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Instructional Plan Template

Mathematics Governor’s Institute 2004

(Download as Microsoft Word documents: group template.doc, Lesson Intro.doc, armspan activity.doc, basketball.doc, LP-PAwealth.doc, Answer key Pa wealth.doc, Review worksheet.doc, Vocabulary.doc)

Names of group members: Tracey Washington, Jon Rush, Carl Ackerman

Topic/Theme: Measures of Central tendencies: Mean, Median, and Mode

Level: Data Analysis

Time Element: 2 block or 3 periods

NCTM Standards Addressed: Measurement and Data Analysis

PA Math Standards Addressed: 2.3, 2.6

Math Assessment Anchors Addressed: M11.E.1 and M11.E.2

Reading Assessment Anchors Addressed: R11.A.2

Objectives: Students will explore the center of distributions with mean, median, and mode through experimentation and data analysis.

Instructional Strategies and Plan (include strategies used to help different types of learners, i.e. auditory, visual, etc):

Review related statistical vocabulary.
Model and instruct how the students should perform the experiment.
Students will perform the experiment and record the data using graph paper.
Analyze results.
Generalize results.

Materials/Resources:

Tape measures or meter sticks.
Graph paper
Worksheets.
Graphing calculator.

Interdisciplinary Connections:

· Reading: Identification of vocabulary. Use the Frayer Model for any words the students do not understand.

· Technology: Create dot plot on graphing calculator

· Other: Social studies- study income/poverty distribution

Assessment Strategies:

· Formative Evaluation (checking student understanding during the lesson):

o Teacher will observe and question students as they conduct the experiment and the analysis

· Summative Evaluation (How will it be determined that the objectives were achieved?):

o Teacher will conduct a post experiment discussion.

o Teacher will evaluate the students’ worksheets

Correctives/Remediation: Students will analyze data from the following activity in order to choose an appropriate measure of central tendency: “Evaluating the L.A. Lakers 2004 team”

Extensions/Enrichment: Students will enter data into graphing calculators to evaluate the median income within the counties of Pennsylvania: “Where is the Wealth in Our Commonwealth”

Special Accommodations (special needs students)

· Description of the Special Needs student selected:

o Severe hearing loss in both ears.

o Language comprehension and expressive skills are 4 years below grade level

o Reading is also 4 years below grade level.

o Basic Math computation skills are above grade level.

o Uses an assistive listening device, but still can not comprehend long verbal explanations.

o She does not use sign language.

o Learns best using visual aids

· Accommodations to use with this student:

o Hand out vocabulary lists before the lesson and allow students to read in advance.

o Seat student close to the front so that he/she can see the teacher’s mouth. Teacher needs to restrict his/her own mobility so that the student with a hearing impairment is in good visual proximity of the teacher at all times.

o Provide a buddy to check with the student to be sure directions are understood or to work on an example of a problem with the student.

o Use an overhead projector to outline points, which you are making in classroom presentations.

o Abbreviate verbal instructions. Speak clearly and avoid long-winded verbalizations when possible.

Lesson Plan: Measures of Central Tendency

Introduction: Main Points on the Measure of Central Tendency

In class, we have been exploring distributions of data, various ways of representing them graphically, and describing their key features verbally. It is often handy to have a single numerical measure to summarize a certain aspect of a distribution. In this topic you will encounter some of the more common measures of the center of a distribution, investigate their properties, apply them to some genuine data, and expose some of their limitations.

We will consider three commonly used measures of the center of a distribution:

· The mean is the ordinary arithmetic average, found by adding up the values of the observations and dividing by the number of observations. The mean can be thought of as the balance point of the distribution.

· The median is the middle observation (once they are arranged in order).

· The mode is the most common value, i.e., the one that occurs most frequently.

· The measure of central tendency is the measure of the location of the middle or the center of a distribution.

Our lesson will begin with an investigation in which students collect data on the length of their classmates’ arm spans. We will build upon the knowledge and skills developed in our arm span lesson by evaluating the points scored by different players on the LA Lakers 2004 team. Finally, our enrichment lesson titled Where is the Wealth in Our Commonwealth, will allow students to examine the Median Household Income in different counties throughout Pennsylvania as reported in the 2000 U.S. Census.[1]

Name: Date:

Arm Span Analysis

In this activity you will measure and analysis the arm span of your classmates and report your findings in order to find the individual who would best represent the center value in the class.

Directions:

Take the measurements, in inches, of each of your classmates’ arm spans in your group.
Record your group’s data on the table below.
Collect and record the data from all the other groups in the class.
Create a dotplot of the data on the graph below.
Answer the questions below.

Answer the following:

What number might you choose if you were asked to select a single number to represent the center of this distribution? Briefly explain how you arrive at this choice.

Calculate the mean of these measurements. Mark this value on the dotplot above with an “x”.

Calculate the median of these measurements. Mark this value on the dotplot above with an “o”.

Find the mode of these measurements. Mark this value on the dotplot above with a “y”.

Which central tendency would you choose to best to represent the center of the distribution of the data? Explain clearly.

Name ________________ Date:

Evaluating the L.A. Lakers 2004 team

Mean ,Median, Mode

On June 8, 2004 the Los Angeles Lakers played the Detroit Pistons in a playoff basketball game. The L.A. Lakers won the game in overtime 99 – 91. During the game, 9 different Laker players were in the game for at least 5 minutes.

1. For the Laker’s team, what was the average (mean) number of points scored per player?

2. In your opinion is this a fair assessment of what the typical player actually scored in the game? Explain.

Here are the stats for those Lakers who played in the game.

3. List the points scored in order from least to greatest.

4. Calculate the following measurements.

Median:

Lower quartile (Q1):

Upper quartile (Q3):

5. Create a box and whiskers graph from the above information

6. What is the mode of the above data

7. Relist your calculations for the Mean, Median, and Mode.

8. If someone asked you what a typical player on the Laker’s team would score,

argue which measure of center (mean, median, mode) would best answer the

question.

9. Extension Question:

The 2004 L.A. Lakers lost the NBA championship series to the Detroit Pistons. The data presented here is limited, but can you make any guesses or statements of why the Lakers might have lost the series?

Laker’s Activity Answer Key

1. mean = 99/9 =11 points per player

2. student answers will vary

3. 0,2,5,7,7,7,9,29,33

4. median=7, lower quartile = 3.5, upper quartile = 19

6. mode = 7

7. mean=11, median = 7, mode = 7

8. Best answers are median and mode since that is where most of the players are

grouped.

9. Various answers possible: Most of the scoring was by Kobe Bryant and S.

O’Neal. However, the median player on the team only scored 7 points. This may mean that the team was not very balanced.

Enrichment Lesson: Where is the wealth in our Commonwealth?

By Carl Ackerman

In this lesson, students will examine the distribution of the median household income among the different counties of Pennsylvania. In the book, Wealth and Our Commonwealth, by William H. Gates and Chuck Collins they write,

The essence of the American experiment is our collective rejection of European hereditary aristocracy and grotesque inequalities of wealth. When Alexis de Tocqueville visited the United States in the mid-nineteenth century, he noted that equality of condition permeated the American spirit: "The American experiment presupposes a rejection of inherited privilege." In the words of novelist John Dos Passos, "rejection of Europe is what America is all about."

This unique disposition of early Americans led to the creation of our great nation based on the spirit of equality. However, does this spirit still persist in our society today? What tools can statisticians use to determine how wealth is distributed in American? Students will analyze the distribution of wealth in Pennsylvania by examining the 2000 U.S. Census data. What can we learn from this data? What problems do we encounter in our analysis? How can we overcome these problems?

Using the data from the 2000 U.S. Census Bureau, create a box and whiskers plot of the median household incomes in the counties of PA. Sketch your box and whiskers plot below.

(Hint: To create a Box - and - Whisker Plot using the TI - 83 Plus calculator, go to http://www.wscc.cc.tn.us/math/jlaprise/Calculator/statplot/statplot.html for a tutorial.)

2. For the median household incomes in the counties of PA, what was the average (mean) household income?

3. In your opinion is this a fair assessment of what the typical household in PA earns each year? Explain your answer.

4. Based on your box and whiskers plot, record the following measurements:

A. Lower extreme:

B. Lower quartile:

C. Median:

D. Upper quartile:

E. Upper extreme:

5. What is the mode of the above data?

6. Re-list your calculations for the Mean, Median, and Mode.

· Mean

· Median

· Mode

7. If someone asked you about the distribution of wealth among the different counties in the state of Pennsylvania, what do you think would be a more accurate measure of center (mean, median, mode) to use for your answer? Why?

8. What are some problems that occur when you examine the averages of averages? Why does this happen? Is this a major problem, when you are looking for an accurate picture of the distribution of wealth in PA?

9. What other sources of data could you use to find a more accurate picture of the distribution of wealth in PA?

10. What further questions does analyzing the distribution of wealth raise?

Read the editorial from the Beaver County Times. Use the Pennsylvania County Selection Map from the U.S. Census and sketch the line described in the article. What does this line show us? Based on your research, write a position paper supporting or refuting the argument posited in the article. Think about the different measures of central tendency. Your argument should be at least five paragraphs typed.

2000 U.S. Census data for Counties in Pennsylvania

Pennsylvania Counties

Median household income, 1999[2]

Pennsylvania Counties

Median household income, 1999

1. Adams

$42,704

40. Luzerne

$33,771

2. Allegheny

$38,329

41. Lycoming

$34,016

3. Armstrong

$31,557

42. Mc Kean

$33,040

4. Beaver

$36,995

43. Mercer

$34,666

5. Bedford

$32,731

44. Mifflin

$32,175

6. Berks

$44,714

45. Monroe

$46,257

7. Blair

$32,861

46. Montgomery

$60,829

8. Bradford

$35,038

47. Montour

$38,075

9. Bucks

$59,727

48. Northampton

$45,234

10. Butler

$42,308

49. Northumberland

$31,314

11. Cambria

$30,179

50. Perry

$41,909

12. Cameron

$32,212

51. Philadelphia

$30,746

13. Carbon

$35,113

52. Pike

$44,608

14. Centre

$36,165

53. Potter

$32,253

15. Chester

$65,295

54. Schuylkill

$32,699

16. Clarion

$30,770

55. Snyder

$35,981

17. Clearfield

$31,357

56. Somerset

$30,911

18. Clinton

$31,064

57. Sullivan

$30,279

19. Columbia

$34,094

58. Susquehanna

$33,622

20. Crawford

$33,560

59. Tioga

$32,020

21. Cumberland

$46,707

60. Union

$40,336

22. Dauphin

$41,507

61. Venango

$32,257

23. Delaware

$50,092

62. Warren

$36,083

24. Elk

$37,550

63. Washington

$37,607

25. Erie

$36,627

64. Wayne

$34,082

26. Fayette

$27,451

65. Westmoreland

$37,106

27. Forest

$27,581

66. Wyoming

$36,365

28. Franklin

$40,476

67. York

$45,268

29. Fulton

$34,882

30. Greene

$30,352

31. Huntingdon

$33,313

32. Indiana

$30,233

33. Jefferson

$31,722

34. Juniata

$34,698

35. Lackawanna

$34,438

36. Lancaster

$45,507

37. Lawrence

$33,152

38. Lebanon

$40,838

39. Lehigh

$43,449

[1]http://quickfacts.census.gov/qfd/states/42/42133.html (U.S. Census Bureau Statistics)

Beaver County Times

Public schools in Pennsylvania on the brink of being divided by wealth and geography

Pennsylvania is becoming two states, one rich and one poor. It is a place of suburban wealth and urban and rural poverty. The impact this income disparity has on public education is profound.

The latest U.S. Census Bureau statistics present a startling contrast in income distribution in Pennsylvania, with the state's wealth being overwhelmingly concentrated in the southeastern corner of the state.

The Philadelphia Inquirer reports that the 10 counties in the commonwealth that have the highest median household income (from $44,609 to $65,295) in Pennsylvania form an arc that starts in Monroe County north of Philadelphia and sweeps south and west to Cumberland County.

It doesn't spread out much more in the next grouping - the counties that were above the median state income of $40,106 (and up to $44,608). Of the nine counties in that category, only two - Butler and Union - weren't contiguous to the high-median counties.

Get a map of Pennsylvania and draw a line from the western edge of south-central Franklin County to the northern tip of northeastern Pike County. Basically, that's where the money is. Seventeen of the commonwealth's 19 richest counties are there.

The other counties - 48 in all - are below the state's median income.

The tilt in income toward the southeast has enormous consequences on the quality of public education. Because the state has steadily cut back on its share of funding in Pennsylvania - it's supposed to provide half but is only coming through with a little more than one-third - local property owners have been left to pick up the tab.

Wealthier school districts can dip into local revenues to make up for the state shortfall. Poor districts don't have the option as easily available to them. As a result, Pennsylvania has some school districts spending more than $13,000 per pupil while others are spending less than $6,000.

People like to argue that money doesn't affect education. But it does directly and indirectly. Directly, children who attend poor school districts don't have access to the educational opportunities that their counterparts in wealthier districts do. On the indirect side, family income is a good predictor of how children will do in school.

Our lawmakers must rectify this inequity by modernizing the way in which public education is funded. The present system, which relies heavily on local property taxes, virtually guarantees the disparity between the haves and have-nots will grow.

If all of Pennsylvania is to grow and prosper, it must have an education system that gives every child in the commonwealth a chance to grow and prosper. Sadly, many children are being denied that opportunity right now - and more will join them if this disparity in education and income is not addressed.

©Beaver County Times/Allegheny Times 2002 6/05/2002 Opinions

Answer Key

Enrichment Lesson: Where is the wealth in our Commonwealth?

By Carl Ackerman

Using the data from the 2000 U.S. Census Bureau, create a box and whiskers plot of the median household incomes in the counties of PA.

(Hint: To create a Box - and - Whisker Plot using the TI - 83 Plus calculator, go to http://www.wscc.cc.tn.us/math/jlaprise/Calculator/statplot/statplot.html for a tutorial.)

2. For the median household incomes in the counties of PA, what was the average (mean) household income?

37,176.97015

3. In your opinion is this a fair assessment of what the typical household in PA earns each year? Explain your answer.

Answers may vary

4. Based on your box and whiskers plot, record the following measurements:

A. Lower extreme: 27,451

B. Lower quartile: 32,212

C. Median: 34,698

D. Upper quartile: 40,838

E. Upper extreme: 65,295

5. What is the mode of the above data?

There is no mode because none of the data is exactly the same.

6. Re-list your calculations for the Mean, Median, and Mode.

· Mean = 37,176.97015

· Median = 34,698

· Mode = none

Answers will vary. The median is probably the most accurate measurement.

Answers will vary. The average of an average throws off the calculation and compounds the estimate of the original average.

9. What other sources of data could you use to find a more accurate picture of the distribution of wealth in PA?

Median Household Income for the State of PA, Tax returns filed, IRS statistics, …

10. What further questions does analyzing the distribution of wealth raise?

Why are their different median household incomes? What are the major jobs in the different counties? In which counties are the major cities located? What are the populations of the different counties? ….

2000 U.S. Census data for Counties in Pennsylvania

Pennsylvania Counties

Median household income, 1999[3]

Pennsylvania Counties

Median household income, 1999

1. Adams

$42,704

40. Luzerne

$33,771

2. Allegheny

$38,329

41. Lycoming

$34,016

3. Armstrong

$31,557

42. Mc Kean

$33,040

4. Beaver

$36,995

43. Mercer

$34,666

5. Bedford

$32,731

44. Mifflin

$32,175

6. Berks

$44,714

45. Monroe

$46,257

7. Blair

$32,861

46. Montgomery

$60,829

8. Bradford

$35,038

47. Montour

$38,075

9. Bucks

$59,727

48. Northampton

$45,234

10. Butler

$42,308

49. Northumberland

$31,314

11. Cambria

$30,179

50. Perry

$41,909

12. Cameron

$32,212

51. Philadelphia

$30,746

13. Carbon

$35,113

52. Pike

$44,608

14. Centre

$36,165

53. Potter

$32,253

15. Chester

$65,295

54. Schuylkill

$32,699

16. Clarion

$30,770

55. Snyder

$35,981

17. Clearfield

$31,357

56. Somerset

$30,911

18. Clinton

$31,064

57. Sullivan

$30,279

19. Columbia

$34,094

58. Susquehanna

$33,622

20. Crawford

$33,560

59. Tioga

$32,020

21. Cumberland

$46,707

60. Union

$40,336

22. Dauphin

$41,507

61. Venango

$32,257

23. Delaware

$50,092

62. Warren

$36,083

24. Elk

$37,550

63. Washington

$37,607

25. Erie

$36,627

64. Wayne

$34,082

26. Fayette

$27,451

65. Westmoreland

$37,106

27. Forest

$27,581

66. Wyoming

$36,365

28. Franklin

$40,476

67. York

$45,268

29. Fulton

$34,882

30. Greene

$30,352

31. Huntingdon

$33,313

32. Indiana

$30,233

33. Jefferson

$31,722

34. Juniata

$34,698

35. Lackawanna

$34,438

36. Lancaster

$45,507

37. Lawrence

$33,152

38. Lebanon

$40,838

39. Lehigh

$43,449

Beaver County Times

Public schools in Pennsylvania on the brink of being divided by wealth and geography

Pennsylvania is becoming two states, one rich and one poor. It is a place of suburban wealth and urban and rural poverty. The impact this income disparity has on public education is profound.

The other counties - 48 in all - are below the state's median income.

Name: Date:

Review and Preteach for Central Tendency.

Use the following data to answer the questions below:

11, 12, 14, 18, 20, 25, 35, 18, 41, 18

List the above data items in order.

What is the meaning of mean, median, and mode?

Find the following:

Mean

Median

Mode

What is Lower and Upper Quartile?

Find the following:

Lower Quartile

Upper Quartile

Draw a dotplot and a whisker plot of the data.

Vocabulary

Mean The sum of the values divided by

the number of values

Median The middle value when arranged from smallest to largest

Mode The value which occurs most frequently

Lower extreme The smallest value

Upper Extreme The largest value

Lower Quartile The value halfway between the lower extreme and the median

Upper Quartile The value halfway between the upper extreme and the median

Measures of Central Measures of the location of the middle

Tendency or the center of a distribution

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[1] Adapted from http://www.public.asu.edu and Rossman – Chance Workshop Statistics.