Unit 4 Frameworks - Student Edition

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Frameworks Student Edition

7th Grade Unit 4: Statistics These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

Mathematics

Georgia Department of Education Common Core Georgia Performance Standards Framework Student Edition

Seventh Grade Mathematics Unit 4

MATHEMATICS GRADE 7 UNIT 4: Statistics Georgia Department of Education

Dr. John D. Barge, State School Superintendent May 2012 Page 2 of 18

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Unit 4 Statistics

TABLE OF CONTENTS Overview ..........................................................................................................................................3 Standards Addressed in this Unit .....................................................................................................4

Key Standards & Related Standards ....................................................................................4 Standards for Mathematical Practice ...................................................................................5

Enduring Understandings.................................................................................................................7 Concepts & Skills to Maintain .........................................................................................................7 Selected Terms and Symbols ...........................................................................................................7 Tasks

Is It Valid? ......................................................................................................................10 Shakespeare vs. Harry Potter .........................................................................................11 Got Friends? ...................................................................................................................15 Travel Times to Work ....................................................................................................17





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OVERVIEW

In this unit students will: use real-life situations to show the purpose for using random sampling to make

inferences about a population. understand that random sampling guarantees that each element of the population

has an equal opportunity to be selected in the sample. compare the random sample to the population, asking questions like, Are all the

elements of the entire population represented in the sample? and Are the elements represented proportionally?

make inferences given random samples from a population along with the statistical measures.

learn to draw inferences about one population from a random sampling of that population.

draw informal comparative inferences about two populations. deal with small populations and determine measures of center and variability for a

population. compare measures of center and variability and make inferences. use graphical representations of data to compare measures of center and

variability. begin to develop understanding of the benefits of the measures of center and

variability by analyzing data with both methods. understand that when they study large populations, random sampling is used as a

basis for the population inference. understand that measures of center and variability are used to make inferences on

each of the general populations. make comparisons for two populations based on inferences made from the

measures of center and variability. Although the units in this instructional framework emphasize key standards and big ideas

at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the eight practice standards should be addressed constantly as well. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under Evidence of Learning be reviewed early in the planning process. A variety of resources should be utilized to supplement this unit. This unit provides much needed content information, but excellent learning activities as well. The tasks in this unit illustrate the types of learning activities that should be utilized from a variety of sources.





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STANDARDS ADDRESSED IN THIS UNIT KEY STANDARDS Apply and extend previous understandings of measurement and interpreting data.

MCC7.SP.1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. MCC7.SP.2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. MCC7.SP.3. Informally assess the degree of visual overlap of two numerical data distributions with similar variability, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. MCC7.SP.4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. RELATED STANDARDS MCC7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 12 mile in each 14 hour, compute the unit rate as the complex fraction 12 14 miles per hour, equivalently 2 miles per hour. MCC7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. MCC7.NS.3 Solve realworld and mathematical problems involving the four operations with rational numbers.





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STANDARDS FOR MATHEMATICAL PRACTICE

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important processes and proficiencies with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Councils report Adding It Up: adaptive reasoning , strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately) and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and ones own efficacy).

1. Make sense of problems and persevere in solving them. In grade 7, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way? 2. Reason abstractly and quantitatively. In grade 7, students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations. 3. Construct viable arguments and critique the reasoning of others. In grade 7, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like How did you get that?, Why is that true? Does that always work?. They explain their thinking to others and respond to others thinking. 4. Model with mathematics. In grade 7, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students explore covariance and represent two





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quantities simultaneously. They use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences, make comparisons and formulate predictions. Students use experiments or simulations to generate data sets and create probability models. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context. 5. Use appropriate tools strategically. Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 7 may decide to represent similar data sets using dot plots with the same scale to visually compare the center and variability of the data. Students might use physical objects or applets to generate probability data and use graphing calculators or spreadsheets to manage and represent data in different forms. 6. Attend to precision. In grade 7, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students define variables, specify units of measure, and label axes accurately. Students use appropriate terminology when referring to rates, ratios, probability models, geometric figures, data displays, and components of expressions, equations or inequalities. 7. Look for and make use of structure. Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist in ratio tables making connections between the constant of proportionality in a table with the slope of a graph. Students apply properties to generate equivalent expressions (i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality), c = 6 by division property of equality). Students compose and decompose two and threedimensional figures to solve real world problems involving scale drawings, surface area, and volume. Students examine tree diagrams or systematic lists to determine the sample space for compound events and verify that they have listed all possibilities. 8. Look for and express regularity in repeated reasoning. In grade 7, students use repeated reasoning to understand algorithms and make generalizations about patterns. During multiple opportunities to solve and model problems, they may notice that a/b c/d = ad/bc and construct other examples and models that confirm their generalization. They extend their thinking to include complex fractions and rational numbers. Students formally begin to make connections between covariance, rates, and representations showing the





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relationships between quantities. They create, explain, evaluate, and modify probability models to describe simple and compound events. ENDURING UNDERSTANDINGS

Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population.

Understand that random sampling tends to produce representative samples and support valid inferences.

Use data from a random sample to draw inferences about a population with an unknown characteristic of interest.

Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

CONCEPTS AND SKILLS TO MAINTAIN In order for students to be successful, the following skills and concepts need to be maintained Analyzing patterns and seeing relationships Represent and interpret data, using addition and subtraction, multiplication and division Data can be represented graphically in a variety of ways. The type of graph is selected to best

represent a particular data set. Measures of center (mean, median, mode) and measures of variation (range, quartiles,

interquartile range) can be used to analyze data. Larger samples are more likely to be representative of a population. SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples.





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Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. The websites below are interactive and include a math glossary suitable for middle school children. Note At the middle school level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks. http://www.amathsdictionaryforkids.com/ This web site has activities to help students more fully understand and retain new vocabulary http://intermath.coe.uga.edu/dictnary/homepg.asp Definitions and activities for these and other terms can be found on the Intermath website. Intermath is geared towards middle and high school students.

Box and Whisker Plot: A diagram that summarizes data using the median, the upper and lowers quartiles, and the extreme values (minimum and maximum). Box and whisker plots are also known as box plots. It is constructed from the five-number summary of the data: Minimum, Q1 (lower quartile), Q2 (median), Q3 (upper quartile), Maximum.

Frequency: the number of times an item, number, or event occurs in a set of data

Grouped Frequency Table: The organization of raw data in table form with classes and frequencies.

Histogram: a way of displaying numeric data using horizontal or vertical bars so that the

height or length of the bars indicates frequency

Inter-Quartile Range (IQR): The difference between the first and third quartiles. (Note that the first quartile and third quartiles are sometimes called upper and lower quartiles.)

Maximum value: The largest value in a set of data.

Mean Absolute Deviation: the average distance of each data value from the mean. The

MAD is a gauge of on average how different the data values are form the mean value.

=

Mean: The average or fair share value for the data. The mean is also the balance

point of the corresponding data distribution.

= = 1 + 2 + 3 +

Measures of Center: The mean and the median are both ways to measure the center for a

set of data.





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Measures of Spread: The range and the mean absolute deviation are both common ways

to measure the spread for a set of data.

Median: The value for which half the numbers are larger and half are smaller. If there are two middle numbers, the median is the arithmetic mean of the two middle numbers. Note: The median is a good choice to represent the center of a distribution when the distribution is skewed or outliers are present.

Minimum value: The smallest value in a set of data.

Mode: The number that occurs the most often in a list. There can more than one mode, or

no mode.

Mutually Exclusive: two events are mutually exclusive if they cannot occur at the same time (i.e., they have not outcomes in common).

Outlier: A value that is very far away from most of the values in a data set.

Range: A measure of spread for a set of data. To find the range, subtract the smallest value from the largest value in a set of data.

Sample: A part of the population that we actually examine in order to gather information.

Simple Random Sampling: Consists of individuals from the population chosen in such a

way that every set of individuals has an equal chance to be a part of the sample actually selected. Poor sampling methods, that are not random and do not represent the population well, can lead to misleading conclusions.

Stem and Leaf Plot: A graphical method used to represent ordered numerical data. Once the data are ordered, the stem and leaves are determined. Typically the stem is all but the last digit of each data point and the leaf is that last digit.





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Is it Valid?

In the summaries below, determine if the sample taken is representative of the population, without bias shown:

1. ABC Family is a television channel that targets families and young adults to view their station. ABC Family regularly posts online poll questions to their website. In 2010, ABC Family polled their viewers to ask about airing Rated R movies after 8pm on their channel. Almost 200,000 people responded, and 85% of them disagreed with airing Rated R movies.

2. Mrs. Jones wants to know how the 5th grade feels about recess time. Mrs. Jones labels every student in the 5th grade with a number. She then draws 50 numbers out of a hat and surveys these students. Mrs. Jones determines that 5th graders would like more recess time than they currently have.

3. The City of Smallville wants to know how its citizens feel about a new industrial park in town. Surveyors stand in the Smallville Mall from 8am-11am on a Tuesday morning and ask people their opinion. 80% of the surveyed people said they disagreed with a new industrial park.

4. The National Rifle Association (NRA) took a poll on their website, www.nra.com, and asked the question, Do you agree with the 2nd Amendment: the Right to Bear Arms? 98% of the people surveyed said Yes, and 2% said No.





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Shakespeare vs. Harry Potter Many skeptics feel that there has been a dumbing down of Americas youth of the past decades. To determine if there is any truth to this claim, we will compare two pieces of literature: Shakespeares Macbeth and JK Rowlings Harry Potter and the Chamber of Secrets. Is there a difference in the length of the words used in a Shakespeare play compared to a Harry Potter book? Today you will sample words from both pieces of literature to determine who used longer words. Below are excerpts from a Shakespeare Novel and a Harry Potter book. Follow the steps below to determine which piece of literature uses longer words.

1. Roll a dice to determine which word to start with. If you roll a 1, start with word number 1 on the Shakespeare excerpt. If you roll a 6, start with word 6 in the passage.

2. Count the number of letters in each word and log it in the table below. Count 50 words

total from Shakespeares Macbeth passage.





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Size of word Count Size of word Count 1 letter 7 letters 2 letters 8 letters 3 letters 9 letters 4 letters 10 letters 5 letters 11 letters 6 letters 12 letters

How Many letters were in each word that was counted? (50 words)

3. Now, roll the dice to see which word you will begin with for the Harry Potter excerpt. Count the number of letters in each word and log it in the table below. Count 50 words total from Harry Potter passage.

Size of word Count Size of word Count 1 letter 7 letters 2 letters 8 letters 3 letters 9 letters 4 letters 10 letters 5 letters 11 letters 6 letters 12 letters

How Many letters were in each word that was counted? (50 words)





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4. Find the mean of the Shakespeare word sample and the Harry Potter word sample. Do this by multiplying the number of letters by the count for that size word. Add these all up, and divide by the total number of words.

Shakespeare mean=________ Harry Potter mean=_______

5. Find the Mean Absolute Deviation of the Shakespeare data.

a) Find the distance that each value is away from the mean.

b) Total these values

c) Divide by the number of values counted

6. Find the Mean Absolute Deviation of the Harry Potter data.

a) Find the distance that each value is away from the mean.

b) Total these values

c) Divide by the number of values counted





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7. Find the five number summary using the data you found for Shakespeare and Harry Potter.

8. Create a box plot for the word counts you found for Shakespeare and Harry Potter:

9. Looking at the box plots and the mean, would you agree that there has been a Dumbing Down of Americas youth over the past decades? Support your answer with numerical data you found in steps #4-8.

10. Do you believe the comparison above could help you conclude the word counts for ALL Harry Potter and Shakespeare Literature? Why or why not?





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Got Friends? Is there a difference between the number of programmed numbers in the number of programmed telephone numbers in girls cell phones and the number in boys cell phones?

1. Do you think there is a difference? Why or why not? When Mrs. Causey, an AP Statistics teacher at Olviedo High School in Seminole County, Florida, polled her students, she got the following data: Males 5 20 26 40 46 47 49 50 51 51 56 57 60 61 68 71 72 72 73 74 75 82 82 84 86 97 100 100 104 104 106 124 171 205 207 232 360 Females 20 46 50 58 62 65 70 72 72 80 86 87 88 90 92 94 94 109 112 114 116 122 125 129 137 137 138 142 142 149 163 170 186 199 204 249

2. Find the 5 number summary for both boys and girls: Males Females Minimum Minimum Q1 Q1 Median Median Q3 Q3 Maximum Maximum

3. Create side-by-side (stacked) box plots for the two sets of data above.





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4. Compare the box plots above. Do you notice a difference in their shape, center, and spread?

5. It is important that you have data integrity. For example, it is important that data be reported accurately and truthfully. Do you think that this is the case here? Do you see any suspicious observations?

6. Can you think of any reason someone might make up a response or stretch the truth in reporting his or her number of programmed telephone numbers?

7. If you DO see a difference between the two groups, can you suggest a possible reason for that difference?

8. Do you think that a study of cell phone programmed numbers for a 7th grade math class would yield similar results? Why or why not?





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Travel Times to Work How long does it take you to get from home to school? Here are the travel times from home to work in minutes for 15 workers in North Carolina, chosen at random by the Census Bureau:

5 10 10 10 10 12 15 20 20 25 30 30 40 40 60

1. Find the mean of the North Carolina travel times. Here are the travel times in minutes of 19 randomly chosen New York workers: 5 10 15 15 15 15 20 20 20 20 30 30 40 40 45 60 60 65 85

2. Find the mean of the New York travel times.

3. Compare the two means. Which state has a longer travel time when comparing the means?

4. Find the median of the North Carolina and New York travel times.

Median of North Carolina = ______ Median of New York= ______

5. Compare the medians. Which state has a longer travel time when comparing the medians?

6. Looking at the New York travel times, which number(s) affect the mean, but not the median?





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7. Find the Mean Absolute Deviation (MAD) of the North Carolina and New York travel

times. What does the MAD tell you about each set of data? North Carolina: a) Find the distance that each value is away from the mean.

b) Total these values.

c) Divide by the total numbers of values in the set New York: a) Find the distance that each value is away from the mean b) Total these values.

c) Divide by the total numbers of values in the set

8. Overall, which measure of center best describes travel time to work, the mean or the median? Why?

OVERVIEW1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.ENDURING UNDERSTANDINGSCONCEPTS AND SKILLS TO MAINTAINShakespeare vs. Harry PotterTravel Times to Work

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Unit 4 Frameworks - Student Edition