unable to complete this part of the
task without more structure.
ü The page may be edited to remove some
of the prizes and present students with fewer choices.
First Steps (pp. 7, 8 of this document)
ü This tool provides students who are not
yet ready to complete the task independently with structured support.
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For further
discussion…
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·
Did you use the experimental or theoretical
probability to calculate the expected cost of the prizes? Why did you make that
choice? How would you deal with unexpected results at the actual carnival?
(Pictures for the following items would not transfer but are easy to make on any clip art program.)
Stuffed animal - $1.25 Bubble Gum - $0.06
Pencil - $0.15 Little Board Game - $1.00
Felt Marker - $0.40 Eraser - $0.12 Mini
Chocolate Bar - $0.25 Small Notebook - $0.50
Helium Balloon- $1.00 Comic Book - $1.50 Lollipop - $0.10 Bouncy Ball - $0.30
Plastic Ring - $0.33 Hot Dog Coupon - $0.75 Sticky Notes - $0.50 Licorice Whip - $0.15
Name
__________________________________
Create
a spinner with 5 sections of different
sizes. What is the theoretical
probability that you will land on each section? Record on the chart below.
(Hint: See how many of the 100 tiny
wedges are in each of your sections.)
Now
test your spinner exactly 100 times.
Keep a tally on the chart. What is the experimental
probability of landing on each section?
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Section
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Theoretical Probability
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Tally
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Experimental Probability
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1:
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2:
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3:
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4:
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5:
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Create a graph to compare the
theoretical and experimental probabilities. What do you notice?
Decide which sections of your spinner will have prizes.
Think about what prizes you could afford to give away.
Remember to think about these questions:
If 100 people play your spinner game,
and each person pays 25¢, how much money will you collect altogether? How much
will it probably cost to give out the prizes those people win?
Make sure you are not spending too much on prizes, or the
committee won’t choose your game!
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Section
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Prize
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Cost for one
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Expected cost for 100 players
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1
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2
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3
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4
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5
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Why do you think your game will make money for the
playground?____________________________________________________________
Student Resources:
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Step right up! Our school is
organizing a Penny Carnival for the neighbourhood to help raise money for a
new community playground. The Carnival Committee will be choosing games for
the event. You are going to design a spinner game, complete with ideas for
prizes, and convince the Carnival Committee to choose your game.
Because
homemade spinners don’t always work the way you expect them to, it will be
important to test your spinner carefully, and compare those results with the
results you expected.
It will also be important to choose
your prizes carefully. People will only have to pay 25¢ to play the games at
the carnival, and the committee won’t pick your game if it’s a money-loser!
As you
create your submission for the committee, you will:
·
Build a spinner with at least 5
sections of different sizes.
·
Determine
the theoretical probability of landing on each section.
·
Collect data to help you determine an experimental probability
for landing on each section.
·
Create
and analyze a graph to compare your experimental data with the theoretical probability. What do
you notice?
·
Determine
the expected cost of prizes. Some questions to consider: Which sections
of your spinner will be prize-winners? What prizes will you offer? How much
will they cost? Why do you expect your spinner game to be a successful
money-maker?
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Student
_____________________________________ Date
_____________________
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Criteria
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Specific Requirements
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Yes
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Not Yet
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Teacher
Comment
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Determine
theoretical probability of outcomes
(Stats and Prob 4)
[PS,
R, V]
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· able to correctly determine the
theoretical probability of landing on each section of the spinner
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Determine experimental probability of
outcomes
(Stats and Prob 4)
[C, PS, R]
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· able to correctly determine the
experimental probability of landing on each section of the spinner
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Level
Criteria
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4
Excellent
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3
Proficient
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2
Adequate
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1
Limited *
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Insufficient/
Blank *
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Create and analyze a graph
(Stats and Prob 3)
[CN, R, V]
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Graphs probability using a purposeful design.
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Graphs probability using a thoughtful design.
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Graphs probability using a basic design.
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No score is awarded because there is
insufficient evidence of student performance based on the requirements of the assessment task.
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Provides an in-depth comparison of probabilities.
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Provides a substantial comparison of
probabilities.
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Provides a simplistic comparison of
probabilities.
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Provides a questionable comparison of
probabilities.
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||
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Determine expected cost of prizes
(Number 2)
[C, R]
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Determines a precise estimate of the cost of prizes.
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Determines a meaningful estimate of the cost of prizes.
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Determines a reasonable estimate of the cost of prizes.
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Determines an inaccurate estimate of the cost of prizes.
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Create submission for committee
[C, R]
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Provides a compelling explanation of why the game should be a successful
money-maker.
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Provides a logical explanation of why the game should be a successful
money-maker.
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Provides a basic explanation
of why the game should be a successful money-maker.
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Provides an unsupported explanation of why the game should be a successful
money-maker.
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* When work is judged to be limited or
insufficient, the teacher makes decisions about appropriate intervention to
help the student improve.
Name
___________________________________ Date ________________
Design a spinner for your game, with at least 5 sections of different sizes.
Title:
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Name
_______________________________________________________
Are you ready to submit your plan to
the committee? Examine your submission, or work together with a partner to check
that you have everything prepared:
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In my submission to the committee, did I…
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Yes
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Not Yet
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… include my spinner, with at least 5
sections of different sizes?
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… determine the theoretical
probability for each section of the spinner?
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… use experimental data from at least
100 trial spins to find the experimental probability for each section of the
spinner?
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… include a graph clearly showing
both the experimental and theoretical probabilities for each section of the
spinner?
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… compare the experimental and
theoretical probabilities, discussing similarities and differences?
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… recommend prizes to go with my
spinner game?
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… clearly explain why I think my
spinner game will be a money-maker, after considering the cost of my prizes?
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Here is one thing I am going to do to improve my final
submission:
_________________________________________________________________________________________________________________________________________________________________________________________________________________
Teacher/ Learning Coach Feedback:
Student
_______________________________________________________________
Coach
_________________________________________________________________
Instructions for the Coach: Using the questions below,
interview a classmate as he or she prepares a submission for the Carnival
Committee. Provide feedback to help your partner be ready to present the
spinner game.
Here
are some questions to ask your partner:
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?
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How did you calculate the theoretical
probabilities for each section of your spinner?
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?
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How did your results for the experimental
probabilities compare to the expected results?
Were there any big surprises? How can you explain
the differences?
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?
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How did you decide on your spinner prizes?
Why do you think your spinner game will make a
profit for the playground fund?
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Instructions
for the Student: Consider the usefulness of the feedback you
received.
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The most helpful suggestion I received was…
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Student Reflection:
Name _________________________________________________________________
Is your submission ready to present to
the Carnival Committee? Choose at least
one area for improvement.
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CRITERIA
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Mark an “x” to show your level of readiness.
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I still have a
lot of
I feel very confident that
work to do on
this part.
this part of my presentation
is ready to go.
I
will improve this part of my submission by…
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My graph is accurate and I’ve made a precise comparison of the theoretical and experimental
probabilities.
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I still have a
lot of
I feel very confident that
work to do on
this part.
this part of my presentation
is
ready to go.
I
will improve this part of my submission by…
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I have a compelling explanation, based on
mathematics, as to why my spinner game will make a profit.
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I still have a
lot of
I feel very confident that
work to do on
this part.
this part of my
presentation
is ready to go.
I
will improve this part of my submission by…
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Alberta Assessment Consortium http://www.aac.ab.ca/grade06/mathematics/the-penny-carnival.html
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