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Welcome & Overview

- Examine Next Generation Sunshine State Benchmarks related to Measures of Central Tendency
- Administer Pretest
- Introduce the module objectives
- Developmental Activities: Measures of Central Tendency – Mean, Mode, Median

Vocabulary development, problem solving, technology, error analysis/misconceptions

- Review content: Journal entry

Next Generation Sunshine state Standards

- MA.6.S.6.1: Determine the measures of central tendency (mean, median, mode) for a given set of data.
- MA.6.S.6.2: Select and analyze the measures of central tendency to represent, describe, analyze, and/or summarize a data set for the purposes of answering questions appropriately
- MA.7.S.6.1: Evaluate the reasonableness of a sample to determine the appropriateness of generalizations made about the population.
- MA.8.S.3.2: Determine and describe how changes in data values impact measures of central tendency.

Objectives

- Participants will:
- Review the vocabulary related to measures of central tendency (MA.6.S.6.1)
- Determine the measures of central tendency (mean, median, mode) for a given set of data. (MA.6.S.6.1)
- Identify problem solving strategies that may be used to develop understanding and familiarity with measures of central tendency. (MA.6.S.6.2)
- Explore how changes in data values and extreme values affect measures of central tendency. (MA.7.S.6.1; MA.8.S.3.2)
- Explore the use of technology and other instructional strategies that may be used to facilitate student understanding of measures of central tendency. (MA.6.S.6.2)
- Develop the ability to communicate mathematically through journal writing and problem solving. (MA.6.S.6.2)

?Measures of Central Tendency?

Brainstorming

- What are measures of

central tendency?

- How are they used in everyday situations?

Measure of central tendency are used to describe the typical member of a population.

- Depending on the type of data, typical could have a variety of “best” meanings.
- Real life examples:
- Most children in the 8th grade are 13 yrs old.
- The median annual family income is $39,600
- Ft. Lauderdale’s average rainfall for the month of July is 6.6 inches.

3 Measures of Central Tendency

- Mean – the arithmetic average. This is used for continuous data.
- Median – a value that splits the data into two halves, that is, one half of the data is smaller than that number, the other half larger. May be used for continuous or ordinal data.
- Mode – this is the category that has the most data. As the description implies it is used for categorical data.

Developing Vocabulary Power

Activity

Describe and complete an activity that may be used to build student vocabulary power.

Terms: central tendency, mean, median, mode, variability, range, set of data, frequency table, numerical data, categorical data, outliers, line plot, circle graph, continuous data,

categorical data, common, middle,

most often

Access the eglossary@glencoe.com

Using manipulative materials

“Manipulating the physical model not only helps [children] understand the formula but also promotes retention.”

“Simply being able to state the algorithm for finding these statistics is not enough. To support the development of data sense, each of these should be developed meaningfully through concrete activities before introducing computation.

(Reys, et. al. 2009; p. 395)

Models for finding the median of an odd/even amount of data in a given set

Find the median of 2, 3, 4, 2, 6.

Participants will use a strip of grid paper that has exactly as many boxes as data values. Have them place each ordered data value into a box. Fold the strip in half. The median is the fold.

Using Manipulative materials Interlocking cubes

- Arrange interlocking/Unifix cubes together in lengths of 3, 6, 6, and 9.
- Describe how you can use the cubes to find the mean, mode, and median.
- Suppose you introduce another length of 10 cubes. Is there any change in

i) the mean,

ii) the median,

iii) the mode?

Finding the mean lengths: Using adding machine tape

- Activity: Return test scores to students on pieces of adding machine tape. The length of each strip is determined by the score (88 cm, and 64 cm). Tape the 2 strips of paper together (add). Write out the addition part on the back - 88, 8…

Fold this in half (divide by 2).

- Try doing it with 3 pieces of strips. Does it work? Explain.

Visualizing the Mode

- Category 4 has the highest bar.
- Category 4 is the mode.
- The mode is the class, not the frequency.

Mean Median & Mode

Purpose: Reinforce concepts through songs.

Click in the link

http://www.youtube.com/watch?v=uydzT_WiRz4

- Would you use this tool in the classroom? Why/why not? Explain.

Using poems, rhymes & Mnemonics

Cheers: Mean, Median, and Mode

Mean (Say in really mean voice and face throughout!)Add all the numbers (Have hand go from waist to neck in increments.)And Divide! (Have same hand slice across the neck!)

Median …MiddleOrder numbers least to greatest (motion hand left to right)Find the middle. (move both hands to middle and clap)

When I say mode you say MostMode….MostMode….Most!Mode is the number that appears most oftenMode…Most oftenMode…Most often!

Understanding The Mean

- The mean is located between the extreme values.
- The mean is influenced by values other than the mean.
- The mean does not necessarily equal one of the values that was summed.
- The mean can be a fraction.
- When you calculate the mean, a value of 0, if it appears, must be taken into account.
- The mean value is representative of the values that were averaged.

Calculating the Mean (Arithmetic Average)

- To find the mean:

add all of the values, then divide by the number of values.

- A student’s score on 4 math tests are: 8, 5, 9, 6. Calculate the mean score.

- Sum of all values

8 + 5 + 9 + 6 = 28

- Number of values = 4
- Mean = 28 ÷4 = 7

Mean score on the math test is 7

Check! 7

8, 5, 9, and 6 or 5, 6, 8, 9

The Median

- The median is a number chosen so that half of the values in the data set are smaller than that number, and the other half are larger.
- To find the median
- List the numbers in ascending order
- If there is a number in the middle (odd number of values) that is the median
- If there is not a middle number (even number of values) take the two in the middle, their average is the median

Finding the Median

- List values in order of size

5, 6, 8, 9

6 8

- Find the average of the selected values.

(6 + 8) ÷ 2 = 7

Median score on the math test is 7

Check! 7

8, 5, 9, and 6 or 5, 6, 8, 9

- To find the median:

list all of the values, in order of size. Select middle value(s)

- A student’s score on 4 math tests are: 8, 5, 9, 6. Calculate the median score.

M E D I A N

Finding Mean, Median and Mode for data sets

- Figure out the mean, median, mode, range, and outlier using a group/card activity Discuss the various measures.
- Materials: 4 sets of activity cards (Nine 4 x 4 Index cards 1, 5, 6, 7, 8, 9, 10, 11, 12, 18) use either 1 or 18 as an outlier; calculators.

The Mode

- The mode is simply the category or value which occurs the most in a data set.
- If a category has more than the others, it is a mode.
- Generally speaking we do not consider more than two modes in a data set. (i.e. Bi-modal)
- No clear guideline exists for deciding how many more entries a category must have than the others to constitute a mode.

outlier effects on measures of central tendency

- Give nine different students a card and have them come to the front of the room and hold their card facing the rest of the class.

Give the remaining seated students a calculator.Follow directions on activity sheet.

Discuss how the outlier affects the measures of central tendency.

Problem Solving

Using different Strategies!

Andy’s results on three tests are: 68, 78, and 88.

1. Find the mean and median score.

2. Explain why the mode is of little value.

3. What score would be needed on the next test to get an average of 81.

4. Describe two different ways you could determine this score.

Investigatingdescriptive statistics

- Using technology

What Does It Mean to Understand the Mean?

Plop It!

- Participants will experiment with the concepts of mean, median, and mode by using a bar graph.
- Participants will change parameters and discover patterns related to mean and median. They can choose their own focus of measure, their own quantity, and their own units.

URL: www.shodor.org/interactivate/activities/plot/what.hmtl

Using Spreadsheets

MA.8.S.3.2: Determine and describe how changes in data values impact measures of central tendency.

- Purpose: use technology to calculate mean, mode, and median of a set of data.
- Task: Make a spreadsheet for a given data set and find the measures of central tendency. (See slide 26 for data.)
- Open a new spreadsheet. Create four columns labeled DATA, MEAN, MEDIAN, and MODE.
- Enter each allowance amount in the DATA column.
- In cell B2, enter =Average(A2:A11). In cell C2, enter =Median(A2:A11). In cell D2, enter =Mode(A2:A11). Each of these will find the mean, mode and median of the data set.
- Analyze the results.

Mrs. Jensen’s 7th grade class was surveyed about how much allowance each student receives each week. The results are shown in the table.

Use this information to make a spreadsheet for the data, and find the mean, median, and mode.

Analyze the results.

- What data value is an extreme for the set? Explain your reasoning.
- Describe how the measures of central tendency would change if the extreme value was not included in the data set.

changes in data values impact on measures of central tendency

- Example: Mrs. Donohue has told her students that she will remove the lowest exam score for each student at the end of the grading period. Sara received grades of 43, 78, 84, 85, 88, 78, and 90 on her exams. What will be the different between the mean, median, and mode of her original grades and the mean, median, and mode of her five grades after Mrs. Donohue removes one grade?

Objectives

- Examine student misconceptions in statistical thinking.
- Use researched information on statistical thinking to create activities geared at helping students develop better understanding of averages.
- Connect Math and Language Arts by providing activities for students to develop their vocabulary skills while simultaneously developing conceptual understandings.
- Construct and analyze histograms, stem-and-leaf plots, and circle graphs.
- Make conjectures about possible relationships from data sets.
- Identify teacher-specific instructional tools and methods for graphically displaying data.
- Explore web-based educational resources designed to reinforce learning of graphical displays and measures of central tendency.

Benchmarks

- MA.6.S.6.2: Select and analyze the measures of central tendency or variability to represent, describe, analyze, and/or summarize a data set for the purposes of answering questions appropriately
- MA.7.S.6.2: Construct and analyze histograms, stem-and-leaf plots, and circle graphs.
- MA.8.S.3.2: Determine and describe how changes in data values impact measures of central tendency.

Misconceptions

“Many middle graders are able to calculate averages but their understanding of the concept of average is shallow.” (Reys, et al., 2009).

Participants will identify misconceptions students might have about statistics.

Connecting Research and Practice

Purpose: to review measures of central tendency, and provide an opportunity for participants to use research findings to inform their instructional practices.

- Task: Participants will read 1 research article on Measures of Central Tendency.
- Participants will summarize the article and discuss the main findings as they apply to the teaching and learning of Measures of Central Tendency.

Difficulties in interpretingthe Mean

“Akira read from a book on Monday, Tuesday and Wednesday. He read an average of 10 pages per day. Circle whether each of the following is possible or not possible.”

- Less than 40% answered all 4 choices correctly.

“Many middle graders are able to calculate averages but of understanding of the concept of average is shallow.” (Reys, et al., 2009, p. 396).

Remediation Activity

Work with a partner to develop an activity to help students better understand the concept of average.

Present your activity to the class with a rationale for selecting this activity.

Misconception about the median

- Problems involving finding the median

Problem: When finding the median of an even-numbered set of data, some students use the mean the data instead of the mean of the two middle numbers.

Describe an activity that may be used to help students overcome this problem.

Focus question/Activity

- How are statistical displays helpful to us in our everyday lives?

Make a list of the different types of graphs that you know. Select one of the graphs, and write about a situation in which you will use that graph to display data.

MISLEADING GRAPHS

Participants compare two Bar graphs with the same data, and discuss why one may be misleading.

Error Analysis

- To look at a student’s sleeping pattern a student made a Line Plot of the number of hours he slept each night for one week. Describe the student’s error and tell which display he should have made.

Using TechnologyConstruct & analyze histograms and circle graphs

Group Activity

- Assign groups of no more than 4per group. Participants use the computer to get a feel for the technology and discuss the usability and benefits of including such tools in the 6-8 math curriculum.
- Circle grapher, and histogram tool

http://illuminations.nctm.org/ActivityDetail.aspx?

Choosing an appropriate displayWhy Me!

Which graph would you use to compare the number of red folders sold by two stores in one week?”

Converting a Bar graphinto a Circle graph

- Instruction: Copy a Bar graph. Write a label on each bar. Cut our each bar from the graph. Tape the ends together (no overlaps) to form a circle.
- Place the circle on a sheet of paper. Trace the circle. Mark where each bar begins and ends around the circle.
- Mark the center of the traced circle. Draw in a radius from each of the lines marked on the circle.
- Color the sections of the circle. Label each section, and title your graph.

ACT IT OUT: STEM AND LEAF PLOTS

- Problem Solving Strategy: Act it out

Given the data set: 7, 51, 25, 47, 42, 55, 50, 26, 44, 55, 26, 33, 39, participants will:

Copy the numbers unto individual index card.

Sort the cards into piles based on place value. (Stem value)

Note what they have in common

Cut one of the stem, place it on a sheet of paper or the table. Cut the remaining leaves with this stem. Add the leaves to the leaf section of the plot.

Repeat for each of the piles.

Discuss the plot by examining the lowest and highest scores, the lengths of each leaf, gaps, tapers, and the median and modal values.

STEM and LEAF Plot

Activity:

- Create the plot from the information presented below.

An example of a stem-and-leaf plot for the data set (34, 30, 38, 42, 67, 68, 68, 56, 54, 34, 82, and 85) is as follows: Legend: 3|234

- Discuss what is the
- median of the data set?
- mode of the data set?

Review Exercise

- What does the data represent?
- What type of central tendency would you use to represent it? Why? Include each of the measures in your justification.
- Sketch the type of graph you would make to represent the data set.
- How can the data representation influence conclusions?

Module Objectives

Participants will:

- Discuss what is meant by the term “measures of variation”
- Develop vocabulary activities to reinforce understanding of concepts of measures of variation
- Examine the 5 point summary or key (Quartiles)
- Select, organize, construct, and analyze Box and Whisper plots (single and double)
- Select, organize, and construct scatter plots and lines of best fit for given data sets
- Examine misconceptions student have about scatter plots
- Make conjectures about possible relationships in the data
- Use technology to create measures of variation

Benchmarks

- MA.6.S.6.1: Determine the measures of central tendency (mean, median, mode) and variability (range) for a given set of data.
- MA.6.S.6.2: Select and analyze the measures of central tendency or variability to represent, describe, analyze, and/or summarize a data set for the purposes of answering questions appropriately
- MA.7.S.6.1: Evaluate the reasonableness of a sample to determine the appropriateness of generalizations made about the population.
- MA.8.S.3.1: Select, organize and construct data displays, including box and whisker plots, scatter plots, and lines of best fit to convey information and make conjectures about possible relationships.
- MA.8.S.3.2: Determine and describe how changes in data values impact measures of central tendency.

Vocabulary Power

- Participants create a graphic organizer to identify the five key measures of variation.

Using a Word Web

- Participants will use a Word Web template to fill in the vocabulary words. (range, lower quartile, outliers, upper quartile, quartiles, interquartile, measures of variation.

Examine web and ask:

- Why are the range and outlier separated from the quartiles?

Measure of Variation

- Used to describe how much the data is spread out.
- The variation in a data set is easily examined in a line plot or graph
- The range is a simple measure of variability. It tells the difference between the highest and lowest values in a given set of data.
- Why measure spread?

Using the Range

- Complete the table. Can you use the mean, mode, or median to compare the two players’ performance? Why/why not?

Range & Quartiles

Mean, the median and the mode are measures of the central tendency of a set of data.

However, such measurements cannot tell us the spread or variation of the data.

Dispersion is the statistical name for the spread or variability of data.

The central quartile is the median

Median

Lower Quartile Upper Quartile

2 3 5 7 10 11 15 16

Interquartile Range

find the lower median. This is the middle of the lower six numbers. The exact centre is half-way between 8 and 9 ... which would be 8.5Now find the upper median. This is the middle of the upper six numbers. The exact centre is half-way between 14 and 14 ... which must be 14

- Rewrite the data in order, from smallest length to largest:
- Find the median of all the numbers. Notice that since there are 13 numbers, the middle one will be the seventh number:
- Find the lower

quartile. This is the

middle of the lower

six numbers. The exact

center is half-way

between 8 and 9 ... (8.5)

- Find the upper quartile.

This is the middle of

the upper six numbers.

The exact centre is

half-way between 14 and

14 ... which must be 14

Effects on the Dispersion with Change in Data

- Removal of a certain from the data.

If the greatest or the least value (assuming both are unique) in a data set is removed, then

- the range will decrease;
- the inter-quartile range may increase or decrease.

If a zero value is inserted in a positive data set, then

- the range will increase;
- the inter-quartile range may increase of decrease

Box and Whisker plots

A box and whisker graph is used to display a set of data so that you can easily see where most of the numbers are.

http://nlvm.usu.edu/en/nav/frames_asid_200_g_3_t_5.html

1

- Line Plot: Quartiles

What are the median, lower and upper quartiles?

Median Upper Q

Lower Q

- x xxxxxxx

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Where do the numbers cluster together and where do they spread out? Explain.

Where are the extremes located in relation to the clustered data?

The difference between the two end-points of the line (represented by the highest and lowest marks) is the range.

The length of the box is the inter-quartile range.

A Box-and-Whisker Plot- It illustrates the spread of a set of data. It provides a graphical summary of the set of data by showing the quartiles and the extreme values of the data.

Alfonso's bowling scores are 125, 142, 165, 138, 176, 102, 156, 130, and 142. Make a box-and-whiskers plot of the data. The box and whiskers plot below represents the bowling scores of Anna. Compare the bowling scores of Alfonso and Anna. Who is a better bowler?

Stem and Leaf/Box and Whisker Plot

Investigation: Drops on a Penny: Graphing Data

http://fcit.usf.edu/FCAT8m/penny1/default.htm

- COLLECT THE DATA
- 1. Guess how many drops of water a Heads-Up penny will hold.
- 2. Count how many drops it will actually hold.
- 3. Record your data.
- GRAPH THE DATA
- Create a Stem-and-Leaf plot of the class data.
- ANALYZE THE DATA
- 1. Use the data to calculate the mean, median, mode and range.
- 2. Use this data to create a box and whiskers plot.

Scatter Plot

It is a scattered plotting of points that may or may not seem to follow some sort of trend or pattern.

- What sort of a trend would they follow?

A fitted line (or line of best fit) of course!

The fitted line is one that is closest to every point in the plot.

If two points are a ways a part, the line must compromise and go between them.

Line of Best Fit

- http://illuminations.nctm.org/ActivityDetail.aspx?ID=146
- This activity allows the user to enter a set of data, plot the data on a coordinate grid, and determine the equation for a line of best fit.
- After reading the instruction, click on Exploration and complete the task.
- Try to explain the changes that occurred when data was removed?

Scatter PlotsDescribing Relationship

- Activity: (Distribute Activity Sheet)

In groups of three, examine the following sample scatter plots. Once you have ’thoroughly’ examine them, rank them in order from strongest to weakest relationship between the variables. Finally, briefly explain your selections in the area provided below.

Be ready to present your findings!

Examining Relationships

3 Types of Relationships

Alternative Activity

Draw the graph to show the correlation between shoe size and height.

This is one of the two things scatter plots can be used to find the answer to:

- Is there a correlation between the variables?
- How can we use that correlation to predict values that we haven’t measured?

Alternative Internet Activity

- Participants will research endangered species such as the Florida Panther or the peregrine falcon. They will use technology to create a scatter plot of the population by year, month, or other time period.
- Participants will answer the following questions:
- What trends do you notice in the population of this animal?
- What measures are being taken to help the animal, and when did they began?
- What effect do you think this had on the population of your animal?
- Based on the graph and the measures you researched, what do you predict the population of your animal to be in ten years?

Scatter Plot: Misconceptions

- Students often fail to learn what situations a newly learned representation is appropriate for, preferring to use the first and simplest representations of data they learn, even in situations where those representations are not useful (Hancock, Kaput, and Goldsmith, 1992).
- Students focused solely on their surface features rather than their structural or informational properties (McGatha, Cobb, and McClain, in press)
- Students’ misconception due to variable error or nominalization error.

Thank You & Farewell

- We sincerely thank you for being creative, and active participants in the instructional sessions. It is the Team’s wish that you will incorporate the ideas technology and strategies shared in your professional practices.
- Great Job!!!

- It’s your turn to show what you know and can do!!
- Best regards
- See you in the classroom soon.

REVIEW EXERCISE: Post Test

Reflections & Suggestions

RESOURCES

- Websites
- Journals
- Books

Circle Grapher: http://illuminations.nctm.org/ActivityDetail.aspx?ID=60

- CTAP Region 4. 6th Grade Matix: Statistics, Data Analysis and Probability. Retrieved fromhttp://www.ctap4.net/projects/middle-school-math/matrices/6th-grade-matrix-statistics-data-analysis-and-probability/126-matrices/150-6th-grade-matix-statistics-data-analysis-and-probability.html
- eGlossay. Retrieved from http://www.glencoe.com/apps/eGlossary612/landing.php
- FCAT 8th grade Math: A staff developmental tool. Retrieved from http://fcit.usf.edu/FCAT8m/default.htm
- FCAT example. Retrieved from http://fcit.usf.edu/FCAT8m/penny1/default.htm
- Histogram Tool: http://illuminations.nctm.org/ActivityDetail.aspx?ID=78
- Justify measure of central tendency. Retrieved fromhttp://wiki.bssd.org/index.php/S%26P:_Justify_measure_of_central_tendency
- Lessons. Retrieved from http://teachers.net/lessons/posts/3789.html
- Math Standards. Retrieved from http://www.floridastandards.org/Resources/PublicPreviewResource1347.aspx
- Mean Median & Mode Math Learning Upgrade.Retrieved fromhttp://www.youtube.com/watch?v=uydzT_WiRz4
- Measures of Central Tendency (Middle School). Retrieved fromhttp://www.beaconlearningcenter.com/lessons/99.htm
- Measures of central Tendency. Retrieved from http://croitz.blogspot.com/2009/12/measures-of-central-tendency.html
- National Library of Virtual Manipulatives. Box Plot.Retrieved fromhttp://nlvm.usu.edu/en/nav/frames_asid_200_g_3_t_5.html
- NCTM website: http://illuminations.nctm.org/ActivityDetail.aspx?ID=146
- Plop It! Retrieved from www.shodor.org/interactivate/activities/plot/what.hmtl
- Retrieved fromhttp://www.google.com/search?hl=en&rls=com.microsoft%3Aen-us%3AIE-Address&rlz=1I7ADBF_en&q=What+are+some+of+the+misconceptions+students+have+about+scatterplots&aq=f&aqi=&aql=&oq=
- Scatter Plots: Describing Relationship. Retrieved fromhttp://www.google.com/search?hl=en&rls=com.microsoft%3Aen-us%3AIE-Address&rlz=1I7ADBF_en&q=What+are+some+of+the+misconceptions+students+have+about+scatterplots&aq=f&aqi=&aql=&oq=

Baker, R., Corbett, A., Koedinger, K., & Schneider, M. (2002). A formative evaluation of a tutor for scatterplot generation: evidence on difficulty factors. Retrieved from http://users.wpi.edu/~rsbaker/BCKSAIED2003Final.pdf

- Capraro, M., Kulm, G., & Capraro, R. (2005). Middle grades: Misconceptions in statistical thinking. School Science and Mathematics,105(4),165 – 175.
- Capraro, R., Kulm, G., Hammer, M., & Capraro, R. (n.d.). The origin and persistence of misconception in statistical thinking. Retrieved from http://www.coe.tamu.edu/~mcapraro/mmcs-published%20articles/PMENA-Misconcep.pdf
- Kraus, S. (2010). Monstrous Methods for Teaching Central Tendency Concepts. Teaching Statistics, 32(1), 21–23.
- Russo, L., & Passanante, M. (2001). Statistics fever. Mathematics Teaching in the Middle School, 6(6), 370 – 374.

Glencoe McGraw-Hill. (2011). Florida Math Connects, Course 3 (Teacher ed.). OH: Glencoe McGraw-Hill.

- Glencoe McGraw-Hill. (2011). Florida Math Connects, Course 2 (Teacher ed.). OH: Glencoe McGraw-Hill.
- Glencoe McGraw-Hill. (2011). Florida Math Connects, Course 1 (Teacher ed.). OH: Glencoe McGraw-Hill.
- Larson, R., Boswell, L. (2010). Big Ideas MATH, Florida Teacher ed. PA: Big Ideas Learning.
- Reys, R., Lindquist, M., Lamdin, D., & Smith, N. (2009). Helping children learn mathematics; Ch. 17, p. 409. NJ: John Wiley & Sons, Inc.

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