ASE MA 04:Geometry, Probability, and Statistics

Steve Schmidtschmidtsj@appstate.edu

abspd.appstate.edu

OverviewThis workshop will assist instructors in making geometry, statistics and probability real so their learners will have the content knowledge to be successful on equivalency exams and in transitioning to college and careers.

Agenda8:30 – 10:00 Geometry

10:00 – 10:15 Break

10:15 – 11:45 Geometry

11:45 – 12:45 Lunch

12:45 – 2:00 Statistics/Data Analysis

2:00 – 2:15 Break

2:15 – 4:00 Is There a Math Brain?

Please Write on this Packet!You can find everything from this workshop at: abspd.appstate.edu Look under: Teaching Resources, Adult Secondary Resources, Math

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016 Page 1

ObjectivesI can:

Explain how to use appropriate activities to teach geometry, probability and statistics

Describe evidence based best practices for teaching math

Apply evidence based math instructional principles

Understand and use ASE standards as a basis for instructional planning

Research Says . . . - Teach math from concrete to representational to abstract (CRA)

Concrete Representational

Volume = length x width x height

Volume = 3 x 3 x 3

Abstract

- Teach math using a problem solving approach with real world application. This will:

Develop students’ beliefs that they are capable of doing mathematics and that it makes sense

Allow an entry point for a wide range of students

See pages 6 – 7 and handouts for ideas

- Learning is social so have students work in pairs/small groups

“The one who does the talking does the learning” Lev Vygotsky

“The best way to learn something is to teach it” Patricia Wolfe (Brain Matters)

- Decide what to teach based on the NC Adult Education Content Standards

See pages 3 to 5

Sources: NIFL, Education Alliance, US Dept. of Ed

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016 Page 2

ASE MA 04: Geometry, Probability, and 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 12/07/2016 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 12/07/2016 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

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Advanced Manufacturing: Straw TowerOne of the mottos in advanced manufacturing is that if you can see it you can build it. Today you will see if you can build a tower out of straws that is strong enough to stand on its own and support weight.

Here are the rules:

1. You can only use the items in your bag – straws and tape

2. Your tower must be self-supporting (you cannot tape the tower to the table)

3. Your tower must be at least 25 cm tall

4. Your tower must be able to hold a ball for at least 10 seconds

5. Before you begin building, you must spend at least 3 minutes planning and draw a sketch

Tower Sketch

Reflection Questions

1. What did you learn about teamwork from this activity?

2. How close did your actual tower come to your plan? Why did you make changes from the plan?

3. Why is it important to plan before starting to build?

4. What made your tower strong enough to hold the ball? Or Why was your tower not strong enough to hold the ball?

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016 Page 6

5. What would you do differently next time if you had to do a similar task?

Geometry: Building Shapes 1. Build a triangle that has three equal sides of 6 centimeters

2. Build a rectangle with a length of 10 centimeters and a width of 4 centimeters

3. Build a square with sides of 2 ½ inches

4. Build a parallelogram with sides of 3 inches

5. Build a trapezoid with two sides of 2 inches, one side of 3 inches, and one side of 4 inches

6. Build a rectangular prism with a length of 9 centimeters, width of 5 centimeters, and height of 4 centimeters

7. Build a pyramid with a square base with 6 centimeter sides and a height of 12 centimeters

8. Build a square with a perimeter of 12 inches

9. Build a rectangle with a perimeter of 30 centimeters

10. Build a triangle with a perimeter of 18 inches

11. Build a square with an area of 4 square inches

12. Build a rectangle with an area of 12 square centimeters

13. Build a parallelogram with an area of 4 square inches

14. Build a triangle with an area of 12 square centimeters

15. Build a trapezoid with an area of 14 square inches

16. Build a rectangular prism with a volume of 24 cubic inches

17. Build a right prism with a triangular base with a volume of 40 cubic inches

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016 Page 7

18. Build a square pyramid with a volume of 18 cubic inches

Statistics: Data tells a story!

Measures of Central Tendency Mean (arithmetic mean)

o Commonly called “average”

o Sum of values ÷ number of values

Median

o Middle value in rank order (if odd # of values)

o Mean of 2 middle values (if even # of values

o Used for skewed data (such as income)

Mode

o Most frequent value

o There may be no mode or multiple modes

Statistics Humor!

Box PlotsBox 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

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016 Page 8

IQR – interquartile range, the difference from Q3 to Q1 Extreme Values – the smallest and largest values in a data s

Making a Box PlotMake 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

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016 Page 9

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

How You Can be Good at Math, and Other Surprising Facts about Learning

1. Is there such a thing as a “math brain”?

2. What happens to your brain when you make mistakes in math?

3. What is the growth mindset?

4. What suggestions does Jo Boaler offer for improving math instruction?

5. What else interested you as you watched this video?

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016 Page 10

Source: Jo Boaler, TED Talk https://www.youtube.com/watch?v=3icoSeGqQtY

Resources

Annenberg Learner. Courses of study in such areas as algebra, geometry, and real-world mathematics. The Annenberg Foundation provides numerous professional development activities or just the opportunity to review information in specific areas of study. http://www.learner.org/index.html

Illuminations. Great lesson plans for all areas of mathematics at all levels from the NationalCouncil of Teachers of Mathematics (NCTM). http://illuminations.nctm.org

Geometry Center (University of Minnesota). This site is filled with information and activities for different levels of geometry. http://www.geom.uiuc.edu/

National Library of Virtual Manipulatives for Math - All types of virtual manipulatives for use in the classroom from algebra tiles to fraction strips. This is a great site for students who need to see the “why” of math. http://nlvm.usu.edu/en/nav/index.html

Real-World MathThe 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

Evidence BaseNational Institute for Literacy. (2010). Algebraic thinking in adult education. Washington, DC:

Author. Retrieved from https://lincs.ed.gov/publications/pdf/algebra_paper_2010V.pdf

The Education Alliance. (2006). Best practices in teaching mathematics. Charleston, W.V: Author. Retrieved from: http://www.gram.edu/sacs/qep/chapter%204/4_1EducationAlliance.pdf

U.S. Department of Education, Office of Vocational and Adult Education. (2014). Math works! Guide. American Institutes for Research. Retrieved from: http://lincs.ed.gov/sites/default/files/Teal_Math_Works_Guide_508.pdf

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016 Page 11

This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 12/07/2016 Page 12