Download Notes Pages (Schmidt) - Adult Basic Skills Professional Development

ASE MA 04:
Geometry, Probability, and Statistics
Steve Schmidt
[email protected]
abspd.appstate.edu
Overview
This workshop will assist instructors in making geometry, statistics and probability real so their
learners will have the content knowledge to be successful on equivalency exams and in
transitioning to college and careers.
Agenda
8:30 – 10:00
Geometry
10:00 – 10:15
Break
10:15 – 11:45
Geometry
11:45 – 12:45
Lunch
12:45 – 2:00
Statistics/Data Analysis
2:00 – 2:15
Break
2:15 – 4:00
Is There a Math Brain?
Please Write on this Packet!
You can find everything from this workshop at: abspd.appstate.edu Look under: Teaching
Resources, Adult Secondary Resources, Math
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
Page 1
Objectives
I can:

Explain how to use appropriate activities to teach geometry, probability and statistics

Describe evidence based best practices for teaching math

Apply evidence based math instructional principles

Understand and use ASE standards as a basis for instructional planning
Research Says . . .
- Teach math from concrete to representational to abstract (CRA)
Volume = length x width x height
Volume = 3 x 3 x 3
Concrete
Abstract
Representational
- Teach math using a problem solving approach with real world application. This will:

Develop students’ beliefs that they are capable of doing mathematics and that it makes
sense

Allow an entry point for a wide range of students

See pages 6 – 7 and handouts for ideas
- Learning is social so have students work in pairs/small groups

“The one who does the talking does the learning” Lev Vygotsky

“The best way to learn something is to teach it”
Patricia Wolfe (Brain Matters)
- Decide what to teach based on the NC Adult Education Content Standards

See pages 3 to 5
Sources: NIFL, Education Alliance, US Dept. of Ed
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
Page 2
ASE MA 04: Geometry, Probability, and Statistics
MA.4.1 Geometry: Understand congruence and similarity.
Objectives
MA.4.1.1 Experiment with
transformations in a plane.
Know precise definitions of
angle, circle, perpendicular
line, parallel line, and line
segment, based on the
undefined notions of point,
line, distance along a line, and
distance around a circular arc.
Example: How would you
determine whether two lines
are parallel or perpendicular?
MA.4.1.2 Prove theorems
involving similarity. Use
congruence and similarity
criteria for triangles to solve
problems and to prove
relationships in geometric
figures.
What Learner Should Know, Understand,
and Be Able to Do
Teaching Notes and Examples
A point has position, no thickness or
distance. A line is made of infinitely many
points, and a line segment is a subset of the
points on a line with endpoints. A ray is
defined as having a point on one end and a
continuing line on the other.
An angle is determined by the intersection
of two rays.
A circle is the set of infinitely many points
that are the same distance from the center
forming a circular are, measuring 360
degrees.
Perpendicular lines are lines in the interest
at a point to form right angles.
Parallel lines that lie in the same plane and
are lines in which every point is equidistant
from the corresponding point on the other
line.
Definitions are used to begin building blocks for proof.
Infuse these definitions into proofs and other
problems. Pay attention to Mathematical practice 3
“Construct viable arguments and critique the
reasoning of others: Understand and use stated
assumptions, definitions and previously established
results in constructing arguments.” Also mathematical
practice number six says, “Attend to precision:
Communicate precisely to others and use clear
definitions in discussion with others and in their own
reasoning.”
Students use similarity theorems to prove
two triangles are congruent.
Students prove that geometric figures
other than triangles are similar and/or
congruent.
Solve Problems using Congruence and Similarity
https://learnzillion.com/lessonsets/668-solveproblems-using-congruence-and-similarity-criteria-fortriangles
Experiment with Transformations in a Plane
http://www.virtualnerd.com/common-core/hsfgeometry/HSG-CO-congruence/A
https://www.illustrativemathematics.org/HSG
MA.4.2 Geometric Measure and Dimension: Explain formulas and use them to solve problems and apply geometric concepts in
modeling situations.
Objectives
What Learner Should Know, Understand,
and Be Able to Do
Teaching Notes and Examples
MA.4.2.1 Explain perimeter,
area, and volume formulas
and use them to solve
problems involving two- and
three-dimensional shapes.
Use given formulas and solve for an
indicated variables within the formulas.
Find the side lengths of triangles and
rectangles when given area or perimeter.
Compute volume and surface area of
cylinders, cones, and right pyramids.
Geometry Lesson Plans
http://www.learnnc.org/?standards=Mathematics-Geometry
MA.4.2.2 Apply geometric
concepts in modeling of
density based on area and
volume in modeling situations
(e.g., persons per square mile,
BTUs per cubic foot).
Use the concept of density when referring
to situations involving area and volume
models, such as persons per square mile.
Understand density as a ratio.
Differentiate between area and volume
densities, their units, and situations in
which they are appropriate (e.g., area
density is ideal for measuring population
density spread out over land, and the
concentration of oxygen in the air is best
measured with volume density).
Explore design problems that exist in local
communities, such as building a shed with maximum
capacity in a small area or locating a hospital for three
communities in a desirable area.
Geometry Problem Solving
http://map.mathshell.org/materials/lessons.php?taski
d=216&subpage=concept
This course is funded by
1
Example: Given the formula 𝑉 = 𝐵𝐻, for the volume
3
of a cone, where B is the area of the base and H is the
height of the. If a cone is inside a cylinder with a
diameter of 12in. and a height of 16 in., find the
volume of the cone.
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
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MA.4.3 Summarize, represent, and interpret categorical and quantitative data on (a) a single count or measurement variable, (b)
two categorical and quantitative variables, and (c) Interpret linear models.
Objectives
MA.4.3.1 Represent data with
plots on the real number line
(dot plots, histograms, and
box plots).
What Learner Should Know, Understand,
and Be Able to Do
Construct appropriate graphical displays
(dot plots, histogram, and box plot) to
describe sets of data values.
Teaching Notes and Examples
Represent Data with Plots
https://learnzillion.com/lessonsets/513-representdata-with-plots-on-the-real-number-line-dot-plotshistograms-and-box-plots
http://www.virtualnerd.com/common-core/hssstatistics-probability/HSS-ID-interpreting-categoricalquantitative-data/A/1
MA.4.3.2 Interpret
differences in shape, center,
and spread in the context of
the data sets, accounting for
possible effects of extreme
data points (outliers).
Understand and be able to use the context
of the data to explain why its distribution
takes on a particular shape (e.g. are there
real-life limits to the values of the data that
force skewness? are there outliers?)
Understand that the higher the value of a
measure of variability, the more spread out
the data set is.
Interpreting Categorical and Quantitative Data
http://www.shmoop.com/common-corestandards/ccss-hs-s-id-3.html
http://www.thirteen.org/get-themath/teachers/math-in-restaurants-lessonplan/standards/187/
Explain the effect of any outliers on the
shape, center, and spread of the data sets.
MA.4.3.3 Summarize
categorical data for two
categories in two-way
frequency tables. Interpret
relative frequencies in the
context of the data (including
joint, marginal, and
conditional relative
frequencies). Recognize
possible associations and
trends in the data.
Create a two-way frequency table from a
set of data on two categorical variables.
Calculate joint, marginal, and conditional
relative frequencies and interpret in
context. Joint relative frequencies are
compound probabilities of using AND to
combine one possible outcome of each
categorical variable (P(A and B)).
Marginal relative frequencies are the
probabilities for the outcomes of one of
the two categorical variables in a twoway table, without considering the other
variable. Conditional relative frequencies
are the probabilities of one particular
outcome of a categorical variable
occurring, given that one particular
outcome of the other categorical variable
has already occurred.
Recognize associations and trends in data
from a two-way table.
Interpreting Quantitative and Categorical data
http://www.ct4me.net/CommonCore/hsstatistics/hss-interpreting-categoricalquantitative-data.htm
MA.4.3.4 Interpret the slope
(rate of change) and the
intercept (constant term) of a
linear model in the context of
the data.
Understand that the key feature of a linear
function is a constant rate of change.
Interpret in the context of the data, i.e. as x
increases (or decreases) by one unit, y
increases (or decreases) by a fixed amount.
Interpret the y-intercept in the context of
the data, i.e. an initial value or a one-time
fixed amount.
Interpreting Slope and Intercepts
http://www.virtualnerd.com/common-core/hssstatistics-probability/HSS-ID-interpreting-categoricalquantitative-data/C/7
Understand that just because two
quantities have a strong correlation, we
cannot assume that the explanatory
(independent) variable causes a change in
the response (dependent) variable. The
best method for establishing causation is
Correlation and Causation
https://learnzillion.com/lessonsets/585-distinguishbetween-correlation-and-causation
MA.4.3.5 Distinguish between
correlation and causation.
This course is funded by
http://www.virtualnerd.com/middlemath/probability-statistics/frequency-tables-lineplots/practice-make-frequency-table
http://ccssmath.org/?page_id=2341
https://learnzillion.com/lessonsets/457-interpret-theslope-and-the-intercept-of-a-linear-model-using-data
https://www.khanacademy.org/math/probability/stati
stical-studies/types-of-studies/v/correlation-and-
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
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conducting an experiment that carefully
controls for the effects of lurking variables
(if this is not feasible or ethical, causation
can be established by a body of evidence
collected over time e.g. smoking causes
cancer).
causality
MA.4.4 Using probability to make decisions.
Objectives
What Learner Should Know, Understand,
and Be Able to Do
Teaching Notes and Examples
M.4.4.1 Develop a probability
distribution for a random
variable defined for a sample
space in which theoretical
probabilities can be
calculated; find the expected
value.
Develop a theoretical probability
distribution and find the expected value.
For example, find the theoretical
probability distribution for the number of
correct answers obtained by guessing on all
five questions of a multiple choice test
where each question has four choices, and
find the expected grade under various
grading schemes.
Probability
http://www.shmoop.com/common-corestandards/ccss-hs-s-md-4.html
M.4.4.2 Develop a probability
distribution for a random
variable defined for a sample
space in which probabilities
are assigned empirically; find
the expected value.
Develop an empirical probability
distribution and find the expected value.
For example, find a current data
distribution on the number of TV sets per
household in the United States, and
calculate the expected number of sets per
household. How many TV sets would you
expect to find in 100 randomly selected
households.
Probability Distribution
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&s
ource=web&cd=7&ved=0CEAQFjAG&url=http%3A%2F
%2Feducation.ohio.gov%2Fgetattachment%2FTopics%
2FOhio-s-New-LearningStandards%2FMathematics%2FHigh_School_Statisticsand-Probability_Model-Curriculum_October20131.pdf.aspx&ei=ec0RVNHBN8UgwSYvYD4Dg&usg=AFQjCNHpyffrA7UVkDyKCXIkYRD
Sw1nsyQ&bvm=bv.74894050,d.eXY
M.4.4.3 Weigh the possible
outcomes of a decision by
assigning probabilities to
payoff values and finding
expected values. Find the
expected payoff for a game of
chance.
Set up a probability distribution for a
random variable representing payoff values
in a game of chance. For example, find the
expected winnings from a state lottery
ticket or a game at a fast-food restaurant.
Expected Value
http://www.youtube.com/watch?v=DAjVAEDil_Q
M.4.4.4 Use probabilities to
make fair decisions (e.g.,
drawing by lots, using a
random number generator).
Make decisions based on expected values.
Use expected values to compare long- term
benefits of several situations.
Using Probability to Make Decisions
http://www.shmoop.com/common-corestandards/ccss-hs-s-md-6.html
M.4.4.5 Analyze decisions and
strategies using probability
concepts (e.g., product
testing, medical testing,
pulling a hockey goalie at the
end of a game).
Explain in context decisions made based on
expected values.
Analyzing Decisions
http://www.ct4me.net/CommonCore/hsstatistics/hss-using-probability-makedecisions.htm
This course is funded by
Using Probability to Make Decisions
https://www.khanacademy.org/commoncore/gradeHSS-S-MD
Weighing Outcomes
https://www.illustrativemathematics.org/illustrations/
1197
Money and Probability
http://becandour.com/money.htm
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
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Advanced Manufacturing: Straw Tower
One of the mottos in advanced manufacturing is that if you can see it you can build it. Today you
will see if you can build a tower out of straws that is strong enough to stand on its own and support
weight.
Here are the rules:
1. You can only use the items in your bag – straws and tape
2. Your tower must be self-supporting (you cannot tape the tower to the table)
3. Your tower must be at least 25 cm tall
4. Your tower must be able to hold a ball for at least 10 seconds
5. Before you begin building, you must spend at least 3 minutes planning and draw a sketch
Tower Sketch
Reflection Questions
1. What did you learn about teamwork from this activity?
2. How close did your actual tower come to your plan? Why did you make changes from the
plan?
3. Why is it important to plan before starting to build?
4. What made your tower strong enough to hold the ball? Or Why was your tower not strong
enough to hold the ball?
5. What would you do differently next time if you had to do a similar task?
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
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Geometry: Building Shapes
1. Build a triangle that has three equal sides of 6 centimeters
2. Build a rectangle with a length of 10 centimeters and a width of 4 centimeters
3. Build a square with sides of 2 ½ inches
4. Build a parallelogram with sides of 3 inches
5. Build a trapezoid with two sides of 2 inches, one side of 3 inches, and one side of 4 inches
6. Build a rectangular prism with a length of 9 centimeters, width of 5 centimeters, and height of 4
centimeters
7. Build a pyramid with a square base with 6 centimeter sides and a height of 12 centimeters
8. Build a square with a perimeter of 12 inches
9. Build a rectangle with a perimeter of 30 centimeters
10. Build a triangle with a perimeter of 18 inches
11. Build a square with an area of 4 square inches
12. Build a rectangle with an area of 12 square centimeters
13. Build a parallelogram with an area of 4 square inches
14. Build a triangle with an area of 12 square centimeters
15. Build a trapezoid with an area of 14 square inches
16. Build a rectangular prism with a volume of 24 cubic inches
17. Build a right prism with a triangular base with a volume of 40 cubic inches
18. Build a square pyramid with a volume of 18 cubic inches
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
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Statistics: Data tells a story!
Measures of Central Tendency

Mean (arithmetic mean)
o Commonly called “average”
o Sum of values ÷ number of values

Median
o Middle value in rank order (if odd # of values)
o Mean of 2 middle values (if even # of values
o Used for skewed data (such as income)

Mode
o Most frequent value
o There may be no mode or multiple modes
Statistics Humor!
Box Plots
Box plots are a handy way to display data broken into four quartiles, each with an equal number
of data values. The box plot doesn't show frequency, and it doesn't display each individual statistic,
but it clearly shows where the middle of the data lies. It's a nice plot to use when analyzing how
your data is skewed.
There are a few important vocabulary terms to know in order to graph a box-and-whisker plot. Here
they are:





Q1 – quartile 1, the median of the lower half of the data set
Q2 – quartile 2, the median of the entire data set
Q3 – quartile 3, the median of the upper half of the data set
IQR – interquartile range, the difference from Q3 to Q1
Extreme Values – the smallest and largest values in a data s
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
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Making a Box Plot
Make a box plot for the geometry test scores given below:
90, 94, 53, 68, 79, 84, 87, 72, 70, 69, 65, 89, 85, 83, 72
Step 1: Order the data from least to greatest.
Step 2: Find the median of the data.
This is also called quartile 2 (Q2).
Step 3: Find the median of the data less than Q2.
Step 4. Find the median of the data greater than Q2.
This is the lower quartile (Q1).
This is the upper quartile (Q3).
Step 5. Find the extreme values: these are the largest and smallest data values. Note: if the data
set contains outliers do not include outliers when finding extreme values. Extreme values = 53 and
94.
Step 6. Create a number line that will contain all of the
data values. It should stretch a little beyond each extreme
value.
Step 7. Draw a box from Q1 to Q3 with a line dividing the box at
Q2. Then extend "whiskers" from each end of the box to the extreme
values.
This plot is broken into four different groups:
the lower whisker, the lower half of the box, the upper half of the box, and
the upper whisker. Since there is an equal amount of data in each group,
each of those sections represent 25% of the data.
Using this plot we can see that 50% of the students scored between 69 and 87 points, 75% of the
students scored lower than 87 points, and 50% scored above 79. If your score was in the upper
whisker, you could feel pretty proud that you scored better than 75% of your classmates. If you
scored somewhere in the lower whisker, you may want to find a little more time to study.
Outliers are values that are much bigger or smaller than the rest of the data. These are
represented by a dot at either end of the plot. Our geometry test example did not have any outliers,
even though the score of 53 seemed much smaller than the rest, it wasn't small enough. In order
to be an outlier, the data value must be: (1) larger than Q3 by at least 1.5 times the interquartile
range (IQR), or (2) smaller than Q1 by at least 1.5 times the IQR.
Source: http://www.shmoop.com/basic-statistics-probability/box-whisker-plots.html
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
Page 9
How You Can be Good at Math, and Other Surprising Facts about Learning
1. Is there such a thing as a “math brain”?
2. What happens to your brain when you make mistakes in math?
3. What is the growth mindset?
4. What suggestions does Jo Boaler offer for improving math instruction?
5. What else interested you as you watched this video?
Source: Jo Boaler, TED Talk
This course is funded by
https://www.youtube.com/watch?v=3icoSeGqQtY
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
Page 10
Resources
Annenberg Learner. Courses of study in such areas as algebra, geometry, and real-world
mathematics. The Annenberg Foundation provides numerous professional development activities
or just the opportunity to review information in specific areas of study.
http://www.learner.org/index.html
Illuminations. Great lesson plans for all areas of mathematics at all levels from the National
Council of Teachers of Mathematics (NCTM). http://illuminations.nctm.org
Geometry Center (University of Minnesota). This site is filled with information and activities for
different levels of geometry. http://www.geom.uiuc.edu/
National Library of Virtual Manipulatives for Math - All types of virtual manipulatives for use in
the classroom from algebra tiles to fraction strips. This is a great site for students who need to see
the “why” of math. http://nlvm.usu.edu/en/nav/index.html
Real-World Math
The Futures Channel
http://www.thefutureschannel.com/algebra/algebra_real_world_movies.php
Real-World Math
Get the Math
http://www.realworldmath.org/
http://www.thirteen.org/get-the-math/
Math in the News
http://www.media4math.com/MathInTheNews.asp
Evidence Base
National Institute for Literacy. (2010). Algebraic thinking in adult education. Washington, DC:
Author. Retrieved from https://lincs.ed.gov/publications/pdf/algebra_paper_2010V.pdf
The Education Alliance. (2006). Best practices in teaching mathematics. Charleston, W.V:
Author. Retrieved from:
http://www.gram.edu/sacs/qep/chapter%204/4_1EducationAlliance.pdf
U.S. Department of Education, Office of Vocational and Adult Education. (2014). Math works!
Guide. American Institutes for Research. Retrieved from:
http://lincs.ed.gov/sites/default/files/Teal_Math_Works_Guide_508.pdf
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016
Page 11