Probability Unit
Phuong, Sue and Vanessa
Understanding the relationship between theoretical and experimental probability,
seventh grade.
Specific standards addressed in math unit: WA math standards; common core math
standards. (list the standards)
Washington State Learning Standards
7.4 Core Content: Probability and Data
B. Determine the theoretical probability to predict experimental outcomes.
Common Core) Statistics and Probability 7.SP.
Investigate chance processes and develop, use, and evaluate
probability models.
5. Understand that the probability of a chance event is a number
between 0 and 1 that expresses the likelihood of the event occurring.
Larger numbers indicate greater likelihood.
6. Approximate the probability of a chance event by collecting data on
the chance process that produces it and observing its long-run relative
frequency, and predict the approximate relative frequency given the
probability.
7. Develop a probability model and use it to find probabilities of events.
Compare probabilities from a model to observed frequencies; if the
agreement is not good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal.
Probability to all outcomes, and use the model to determine
probabilities of events.
Why this topic is important for students to know and for you to teach (beyond the
standards link).
Here is why it is important to teach:



develop simulation approach to solve real-world problems--experimental approach to
probability (VDW 464)
"it is significantly more intuitive. Results begin to make sense and do not come from
some abstract rule--experimental approach to probability (VDW 464)
"it provides an experiential background for examining theoretical model (VDW 464)
Here is why it is important for students to know:
Probability Unit



Phuong, Sue and Vanessa
"Informed citizens need to be numerate in data and chance and need to know how to
decipher and make sense out of information that is presented in newspapers, medical
reports, consumer reports and environmental studies (NCTM ).
To make well informed decisions rather than relying on intuition or guessing,
understanding probability enables people to be critical consumers of data.
For example, probability (chances) are all around us, e.g., weather, risks and other
probability ideas are very prevalent in today's world (VDW 456)
Description of learning issues related to this topic :



Students may believe theoretical probability is the true probability, and should be
reflected in each trial. Otherwise they feel they are getting incorrect results.
Students may have difficulty defining sample space, understanding all the possible
outcomes that can occur (Swedish dice article).
Students may have trouble conceptualizing that the probability of an event must be
between 0 and 1 inclusive.
Key understandings





Student will develop understanding the relationship of theoretical and experimental
probability
Student will develop understanding of a probability continuum from impossible to
certain. Then students will connect the probability's vocabulary to a numerical continuum
from 0 to 1 and from 0% to 100%.
Student will develop a understanding of "law of large number--the phenomenon that the
relative frequency becomes a closer approximation of the actual or the theoretical
probability as the size of the data set (sample) increases." (VDW 462)
For simple experiment. "Chance has no memory. The outcomes of prior trials have no
impact on the next." (After 5 heads in a row, the chance of a head is still .5)
The representation of the probability of an event can take the form of fractions, percent or
decimals, and can move between these representations.
Common Confusions



"Law of small number--a misconception commonly think that a probability should play
out in the short term" (VDW 462). For example, students think that if a coin has had a
series of heads, it is more likely to have several tails (contradicting "chance has no
memory.")
Manipulating between the representations can cause students difficulties.
Independent versus dependent variables.
Developmental Milestones
Probability Unit




Phuong, Sue and Vanessa
As probabilities fall in the range from 0 to 1, understanding of fractions and percentages
is vital.
The more experimental trials conducted, the closer we approach the theoretical
probability, unless there is some type of experimental error.
Being able to identify all the possible outcomes in a sample space in order to correctly
find the probability.
Intuition is not the same as probability.
Specific Strategies



Introduce the spectrum of probability from impossible to certain
Use an experimental approach
Link experimental data to theoretical probabilities.
Learning goals of the mini-unit:
What do you want the students to know and be able to do? How do these goals relate to
specific learning issue


Students will know that probabilities fall in the range from impossible to certain and that
translates into anything from 0 to 1 (fractions, ratios or percent all work).
Students will understand and be able to define all possible outcomes, sample space.
Students will understand how experimental results relate to theoretical probability and
reasons why they may be different.
Brief Lesson descriptions: Provide brief descriptions of your three lesson ideas and how
they connect to each other.
Lesson idea #1
Student will group the following items (events) into one of the these categories: Impossible,
Possible or Certain
Number line from zero to 1, place real life example on the number line, based on the probability
from 0 to 1.
Spinner activity to generate data for the class to work with.(VDW, NCTM)
Lesson idea #2:
Using a two spinner activity taken from Making the most out of chance. The activity is titled, Is
it a fair game? The students will map the outcomes and sample space, where certain outcomes
are more likely (higher probability). Baker, M., & Chick, H. (2007), Ely, R. E., & Cohen, J.
(2010).
Lesson idea #3:
Involves another two event probability. The first event is flipping a fair coin followed by rolling
a die. Students will map experimental outcome within the sample space.
Lessons goals: For lesson #1, we will first introduce the students to real life situations involving
probability, and have them categorize them on a number line. At this time, students will get
Probability Unit
Phuong, Sue and Vanessa
practice making connections between real-life situations, and the representation of its probability
(decimal, fraction and percentage). Lesson #2 will narrow the focus on all possible outcomes or
sample space. Student will learn how to conduct experimental probability and make conjectures
to and comparisons with theoretical probability using spinners. For lesson #3, we are continuing
with practice on making connections between experimental probability and theoretical
probability, and we are working on how we can connect the lessons with students’ families and
communities.
Rationale behind our Unit: Besides using VDW as a reference, we think it is important for
students to be able to make an intuitive judgment of the likeliness of an event. Shaughnessy
(2003) agrees, “informed citizens” must be able to read and interpret such data. Our remaining
activities focus on students learning probability hands-on (experimental probability,) so they can
make the critical conceptual connection to theoretical probability as supported by VDW, Baker
& Chick (2007), and Ely & Cohen (2010).
Summative assessment:
1. Please group the following items into one of the three categories: Impossible, Possible or
Certain:
 The sun will rise tomorrow.
 The next baby born in Puyallup will be a girl.
 It will rain snails tomorrow.
 The next US president will be over 21 years old.
 It will rain tomorrow.
 Trees will talk to us this afternoon.
 Five students will be absent tomorrow.
 You will get an A in math this year.
Bonus credit if you add an original item of your own to each category.
2. We want to use the matching plot line with spinners (from VDW page 458). We will make our
version for the final draft.
3. We want them to find the sample space of a two event experiment such as flipping a coin and
rolling a fair die.
Probability Unit
Phuong, Sue and Vanessa
Probability Mini-unit Assessment
Levels/Criteria
Beginning
Approaching
Meeting
Exceeding
1. Student
understands
the scale from
impossible to
certain.
Student does
not
conceptualize
the
connections
between real
life events
and probable
outcomes.
Student is
able to make
some
connections
between real
life events
and probable
outcomes.
Student is
able to make
connections
between real
life events
and probable
outcomes.
Student is able to
make some
connections
between real life
events and
probable
outcomes and
provide examples
from their own
lives.
Student
understands
how
experimental
results relate to
theoretical
probability and
the reasons
why they may
be different.
Student does
not see the
connection
between
experimental
and
theoretical
probability.
Student is
able to see
some
connection
between
experimental
and
theoretical
probability.
Student is
able to see
make
connections
between
experimental
and
theoretical
probability.
Student is able to
see make
connections
between
experimental and
theoretical
probability and
explain
differences.
Student
understands
and is able to
define all
possible
outcomes in a
sample space.
Student does
not
understand
the concept of
sample
space.
Student is
able to
conceptualize
sample space
and can
identify some
possible
outcomes.
Student is
able to
conceptualize
sample space
and can
identify all
possible
outcomes.
Student is able to
conceptualize
sample space and
can identify all
possible
outcomes.
Student can also
use fractions,
ratios or
percentages to
identify the
probability of all
possible
outcomes.
Score/Lev
el
Probability Unit
Phuong, Sue and Vanessa
Reference
Baker, M., & Chick, H. (2007). Making the Most of Chance. Australian Primary Mathematics
Classroom, 12(1), 8-13
Ely, R. E., & Cohen, J. (2010). Put the Right Spin on Student Work. Mathematics Teaching In the
Middle School, 16(4), 208-215.
Lovell, R. (1993). Probability Activities for problem solving and skills reinforcement. Key
Curriculum Press, Berkley, CA.
Shaughnessy, J. M. (2003). Research on Students’ Understandings of Probability. A Research
Companion to Principles and Standards of School Mathematics. National Council of Teachers of
Mathematics, 216 – 224.