ASE MA 4: Geometry, Probability and Statistics
Dianne B. Barber () & William D. Barber ()
Appalachian State University, Boone, NC
Agenda:
8:30 – 9:45 Introduction & Overview
9:45 – 10:15 Probability
10:00 – 10:15 Break
10:15 – 11:45 Probability & Statistics
11:45 – 12:45 Lunch
12:45 – 2:15 Statistics & Geometry
2:15 – 2:30 Break
2:30 – 3:55 Geometry
3:55 – 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
Standards for Mathematical Practices
This course is funded by ASE MA 4 Geometry, Probability, and Statistics; Revised 03/21/16 Page 1
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 1
This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 18
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 Plane
http://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 Similarity
https://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 Plans
http://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 Solving
http://map.mathshell.org/materials/lessons.php?taskid=216&subpage=concept
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 Plots
https://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 Data
http://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 data
http://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 Intercepts
http://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 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). / Correlation and Causation
https://learnzillion.com/lessonsets/585-distinguish-between-correlation-and-causation
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. / Probability
http://www.shmoop.com/common-core-standards/ccss-hs-s-md-4.html
Using Probability to Make Decisions
https://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 Distribution
http://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 Value
http://www.youtube.com/watch?v=DAjVAEDil_Q
Weighing Outcomes
https://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 Decisions
http://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 Decisions
http://www.ct4me.net/Common-Core/hsstatistics/hss-using-probability-make-decisions.htm
Money and Probability
http://becandour.com/money.htm
ASE MA 4: Geometry, Probability, and Statistics – Instructor Checklist
MA.4.1 Geometry: Understand congruence and similarity.
Objectives / Curriculum – Materials Used / Notes
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.
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).