WELCOME BACK TO CAMP!
What’s golden?
Agenda
• Warm-Up for statistical and
probabilistic thinking
• Norms for our PD
• Recapping from lesson study
• NCTM (2007) Teaching
Standards: Worthwhile task
• Break
• Statistics Task
• Lunch
• Exploring Statistics and
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Probability Content Standards
Standards for Mathematical
Practice
Synthesizing from the Day
Student evaluation
Closure
Warm-up Activity in Stats and Prob
Are you a Good Timer?
•Quick Experiment:
– Close your eyes
– When you hear the “START”,
begin counting off seconds in your head
– When you hear the “STOP”, write down the number you
reached
Are you a Good Timer?
• Graph the results-univariate and bivariate options
• What do we see?
• Can you guess what was the exact number of
seconds?
• How consistent were the estimates?
• What does the y-intercept of the regression line
tell us? Slope?
Are you a Good Timer?
•Experimental Design Options:
1. Announce a #seconds; everyone counts until the timer
goes off / we say Stop.
2. Everyone counts until they reach 30, then opens their eyes
and looks at a clock and writes down how many seconds
actually elapsed.
3. Like #1 but don’t announce #seconds ahead of time.
Everyone counts until we say Stop
•Use the same #seconds the 2nd time?
•Tell what the #seconds was after the 1st time?
Evolving Norms for this PD
• We will persist with every problem and examine it from multiple
perspectives.
• We will be ready for class and use our class time effectively.
• We will keep our focus on learning and use technology for personal reasons
during breaks.
• We will be respectful of each other’s time and space and work efficiently.
• We will actively participate by (a) listening to each other, (b) giving others
our attention, (c) not speaking when someone else is talking, and (d)
regularly sharing our ideas in class.
• If we disagree with someone or are unclear, we will ask a question about his
or her idea and describe why we disagree or are confused.
• We will ask questions when we do not understand something. We will
comment on others’ ideas rather than the person.
Evolving Norms for this PD
• We will be ready for class and use our class time effectively.
• We will keep our focus on learning and use technology for personal reasons
during breaks.
• We will be respectful of each other’s time and space and work efficiently.
• We will actively participate by (a) listening to each other, (b) giving others
our attention, (c) not speaking when someone else is talking, and (d)
regularly sharing our ideas in class.
• If we disagree with someone or are unclear, we will ask a question about his
or her idea and describe why we disagree or are confused.
• We will ask questions when we do not understand something. We will
comment on others’ ideas rather than the person.
Evolving Norms for this PD
O We will take advantage of opportunities to share ideas and gather
feedback through presentations.
O We will encourage one another to share ideas.
O We will show our appreciation to one another after a presentation
by applause.
O If we disagree with someone or are unclear about their ideas related
to mathematics content and pedagogy, we will ask a question about
his or her idea and describe why we disagree or are confused.
O We will ask questions when we do not understand something about
mathematics content and pedagogy.
O We will comment on others’ ideas about mathematics content and
pedagogy rather than the person.
Evolving Norms for this PD
• We will always look for another approach to solve problems.
• We will use pictures, graphs, tables, symbols, numbers,
manipulatives, and/or words to assist us while doing mathematics.
• We will persist with every problem and examine it from multiple
perspectives.
• We will be mathematically precise whenever possible.
• We will explain and justify our ideas in a way that everyone can
understand.
Expectation for technology use
• Please limit the use of technology for the use
of chatting, phone calls, and texts strictly to
break times as well as before and after class
out of respect for the nature of our
collaboration and thinking together.
Lessons learned from Lesson Study
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What did you learn from the experience?
What surprised you?
What did you like about the experience?
How would improve upon those aspects that
you did not like?
• …thoughts for moving forward.
NCTM Standards for Teaching and Learning as
Related to the Common Core State Standards
(2007)
1. Knowledge of Mathematics and General Pedagogy
2. Knowledge of Student Mathematical Learning
3. Worthwhile Mathematical Tasks
4. Learning Environment
5. Discourse
6. Reflection on Student Learning
7. Reflection on Teaching Practice
Frayer models: Worthwhile task
Break
STATISTICAL ASSOCIATION
2 TYPES WE WILL FOCUS ON:
•BIVARIATE CATEGORICAL: Is there an
association between gender and whether you
have a part-time job?
•BIVARIATE QUANTITATIVE: Is there an
association between a golf ball’s drop height and
bounce height?
Categorical Association
• CCSS.Math.Content.8.SP.A.4 Understand that patterns of
association can also be seen in bivariate categorical data by
displaying frequencies and relative frequencies in a two-way
table. Construct and interpret a two-way table summarizing
data on two categorical variables collected from the same
subjects. Use relative frequencies calculated for rows or
columns to describe possible association between the two
variables. For example, collect data from students in your class
on whether or not they have a curfew on school nights and
whether or not they have assigned chores at home. Is there
evidence that those who have a curfew also tend to have
chores?
Categorical Association
What numerical analysis could be done to explore the data?
Categorical Association
Job Experience
Male Female Total
Never had a part44%
60%
52%
time job
Had a part-time job
31%
25%
28%
during summer only
Had a part-time job
25%
15%
20%
but not only during
summer
Total
100%
100% 100%
Categorical Association
What graphical representations could we make to display our
numerical analyses?
Segmented Bar Chart
Categorical Association
Is there an association between gender and job
experience for the students in this sample?
Justify your response.
How does the knowledge needed to approach
these tasks connect with other topics in the
mathematics curriculum?
Teaching Categorical Association
• 2 tasks: smoking and drug
• Sample student responses
Teaching Categorical Association
In summary, these are the common student misconceptions
students have when analyzing categorical data for associations:
• Lack of proportional reasoning
• Deterministic: absolute, everyone has to
follow
• Unidirectional: direct only
• Localist: look at only one cell or one
conditional distribution
• Use of intuition and ignoring the data
LUNCH
Quantitative Association
8.SP.A.2: Know that straight lines are widely used to
model relationships between two quantitative
variables. For scatter plots that suggest a linear
association, informally fit a straight line, and informally
assess the model fit by judging the closeness of the
data points to the line.
8.SP.A.3: Use the equation of a linear model to solve
problems in the context of bivariate measurement
data, interpreting the slope and intercept.
Quantitative Association: Graphing
Quantitative Association
STEW Lesson plan “What Fits?”
http://www.amstat.org/education/stew/
•Using the piece of spaghetti, determine the line
of best fit for the data shown in the scatterplot.
•Be cognizant of your thoughts as you decide
where to place the line.
Quantitative Association
• What things did you consider when you were
deciding where to place it?
• Why did you choose to put the line there?
• What is your definition of the line of best fit?
Quantitative Association
Student conceptions study
Quantitative Association
Handout: Part Two Instructions
Some sample student responses to this task are on the
following page. For each student’s response, analyze
his/her criterion. Will the criterion always work to
produce a line that accurately models any data set? If
it will, explain why. If it won’t, draw at least one
example of a scatterplot with the line placed using that
criterion and explain why the criterion produces a poor
line of best fit.
Quantitative Association: Slope
The equation of the regression line for the golf
ball task is:
Bounce height = 0.7 Drop Height -3.4
Interpret the slope of this line in context.
Quantitative Association: Slope
y
3
x2
5
Bounce height = 0.7 Drop Height -3.4
Quantitative Association: Slope
No association case
Quantitative Association: Slope
• Algebra: slope =
how much y will change for a 1-unit change in x
• Statistics: slope of regression line =
AVERAGE DIFFERENCE in y per 1-unit DIFFERENCE in x
• AVERAGE: no guarantee that y will change exactly that much.
• DIFFERENCE: saying “change” might give the impression that
we are changing the x value of a data point (putting someone
on a stretching machine) instead of comparing two different x
values (two people of different heights)
Quantitative Association: y-intercept
• Predicted value for response variable (y) when
predictor variable (x) is 0
Bounce height = 0.7 Drop Height -3.4
• Huh? How can the bounce height be -3.4 cm?
Quantitative Association: Slope
• Statistical model: usually not of interest;
fitting to the data at hand which aren’t
necessarily close to y-axis; when fitting line of
best fit, it is not the starting point
• Mathematical function: usually of interest;
often starting point when graphing
Quantitative Association
Learning mathematical lines and statistical lines of best fit
Students’ previous study of lines and slope with mathematical
functions can create cognitive obstacles to
•Plotting statistical data
•Making lines that don’t go through all of the points
•Understanding what the line of best fit is
•Developing criteria for the line of best fit
•Correctly interpreting the slope and y-intercept of best fit lines
in a statistical setting.
Synthesizing from the Day
1. Describe the SMPs that you engaged in during
today’s tasks.
2. What evidence might you notice when reviewing
tasks that suggests it meets expectations for a
worthwhile task?
Student Evaluation
PSM6, PSM7, and PSM8
Take Care
• Next meeting is Friday January 16 @ Lima HS.
• Please ask any questions to Dr. Stats and/or
Dr. Math.
• Most importantly, keep in touch!