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ASE MA 4: Geometry, Probability and Statistics Dianne B. Barber ([email protected]) & William D. Barber ([email protected]) Appalachian State University, Boone, NC Agenda: 8:30 – 9:45 9:45 – 10:15 10:00 – 10:15 10:15 – 11:45 11:45 – 12:45 12:45 – 2:15 2:15 – 2:30 2:30 – 3:55 3:55 – 4:00 Introduction & Overview Probability Break Probability & Statistics Lunch Statistics & Geometry Break Geometry Certificates & Evaluations Overview: This training will assist instructors in making geometry, statistics and probability real so their learners will have the content knowledge to be successful on equivalency exams and in transitioning to college and careers. Objectives: Understand and use ASE standards as a basis for instructional planning Teach using best practices Use technology to enhance teaching and learning Know where to locate supplemental resources NCCCS College & Career Readiness Adult Education Content Standards Standards for Mathematical Practices (page 2) ASE MA04 Content Standards (pages 3-5) ASE MA04 Instructor Checklist (page 6) ASE MA04 Student Checklist (pages 7-8) Design learning around logical and consistent progression Teach fewer concepts with more depth of learning Teach conceptual understanding, procedural skill and fluency, Application This course is funded by ASE MA 4 Geometry, Probability, and Statistics; Revised 03/21/16 Page 1 Standards for Mathematical Practices 1. Makes sense of problems and perseveres in solving them ☐ Understands the meaning of the problem and looks for entry points to its solution ☐ Analyzes information (givens, constrains, relationships, goals) ☐Designs a plan ☐Monitors and evaluates the progress and changes course as necessary ☐ Checks their answers to problems and ask, “Does this make sense?” 2. Reason abstractly and quantitatively ☐Makes sense of quantities and relationships ☐ Represents a problem symbolically ☐ Considers the units involved ☐ Understands and uses properties of operations 3. Construct viable arguments and critique the reasoning of others ☐ Uses definitions and previously established causes/effects (results) in constructing arguments ☐Makes conjectures and attempts to prove or disprove through examples and counterexamples ☐ Communicates and defends their mathematical reasoning using objects, drawings, diagrams, actions ☐ Listens or reads the arguments of others ☐Decide if the arguments of others make sense ☐ Ask useful questions to clarify or improve the arguments 4. Model with mathematics. ☐ Apply reasoning to create a plan or analyze a real world problem ☐ Applies formulas/equations ☐Makes assumptions and approximations to make a problem simpler This course is funded by ☐ Checks to see if an answer makes sense and changes a model when necessary 5. Use appropriate tools strategically. ☐ Identifies relevant external math resources and uses them to pose or solve problems ☐Makes sound decisions about the use of specific tools. Examples may include: ☐ Calculator ☐ Concrete models ☐Digital Technology ☐Pencil/paper ☐ Ruler, compass, protractor ☐ Uses technological tools to explore and deepen understanding of concepts 6. Attend to precision. ☐ Communicates precisely using clear definitions ☐Provides carefully formulated explanations ☐ States the meaning of symbols, calculates accurately and efficiently ☐ Labels accurately when measuring and graphing 7. Look for and make use of structure. ☐ Looks for patterns or structure ☐ Recognize the significance in concepts and models and can apply strategies for solving related problems ☐ Looks for the big picture or overview 8. Look for and express regularity in repeated reasoning ☐ Notices repeated calculations and looks for general methods and shortcuts ☐ Continually evaluates the reasonableness of their results while attending to details and makes generalizations based on findings ☐ Solves problems arising in everyday life Adapted from Common Core State Standards for Mathematics: Standards for Mathematical Practice ASE MA 4 Geometry, Probability, and Statistics; Revised 03/21/16 Page 2 ASE MA 04: Geometry, Probability, and Statistics MA.4.1 Geometry: Understand congruence and similarity. Objectives MA.4.1.1 Experiment with transformations in a plane. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Example: How would you determine whether two lines are parallel or perpendicular? MA.4.1.2 Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. What Learner Should Know, Understand, and Be Able to Do Teaching Notes and Examples A point has position, no thickness or distance. A line is made of infinitely many points, and a line segment is a subset of the points on a line with endpoints. A ray is defined as having a point on one end and a continuing line on the other. An angle is determined by the intersection of two rays. A circle is the set of infinitely many points that are the same distance from the center forming a circular are, measuring 360 degrees. Perpendicular lines are lines in the interest at a point to form right angles. Parallel lines that lie in the same plane and are lines in which every point is equidistant from the corresponding point on the other line. Definitions are used to begin building blocks for proof. Infuse these definitions into proofs and other problems. Pay attention to Mathematical practice 3 “Construct viable arguments and critique the reasoning of others: Understand and use stated assumptions, definitions and previously established results in constructing arguments.” Also mathematical practice number six says, “Attend to precision: Communicate precisely to others and use clear definitions in discussion with others and in their own reasoning.” Students use similarity theorems to prove two triangles are congruent. Students prove that geometric figures other than triangles are similar and/or congruent. Solve Problems using Congruence and Similarity https://learnzillion.com/lessonsets/668-solveproblems-using-congruence-and-similarity-criteria-fortriangles Experiment with Transformations in a Plane http://www.virtualnerd.com/common-core/hsfgeometry/HSG-CO-congruence/A https://www.illustrativemathematics.org/HSG MA.4.2 Geometric Measure and Dimension: Explain formulas and use them to solve problems and apply geometric concepts in modeling situations. Objectives What Learner Should Know, Understand, and Be Able to Do Teaching Notes and Examples MA.4.2.1 Explain perimeter, area, and volume formulas and use them to solve problems involving two- and three-dimensional shapes. Use given formulas and solve for an indicated variables within the formulas. Find the side lengths of triangles and rectangles when given area or perimeter. Compute volume and surface area of cylinders, cones, and right pyramids. Geometry Lesson Plans http://www.learnnc.org/?standards=Mathematics-Geometry MA.4.2.2 Apply geometric concepts in modeling of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Use the concept of density when referring to situations involving area and volume models, such as persons per square mile. Understand density as a ratio. Differentiate between area and volume densities, their units, and situations in which they are appropriate (e.g., area density is ideal for measuring population density spread out over land, and the concentration of oxygen in the air is best measured with volume density). Explore design problems that exist in local communities, such as building a shed with maximum capacity in a small area or locating a hospital for three communities in a desirable area. Geometry Problem Solving http://map.mathshell.org/materials/lessons.php?taski d=216&subpage=concept This course is funded by 1 Example: Given the formula 𝑉 = 𝐵𝐻, for the volume 3 of a cone, where B is the area of the base and H is the height of the. If a cone is inside a cylinder with a diameter of 12in. and a height of 16 in., find the volume of the cone. ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 3 MA.4.3 Summarize, represent, and interpret categorical and quantitative data on (a) a single count or measurement variable, (b) two categorical and quantitative variables, and (c) Interpret linear models. Objectives MA.4.3.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). What Learner Should Know, Understand, and Be Able to Do Construct appropriate graphical displays (dot plots, histogram, and box plot) to describe sets of data values. Teaching Notes and Examples Represent Data with Plots https://learnzillion.com/lessonsets/513-representdata-with-plots-on-the-real-number-line-dot-plotshistograms-and-box-plots http://www.virtualnerd.com/common-core/hssstatistics-probability/HSS-ID-interpreting-categoricalquantitative-data/A/1 MA.4.3.2 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Understand and be able to use the context of the data to explain why its distribution takes on a particular shape (e.g. are there real-life limits to the values of the data that force skewness? are there outliers?) Understand that the higher the value of a measure of variability, the more spread out the data set is. Interpreting Categorical and Quantitative Data http://www.shmoop.com/common-corestandards/ccss-hs-s-id-3.html http://www.thirteen.org/get-themath/teachers/math-in-restaurants-lessonplan/standards/187/ Explain the effect of any outliers on the shape, center, and spread of the data sets. MA.4.3.3 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Create a two-way frequency table from a set of data on two categorical variables. Calculate joint, marginal, and conditional relative frequencies and interpret in context. Joint relative frequencies are compound probabilities of using AND to combine one possible outcome of each categorical variable (P(A and B)). Marginal relative frequencies are the probabilities for the outcomes of one of the two categorical variables in a twoway table, without considering the other variable. Conditional relative frequencies are the probabilities of one particular outcome of a categorical variable occurring, given that one particular outcome of the other categorical variable has already occurred. Recognize associations and trends in data from a two-way table. Interpreting Quantitative and Categorical data http://www.ct4me.net/CommonCore/hsstatistics/hss-interpreting-categoricalquantitative-data.htm MA.4.3.4 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Understand that the key feature of a linear function is a constant rate of change. Interpret in the context of the data, i.e. as x increases (or decreases) by one unit, y increases (or decreases) by a fixed amount. Interpret the y-intercept in the context of the data, i.e. an initial value or a one-time fixed amount. Interpreting Slope and Intercepts http://www.virtualnerd.com/common-core/hssstatistics-probability/HSS-ID-interpreting-categoricalquantitative-data/C/7 Understand that just because two quantities have a strong correlation, we cannot assume that the explanatory (independent) variable causes a change in the response (dependent) variable. The best method for establishing causation is Correlation and Causation https://learnzillion.com/lessonsets/585-distinguishbetween-correlation-and-causation MA.4.3.5 Distinguish between correlation and causation. This course is funded by http://www.virtualnerd.com/middlemath/probability-statistics/frequency-tables-lineplots/practice-make-frequency-table http://ccssmath.org/?page_id=2341 https://learnzillion.com/lessonsets/457-interpret-theslope-and-the-intercept-of-a-linear-model-using-data https://www.khanacademy.org/math/probability/stati stical-studies/types-of-studies/v/correlation-and- ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 4 conducting an experiment that carefully controls for the effects of lurking variables (if this is not feasible or ethical, causation can be established by a body of evidence collected over time e.g. smoking causes cancer). causality MA.4.4 Using probability to make decisions. Objectives What Learner Should Know, Understand, and Be Able to Do Teaching Notes and Examples M.4.4.1 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. Develop a theoretical probability distribution and find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple choice test where each question has four choices, and find the expected grade under various grading schemes. Probability http://www.shmoop.com/common-corestandards/ccss-hs-s-md-4.html M.4.4.2 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. Develop an empirical probability distribution and find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households. Probability Distribution http://www.google.com/url?sa=t&rct=j&q=&esrc=s&s ource=web&cd=7&ved=0CEAQFjAG&url=http%3A%2F %2Feducation.ohio.gov%2Fgetattachment%2FTopics% 2FOhio-s-New-LearningStandards%2FMathematics%2FHigh_School_Statisticsand-Probability_Model-Curriculum_October20131.pdf.aspx&ei=ec0RVNHBN8UgwSYvYD4Dg&usg=AFQjCNHpyffrA7UVkDyKCXIkYRD Sw1nsyQ&bvm=bv.74894050,d.eXY M.4.4.3 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. Find the expected payoff for a game of chance. Set up a probability distribution for a random variable representing payoff values in a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. Expected Value http://www.youtube.com/watch?v=DAjVAEDil_Q M.4.4.4 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Make decisions based on expected values. Use expected values to compare long- term benefits of several situations. Using Probability to Make Decisions http://www.shmoop.com/common-corestandards/ccss-hs-s-md-6.html M.4.4.5 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Explain in context decisions made based on expected values. Analyzing Decisions http://www.ct4me.net/CommonCore/hsstatistics/hss-using-probability-makedecisions.htm This course is funded by Using Probability to Make Decisions https://www.khanacademy.org/commoncore/gradeHSS-S-MD Weighing Outcomes https://www.illustrativemathematics.org/illustrations/ 1197 Money and Probability http://becandour.com/money.htm ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 5 ASE MA 4: Geometry, Probability, and Statistics – Instructor Checklist MA.4.1 Geometry: Understand congruence and similarity. Objectives Curriculum – Materials Used MA.4.1.1 Experiment with transformations in a plane. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MA.4.1.2 Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Notes MA.4.2 Geometric Measure and Dimension: Explain formulas and use them to solve problems and apply geometric concepts in modeling situations. Objectives Curriculum – Materials Used Notes MA.4.2.1 Explain perimeter, area, and volume formulas and use them to solve problems involving two- and three-dimensional shapes. MA.4.2.2 Apply geometric concepts in modeling of density based on area and volume in modeling. MA.4.3 Summarize, represent, and interpret categorical and quantitative data on (a) a single count or measurement variable, (b) two categorical and quantitative variables, and (c) Interpret linear models. Objectives Curriculum – Materials Used Notes MA.4.3.1 Represent data with plots on the real number line. MA.4.3.2 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). MA.4.3.3 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data. Recognize possible associations and trends in the data. MA.4.3.4 Interpret the slope and the intercept of a linear model in the context of the data. MA.4.3.5 Distinguish between correlation and causation. MA.4.4 Using probability to make decisions. Objectives M.4.4.1 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. M.4.4.2 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. M.4.4.3 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. Find the expected payoff for a game of chance. Curriculum – Materials Used Notes M.4.4.4 Use probabilities to make fair decisions. M.4.4.5 Analyze decisions and strategies using probability concepts. This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 6 ASE MA 04: Geometry, Probability, and Statistics – Student Checklist MA.4.1 Geometry: Understand congruence and similarity. Learning Targets Mastery Level % Date I can define an angle based on my knowledge of a point, line, distance along a line, and distance around a circular arc. I can define a circle based on my knowledge of a point, line, distance along a line, and distance around a circular arc. I can define perpendicular lines based on my knowledge of a pint, line, distance, along a line, and distance around a circular arc. I can define a line segment based on my knowledge of a pint, line, distance, along a line, and distance around a circular arc. I can define parallel lines based on my knowledge of a pint, line, distance, along a line, and distance around a circular arc. I can describe the relationship between similarity and congruence. I can set up and write equivalent ratios. I can identify corresponding angles and sides of two triangles. I can determine a scale factor and use it in a proportion. I can solve geometric problems using congruence and similarity. I can prove relationships using congruence and similarity. MA.4.2 Geometric Measure and Dimension: Explain formulas and use them to solve problems and apply geometric concepts in modeling situations. Learning Targets Mastery Level % Date I can apply the formula for the perimeter of a rectangle to solve problems. I can apply the formula for the area of a rectangle to solve problems. I can apply the formula for the area of a triangle to solve problems. I can apply the formula for the volume of a cone to solve problems. I can apply the formula for the volume of a cylinder to solve problems. I can apply the formula for the volume of a pyramid to solve problems. I can apply the formula for the volume of a sphere to solve problems. I can find the surface area of right circular cylinders. I can find the surface area of rectangular prisms. I can use geometric shapes, their measures, and their properties to describe objects. I can apply concepts of density based on area and volume in modeling situations. MA.4.3 Summarize, represent, and interpret categorical and quantitative data on (a) a single count or measurement variable, (b) two categorical and quantitative variables, and (c) Interpret linear models. Learning Targets Mastery Level % Date I can classify data as either categorical or quantitative. I can identify an appropriate scale needed for the data display. I can identify an appropriate number of intervals for a histogram. I can identify an appropriate width for intervals in a histogram. I can select an appropriate data display for real-world data. This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 7 ASE MA 4: Geometry, Probability, and Statistics – Student Checklist, Page 2 I can construct dot plots. I can create a frequency table. I can construct histograms. I can construct box plots. I can identify a data set by its shape and describe the data set as symmetric or skewed. I can use the outlier rule (e.g., Q1 – 1.5 x IQR and Q3 + 1.5 IQR) to identify outliers in a data set. I can analyze how adding/removing an outlier affects measures of center and spread. I can interpret differences in shape, center and spread in the context of data sets. I can compare and contrast two or more data sets using shape, center, and spread. I can organize categorical data in two-way frequency tables. I can interpret joint frequencies and joint relative frequencies in the context of the data. I can interpret marginal frequencies and marginal relative frequencies in the context of the data. I can interpret conditional frequencies and conditional relative frequencies in the context of the data. I can recognize possible associations between categorical variables in a two-way frequency or relative frequency table. I can determine the y-intercept graphically and algebraically. I can determine the rate of change by choosing two points. I can determine the equation of a line using data points. I can interpret the slope in the context of the data. I can interpret the y-intercept in the context of the data. I can differentiate between causation and correlation/association. I can interpret paired data to determine whether correlation implies causation/association. MA.4.4 Using probability to make decisions. Learning Targets Mastery Level % Date I can develop a probability distribution for a random variable defined for a sample space of theoretical probabilities. I can calculate theoretical probabilities and find expected values. I can develop a probability distribution for a random variable for a sample space of empirically assigned probabilities. I can assign probabilities empirically and find expected values. I can weigh the possible outcomes of a decision and find expected values. I can assign probabilities to payoff values and find expected values. I can evaluate strategies based on expected values. I can compare strategies based on expected values. I can explain the difference between theoretical and experimental probability. I can compute theoretical and experimental probability. I can determine the fairness of a decision based on the available data. I can determine the fairness of a decision by comparing theoretical and experimental probability. I can use counting principles to determine the fairness of a decision. I can analyze decisions and strategies related to product testing, medical testing, and sports. This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 8 Best Practices in Teaching Mathematics Instructional Element Curriculum Design Recommended Practices Ensure mathematics curriculum is based on challenging content Ensure curriculum is standards- based Clearly identify skills, concepts and knowledge to be mastered Ensure that the mathematics curriculum is vertically and horizontally articulated Provide professional development which focuses on: o Knowing/understanding standards o Using standards as a basis for instructional planning o Teaching using best practices o Multiple approaches to assessment Develop/provide instructional support materials such as curriculum maps and pacing guides Establish math leadership teams and provide math coaches Professional Development for Teachers Technology Provide professional development on the use of instructional technology tools Provide student access to a variety of technology tools Integrate the use of technology across all mathematics curricula and courses Manipulatives Use manipulatives to develop understanding of mathematical concepts Use manipulatives to demonstrate word problems Ensure use of manipulatives is aligned with underlying math concepts Focus lessons on specific concept/skills that are standards- based Differentiate instruction through flexible grouping, individualizing lessons, compacting, using tiered assignments, and varying question levels Ensure that instructional activities are learner-centered and emphasize inquiry/problem-solving Use experience and prior knowledge as a basis for building new knowledge Use cooperative learning strategies and make real life connections Use scaffolding to make connections to concepts, procedures and understanding Ask probing questions which require students to justify their responses Emphasize the development of basic computational skills Ensure assessment strategies are aligned with standards/concepts being taught Evaluate both student progress/performance and teacher effectiveness Utilize student self-monitoring techniques Provide guided practice with feedback Conduct error analyses of student work Utilize both traditional and alternative assessment strategies Ensure the inclusion of diagnostic, formative and summative strategies Increase use of open-ended assessment techniques Instructional Strategies Assessment Source: Best Practices in Teaching Mathematics, Spring 2006. The Education Alliance, Charleston, West Virginia. Website: www.educationalliance.org Problem Solving and Value of Teaching with Problems Places students’ attention on mathematical ideas Develops “mathematical power” Develops students’ beliefs that they are capable of doing mathematics and that it makes sense Provides ongoing assessment data that can be used to make instructional decisions Allows an entry point for a wide range of students This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 9 Probability Experimental versus Theoretical Probability – See handout Probability Rules Multiplication rule o Probability of both with two independent events o P(A and B) = P(A) X P(B) o Intersection o Examples: P(both dice < 6) P(first and second child both girls) P(both were born on Tuesday) Addition rule o Used to find probability of “at least one” when events are independent o P(A or B) = P(A) + P(B) – P(A and B) o Examples o P(at least one girl in two children) o P(at least one 6 when rolling 2 dice) o P(at least one of two people was born on Tuesday) Contextualized Example: When both parents have rh+ blood but are carriers of negative blood, each of their children has a 25% probability of having rh- blood. If these parents have non-identical twins 1. What is the probability that both will have rh- blood? 2. What is the probability that at least one will be rh-? Real Life Decisions & Probability Extension Activity #1: Probability in Advertising Ask students to look at newspapers and magazines for examples of how numbers are used in advertisements. For example, it is not unusual to see something like "two-thirds less fat than the other leading brand" or "four out of five dentists recommend Brand T gum for their patients who chew gum." Why do advertisers use numbers like these? What information are they trying to convey? Do students think that the numbers give accurate information about a product? Why or why not? Extension Activity #2: They Said What? Ask students to look at newspapers or magazines for examples of how politicians, educators, environmentalists, or others use data such as statistics and probability. Then have them analyze the use of the information. Why did the person use data? What points were effectively made? Were the data useful? Did the data strengthen the argument? Have students provide evidence to support their ideas. This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 10 Statistics – Working with Data Qualitative/categorical Data o Nominal o Ordinal Quantitative/numerical Data o Interval vs. ratio o Discrete vs. continuous Representing Data Identifying Misleading Graphs: Video - http://www.youtube.com/watch?v=ETbc8GIhfHo#t=11 o o o o o o o o no title no labels on the axes no key – when one is required vertical axis doesn’t start at 0 different sized graphing icons or bars unequal intervals on any axis “broken” or “squished” axis distorted image (sloped circle) Graphing Activity: As a group, examine the graph you have been assigned. Is it misleading? Explain why. How could you make it a better graph? Be prepared to share your groups results with the class. This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 11 Scatterplots Plot data points on a graph Shows relationships between variables Correlation Positive vs. negative DOES NOT prove cause/effect Lurking (confounding) variables Independent & Dependent Variables If there was a cause/effect relationship, the cause would be the independent variable Researcher controls independent variable, then evaluates/measures dependent. Independent/dependent variables may not be obvious in observational studies o Or both may be dependent on something else Correlation Contextualized This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 12 Measures of Central Tendency Mean (arithmetic mean) o Commonly called “average” o Sum of values ÷ number of values Median o Middle value in rank order (if odd # of values) o Mean of 2 middle values (if even # of values o Used for skewed data (such as income) Mode o Most frequent value o There may be no mode or multiple modes Measures of Spread Range (R) Highest value – lowest value Interquartile range (IQR) = Q3 – Q1 o Range of the middle 50% of data o Q1 – separates the lowest 25% of data o Q3 – separates the highest 25% of data Standard deviation Boxplots Label each of the following on the boxplot above: * Q1 - median of the lower half of the data set Q2 - median of the data set Q3 - median of the upper half of the data set Lowest and highest values not more than 1.5 IQR from box Outliers This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 13 Drawing a Boxplot: Use the data below to graph a boxplot 90, 94, 53, 68, 79, 84, 87, 72, 70, 69, 65, 89, 85, 83, 72 1. Order data 2. Find Q2, Q1, Q3 3. Find IQR 4. Calculate 1.5 IQR to determine outliers, if any 5. Draw and label x-axis in the space provided below 6. Draw boxplot above x-axis using information found in #2-4 Uses of Boxplots May indicate skewed distribution Comparison of IQRs Comparison of side-by-side boxplots This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 14 Directions for Making a Box Plot Box plots are a handy way to display data broken into four quartiles, each with an equal number of data values. The box plot doesn't show frequency, and it doesn't display each individual statistic, but it clearly shows where the middle of the data lies. It's a nice plot to use when analyzing how your data is skewed. There are a few important vocabulary terms to know in order to graph a box-and-whisker plot. Here they are: Q1 – quartile 1, the median of the lower half of the data set Q2 – quartile 2, the median of the entire data set Q3 – quartile 3, the median of the upper half of the data set IQR – interquartile range, the difference from Q3 to Q1 Extreme Values – the smallest and largest values in a data set Make a box plot for the geometry test scores given below: 90, 94, 53, 68, 79, 84, 87, 72, 70, 69, 65, 89, 85, 83, 72 Step 1: Order the data from least to greatest. Step 2: Find the median of the data. This is also called quartile 2 (Q2). Step 3: Find the median of the data less than Q2. This is the lower quartile (Q1). Step 4. Find the median of the data greater than Q2. This is the upper quartile (Q3). Step 5. Find the extreme values: these are the largest and smallest data values. Note: if the data set contains outliers do not include outliers when finding extreme values. Extreme values = 53 and 94. Step 6. Create a number line that will contain all of the data values. It should stretch a little beyond each extreme value. Step 7. Draw a box from Q1 to Q3 with a line dividing the box at Q2. Then extend "whiskers" from each end of the box to the extreme values. This plot is broken into four different groups: the lower whisker, the lower half of the box, the upper half of the box, and the upper whisker. Since there is an equal amount of data in each group, each of those sections represent 25% of the data. Using this plot we can see that 50% of the students scored between 69 and 87 points, 75% of the students scored lower than 87 points, and 50% scored above 79. If your score was in the upper whisker, you could feel pretty proud that you scored better than 75% of your classmates. If you scored somewhere in the lower whisker, you may want to find a little more time to study. Outliers are values that are much bigger or smaller than the rest of the data. These are represented by a dot at either end of the plot. Our geometry test example did not have any outliers, even though the score of 53 seemed much smaller than the rest, it wasn't small enough. In order to be an outlier, the data value must be: (1) larger than Q3 by at least 1.5 times the interquartile range (IQR), or (2) smaller than Q1 by at least 1.5 times the IQR. Source: http://www.shmoop.com/basic-statistics-probability/box-whisker-plots.html This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 15 Geometry – Big Ideas Spatial visualization, scale factors Perimeter, circumference and area of 2-dimensional figures Surface area and volume of 3-dimensional figures Seeking relationships Making convincing arguments Visualization Recognize and name shapes Students often do not recognize properties or if they do, do not use them for sorting or recognition Students may not recognize shape in different orientation Implications for Instruction Provide activities that have students Sort, identify and describe shapes Use manipulatives, build and draw shapes See shapes in different orientations and sizes Define properties, make measurements, recognize patterns Explore what happens if a measurement or property changes Follow informal proofs Vocabulary (handout) Surface Area What is surface area? Surface area measures the combined surfaces of a 3-dimensional shape It is measured using squares Units include in2, ft2, yd2, mi2 or metric units such as mm2, cm2, m2, km2 This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 16 3 in h=6 r=5 Contextualized: How much sheet metal will be needed to construct a water tank in the shape of a right circular cylinder thati s 30 feet long and 8 feet in diameter? This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 17 Resources for Teaching Mathematics Free Resources for Educational Excellence. Teaching and learning resources from a variety of federal agencies. This portal provides access to free resources. http://free.ed.gov/index.cfm Annenberg Learner. Courses of study in such areas as algebra, geometry, and real-world mathematics. The Annenberg Foundation provides numerous professional development activities or just the opportunity to review information in specific areas of study. http://www.learner.org/index.html Illuminations. Great lesson plans for all areas of mathematics at all levels from the National Council of Teachers of Mathematics (NCTM). http://illuminations.nctm.org Khan Academy. A library of over 2,600 videos covering everything from arithmetic to physics, finance, and history and 211 practice exercises. http://www.khanacademy.org/ The Math Dude. A full video curriculum for the basics of algebra. http://www.montgomeryschoolsmd.org/departments/itv/MathDude/MD_Downloads.shtm Geometry Center (University of Minnesota). This site is filled with information and activities for different levels of geometry. http://www.geom.uiuc.edu/ National Library of Virtual Manipulatives for Math - All types of virtual manipulatives for use in the classroom from algebra tiles to fraction strips. This is a great site for students who need to see the “why” of math. http://nlvm.usu.edu/en/nav/index.html Teacher Guide for the TI-30SX MultiView Calculator – A guide to assist you in using the new calculator, along with a variety of lesson plans for the classroom. http://education.ti.com/en/us/guidebook/details/en/62522EB25D284112819FDB8A46F90740/30 x_mv_tg http://education.ti.com/calculators/downloads/US/Activities/Search/Subject?s=5022&d=1009 Mometrix Academy Free videos for math concepts http://www.mometrix.com/academy/basics-of-functions/ Real-World Math The Futures Channel http://www.thefutureschannel.com/algebra/algebra_real_world_movies.php Real-World Math http://www.realworldmath.org/ Get the Math http://www.thirteen.org/get-the-math/ Math in the News http://www.media4math.com/MathInTheNews.asp Please join us at Appalachian State University for Institute 2016: Shifts in Instruction for WIOA Implementation May 23-26, 2016 or May 30 – June 2, 2016 Register at www.abspd.appstate.edu. Earn 3 hours of graduate credit. This course is funded by ASE MA04 Geometry, Probability, and Statistics; Revised 03/03/2016 Page 18