Common Core State Standards 6 – 8 Mathematics

Presented  by      NC  Department  of  Public  Instruc8on  

Statistics and Probability Progression

6th grade   Statistical thinking  Measures of central tendency (median and

mean)  Measures of variability (Mean Absolute

Deviation (M.A.D.) and Inter-quartile Range)  Dot plots, box plots and histograms

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7th grade  Compare two distributions using measures of

center and measures of variability   Probability

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Statistics and Probability Progression

8th grade   Scatter plots   Linear models – no linear regression   Two-way tables for categorical data

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Statistics and Probability Progression

High School Math One   Interpret shape, center, spread and outliers   Incorporate technology   Standard deviation   Scatter plots, linear models, LSRL and

correlation coefficient   Two-way tables (joint, marginal, and

conditional relative frequencies)

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Statistics and Probability Progression

Finding the Median from a Dot Plot

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If you only know the MEDIAN…

Turn and Talk: What information can you give about the

data set?

What information can’t you give about the data set?

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Turn and Talk

Remove two values from the original data set so that:

a.  The median stays the same b.  The median increases c.  The median decreases

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Finding the Median from a Dot Plot

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Turn and Talk

Add two values to the original data set so that

a.  The median stays the same b.  The median increases c.  The median decreases

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What does it mean to understand the “mean”?

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Mean As A Fair Share •  Use snap cubes to represent the number

of letters in each of the birth places on the red card.

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•  Manipulate the snap cubes so that each city has the same number of letters.

•  What strategies did you use to determine this amount?

What If… 1.  Were there any “a-ha” moments?

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2.  Reference Practice 3: Construct viable arguments and critique the reasoning of others. Give examples of how this practice was evidenced with this task.

3. Reference Practice 6: Attend to precision. Give examples of how this practice was evidenced with this task.

More “MEAN” Problems Susan has four 20-point projects for math class.

Susan’s scores on the first 3 project are shown below: Project 1: 18 Project 2: 15 Project 3: 16 Project 4: ???

What does she need to make on Project 4 so that her average for the four projects is 17? Explain how you found your answer.

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BREAK

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Seven people were asked, “How many keys are on your key ring?”

One result from the poll is that the average number of keys was 4.

What are possibilities for the distribution for the number of keys for seven people with the requirement that the mean number of keys be 4?

Use the number line to create a dot plot with 7 data points and a mean of 4.

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Summing Up

1.  What strategies did you use to create your data set?

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2.  Reference Practice 1: Make sense of problems and persevere in solving them. Give examples of how this practice was evidenced with this task.

If you only know the MEAN and the number of data points…

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Turn and Talk: What information can you give about the

data set?

What information can’t you give about the data set?

Differing Distributions

Which distribution shows data values that differ the least from the mean of 4?

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Which distribution shows data values that differ the most from the mean of 4?

How would you order the other distributions from least to most difference from a mean of 4?

Finding the Mean Absolute Deviation (M.A.D.)

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Mean 4 + 4 + 4 = 12 -3 + (-3) + (-3) + (-3) = -12

|-3| + |-3| + |-3| + |-3| + |4| + |4| + |4| = 24 24/7 = 3.43

In the Classroom…

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Based  on  this  informa8on,  would  you  revise  the  order  of  the  distribu8ons?  

Check Your Answers •  Distribution 1: M.A.D. = 0

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•  Distribution 7: M.A.D. = 1.42 •  Distribution 3: M.A.D. = 1.71 •  Distribution 5: M.A.D. = 2.00 •  Distribution 2 and 6: M.A.D. = 2.57 •  Distribution 4: M.A.D. = 2.86 •  Distribution 8: M.A.D. = 3.43

Summing Up •  What does a small M.A.D. tell us about a

data set? •  What does a large M.A.D. tell us about a

data set? •  What does a M.A.D. of 2.89 tell us about

the distribution of keys? •  How could you change the data set to

increase the M.A.D.?

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Finding the M.A.D.

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DATA  SET   DEVIATION  FROM  THE  MEAN  (19.6)  

ABSOLUTE  DEVIATION  

3  

7  

5  

13  

20  

23  

39  

23  

40  

23  

Finding the M.A.D.

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DATA  SET   DEVIATION  FROM  THE  MEAN  (19.6)  

ABSOLUTE  DEVIATION    

3   -­‐16.6  

7   -­‐12.6  

5   -­‐14.6  

13   -­‐6.6  

20   0.4  

23   3.4  

39   19.4  

23   3.4  

40   20.4  

23   3.4  

Finding the M.A.D.

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DATA  SET   DEVIATION  FROM  THE  MEAN  (19.6)  

ABSOLUTE  DEVIATION  

3   -­‐16.6   16.6  

7   -­‐12.6   12.6  

5   -­‐14.6   14.6  

13   -­‐6.6   6.6  

20   0.4   0.4  

23   3.4   3.4  

39   19.4   19.4  

23   3.4   3.4  

40   20.4   20.4  

23   3.4   3.4  

100.8/10  M.A.D.  =  10.08  

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Mean as the Balancing Point

       4                          5                        6                    7                          8                      9                    10                  11                12            13  

Mean

3  

5  

5  

1  

3  

2  

3  

2  

2  

Middle School Standards

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Statistics and Probability 6.SP  

6th Grade Statistics: Develop understanding of statistical variability. 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.

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6th Grade Statistics: Summarize and describe distributions. 4.  Display numerical data in plots on a number line, including dot plots,

histograms, and box plots.

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

Middle School Standards Statistics and Probability 6.SP  

Middle School Standards

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Statistics and Probability 7.SP  

7th Grade Statistics: 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.

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

Heights of Basketball Players vs.

Heights of Soccer Players

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Basketball Players: 185, 185, 185, 185, 185, 185, 185, 185, 185 Soccer Players: 175, 175, 175, 175, 175, 175, 175, 175, 175

                 174        175        176        177        178        179      180      181        182      183      184      185          186  

Heights of Basketball Players vs.

Heights of Soccer Players

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Basketball Players: 175, 180, 182, 184, 185, 186, 187, 190, 195 Soccer Players: 165, 170, 172, 174, 175, 176, 177, 180, 185

                 165                        170                          175                      180                    185                      190                      195                      200          

Basketball  Players  

Soccer  Players  

In 7th grade, students understand…

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1.  A  full  understanding  of  the  data  requires  considera8on  of  the  measures  of  variability  as  well  as  mean  or  median.  

2.  Variability  is  responsible  for  the  overlap  of  two  data  sets  and  that  an  increase  in  variability  can  increase  the  overlap.  

3.  Median  is  paired  with  the  interquar8le  range  and  mean  is  paired  with  the  mean  absolute  devia8on  .  

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