Card Game
This problem gives you the chance to:
• figure out and explain probabilities
Mrs Jakeman is teaching her class about probability.
She has ten cards, numbered 1 to 10.
She mixes them up and stands them on a shelf so that the numbers do not show.
Mrs. Jakeman turns the cards around one at a time.
Students have to guess whether the next card will have a higher or a lower number than the one just
turned.
The first card turned is the number 3.
3
1. Would you expect the next number to be higher than 3 or lower?
___________
Explain why you made this decision.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
The second card is number 10.
3 10
2. What is the probability that the next card will be a higher number than 10?
____________
Explain how you know.
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
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Card Game Test 6
The third card is number 4.
3 10 4
3. What is the probability that the next number is higher than 4?
Show your work.
The fourth card is number 7.
3 10 4
__________________
7
4. What is the probability that the next number is lower than 7?
Show your work.
The fifth card is the number 1.
3 10 4
_____________
7
1
When the sixth card is turned the probability that the next card is higher is the same as the
probability that it is lower.
5. What must the sixth card be?
______________
Explain how you figured it out.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
9
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Card Game Test 6
Task 2: Card Game
Rubric
The core elements of performance required by this task are:
• figure out and explain probabilities
points
section
points
and gives a correct explanation such as:
There are more cards higher that 3 than lower than 3.
1
1
Gives a correct answer: 0 or impossible
1
Based on these, credit for specific aspects of performance should be assigned as follows
1.
2.
3
Gives a correct answer: higher
2
Gives a correct explanation such as:
All the cards are lower than 10 so it is impossible for the next card to be
higher.
1
Gives a correct answer: 5/7 or equivalent (71%)
1
Shows correct work such as:
5,6,7,8,9 There are five higher numbers
1
2
4.
5.
Gives a correct answer: 4/6 or equivalent (66.6%)
1
Shows correct work such as:
1,2,5,6 There are four lower numbers
1
Gives a correct answer: 6
1
Gives a correct explanation such as:
The cards left are 2, 5, 6, 8 and 9
The middle one of these is the 6 leaving two higher and two lower.
1
2
Total Points
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2
9
Card Game Test 6
Card Game
Work the task. Look at the rubric. What are the big ideas about probability a student
needs to understand to work this task? What strategies might the student use to identify
the sample space?
What is the classroom norm for providing justification or explanation for work?
Look at student work for part 2. How many of your students thought the probability
would be lower rather than 0% chance of higher? _____________
Now look at student work for part 3. How many of your students put:
5/7
1/2
6/10 Words: good,
7
Higher
2/5
yes, likely
7/10
Other
How do you think students got some of these answers? How are the misconceptions
different?
Now look at the work for part 4. How many of your students put:
4/6 or 2/5 or Lower
Words:
6/10
1/2
3/10
2/3
4/10
likely, good,
it’s a pattern
A
whole
number
Other
How do you think students got some of these answers? What is the logic of their
misconceptions? What don’t they understand about probability?
Part 5, was understanding and interpreting the remaining sample space. Look at student
work. How many of your students put:
6
Lower
Higher
2
5
8
No
Other
response
What made this part difficult for students?
Now look at all the work. What were some of the best ways that students made models
to help them think about sample space. How could you use these examples with your
class to help them design better models for themselves?
Look at your textbook. What opportunities do students have to think about and make
models for sample space? Does your textbook give students tasks involving a diminishing
sample space or probability without replacement? If this is the only year that students are
exposed to probability, do you think that your textbook provides adequate time for
students to build a deep understanding of probability? What might you change in your
program for next year?
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Looking at Student Work on Card Game
Student A is able to model the sample space and define which parts of the sample space
contribute to the probability. Notice the clarity of explanations.
Student A
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Student B also makes an interesting model for the sample space. Can you identify what
the numbers on the top and bottom of the model represent? Student B makes the common
mistake of continuing to think about probability, instead of what finding a card will
produce a 50/50 chance of the following card being higher or lower.
Student B
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Student C is able to produce good models for part 3 and 4 of the task. The student gives
odds instead of probability. Odds are expressed as favorable to unfavorable outcomes.
Probability is expressed as favorable outcomes out of total possible outcomes.
Student C
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Student D understands how to express probability, but doesn’t understand that the size of
the sample space decreases as each card is drawn and not replaced back into the deck.
Student D
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Some students, like Student E, try to find a pattern. They are missing the basic
understanding that all remaining cards are equally likely.
Student E
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Student F is also looking at patterns. This process focuses Student F on looking at
outcomes as either higher or lower, rather than quantifying how many are higher or how
many are lower.
Student F
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Student G is able to model and describe the sample space for the task, but doesn’t
understand how to express a probability numerically.
Student G
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Student H expresses the situation as a ratio of used cards: unused cards. Looking at the
explanation in part 2, the student is not thinking about what the actual unused cards might
be.
Student H
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Student I attempts to model the situation. The student is thinking about used cards out of
total cards; a fractional model rather than a probability model. How do we help students
develop their skills for modeling mathematical situations?
Student I
Student J does not understand the idea that probability is thinking about something that
hasn’t happened yet, when there are a variety of possibilities. Once something has
happened it is no longer a probability, but a fact. How do students learn these basic
underpinnings of probability? What types of experiences do they need? What questions
help students to examine these basic ideas?
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Student J
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Student K seems to have no idea about probability. Instead the student just gives number
sentences. This student seems not to expect mathematics to make sense, but to be a series
of procedures. What questions might you want to ask this student? What might be good
first experiences to introduce the idea of probability to this student?
Student K
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6th Grade
Student Task
Core Idea 5
Probability
Task 2
Card Game
Figure out and explain probabilities. Work with odds in a practical
situation by determining the sample space.
Demonstrate understanding and use probability in problem
situations.
• Determine theoretical and experimental probabilities and use
these to make predictions about events.
• Represent probabilities as ratios, proportions, decimals or
percents.
• Represent sample space for a given event in an organized way.
Based on teacher observation, this is what sixth graders know and are able to do:
• Describe whether the next number would be higher or lower for part 1.
• Describe the probability for getting a card higher than 10 and why it is
impossible.
Areas of difficulties for sixth graders:
• Expressing probabilities numerically instead of verbally
• Defining sample space in a series of events with no replacement
• Understanding that events of chance are not patterns
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The maximum score available for this task is 9 points.
The minimum score needed for a level 3 response, meeting standards, is 4 points.
A majority of students, 80%, could explain that if 3 was the first card drawn from a deck of 10 the
next number would be higher because there are more high cards left in the deck than low cards.
More than half the students could also give the probability for drawing a card higher than 10 and
explain why this was true. Almost 42% could also draw or explain a sample space for part 3 or 4.
8% of the students could meet all the demands of the task including thinking about a series of
events with no replacement and describing the probability numerically, and finding a number
from the remaining sample space that would make the probability of getting higher or lower card
on the next turn equally likely. 20% of the students scored no point on this task. All of the
students in the sample with this score attempted the task.
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Card Game
Points
Understandings
All the students in the sample
0
1
3
4
5
7
9
Misunderstandings
Students couldn’t explain why the next
attempted the task.
card would most likely be higher. 35% said
higher with an incorrect explanation or no
explanation.
Students could explain why the
Students had difficulty explaining the
next number would be higher in probability for getting a card higher than 10
part 1.
in part 2. 24% put lower for the probability.
Students could solve part 1 and 2 Students had difficulty defining the sample
of the task.
space in part 3 and 4. 12% thought the
probability in part 3 was 1/2. 9% used
words like yes, good, likely. 17% just said
higher without giving a numerical value for
part 3. 7% thought the answer would be
6/10 considering only the unused cards out
of the total cards. 4% thought 4/10,
considering the used cards to total cards.
Both of the last responses ignore the
shrinking sample size.
Students could solve part 1,
In part 4, 20% thought the answer was just
explain part 2, and draw a
lower. 11% gave the answer 2/5 or 4/10.
sample space for part 3 and 4.
Students could solve parts 1, 2,
and 5.
Students missed all of part 5. 24% thought
the answer was higher. 13% thought the
answer was 5. 5% thought the answer was
2 and 5% thought the answer was 8.
Students could meet all the
demands of the task including
thinking about a series of events
with no replacement and
describing the probability
numerically, and finding a
number from the remaining
sample space that would make
the probability of getting higher
or lower card on the next turn
equally likely.
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Implications for Instruction
Students should be able to think about sample space, what cards are available and then
find a numerical probability of getting a higher or lower number. Many students think
that probability is about a pattern. If one card is high, the next will be low. If I toss a
coin and get heads, then the next time I toss a coin I will get tails. They don’t have
enough experience with chance to understand that all events are possible, but some are
more likely than others. Students need opportunities to experiment with chance
opportunities, like rolling die, pulling cards, tossing cards. Computer generated
experiments that can show them the results of probability experiments over large
numbers are also helpful to build general intuitions or sense about probability. This
ground word needs to be developed, before trying to formalize the information with
algorithms. Many of the games students play electronically, obscure how probability
plays in the outcomes. So even more direct experiences need to happen in the classroom
and talked about explicitly.
Students need to be able to define sample space: what are all the possible outcomes. They
should be familiar with tools, like tree diagrams or organized lists, to help them record.
This is the grade level where probability should be mastered to prepare for the high
school exit exam.
Ideas for Action Research – Ideas for Re-engagement
One useful strategy when student work does meet your expectations is to use student
work to promote deeper thinking about the mathematical issues in the task. In planning
for re-engagement it is important to think about what is the story of the task, what are the
common errors and what are the mathematical ideas I want students to think about more
deeply. Then look through student work to pick key pieces of student work to use to pose
questions for class discussion. Often students will need to have time to rework part of the
task or engage in a pair/share discussion before they are ready to discuss the issue with
the whole class. This reworking of the mathematics with a new eye or new perspective is
the key to this strategy.
Students thinking on Card Game showed several errors on expressing probability and
defining sample space. Look at your student papers. What are some of the key
mathematical ideas around probability that your students had difficulty with? What
strategies did students use to solve the problem? What questions could you pose to plan a
class discussion to get students to refocus on the key mathematics? For example, you
might pose this question for the class:
Fred says, “ I think we use this model to help us solve the problem.
What is Fred showing in his model? Can you explain what he is doing?
How does this question get all students rethinking the mathematics of the task? How did
this prompt stimulate class discussion? What big mathematical ideas came up?
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For further questions or prompts, consider:
Kira says, “I made my model slightly differently.”
Does this make sense?
How are the models the same? How are they different?
How does this prompt build on the thinking from the first task? How does this help build
to writing a numerical value for probability?
Now pose a question, like:
Samantha says, “I think the probability is 5 to 2.”
George says, “I disagree. I think the probability is 4/10.”
Lisa says, “I think the probability is 5/7. “
Who is correct? Why does that make sense?
The heart of the process of re-engagement is in the discussion, controversy, and
convincing of the big mathematical ideas. This is where students have the opportunity to
clarify their own thinking, confront their misconceptions to see the errors in logic, use
mathematical vocabulary for a purpose, and make generalizations and connections.
What other issues from this task might you want to raise for class discussion?
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