ASE MA 4: Geometry, Probability and Statistics
Dianne B. Barber and William D. Barber
Appalachian State University
Overview:
This training will assist instructors in making geometry, statistics and probability real so their learners will have
the content knowledge to be successful in high school courses, 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 ASE Credentials

General Credential
o


2 courses from each of 4 areas
Math/Science Specialty
o
4 math courses
o
4 science courses
Language Arts/Social Studies Specialty
o
4 language arts courses
o
4 social studies courses
Content Standards for Geometry, Statistics & Probability

GED 2014 Assessment Targets (page 2)

Adult Secondary Education Content Standards (handout)

Standards for Mathematical Practices (page 3)
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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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
☐ 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 11/12/13
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Standards Shift 1: Focus on…

Key ideas, understandings, and skills

Deep learning of concepts stressed

Fewer concepts with more depth
Standards Shift 2 - Coherence…

Designing learning around coherent progressions level to level
o
Coherence for mastery
o
All roads lead to algebraic thinking – abstract reasoning
Standards Shift 3 - Rigor…

Pursuing conceptual understanding, procedural skill and fluency,
and application

Increasing
Webb’s Depth of Knowledge (page 5)
Level 1: recall or recognize a fact, term, or procedure
Level 2: use conceptual knowledge, procedures, or multiple steps
Level 3: develop a plan or sequence, more complex, more than one possible answer
Blooms Taxonomy versus Webb’s Depth of Knowledge (page 6)
Best Practices in Teaching Mathematics (page 7)

Curriculum Design

Professional Development

Technology

Manipulatives

Instructional Strategies

Assessment
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Webb’s Depth of Knowledge
In 1997, Norman Webb developed a process and criteria for systematically analyzing the alignment between
instructional standards and standardized assessments. Webb’s work grew out of research on studying different
state assessments and their alignment with various state standards. Psychometricians and test developers use
Webb’s Depth of Knowledge (DOK) as a way to design and evaluate different assessment tasks. It is Webb’s DOK
that is used by the 2014 GED® test. It is important to recognize that Webb’s Depth of Knowledge:


Is descriptive; it is not a taxonomy
Focuses on how deeply a student has to know the content in order to respond
DOK provides instructors with a vocabulary and frame of reference when thinking about how students engage
with course content and a common language to understand the cognitive demand of the 2014 GED® test.’s
Instructor Handbook for GED® Preparation42OK Level DOK Definition DOK Examples
Webb’s Depth of Knowledge Levels
DOK Level
DOK-1
Recall and
Reproduction
DOK Definition
DOK Examples
Recall of a fact, term, principle,
concept, or perform a routine
procedure.
DOK-2
Use of information, conceptual
knowledge, select appropriate
Basic
procedures for a task, two or more
Application of steps with decision points along the
Skills/Concepts way, routine problems,
organize/display data, interpret/use
simple graphs.
DOK-3
Strategic
Thinking
DOK-4
Extended
Thinking
Recall elements and details of story; structure, such
as sequence of events, character, plot and setting;
Conduct basic mathematical calculations;
Label locations on a map; Represent in words or
diagrams a scientific concept or relationship.
Perform routine procedures like measuring length or
using punctuation marks correctly; Describe the
features of a place or people.
Identify and summarize the major events in a
narrative; Use context cues to identify the meaning of
unfamiliar words; Solve routine multiple-step
problems; Describe the cause/effect of a particular
event; Identify patterns in events or behavior;
Formulate a routine problem given data and
conditions; Organize, represent, and interpret data.
Requires reasoning, developing a
plan or sequence of steps to
approach problem; requires some
decision-making and justification;
abstract, complex, or non-routine;
often more than one possible
answer.
Support ideas with details and examples; Use voice
appropriate to the purpose and audience; Identify
research questions and design investigations for a
scientific problem; Develop a scientific model for a
complex situation; Determine the author's purpose
and describe how it affects the interpretation of a
reading selection; Apply a concept in other contexts.
Requires investigation or
application to real world; requires
time to research, problem solve,
and process multiple conditions of
the problem or task; non-routine
manipulations, across
disciplines/content areas/multiple
sources.
A product or a project that requires specifying a
problem, designing and conducting an experiment,
analyzing its data, and reporting results/solutions;
Apply mathematical model to illuminate a problem or
situation; Analyze and synthesize information from
multiple sources; Describe and illustrate how
common themes are found across texts from
different cultures; Design a mathematical model to
inform and solve a practical or abstract situation.
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Bloom Taxonomy vs. Webb Depth of Knowledge
You may be more familiar with Bloom’s Taxonomy. The following chart provides a comparison of the cognitive
complexity of Bloom’s Taxonomy and Webb’s Depth of Knowledge.
Bloom’s Taxonomy Webb’s Depth of Knowledge
Bloom’s Taxonomy
Knowledge
The recall of specifics and universals, involving
little more than bringing to mind the appropriate
material.
Webb’s DOK
Recall
Recall of a fact, information, or procedure (e.g.,
What are 3 critical skill cues for the overhand
throw?)
Comprehension
The ability to process knowledge on a low level
such that the knowledge can be reproduced or
communicated without a verbatim repetition.
Application
The use of abstractions in concrete situations.
Basic Application of Skill/Concept
Use of information, conceptual knowledge,
procedures, two or more steps, etc. (e.g.,
Explain why each skill cue is important to the
overhand throw. By stepping forward you are able
to throw the ball further.)
Analysis
The breakdown of a situation into its component
parts.
Strategic Thinking
Requires reasoning, developing a plan or sequence
of steps; has some complexity; more than one
possible answer; generally takes less than 10
minutes to do (e.g., Design 2 different plays in
basketball and explain what different skills are
needed and when the plays should be carried out.)
Synthesis and Evaluation
Putting together elements and parts to form a
whole and then making value judgments about the
method.
Extended Thinking
Requires an investigation; time to think and
process multiple conditions of the problem or task;
and more than 10 minutes to do nonroutine
manipulations (e.g., Analyze 3 different tennis,
racquetball, and badminton strokes for similarities,
differences, and purposes. Then, discuss the
relationship between the mechanics of the stroke
and the strategy for using the stroke during game
play.)
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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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

Instructional
Strategies






Assessment







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
Source: Best Practices in Teaching Mathematics, Spring 2006. The Education Alliance, Charleston, West Virginia. Website:
www.educationalliance.org
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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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
Polya’s Problem Solving Strategy
Understand the Problem
 What am I given? (facts/ information/data)
 What am I asked to find?
 How can I make sense of the information
given to me?
 What can I infer from the given data?
Devise a Plan
 Which strategy should I use? (Look for
patterns, draw a picture, make a list, table
or chart, work backward, guess and check,
write an equation, use objects, consider all
possibilities)
 Have I solved similar problem before?
Act: Carry out the Plan
 Which strategy is the most suitable?
 Have I shown all the necessary
steps/labeling?
 If the plan does not seem to be working,
then start over and try another approach.
Check: Look Back
 Have I answered the question?
 Is the answer reasonable?
 Is the answer accurate?
 Can I work backwards/use another method
to check my answer
 Justify my answer
SOLVE
Study the problem - What am I trying to find?
Organize the facts - What do I know?
Line up a plan - What steps will I take?
Verify your plan with action – How will I carry out my plan?
Examine the results – Does my answer make sense? If not, rework. Always double check!
You try it!
The sum of the interior angles of a triangle is 180°, of a quadrilateral is 360° and of a pentagon is 540°. Assuming
this pattern continues, find the sum of the interior angles of a dodecagon (12 sides).
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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2014 GED Math Test
Overview
 90 minutes, 46 items
 One test, 2 sections
o ≈12 minutes non-calculator (1st 5 items)
o ≈78 minutes calculator available
 TI 30 XS Virtual Calculator
 Scaled scores range from ≈ 100 to 200
 High school equivalency passing score > 150
Content
 45% Quantitative Problem Solving
o Number operations
o Geometric thinking
 55% Algebraic Problem Solving
 Presented in academic and workforce contexts
 Statistics and data interpretation standards are included in other tests
 Integration of mathematical practices
 Content Matrix on page 11
Technology-Enhanced Items
 Multiple choice
 Fill-in-the-blank


Hot-spot
Drag-and-drop

Drop-down
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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GED Calculator Reference Sheet (page 12)
GED Formula Sheet (page 13)
GED Symbol Selector Tool
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Embrace Technology – Computer Skills
Word Processing Skills
 Basic Keyboarding
 Cut
 Copy
 Paste
 Undo/Redo
 Insert
 Enter-hard return
 Spacing
 Backspacing
 Highlight
Directional Tools
 Previous/Next
 Close
 Minimize
 Page Tabs
Resource Tools
 Virtual Calculator
 Calculator Reference Page
 Formula Page
 AE Symbol
 Item Review/Flagging
Pre-Adult Secondary Education Standards for Statistics and Probability








Understand that a data set has a distribution that can be described by its center, spread and shape.
Distinguish measures of center from measures of variability. Choose appropriate measures of center and
distribution.
Construct and interpret dot plots, histograms, box plots and scatterplots.
Draw inferences about a population and informal comparative inferences about two populations.
Solve problems related to slope and intercept.
Construct and interpret two-way tables summarizing two categorical variables.
Develop, use, and evaluate probability models.
Find probabilities of compound events
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Conditional Probability and Rules of Probability
Standards
 Distinguish and use independent probabilities and conditional probabilities

Use probability rules to compute probabilities of compound events in a uniform probability model
o
Addition rule
o
Multiplication rule
o
Use permutations and combinations to solve problems
Sample Space

One coin
S = {H, T}

One fair die
S = {1, 2, 3, 4, 5, 6}

Two coins
S = {HH, HT, TH, TT}

How many events are in sample space for rolling 2 fair dice? How would you list them?
Finding Probabilities from Sample Space
1. P(at least one H on two coins) =
2. P(sum on two dice is 12) =
3. P(sum on two dice is 11) =
4. P(sum on two dice is 7) =
5. P(at least one 6 on two dice) =
Probability of “Failure”

Define success, any other outcome is failure

P(success) + P(failure) = 1 or 100%

Complement

Examples:
P(not A) = 1 - P(A)
o
P(child not born on Sunday) =
o
P(first card not heart) =
Independent vs. Dependent Events in Probability

Does result for event A affect P(B)?

Sampling with replacement vs. without

Is flipping 2 coins different from flipping one coin twice?

“Lack of memory” property
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Probability of Both with Two Independent Events

P(A and B) = P(A) X P(B)

Intersection in sample space – Venn diagram

Examples:

o
P(first and second child both girls) =
o
P(both dice < 6) =
o
P(both were born on Tuesday) =
Extends to multiple events
o
P(6 on all five dice) =
o
P(“Yahtzee”) =
Conditional Events

P(A and B) = P(A) X P(B|A)

Used when sampling without replacement

Examples:

o
P(first two cards are aces) =
o
P(both socks will be black) =
Can we use above equation for independent events?
Addition Rule

Used to find probability of “at least one” when events are independent

Union in sample space

P(A or B) = P(A) + P(B) – P(A and B)

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)
Permutations

ORDER MATTERS

Examples: Class rankings, Phone numbers, Zip codes, Car license plates, Series
numbers on products, Lock combinations (IRONIC)

Sampling with replacement

Sampling without replacement
P(N,r)=Nr
P(N,r)=N!÷(N-r)!
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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
How many orders could result from
o
flipping a coin 3 times?
o
rolling a die 3 times?
o
ranking top 2 of 5 students?
o
five-number zip codes?
Combinations

ORDER DOES NOT MATTER

Examples: Winning free dinner (3 people), Teacher taking attendance, Voting (no matter who votes
first), Making a sandwich (no matter in what order the toppings are), Selecting a college course schedule
Combinations for Sampling Without Replacement
𝑁!

𝐶(𝑁, 𝑟) = (𝑁−𝑟)!𝑟!

Examples: How many
1. 5 card hands can be chosen from a 52-card deck?
2. groups of 3 from this class can be chosen for a free dinner tonight?
3. 4-person committees from 10 club members?
Finding Probabilities using Combinations and Permutations
Examples: Find probability that…
1.
a poker hand has 4 aces and 1 king
2.
first 3 participants signing in today will win free dinners
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Interpreting Data
Standards



Use mean and standard deviation to fit data to a Normal Distribution and estimate population
percentages
Summarize, represent, and interpret data on two categorical and two quantitative variables
Interpret linear models
o Find the correlation coefficient using technology
o Distinguish between correlation and causation
Data and Data Types

Data are…
o Singular is “datum”
 Types of data
o Qualitative/categorical
o Quantitative/numerical
 Types of quantitative data
o Ordinal vs. interval/ratio
o Discrete vs. continuous
Representing Data
Misleading Graphs
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Methods of Data Collection

Surveys

Observational studies

Controlled experiments
o
Experimental vs. control group
o
Avoiding placebo effect
Measures of Central Tendency

Mode
o
Most frequent value
o
There may be no mode or multiple modes

Mid-range = mean of low and high 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)
Mean (arithmetic mean)
o
Commonly called “average”
o
Sum of values ÷ number of values
Measures of Spread

Range (R)
o
Highest value – lowest value

Interquartile range (IQR) = Q3 – Q1

Standard deviation (σ)
o
Sample standard deviation
s=
å(x - x)
2
n -1
Margin of Error (ME or MOE)

For means

For proportions

Interpreting survey results
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Normal Distribution
 Characteristics
 Empirical rules
 Robust
1. Calculate the percent for each segment in the distribution above.
2. Given IQ Parameters: Mean = 100, Standard Deviation = 15, label the x-axis.
3. What % of IQs are between 100 and 115?
4. What % of IQs are between 85 and 115?
5. What % of IQs are between 70 and 115?
6. What % of IQs are greater than 115?
7. What is the probability an IQ is below 70?
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Bivariate Frequency Tables

Aka “contingency tables”

Row and column totals

Probabilities for A, B, A and B, A|B, B|A
Examples:
Let A = math/science
Let B = 8th grade
1. P(A) =
2. P(B) =
3. P(A and B) =
4. P(A|B) =
5. P(B|A) =
Scatterplots
 Plot data points on a graph
 Obvious relationships between variables
 Line of best fit
 Slope and Y-intercept
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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1. Make a scatter plot of the data provided. Be sure to label the grid appropriately.
2. Draw the line of best fit.
3. Calculate the slope of the line of best fit. Use mathematics to explain how you determined your
answer. Use words, symbols, or both in your explanation. Explain what this slope means in the
context of the problem.
4. Calculate the y-intercept of the line of best fit. Use mathematics to explain how you determined
your answer. Use words, symbols, or both in your explanation. Explain what the y-intercept
means in the context of this problem.
5. Write the equation for the line of best fit.
6. Do all data points follow this trend? Use mathematics to explain your answer. Use words,
symbols, or both in your explanation.
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Making Inferences and Justifying Conclusions
Standards
 Understand and evaluate random processes

Make inferences and justify conclusions

Understand and develop Margin of Error

Use data from a randomized experiment to compare two treatments

Evaluate reports based on data
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.
 In observational studies which is which may not be obvious
o Or both may be dependent on something else
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Using Probability to Make Decisions
Standards


Calculate expected values and use them to solve problems
Use probability to evaluate outcomes of decisions
o Find the expected payoff for a game of chance
o Use probabilities to make fair decisions
o Analyze decisions and strategies (such as in product testing and medical testing)
Parameters and Statistics

Samples of populations

Statistical inference

Types of samples
o
Simple random
o
Stratified random
o
Systematic “random”
o
Convenience
o
Cluster
o
Multistage
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

Draw box from Q1 to Q3  IQR = Q3- Q1

Extreme values - lowest and highest values not more than 1.5 IQR from box

Outliers
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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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
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
Page 25
Graphing 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.
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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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
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:

larger than Q3 by at least 1.5 times the interquartile range (IQR), or

smaller than Q1 by at least 1.5 times the IQR.
Source: http://www.shmoop.com/basic-statistics-probability/box-whisker-plots.html
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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Geometry
On a blank piece of paper, draw as realistically as possible:
1. Circles the size of a penny, dime, & quarter.
2. A rectangle the size of a dollar bill.
3. A rectangle the size of a large paper clip.
4. A rectangle the size of a credit card.
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
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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3 in
h=6
r=5
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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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
Algebra 4 All. A website from Michigan Virtual University with an interactive site for using algebra tiles to solve various
types of problems. http://a4a.learnport.org/page/algebra-tiles
Working with Algebra Tiles. An online workshop that provides the basics of using algebra tiles in the classroom.
http://mathbits.com/MathBits/AlgebraTiles/AlgebraTiles.htm
Teaching Algebra Using Algebra Tiles. An instructor site that provides information on teaching algebra, as well as basic
algebraic concepts. http://www.jamesrahn.com/homepages/algebra_tiles.htm
Key Elements to Algebra Success 46 lessons, homework assignments, and videos. http://ntnmath.keasmath.com/
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
ASE MA 4 Geometry, Probability, and Statistics; Revised 11/12/13
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