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NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Statistics and Probability in Grades 6 - 11 A Story of Ratios A Story of Functions © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Session Objectives • • • • • Explore the distinction between mathematical thinking and statistical thinking. Examine the development of the statistics and probability content over grades 6 – 11. Introduce overarching themes that provide coherence in the statistics and probability content across the grades. Illustrate development of statistical thinking across the grades with a trajectory of lesson activities. Explore other dimensions in the development of statistical thinking across the grades. 2 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Statistical Thinking vs Mathematical Thinking Grade 6 – 11 Statistics and Probability—Overarching Themes Linking the Two Statistics Themes Example Trajectory Other Developmental Dimensions Questions and Discussion 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Consider the map of counties shown below. The number in each county is last month’s incidence rate for a disease in cases per 100,000 population. What do you think is going on? (Dick Schaeffer, 2005) 4 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Statistical Thinking Versus Mathematical Thinking Mathematical Thinking • Explain patterns • Often a deterministic way of thinking Statistical Thinking • Search for patterns in the presence of variability • Acknowledge role of chance variation (distinguish “signal” from “noise”) 5 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Statistical Thinking Statistical thinking is HARD! Needs to be developed and nurtured over time, much like Mathematical thinking. Is a skill that is critical to making informed decisions based on empirical evidence. 6 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Statistical Thinking vs Mathematical Thinking Grade 6 – 11 Statistics and Probability—Overarching Themes Linking the Two Statistics Themes Example Trajectory Other Developmental Dimensions Questions and Discussion 7 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Developing Statistical Thinking Across the Grades 8 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Developing Statistical Thinking Across the Grades 9 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Developing Statistical Thinking Across the Grades Common Core Challenges: • Achieving coherence across the grades. • Long time gaps—many necessary connections skip years! • New content for most teachers. 10 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Three Overarching Themes • Variability • Learning from Data (The Investigative Process) • Probability 11 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Variability • • • • Anticipating variability Describing variability Understanding variability Drawing conclusions in the presence of variability 12 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Learning from Data The Investigative Process • Pose a statistical question A statistical question is one that can be answered by collecting data and where there will be variability in the data. • Collect data • Summarize and describe the data distribution • Draw conclusions based on data 13 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Probability • Understanding probability • Foundation for “ruling out chance” 14 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Statistical Thinking vs Mathematical Thinking Grade 6 – 11 Statistics and Probability—Overarching Themes Linking the Two Statistics Themes Example Trajectory Other Developmental Dimensions Questions and Discussion 15 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Linking the Two Statistics Themes Step Learning from Data The Role of Variability 1 Pose a statistical question Anticipating variability (in the data) 2 Collect data Planning for variability (recognizing the need for a representative sample or comparable experimental groups) 3 Summarize and describe the data distribution Understanding variability (looking at the data distribution using graphical and numerical summaries) 4 Draw conclusions based on data Taking variability into account when drawing conclusions (sampling variability) 16 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Themes Across the Grades Grade 6: Focus on Steps 1 and 3 (for census data), Step 4 in an informal way Grade 7: Continue with Step 1, adds Step 2 (selecting a sample) and Step 3 (for sample data) Grade 8: Focus on Step 3 in the context of relationships between two variables Grade 9: Focus moves to Step 4 in an informal way Grade 11: Formalizes Step 4. 17 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The Coherence Challenge Keeping the overarching themes in mind given • Spread over grade levels • Long gaps between statistics modules 18 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Statistical Thinking vs Mathematical Thinking Grade 6 – 11 Statistics and Probability—Overarching Themes Linking the Two Statistics Themes Example Trajectory Other Developmental Dimensions Questions and Discussion 19 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Example Trajectory: Learning About a Population Grade 6: Describing and comparing populations Grade 7: Sampling from a population, concept of sampling variability Grade 11: Drawing conclusions about a population based on data from a sample. 20 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 6 —Example 1 (Lesson 19) In 2012, Major League Baseball was comprised of two leagues: an American League of 14 teams, and a National League of 16 teams. It is believed that the American League teams would generally have higher values of certain offensive statistics such as "batting average" and "on-base percentage." (Teams want to have high values of these statistics.) Use the following side-by-side box plots to investigate these claims. 21 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 6 Example 1 Continued 1. Was the highest American League team "batting average" very different from the highest National League team "batting average"? If so, approximately how large was the difference and which league had the higher maximum value? 2. Was the range of American League team "batting averages" very different or only slightly different from the range of National League team "batting averages"? 3. Which league had the higher median team "batting average"? Given the scale of the graph and the range of the datasets, does the difference between the median values for the two leagues seem to be small or large? Explain why you think it is large or small. 22 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 6 Example 1 Continued 4. 5. 6. Based on the box plots below for "on-base percentage," which 3 summary values (from the 5-number summary) appear to be the same or virtually the same for both leagues? Which league's data set appears to have less variability? Explain. Respond to the original statement: "It is believed that the American League teams would generally have higher values of … 'on-base percentage.'" Do you agree or disagree based on the graphs above? Explain. 23 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 6 —Example 2 (Lesson 10) Decision making by comparing distributions is an important function of statistics. Recall that Robert is trying to decide whether to move to New York City or to San Francisco based on temperature. Comparing the center, spread, and shape for the two temperature distributions could help him decide. Which city should he choose if he loves hot weather and really dislikes cold weather? What measure of the data would justify your decision? Why did you choose that measure? 24 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 6 Examples • Both activities require students to consider the role of variability in trying to answer the questions posed. • Both activities will be challenging for grade 6 students—this is a new way of thinking about data that requires them to think about data as a distribution rather than in terms of individual points. This is a big conceptual leap. 25 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 7 Example—Casey at the Bat 1. 2. Suppose you wanted to learn about the lengths of the words in the poem Casey at the Bat. You plan to select a sample of words from the poem and use these words to answer the following statistical question: On average, how long is a word in the poem? What is the population of interest here? Look at the poem, Casey at the Bat by Ernest Thayer, and select eight words you think are representative of words in the poem. Record the number of letters in each word you selected. Find the mean number of letters in the words you chose. 26 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Casey at the Bat Continued 3. 4. A random sample is a sample in which every possible sample of the same size has an equal chance of being chosen. Do you think the set of words you wrote down was random? Why or why not? Working with a partner, follow your teacher’s instruction for randomly choosing eight words. Begin with the title of the poem and count a hyphenated word as one word. 1. 2. Record the eight words you randomly selected and find the mean number of letters in those words. Compare the mean of your random sample to the mean you found in Exercise 4. 27 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Casey at the Bat Continued 28 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Casey at the Bat Continued 5. As a class, compare the means from exercise 4 and the means from exercise 6. Your teacher will provide a chart to compare the means. Record your mean from exercise 4 and your mean for exercise 6 on this chart. 29 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Casey at the Bat Continued 6. 7. 8. 9. Do you think means from exercise 4 or the means from exercise 6 are more representative of the mean of all of the words in the poem? Explain your choice. The actual mean of the words in the poem Casey At The Bat is 4.2 letters. Based on the fact that the population mean is 4.2 letters, are the means from Exercise 4 or means from Exercise 6 a better representation the mean of the population. Explain your answer. How did population mean of 4.2 letter compare to the mean of your random sample from exercise 6 and to the mean you found in exercise 4? Summarize how you would estimate the mean number of letters in the words of another poem based on what you learned in the above exercises. 30 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 7 Example • Illustrates the need for random sampling • Develops the concept of sampling variability 31 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 11 Example—Margin of Error A newspaper in New York took a random sample of 500 people from New York City and found that 300 favored a certain candidate for governor of the state. A second newspaper polled 1000 people in upstate New York and found that 550 people favored the opposing candidate. Explain how you would interpret the results. 32 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Margin of Error Continued Possible answer: In New York City, the proportion of people who favor the candidate is 0.60 +/- 0.044% or from 0.556 to 0.644 of the people. In upstate New York, the proportion of people who favor the candidate is 0.55 +/- 0.032 or from 0.518 to 0.592. Because the margins of error for the proportion that favor the candidate produce intervals that overlap, you cannot really say that the proportion who prefer this candidate is different for people in New York City and people in upstate New York. 33 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Statistical Thinking vs Mathematical Thinking Grade 6 – 11 Statistics and Probability—Overarching Themes Linking the Two Statistics Themes Example Trajectory Other Developmental Dimensions Questions and Discussion 34 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Other Developmental Dimensions • Census of entire population (G6) → Sample from a population (G7, 9, 11) • Data on one variable (G6, 7) → Two variables with focus on relationship between variables (G8 (informal), 9, 11) • One group (G6) → Comparing two groups (G7 (informal) 9, 11) • Data from sampling (observational studies) → data from experiments (G11) 35 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Where It All Leads Activity from last lessons of Grade 11 illustrates where we hope students will be at the end of their Grades 6 – 11 statistics journey. Imagine that 10 tomatoes of varying shapes and sizes have been placed in front of you. These 10 tomatoes (all of the same variety) have been part of a nutrient experiment where the application of a nutrient is expected to yield larger tomatoes that weigh more. All 10 tomatoes have been grown under similar conditions (soil, water, sunlight, etc.) except that 5 of the tomatoes received the additional nutrient supplement. Using the weight data of these 10 tomatoes, you wish to examine the claim that the nutrient yields heavier tomatoes on average. 36 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions More Tomatoes 37 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions More Tomatoes Question of Interest: Is the difference in the two group means due to the nutrient, or might it just be due to chance? That is, could it just be due to the expected differences that would occur just because of random assignment of tomatoes to groups? 38 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions More Tomatoes The reasoning: Suppose that the nutrient had no effect. Then the big tomato that weighed 9.1 ounces would have weighed 9.1 ounces with or without the nutrient. If this is the case, the difference in the treatment and the control group means is just a consequence of the random assignment. Is this a plausible explanation for the difference? 39 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions More Tomatoes Let’s investigate: If there were no treatment effect, what kind of differences would be expected just by chance? 1. Write the 10 weights on the cards provided. 2. Turn the cards over, mix well, and divide them into two groups (group A and group B). 3. Calculate the group A mean, the group B mean, and the difference in the two means. 40 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions More Tomatoes If this process was repeated many times, we would have a sense of what “chance differences” would look like if there is no treatment effect. 41 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions More Tomatoes Drawing conclusions: The observed difference in sample means is 2.24. Is consistent with chance? 42 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Tomato Example Illustrates 1. The investigative process 2. The role of variability in decision making 3. The link between statistics and probability 43 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Thanks Thank you for your attention and participation. We hope this overview of statistics and probability in Grades 6 – 11 has provided some perspective on the content of these modules. Questions and Discussion 44 © 2012 Common Core, Inc. All rights reserved. commoncore.org