Download Notes Pages - Adult Basic Skills Professional Development

ASE MA 4: Geometry, Probability and Statistics
Dianne B. Barber ([email protected]) & William D. Barber ([email protected])
Appalachian State University, Boone, NC
Agenda:
8:30 – 9:45
9:45 – 10:15
10:00 – 10:15
10:15 – 11:45
11:45 – 12:45
12:45 – 2:15
2:15 – 2:30
2:30 – 3:55
3:55 – 4:00
Introduction & Overview
Probability
Break
Probability & Statistics
Lunch
Statistics & Geometry
Break
Geometry
Certificates & Evaluations
Overview:
This training 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.
Objectives:
 Understand and use ASE standards as a basis for instructional planning

Teach using best practices

Use technology to enhance teaching and learning

Know where to locate supplemental resources
NCCCS College & Career Readiness Adult Education Content Standards
 Standards for Mathematical Practices (page 2)

ASE MA04 Content Standards (pages 3-5)

ASE MA04 Instructor Checklist (page 6)

ASE MA04 Student Checklist (pages 7-8)

Design learning around logical and consistent progression

Teach fewer concepts with more depth of learning

Teach conceptual understanding, procedural skill and fluency,

Application
This course is funded by
ASE MA 4 Geometry, Probability, and Statistics; Revised 03/21/16
Page 1
Standards for Mathematical Practices
1. Makes sense of problems and perseveres in
solving them
☐ Understands the meaning of the problem and
looks for entry points to its solution
☐ Analyzes information (givens, constrains,
relationships, goals)
☐Designs a plan
☐Monitors and evaluates the progress and
changes course as necessary
☐ Checks their answers to problems and ask,
“Does this make sense?”
2. Reason abstractly and quantitatively
☐Makes sense of quantities and relationships
☐ Represents a problem symbolically
☐ Considers the units involved
☐ Understands and uses properties of
operations
3. Construct viable arguments and critique the
reasoning of others
☐ Uses definitions and previously established
causes/effects (results) in constructing
arguments
☐Makes conjectures and attempts to prove or
disprove through examples and
counterexamples
☐ Communicates and defends their mathematical
reasoning using objects, drawings, diagrams,
actions
☐ Listens or reads the arguments of others
☐Decide if the arguments of others make sense
☐ Ask useful questions to clarify or improve the
arguments
4. Model with mathematics.
☐ Apply reasoning to create a plan or analyze a
real world problem
☐ Applies formulas/equations
☐Makes assumptions and approximations to
make a problem simpler
This course is funded by
☐ Checks to see if an answer makes sense and
changes a model when necessary
5. Use appropriate tools strategically.
☐ Identifies relevant external math resources
and uses them to pose or solve problems
☐Makes sound decisions about the use of
specific tools. Examples may include:
☐ Calculator
☐ Concrete models
☐Digital Technology
☐Pencil/paper
☐ Ruler, compass, protractor
☐ Uses technological tools to explore and
deepen understanding of concepts
6. Attend to precision.
☐ Communicates precisely using clear definitions
☐Provides carefully formulated
explanations
☐ States the meaning of symbols, calculates
accurately and efficiently
☐ Labels accurately when measuring and
graphing
7. Look for and make use of structure.
☐ Looks for patterns or structure
☐ Recognize the significance in concepts and
models and can apply strategies for solving
related problems
☐ Looks for the big picture or overview
8. Look for and express regularity in repeated
reasoning
☐ Notices repeated calculations and looks for
general methods and shortcuts
☐ Continually evaluates the reasonableness of
their results while attending to details and
makes generalizations based on findings
☐ Solves problems arising in everyday life
Adapted from Common Core State Standards for Mathematics:
Standards for Mathematical Practice
ASE MA 4 Geometry, Probability, and Statistics; Revised 03/21/16
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 03/03/2016
Page 3
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 03/03/2016
Page 4
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 03/03/2016
Page 5
ASE MA 4: Geometry, Probability, and Statistics – Instructor Checklist
MA.4.1 Geometry: Understand congruence and similarity.
Objectives
Curriculum – Materials Used
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.
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.
Notes
MA.4.2 Geometric Measure and Dimension: Explain formulas and use them to solve problems and apply geometric concepts in
modeling situations.
Objectives
Curriculum – Materials Used
Notes
MA.4.2.1 Explain perimeter, area, and volume
formulas and use them to solve problems involving
two- and three-dimensional shapes.
MA.4.2.2 Apply geometric concepts in modeling of
density based on area and volume in modeling.
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
Curriculum – Materials Used
Notes
MA.4.3.1 Represent data with plots on the real
number line.
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).
MA.4.3.3 Summarize categorical data for two
categories in two-way frequency tables. Interpret
relative frequencies in the context of the data.
Recognize possible associations and trends in the
data.
MA.4.3.4 Interpret the slope and the intercept of a
linear model in the context of the data.
MA.4.3.5 Distinguish between correlation and
causation.
MA.4.4 Using probability to make decisions.
Objectives
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.
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.
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.
Curriculum – Materials Used
Notes
M.4.4.4 Use probabilities to make fair decisions.
M.4.4.5 Analyze decisions and strategies using
probability concepts.
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 6
ASE MA 04: Geometry, Probability, and Statistics – Student Checklist
MA.4.1 Geometry: Understand congruence and similarity.
Learning Targets
Mastery
Level %
Date
I can define an angle based on my knowledge of a point, line, distance along a line, and distance around a
circular arc.
I can define a circle based on my knowledge of a point, line, distance along a line, and distance around a
circular arc.
I can define perpendicular lines based on my knowledge of a pint, line, distance, along a line, and distance
around a circular arc.
I can define a line segment based on my knowledge of a pint, line, distance, along a line, and distance around
a circular arc.
I can define parallel lines based on my knowledge of a pint, line, distance, along a line, and distance around a
circular arc.
I can describe the relationship between similarity and congruence.
I can set up and write equivalent ratios.
I can identify corresponding angles and sides of two triangles.
I can determine a scale factor and use it in a proportion.
I can solve geometric problems using congruence and similarity.
I can prove relationships using congruence and similarity.
MA.4.2 Geometric Measure and Dimension: Explain formulas and use them to solve problems and apply geometric concepts in
modeling situations.
Learning Targets
Mastery
Level %
Date
I can apply the formula for the perimeter of a rectangle to solve problems.
I can apply the formula for the area of a rectangle to solve problems.
I can apply the formula for the area of a triangle to solve problems.
I can apply the formula for the volume of a cone to solve problems.
I can apply the formula for the volume of a cylinder to solve problems.
I can apply the formula for the volume of a pyramid to solve problems.
I can apply the formula for the volume of a sphere to solve problems.
I can find the surface area of right circular cylinders.
I can find the surface area of rectangular prisms.
I can use geometric shapes, their measures, and their properties to describe objects.
I can apply concepts of density based on area and volume in modeling situations.
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.
Learning Targets
Mastery
Level %
Date
I can classify data as either categorical or quantitative.
I can identify an appropriate scale needed for the data display.
I can identify an appropriate number of intervals for a histogram.
I can identify an appropriate width for intervals in a histogram.
I can select an appropriate data display for real-world data.
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 7
ASE MA 4: Geometry, Probability, and Statistics – Student Checklist, Page 2
I can construct dot plots.
I can create a frequency table.
I can construct histograms.
I can construct box plots.
I can identify a data set by its shape and describe the data set as symmetric or skewed.
I can use the outlier rule (e.g., Q1 – 1.5 x IQR and Q3 + 1.5 IQR) to identify outliers in a data set.
I can analyze how adding/removing an outlier affects measures of center and spread.
I can interpret differences in shape, center and spread in the context of data sets.
I can compare and contrast two or more data sets using shape, center, and spread.
I can organize categorical data in two-way frequency tables.
I can interpret joint frequencies and joint relative frequencies in the context of the data.
I can interpret marginal frequencies and marginal relative frequencies in the context of the data.
I can interpret conditional frequencies and conditional relative frequencies in the context of the data.
I can recognize possible associations between categorical variables in a two-way frequency or relative
frequency table.
I can determine the y-intercept graphically and algebraically.
I can determine the rate of change by choosing two points.
I can determine the equation of a line using data points.
I can interpret the slope in the context of the data.
I can interpret the y-intercept in the context of the data.
I can differentiate between causation and correlation/association.
I can interpret paired data to determine whether correlation implies causation/association.
MA.4.4 Using probability to make decisions.
Learning Targets
Mastery
Level %
Date
I can develop a probability distribution for a random variable defined for a sample space of theoretical
probabilities.
I can calculate theoretical probabilities and find expected values.
I can develop a probability distribution for a random variable for a sample space of empirically assigned
probabilities.
I can assign probabilities empirically and find expected values.
I can weigh the possible outcomes of a decision and find expected values.
I can assign probabilities to payoff values and find expected values.
I can evaluate strategies based on expected values.
I can compare strategies based on expected values.
I can explain the difference between theoretical and experimental probability.
I can compute theoretical and experimental probability.
I can determine the fairness of a decision based on the available data.
I can determine the fairness of a decision by comparing theoretical and experimental probability.
I can use counting principles to determine the fairness of a decision.
I can analyze decisions and strategies related to product testing, medical testing, and sports.
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 8
Best Practices in Teaching Mathematics
Instructional Element
Curriculum Design
Recommended Practices




Ensure mathematics curriculum is based on challenging content
Ensure curriculum is standards- based
Clearly identify skills, concepts and knowledge to be mastered
Ensure that the mathematics curriculum is vertically and horizontally
articulated

Provide professional development which focuses on:
o Knowing/understanding standards
o Using standards as a basis for instructional planning
o Teaching using best practices
o Multiple approaches to assessment
Develop/provide instructional support materials such as curriculum maps and pacing
guides Establish math leadership teams and provide math coaches
Professional
Development for
Teachers

Technology



Provide professional development on the use of instructional technology tools
Provide student access to a variety of technology tools
Integrate the use of technology across all mathematics curricula and courses
Manipulatives



Use manipulatives to develop understanding of mathematical concepts
Use manipulatives to demonstrate word problems
Ensure use of manipulatives is aligned with underlying math concepts







Focus lessons on specific concept/skills that are standards- based
Differentiate instruction through flexible grouping, individualizing lessons, compacting,
using tiered assignments, and varying question levels
Ensure that instructional activities are learner-centered and emphasize
inquiry/problem-solving
Use experience and prior knowledge as a basis for building new knowledge
Use cooperative learning strategies and make real life connections
Use scaffolding to make connections to concepts, procedures and understanding
Ask probing questions which require students to justify their responses
Emphasize the development of basic computational skills








Ensure assessment strategies are aligned with standards/concepts being taught
Evaluate both student progress/performance and teacher effectiveness
Utilize student self-monitoring techniques
Provide guided practice with feedback
Conduct error analyses of student work
Utilize both traditional and alternative assessment strategies
Ensure the inclusion of diagnostic, formative and summative strategies
Increase use of open-ended assessment techniques

Instructional
Strategies
Assessment
Source: Best Practices in Teaching Mathematics, Spring 2006. The Education Alliance, Charleston, West Virginia. Website: www.educationalliance.org
Problem Solving and Value of Teaching with Problems





Places students’ attention on mathematical ideas
Develops “mathematical power”
Develops students’ beliefs that they are capable of doing mathematics and that it makes sense
Provides ongoing assessment data that can be used to make instructional decisions
Allows an entry point for a wide range of students
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 9
Probability
Experimental versus Theoretical Probability – See handout
Probability Rules


Multiplication rule
o
Probability of both with two independent events
o
P(A and B) = P(A) X P(B)
o
Intersection
o
Examples:

P(both dice < 6)

P(first and second child both girls)

P(both were born on Tuesday)
Addition rule
o
Used to find probability of “at least one” when events are independent
o
P(A or B) = P(A) + P(B) – P(A and B)
o
Examples
o
P(at least one girl in two children)
o
P(at least one 6 when rolling 2 dice)
o
P(at least one of two people was born on Tuesday)
Contextualized Example: When both parents have rh+ blood but are carriers of negative blood, each of
their children has a 25% probability of having rh- blood. If these parents have non-identical twins
1. What is the probability that both will have rh- blood?
2. What is the probability that at least one will be rh-?
Real Life Decisions & Probability
Extension Activity #1: Probability in Advertising
Ask students to look at newspapers and magazines for examples of how numbers are used in advertisements. For example,
it is not unusual to see something like "two-thirds less fat than the other leading brand" or "four out of five dentists
recommend Brand T gum for their patients who chew gum." Why do advertisers use numbers like these? What information
are they trying to convey? Do students think that the numbers give accurate information about a product? Why or why
not?
Extension Activity #2: They Said What?
Ask students to look at newspapers or magazines for examples of how politicians, educators, environmentalists, or others
use data such as statistics and probability. Then have them analyze the use of the information. Why did the person use
data? What points were effectively made? Were the data useful? Did the data strengthen the argument? Have students
provide evidence to support their ideas.
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 10
Statistics – Working with Data

Qualitative/categorical Data
o Nominal
o Ordinal

Quantitative/numerical Data
o Interval vs. ratio
o Discrete vs. continuous
Representing Data
Identifying Misleading Graphs: Video - http://www.youtube.com/watch?v=ETbc8GIhfHo#t=11
o
o
o
o
o
o
o
o
no title
no labels on the axes
no key – when one is required
vertical axis doesn’t start at 0
different sized graphing icons or bars
unequal intervals on any axis
“broken” or “squished” axis
distorted image (sloped circle)
Graphing Activity: As a group, examine the graph you have been assigned.
Is it misleading? Explain why.
How could you make it a better graph?
Be prepared to share your groups results with the class.
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 11
Scatterplots

Plot data points on a graph

Shows relationships between variables
Correlation
 Positive vs. negative

DOES NOT prove cause/effect

Lurking (confounding) variables
Independent & Dependent Variables

If there was a cause/effect relationship, the cause would be the independent variable

Researcher controls independent variable, then evaluates/measures dependent.

Independent/dependent variables may not be obvious in observational studies
o
Or both may be dependent on something else
Correlation Contextualized
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 12
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
Measures of Spread

Range (R)  Highest value – lowest value

Interquartile range  (IQR) = Q3 – Q1

o
Range of the middle 50% of data
o
Q1 – separates the lowest 25% of data
o
Q3 – separates the highest 25% of data
Standard deviation
Boxplots
Label each of the
following on the
boxplot above:





*
Q1 - median of the lower half of the data set
Q2 - median of the data set
Q3 - median of the upper half of the data set
Lowest and highest values not more than 1.5 IQR from box
Outliers
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 13
Drawing a Boxplot: Use the data below to graph a boxplot
90, 94, 53, 68, 79, 84, 87, 72, 70, 69, 65, 89, 85, 83, 72
1. Order data
2. Find Q2, Q1, Q3
3. Find IQR
4. Calculate 1.5 IQR to determine outliers, if any
5. Draw and label x-axis in the space provided below
6. Draw boxplot above x-axis using information found in #2-4
Uses of Boxplots
 May indicate skewed distribution
 Comparison of IQRs
 Comparison of side-by-side boxplots
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 14
Directions for Making a Box Plot
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 set
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.
This is the lower quartile (Q1).
Step 4. Find the median of the data greater than Q2. 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 03/03/2016
Page 15
Geometry – Big Ideas

Spatial visualization, scale factors

Perimeter, circumference and area of 2-dimensional figures

Surface area and volume of 3-dimensional figures

Seeking relationships

Making convincing arguments
Visualization

Recognize and name shapes

Students often do not recognize properties or if they do, do not use them for sorting or recognition

Students may not recognize shape in different orientation
Implications for Instruction
Provide activities that have students

Sort, identify and describe shapes

Use manipulatives, build and draw shapes

See shapes in different orientations and sizes

Define properties, make measurements, recognize patterns

Explore what happens if a measurement or property changes

Follow informal proofs
Vocabulary (handout)
Surface Area
What is surface area?

Surface area measures the combined surfaces of a 3-dimensional shape

It is measured using squares

Units include in2, ft2, yd2, mi2 or metric units such as mm2, cm2, m2, km2
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 16
3 in
h=6
r=5
Contextualized:
How much sheet metal will be needed to construct a water tank in the shape of a right circular cylinder
thati s 30 feet long and 8 feet in diameter?
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 17
Resources for Teaching Mathematics
Free Resources for Educational Excellence. Teaching and learning resources from a variety of federal agencies. This portal
provides access to free resources. http://free.ed.gov/index.cfm
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
Khan Academy. A library of over 2,600 videos covering everything from arithmetic to physics, finance, and history and 211
practice exercises. http://www.khanacademy.org/
The Math Dude. A full video curriculum for the basics of algebra.
http://www.montgomeryschoolsmd.org/departments/itv/MathDude/MD_Downloads.shtm
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
Teacher Guide for the TI-30SX MultiView Calculator – A guide to assist you in using the new calculator, along with a variety
of lesson plans for the classroom.
http://education.ti.com/en/us/guidebook/details/en/62522EB25D284112819FDB8A46F90740/30 x_mv_tg
http://education.ti.com/calculators/downloads/US/Activities/Search/Subject?s=5022&d=1009
Mometrix Academy Free videos for math concepts
http://www.mometrix.com/academy/basics-of-functions/
Real-World Math
The Futures Channel http://www.thefutureschannel.com/algebra/algebra_real_world_movies.php
Real-World Math http://www.realworldmath.org/
Get the Math http://www.thirteen.org/get-the-math/
Math in the News http://www.media4math.com/MathInTheNews.asp
Please join us at Appalachian State University
for
Institute 2016: Shifts in Instruction for WIOA
Implementation
May 23-26, 2016
or
May 30 – June 2, 2016
Register at www.abspd.appstate.edu. Earn 3 hours of graduate credit.
This course is funded by
ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016
Page 18