Transcript Probability

Chapter 17
To a c c o m p a n y H e l p i n g C h i l d re n L e a r n M a t h C d n E d , R e y s e t a l .
© 2 0 1 0 J o h n Wi l e y & S o n s C a n a d a L t d .
Guiding Questions
1. How do data analysis, statistics, and probability
help children develop critical thinking skills?
2. How can you help children develop skills in
analyzing data?
3. What descriptive statistics are appropriate to
introduce in the elementary grades? What are
some examples of ways they can be introduced?
4. What are some common misconceptions young
students have about probability?
Why Teach Statistics and
Probability?
• Children encounter ideas of statistics and probability
outside of school every day.
• Data analysis, statistics and probability provide
connections to other mathematics topics or school
subjects.
• Data analysis, statistics and probability provide
opportunities for computational activity in a
meaningful context.
• Data analysis, statistics, and probability provide
opportunities for developing critical thinking skills.
Three Steps of Data Analysis
1. Pose a question and collect data.
2. Display collected data.
3. Analyze data and communicate results.
Formulating Questions and
Collecting Data
• There is real benefit to having students
identify their own questions or problems, for
they take ownership of the investigation and
their motivation will be high.
• Once students have identified a suitable
question, they will need to plan how to collect
the data to answer the question.
• Students must clearly communicate with
others to negotiate the details of the
investigation.
Surveys
• Survey data result from collecting information.
• These data may range from a national census
poll or observing cars pass the window, to
simply tallying the ages of students in a class.
• A wide array of data are available from the
Statistics Canada website.
Surveys (cont.)
• The actual data used depends on student
interest and maturity.
– To sharpen data-collecting techniques, students
may consider the following questions:
• What questions will this survey answer?
• Where should I conduct my survey?
• When should I conduct the survey?
Experiments
• Experiments may be somewhat more
advanced than surveys.
• When students conduct experiments, in
addition to using observation and recording
skills, they often incorporate the use of the
scientific method
Simulations
• Although a simulation is similar to an
experiment, random number tables or devices
such as coins, dice, spinners, or computer
programs are also used to model real-world
occurrences.
• Sampling is another method of data collection
that students can simulate
– The whole group you are studying is called the
population.
– A sample is a sub-set of the population.
Analyzing Data: Graphical
Organization
• After data has been collected, graphs are
often used to display data and help others
digest the results
What do you notice about these graphs?
Quick and Easy Graphing Methods
Plots
• A plot is another type of graph used to visually
display data.
• Line plots A line plot may be used to quickly
display numerical data with a small range.
Plots
• Stem-and-leaf plots divide the data into tens and
ones and arrange it in numerical order.
Plots
• Box plots (also called a
box-and-whisker plot)
summarize data and
provide a visual means
of showing variability—
the spread of the data.
Picture Graphs
Bar Graph
• Bar graphs are used mostly for discrete or
separate and distinct data; the bars represent
these data
Histograms
• Although a histogram looks like a bar graph, a
histogram is used with continuous data, not discrete
data. Therefore the data are represented with
connected bars, each representing an interval.
Pie Graphs
• A pie graph is a circle
representing the whole,
with wedges reporting
percentages of the
whole.
Line Graphs
•
•
•
•
Line graphs are effective for showing trends over time.
The data are continuous rather than discrete.
Data can occur between points with continuous data.
Change is accurately represented with linear functions rather
than some other curve.
Graphical Roundup
• Each type of graph deserves instructional attention
as students examine ways to display their data.
• Children need experience constructing them and
interpreting information that is represented.
• The availability of graphing calculators and graphing
programs allows for easy construction of a variety of
graphs.
• It is vital that children are taught how to interpret
and understand graphs.
Analyzing Data: Descriptive
Statistics
• Another way to analyze data is to use
descriptive statistics.
Measure of Variation: Range
• Range is the variation of a set of data or how
spread out the data are.
• Once the range has been introduced in the
elementary grades, middle school students
often learn to measure variability with
variance or standard deviations.
Measures of Central Tendency or
Averages
• Any number that is used to describe the
centre or middle of a set of values is called an
average of those values.
• Many different averages exist, but we will look
at the three main ones:
– Mode: the value that occurs most frequently in a
collection of data
– Median: the middle value in a set of ordered data
– Mean: the arithmetic average
Data Sense
• Reading the data. The student is able to answer specific
questions for which the answer is prominently displayed.
For example, “Which player averaged the most points?”
• Reading between the data. The student is able to find
relationships in the data such as comparison, and is able
to operate on the data. For example, “How many players
had a median less than their mean?”
• Reading beyond the data. The student is able to predict
or make inferences. For example, “Which player had the
greatest range? The smallest range? What do these
numbers tell you about the player?”
Misleading Graphs
• These different graphs of the
same data demonstrate how
graphs can distort and
sometimes misrepresent
information.
Communicating Results
• Once data have been collected, analyzed, and
interpreted it is appropriate for students to
communicate their findings.
• Just as in problem solving, students should be
encouraged to look back at their results.
• Communication can help students clarify their
ideas during this process.
Probability
• Probability is used to predict the chance of
something happening. The terms chance and
probability are often applied to those
situations where the outcome cannot be
completely determined in advance.
Probability of an Event
• The probability of tossing a head on a coin is
1/2.
• The probability of rolling a four on a standard
six-sided die is 1/6 .
• The probability of having a birthday on
February 30 is 0.
• In the previous examples, tossing a head,
rolling a four, and having a birthday on
February 30 are events or outcomes.
Probability of an Event
• Probability assigns a number (from zero to one) to an
event.
• Long before children are ready to calculate
probability of specific events, it is important that you
introduce and discuss terms such as “certain,”
“uncertain,” “impossible,” “likely,” and “unlikely.”
Sample Space
• The sample space for a probability problem
represents all possible outcomes.
What is the
sample space
for this spinner?
Randomness
• When something is random, it means that it is
not influenced by any factors other than
chance.
• It is helpful to have children discuss
randomness in a specific context such as
drawing a name out of a hat.
– Through this notions of fair and unfair will be
developed in a meaningful way.
Independence of Events
• If two events are independent, one event in
no way affects the outcome of the other.
• If a coin is tossed, lands on heads, and then is
rolled again, it is still equally likely to land on
heads or tails.
• If the same result occurs repeatedly, students
begin to doubt what they have been taught
about probability and assume that a different
outcome is inevitable.
Misconceptions about Probability
• Young children often hold common
misconceptions about various aspects of
probability.
• The more opportunities you give them to
explore a variety of probability notions
through hands-on activities, the better they
will be able to develop and evaluate
inferences and predictions that are based on
data and apply basic concepts of probability.
A Probability Activity
• The following slides contain a probability
activity which demonstrates to children how
the more times a sample is taken, the closer
you will get to knowing the actual population.
• This activity is suitable for students at all grade
levels.
Mystery Bags
• Open your mystery bag, but do not look inside.
• Shake the contents, reach in and pull out one
colour tile. Record the colour of the tile chosen
and return it to the bag.
• Repeat this process 9 more times, recording
each result.
• Based on these 10 trials, how many of each
colour of tile are in your bag? (Remember there
are a total of 12 tiles in the bag.)
Mystery Bags
• Conduct 10 more trials using the same process
of random selection from the bag.
• Now, based on 20 trials, how many of each
colour does your bag contain? Is this different
from your previous guess?
Mystery Bags
• Based on your experiment, what is the
probability that the tile chosen will be red?
yellow? blue? green?
• Open the bag and look at the contents. How
well did your experimentation reveal the
actual contents of the bag?
Copyright
Copyright © 2010 John Wiley & Sons Canada, Ltd. All rights reserved.
Reproduction or translation of this work beyond that permitted by
Access Copyright (The Canadian Copyright Licensing Agency) is
unlawful. Requests for further information should be addressed to the
Permissions Department, John Wiley & Sons Canada, Ltd. The
purchaser may make back-up copies for his or her own use only and
not for distribution or resale. The author and the publisher assume no
responsibility for errors, omissions, or damages caused by the use of
these programs or from the use of the information contained herein.