ASE MA 4: Geometry, Probability and StatisticsDianne B. Barber (barberdb@appstate.edu) & William D. Barber (barberwd@appstate.edu)

Appalachian State University, Boone, NC

Agenda:8:30 – 9:45 Introduction & Overview9:45 – 10:15 Probability 10:00 – 10:15 Break10:15 – 11:45 Probability & Statistics11:45 – 12:45 Lunch12:45 – 2:15 Statistics & Geometry2:15 – 2:30 Break2:30 – 3:55 Geometry3:55 – 4:00 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☐ 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

This course is funded by ASE MA 4 Geometry, Probability, and Statistics; Revised 03/21/16 Page 2

ASE MA 04: Geometry, Probability, and StatisticsMA.4.1 Geometry: Understand congruence and similarity.

Objectives What Learner Should Know, Understand, and Be Able to Do Teaching Notes and Examples

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?

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.”

Experiment with Transformations in a Planehttp://www.virtualnerd.com/common-core/hsf-geometry/HSG-CO-congruence/A

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.

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 Similarityhttps://learnzillion.com/lessonsets/668-solve-problems-using-congruence-and-similarity-criteria-for-triangles

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 Planshttp://www.learnnc.org/?standards=Mathematics--Geometry

Example: Given the formula V=13BH , for the

volume 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.

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 Solvinghttp://map.mathshell.org/materials/lessons.php?taskid=216&subpage=concept

This course is funded by 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 What Learner Should Know, Understand, and Be Able to Do Teaching Notes and Examples

MA.4.3.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

Construct appropriate graphical displays (dot plots, histogram, and box plot) to describe sets of data values.

Represent Data with Plotshttps://learnzillion.com/lessonsets/513-represent-data-with-plots-on-the-real-number-line-dot-plots-histograms-and-box-plots

http://www.virtualnerd.com/common-core/hss-statistics-probability/HSS-ID-interpreting-categorical-quantitative-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.

Explain the effect of any outliers on the shape, center, and spread of the data sets.

Interpreting Categorical and Quantitative Datahttp://www.shmoop.com/common-core-standards/ccss-hs-s-id-3.html

http://www.thirteen.org/get-the-math/teachers/math-in-restaurants-lesson-plan/standards/187/

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 two-way 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 datahttp://www.ct4me.net/Common-Core/hsstatistics/hss-interpreting-categorical-quantitative-data.htm

http://www.virtualnerd.com/middle-math/probability-statistics/frequency-tables-line-plots/practice-make-frequency-table

http://ccssmath.org/?page_id=2341

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 Interceptshttp://www.virtualnerd.com/common-core/hss-statistics-probability/HSS-ID-interpreting-categorical-quantitative-data/C/7

https://learnzillion.com/lessonsets/457-interpret-the-slope-and-the-intercept-of-a-linear-model-using-data

MA.4.3.5 Distinguish between correlation and causation.

Understand that just because two quantities have a strong correlation, we cannot assume that the explanatory (independent) variable causes a change in

Correlation and Causationhttps://learnzillion.com/lessonsets/585-distinguish-between-correlation-and-causation

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 4

the response (dependent) variable. The best method for establishing causation is 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).

https://www.khanacademy.org/math/probability/statistical-studies/types-of-studies/v/correlation-and-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.

Probabilityhttp://www.shmoop.com/common-core-standards/ccss-hs-s-md-4.html

Using Probability to Make Decisionshttps://www.khanacademy.org/commoncore/grade-HSS-S-MD

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 Distributionhttp://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=7&ved=0CEAQFjAG&url=http%3A%2F%2Feducation.ohio.gov%2Fgetattachment%2FTopics%2FOhio-s-New-Learning-Standards%2FMathematics%2FHigh_School_Statistics-and-Probability_Model-Curriculum_October2013-1.pdf.aspx&ei=ec0RVNHBN8-UgwSYvYD4Dg&usg=AFQjCNHpyffrA7UVkDyKCXIkYRDSw1nsyQ&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 Valuehttp://www.youtube.com/watch?v=DAjVAEDil_Q

Weighing Outcomeshttps://www.illustrativemathematics.org/illustrations/1197

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 Decisionshttp://www.shmoop.com/common-core-standards/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 Decisionshttp://www.ct4me.net/Common-Core/hsstatistics/hss-using-probability-make-decisions.htm

Money and Probabilityhttp://becandour.com/money.htm

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 5

ASE MA 4: Geometry, Probability, and Statistics – Instructor ChecklistMA.4.1 Geometry: Understand congruence and similarity.

Objectives Curriculum – Materials Used NotesMA.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.

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 NotesMA.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 NotesMA.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 Curriculum – Materials Used Notes

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.

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 6

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 7

ASE MA 04: Geometry, Probability, and Statistics – Student ChecklistMA.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 8

ASE MA 4: Geometry, Probability, and Statistics – Student Checklist, Page 2I 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 9

Best Practices in Teaching Mathematics

Instructional Element Recommended Practices

Curriculum Design

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

Professional Development for Teachers

Provide professional development which focuses on:o Knowing/understanding standardso Using standards as a basis for instructional planningo Teaching using best practiceso Multiple approaches to assessment

Develop/provide instructional support materials such as curriculum maps and pacing guides Establish math leadership teams and provide math coaches

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

InstructionalStrategies

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

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

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 10

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 AdvertisingAsk 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 11

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 12

Statistics – Working with Data Qualitative/categorical Data

o Nominalo Ordinal

Quantitative/numerical Datao Interval vs. ratioo Discrete vs. continuous

Representing Data

Identifying Misleading Graphs: Video - http://www.youtube.com/watch?v=ETbc8GIhfHo#t= 11

o no titleo no labels on the axeso no key – when one is requiredo vertical axis doesn’t start at 0o different sized graphing icons or barso unequal intervals on any axiso “broken” or “squished” axiso 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 13

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 14

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 15

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 16

*

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 17

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

Directions for Making a Box Plot

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 18

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 19

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 20

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 21

3 in

h = 6r = 5

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 NationalCouncil 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_tghttp://education.ti.com/calculators/downloads/US/Activities/Search/Subject?s=5022&d=1009

Mometrix Academy Free videos for math conceptshttp://www.mometrix.com/academy/basics-of-functions/

Real-World Math

The Futures Channel http://www.thefutureschannel.com/algebra/algebra_real_world_movies.phpReal-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, 2016Register 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 22