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Data and Statistics for Middle School

1

Introduction to Statistics

Your PLC has been asked to determine the effects of two different math programs. The programs were assigned to Teacher A and Teacher B. The following table shows the final test scores of the last test in two classes. Which class did better on the test? Which program had better results? Use as many pieces of evidence that you know about to provide evidence for your claim.

Teacher A Teacher B

100 100

95 100

67 98

85 97

45 45

80 46

77 50

77 73

75 65

50 73

100 85

80 94

60 58

70 52

84 57

82 85

90 75

94 80

47 70

60 66


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Statistics in 6-8th Grade Common Core Standards

Statistics and Probability 6.SP Develop understanding of statistical variability. 1. Recognize a statistical question as one that anticipates variability in the data related to the question

and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Summarize and describe distributions. 4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

5. Summarize numerical data sets in relation to their context, such as by:

a. Reporting the number of observations.

b. Describing the nature of the attribute under investigation, including how it was measured and

its units of measurement.

c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.


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Statistics and Probability 7.SP Use random sampling to draw inferences about a population. 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.

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.

Draw informal comparative inferences about two populations. 3. 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. 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.

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.

Investigate chance processes and develop, use, and evaluate probability models. 5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the

likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.


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a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

a. Understand that, just as with simple events, the probability of a compound event is the fraction

of outcomes in the sample space for which the compound event occurs.

b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.

c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

Statistics and Probability 8.SP Investigate patterns of association in bivariate data. 1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of

association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?


5

Answer the following questions in your group: 1. Which standards are completely new to you? 2. Which standards do you think will be particularly difficult for students?


6 Retrieved from http://ime.math.arizona.edu/progressions/ https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf

Progressions

http://ime.math.arizona.edu/progressions/

https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf


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Mathematical Practices


20

American Statistical Association. (2007). Guidelines for assessment and instruction in statistics education (GAISE) report. Alexandria, VA: Author. Retrieved from http://www.amstat.org/education/gaise/GAISEPreK-12_Full.pdf

Framework of Statistical Literacy

The GAISE report published by the American Statistical Association was referenced in Developing Essential Understanding of Statistics: Grades 6-8. Excerpts from the report will be used in this professional development. The full document can be accessed via http://tinyurl.com/k9mqkev or

http://www.amstat.org/education/gaise/GAISEPreK-12_Full.pdf


http://tinyurl.com/k9mqkev


21


Framework of Statistical Literacy




22





23





24 Identifying Statistical Questions; Retrieved from Illustrative Mathematics: https://www.illustrativemathematics.org/content-standards/6/SP/A/1/tasks/703

What is a statistical question?

Which of the following are statistical questions? (A statistical question is one that can be answered by collecting data and where there will be variability in that data.)

a. How many days are in March? b. How old is your dog? c. How old are the dogs on this street? d. What percent of people like watermelons? e. Do you like watermelons? f. How many bricks are in this wall? g. What was the highest temperature today in town?

https://www.illustrativemathematics.org/content-standards/6/SP/A/1/tasks/703


25 Identifying Statistical Questions; Retrieved from Illustrative Mathematics: https://www.illustrativemathematics.org/content-standards/6/SP/A/1/tasks/703

What is a statistical question? – Revisited

For the questions, you deemed to be statistical questions, explain ways you would collect data to answer the question.



26

Spinner Activity

This page is left blank for calculations, examples or notes.

http://illuminations.nctm.org/adjustablespinner/ or http://tinyurl.com/pyjpkcs or

http://illuminations.nctm.org/adjustablespinner/

http://illuminations.nctm.org/adjustablespinner/

http://tinyurl.com/pyjpkcs

http://tinyurl.com/pyjpkcs


27 Probability Games, Retrieved from: http://map.mathshell.org/materials/lessons.php?taskid=596&subpage=concept http://tinyurl.com/lvzz7sj

Probability Games Activity Formative Assessment Lesson

Record any information here that may be helpful for you to remember when using this FAL with students: Create a statistical question using the data you collected in the Probability Games Activity. Is this the appropriate method of creating a statistical question? Why or why not?

http://map.mathshell.org/materials/lessons.php?taskid=596&subpage=concept

http://tinyurl.com/lvzz7sj




Race Results Sheet

Explain your results. Are they different from what you expect? Why is this?






Finishing Places of Horses





31

Grand Canyon Temperature Graphing Activity

Temperatures were taken over a 15-day period at the Grand Canyon and are shown in the table below. Using the data, create the following graphs.

98 99 98 98 117

87 87 89 84 83

70 79 78 83 95

1. Find the mean, median, mode, and range. Mean = _________ Mode = _________ Median = __________ Range = _________ 2. Create a frequency table.

3. Create a dot plot.


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4. Create a histogram. 5. Create a box plot.


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This page is left blank for calculations, examples or notes.


34

Day 2

Let’s review information from the first day. 1. What are the steps to statistical problem solving? 2. How can we use different measures of center to find out more information about a

dataset?


35 Purposeful Pedagogy retrieved from http://commoncore.aetn.org/mathematics/ccss-mathematics-4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf

Purposeful Pedagogy

http://commoncore.aetn.org/mathematics/ccss-mathematics-4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf



42 This activity was retrieved from http://www.regentsprep.org/regents/math/algebra/AD3/FruitLoopsActivity.pdf (Copyright © Regents Exam Prep Center)

Froot Loops Activity Name________________________________________________________________ Materials: Fruit Loops, paper towels, and Froot Loops Activity handout Query: How many Fruit Loops are there in a “handful”? Instructions: 1. Each student grabs a handful of Froot Loops and counts them. 2. Record the class data using the table below.

Using your class data: 1. Find the measures of central tendency. Mean = _________ Mode = _________ Median = __________

Range = __________

Student # Cereal # Student # Cereal #

1 16

2 17

3 18

4 19

5 20

6 21

7 22

8 23

9 24

10 25

11 26

12 27

13 28

14 29

15 30

http://www.regentsprep.org/regents/math/algebra/AD3/FruitLoopsActivity.pdf



2. Which measure of central tendency do you feel best represents the number of pieces of

cereal in a handful for this class? Explain. 3. The Principal, a former basketball player, comes into class and takes a handful. With his

entry, the mean increases. What can be said about the number of cereal pieces in the Principal’s handful?

4. Peggy Sue comes into class and grabs a handful. With her entry, the median does not change.

What can be said about the number of cereal pieces in Peggy Sue’s handful? 5. Mrs. Smith, the librarian, and her pre-school daughter, Ashley, come in and grab handfuls.

When Mrs. Smith’s entry is added, the median decreases. What can be said about the number of cereal pieces in Mrs. Smith’s handful?

6a. Little Ashley’s entry is added. Ashley is a very small little girl. With her entry, what would

you predict would happen to the mean? 6b. Again, with Ashley’s entry, what would you predict would happen to the median?




7. Specify the five statistical summary for your class data:

minimum = ______________ maximum = ______________ 1st quartile = ______________ 2nd quartile = ______________ 3rd quartile = ______________

8. Construct a box plot for the class data.




Froot Loop Activity Extension (for Pre-AP, Honors or other Advanced classes)

1. Predict what color you think will occur most often. Explain your reasoning. 2. Use your handfuls of Froot Loops to answer the following statistical question: What are the

ratios of colors in a typical box of Froot Loops?

3. Create a display for your data.

4. Determine the ratio of each color using percentages.

5. Compare your results to the results of other groups. Are your results similar? Explain:

6. What could you do to make your predictions more valid?



46

A New Park Does Little Rock need a new city park? To seek support in determining if Little Rock needs a new park, 20 people were asked if they would vote for the park. Their gender and political party association were also collected. Using the data, create a table to organize the data so that it could be used to make a decision.

Gender Political

Party Vote

M D Y

M D N

F R Y

M R Y

F R Y

F R Y

F D N

M R Y

F D N

M R N

M R N

M R Y

F R N

M R Y

F D Y

M D Y

F D Y

M R Y

F D Y

M R N


47

This page is for calculations, notes and explanations.


48 This activity was modified from S-ID Used Subaru Foresters I; retrieved from Illustrative Mathematics: https://www.illustrativemathematics.org/content-standards/HSS/ID/B/6/tasks/941

Value of Used Subaru Forester I

Jane wants to sell her Subaru Forester, but doesn’t know what the listing price should be. She checks on craigslist.com and finds 22 Subarus listed. The table below shows age (in years), mileage (in miles), and listed price (in dollars) for these 22 Subarus. (Collected on June 6th, 2012 for the San Francisco Bay Area.)

Age Mileage Price

8 109,428 12,995

5 84,804 14,588

3 55,321 20,994

3 57,474 18,991

1 11,696 19,981

13 125,260 6,888

10 67,740 9,888

11 97,500 6,950

6 36,967 19,700

12 148,000 3,995

2 29,836 18,990

3 32,349 21,995

10 161,460 5,995

4 68,075 12,999

3 30,007 22,900

8 66,000 13,995

10 93,450 8,488

3 35,518 22,995

3 30,047 20,850

8 107,506 11,988

11 89,207 8,995

13 141,235 5,977

https://www.illustrativemathematics.org/content-standards/HSS/ID/B/6/tasks/941



1. Make appropriate plots with well-labeled axes that would allow you to see if there is a relationship between price and age and between price and mileage. Describe the direction, strength and form of the relationships that you observe. Does either mileage or age seem to be a good predictor of price?




2. If appropriate, describe the strength of each relationship using the correlation coefficient. Do the values of the correlation coefficients agree with what you see in the plots?

3. Find the equation that describes each of the relationships.

4. If Jane’s car is 9 years old with 95000 miles on it, what listing price would you suggest? Explain how you arrived at this price.



51

Central Tendency and Shapes of Data

For each set of data, find the mean, median, and mode.

A. 3, 6, 8, 8, 5, 2, 3, 6, 4

Mean_____ Median_____ Mode______ B. 5, 6, 5, 6, 4, 5, 4, 5, 5

Mean_____ Median_____ Mode______ C. 10, 10, 1, 1, 1, 1, 1, 10, 10

Mean_____ Median_____ Mode______ Create a dot plot for each set: Dataset A: Dataset B: Dataset C:

http://etc.usf.edu/clipart/41600/41660/1-10_41660_md.gif




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1. What do each of the data sets have in common? 2. Compare the “shapes” of the data on the dot plots. What do you see? 3. What does this tell you about using Mean, Median and Mode as descriptors for data? 4. Why would the median be a better descriptor of a data set than the mean in some cases? 5. Create a data set where the median is a better descriptor than the mean:


53

Mean Absolute Deviation (MAD)

CCSS will require students to not only calculate mean absolute deviation, but to also understand, interpret and describe how it helps to identify variability in data. To calculate mean absolute deviation:

1. Find the mean of the data set. 2. Calculate the difference between the mean and each data point. 3. Take the absolute value of each difference. 4. Add the differences together and find the mean.

Example Data set: 5, 7, 4, 13, 4, 10, 10, 7, 3 Mean: 7

Data point Difference from mean Absolute value of difference

3 -4 4 4 -3 3 4 -3 3 5 -2 2 7 0 0 7 0 0

10 3 3 10 3 3 13 6 6

Sum of the absolute value of the differences= 24 24/9 = 2.67, which is MAD


54

Now, go back and find the mean absolute deviation for the three sets of data in the Central Tendency and Shapes of Data activity. How does the MAD help you describe the data set more accurately than just mean or median? Dataset A: 3, 6, 8, 8, 5, 2, 3, 6, 4 Dataset B: 5, 6, 5, 6, 4, 5, 4, 5, 5 Dataset C: 10, 10, 1, 1, 1, 1, 1, 10, 10


55

The Best Basketball Shooter Award

Coach Jameson has a problem. He has to pick one of his basketball players to receive the Best Shooter Award at the athletic celebration next week. He has it narrowed down to 2 players, Jamal and Steven. Here is the list of games and scores each player made:

Jamal’s Points Scored

Steven’s Points Scored

Game 1 11 10

Game 2 10 13

Game 3 14 28 Game 4 12 0

Game 5 8 4

Game 6 4 8

Game 7 15 10 Game 8 14 16

Game 9 13 16

Game 10 8 4

Game 11 9 10

Game 12 14 0

Game 13 11 20

Game 14 15 14

Game 15 10 15 1. What is Coach Jameson’s problem? What measure of center do you think he is using?

2. Create a statistical question for Coach Jameson’s problem:


56

The coach decided to go talk to Mrs. Smith, the math teacher, to find out if there was another way to determine which player should get the award. Mrs. Smith told Coach Jameson that he should try Mean Absolute Deviation. 3. Why did Mrs. Smith suggest using Mean Absolute Deviation? 4. Determine the Mean Absolute Deviation for each set of scores. 5. Which player would you recommend get the award? Explain your reasoning including the

mathematical reasoning you used to find your answer:


57

Returning to the First Dataset

Now that you know more about statistical analysis, let’s return to the first dataset. Which teacher has better scores? Why? Be sure to explain using evidence.

Teacher A Teacher B

100 100

95 100

67 98

85 97

45 45

80 46

77 50

77 73

75 65

50 73

100 85

80 94

60 58

70 52

84 57

82 85

90 75

94 80

47 70

60 66


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This page is left blank for calculations, notes and explanations.


59

Variance and Standard Deviation

Variance and Standard Deviation give us another piece of information about datasets. Here are the steps:

1) Find the mean of your dataset. 2) Subtract the mean from each datapoint (this is the deviation from the mean). 3) Square each deviation. 4) Take the sum of all the squared deviation scores. 5) A) For a population, you will divide the sum of the squared deviation scores by the number of

data points in your dataset. This is the variation. B) If your dataset is a sample from a population, you have to use a correction. You will subtract one from the number of data points in your dataset. This is the variation.

6) Take the square root of the variance. This is the standard deviation.

Here is an example:

Score

Deviation from mean

Squared Deviation

7 7-4=3 9

6 6-4=2 4

4 4-4=0 0

2 2-4=-2 4

1 1-4=-3 3

Mean=4 Sum: 26

Since this is a sample, we need to subtract one from the total number of data points before we divide. Variance: 26/4 = 6.5

Standard deviation: √6.5 = 2.55


60

Now, find the variance and standard deviation for all three datasets from the Central Tendency and Shapes of Data activity. Dataset A: 3, 6, 8, 8, 5, 2, 3, 6, 4 Dataset B: 5, 6, 5, 6, 4, 5, 4, 5, 5 Dataset C: 10, 10, 1, 1, 1, 1, 1, 10, 10 How does standard deviation help you compare data? How is it different from the mean absolute deviation?


61

Comparisons of data with the same mean, but different standard deviations:

Describe the different datasets. The normal curve:


62

The normal curve shows that approximately 68% of data points fall between +1 and -1 standard deviation. We use this to figure out which scores are less normal than the rest.

Teacher A Teacher B

100 100

95 100

67 98

85 97

45 45

80 46

77 50

77 73

75 65

50 73

100 85

80 94

60 58

70 52

84 57

82 85

90 75

94 80

47 70

60 66

The following tables are descriptive statistics from the datasets above (these are the same datasets we used yesterday). These were found using Excel. How does using variance and standard deviation help you compare data sets?

Teacher A Teacher B

Mean 75.9 Mean 73.45

Standard Error 3.754225689 Standard Error 4.143145893

Median 78.5 Median 73

Mode 100 Mode 100

Standard Deviation 16.78940769 Standard Deviation 18.52871172

Sample Variance 281.8842105 Sample Variance 343.3131579

Range 55 Range 55

Minimum 45 Minimum 45

Maximum 100 Maximum 100

Sum 1518 Sum 1469

Count 20 Count 20


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With a partner, one of you develop a normal curve for the class of Teacher A and the other develop a normal curve for the class of Teacher B. Which scores fall into the top 5%? Which are in the bottom 5%?


64

Fizz, Fizz

This activity was adapted from the Antacids in Developing Essential Understanding of Statistics: Grades 6-8, pgs. 42-50. Name________________________________________________________________ Materials: Two brands of effervescent tablets, stop watch, like cups to dissolve the tablets, enough water at the same temperature to fill the cups equally Scenario: Cold medicine comes in different forms: tablets, capsules, liquids, and effervescent tablets. Effervescent tablets are tablets, that when dropped in water, dissolve. You then drink the solution. If you are sick, you want to take the medicine as soon as possible to start relieving your cold. Instructions: In this activity, you want the students to engage in all four components of statistical problem solving: 1) write a statistical question, 2) collect data, 3) analyze data, and 4) interpret data. Depending on the level of your students more guidance may need to be provided for the components. Divide students into groups of 2-3 students. There will need to be someone recording the data and taking measurements. These jobs can be shared between students, but this could cause variability in the data collected. Not all students may interpret “dissolve” time the same. Use this as a discussion topic at the end when groups are sharing their results. Distribute materials to groups. When providing students with the effervescent tables, do not let them know which brand they are receiving. Since you have selected the cups and water, it is important to talk about how this reduces variability. The discussion could be done while you are walking around listening to the groups as they determine how they are going to collect data or it could be done when groups are sharing their results. At the end of the activity, be sure to discuss the following with the students if not previously addressed.

Students were not told which brand they were observing. Why?

Would it have been better to have one student to measure the time? Explain.

Explain why using the same “cup/glass” was important.

Would different amounts of water used to dissolve the tablet affected the outcome?

Would having water of different temperatures make a difference in dissolve times? Explain.

What are other factors that could affect the results of the fizz time? Why?


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This page is left blank for calculations, notes and explanations.


66

Day 1 Exit Slip

1. What are 3 things you have learned?

2. What are 2 things you still don’t understand?

3. What was your favorite activity today?


67

Day 2 Exit Slip

1. What are 3 things you have learned?

2. What are 2 things you still don’t understand?

3. What was your favorite activity today?


68

Materials List

Activity Teacher Two Sets of Test Scores – pg. 1

Data & Statistics Grades 6-8 Standards – pgs. 2-5

Progressions for Probability & Statistics – pgs. 6-18

Purposeful Pedagogy – pgs. 19-25

Statistical Literacy Framework – pgs. 26-27

Collecting Data – pgs. 28-29

What is a Statistical Question – pg. 31 1 per student

What is a Statistical Question (Revisited) – pg. 32 1 per student

Spinner Activity – pg. 33 1 per student

Probability Games – pgs. 34-37

Probability Games Recording Sheet– pg. 34 1 per student

Race Card – pg. 35 1 per student

Race Results Sheet – pg. 36 1 per student

Finish Places of Horses – pg. 37 1 per student

Counters 11 per group

A pair of die 1 pair per group

Grand Canyon Temperatures – pgs. 38-40 1 per student

Opening Activity Day 2 – pg. 41

Froot Loops – pgs. 42-45

Paper towels

Plastic Gloves or Hand-sanitizer Enough for class

Boxes of Froot Loops 1 box per 20 students

Recording Handout – pgs. 42-44 1 per student

Extension Activity – pg. 45 1 per student - optional

A New Park – pgs. 46-47 1 per student

Used Subaru Forester I – pgs. 48-50 1 per student

Central Tendency & Shapes of Data – pgs. 51-52 1 per student

Mean Absolute Deviation – pgs. 53-54 1 per student

Best Basketball Shooter Award – pgs. 55-56 1 per student

Return to Two Sets of Test Scores – pgs. 57-58 1 per student

Variance and Standard Deviation – pgs. 59-63 1 per student

Fizz, Fizz – pgs.64-67 1 per student

Pitcher to pour water

Glass with water 1 per group ( all glasses should the same and filled equally with water of same temperature)

Two brands of effervescent tablets 1 of Brand A per group for half of the groups 1 of Brand B per group for other half of groups


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Resources


Common Core State Standards retrieved from http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

Fizz, Fizz activity was adapted from the Antacids in Developing Essential Understanding of Statistics: Grades 6-8, pgs. 42-50. Froot Loops activity was retrieved from http://www.regentsprep.org/regents/math/algebra/AD3/FruitLoopsActivity.pdf (Copyright © Regents Exam Prep Center) Identifying Statistical Questions; Retrieved from Illustrative Mathematics:

https://www.illustrativemathematics.org/content-standards/6/SP/A/1/tasks/703 Mathematical Practices retrieved from http://www.corestandards.org/Math/Practice/ Probability Games, retrieved from: http://map.mathshell.org/materials/lessons.php?taskid=596&subpage=concept Progression Documents retrieved from http://achievethecore.org/content/upload/Draft%206%E2%80%938%20Progression%20on%20Statistics%20and%20Probability.pdf Purposeful Pedagogy retrieved from http://commoncore.aetn.org/mathematics/ccss-mathematics-4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf Used Subaru Foresters I activity was modified from S-ID Used Subaru Foresters I; retrieved from Illustrative Mathematics: https://www.illustrativemathematics.org/content-standards/HSS/ID/B/6/tasks/941


http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf



http://www.corestandards.org/Math/Practice/


http://achievethecore.org/content/upload/Draft%206%E2%80%938%20Progression%20on%20Statistics%20and%20Probability.pdf

http://achievethecore.org/content/upload/Draft%206%E2%80%938%20Progression%20on%20Statistics%20and%20Probability.pdf


