Grade 6 Supporting Idea 6: Data Analysis

Grade 6 Supporting Idea 6: Data Analysis

Grade 6 Supporting Idea: Data Analysis
MA.6.S.6.1 Determine the measures of central tendency (mean, median, and 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.

Green: adequate content coverage
Yellow: not mentioned in the text of the benchmark, but in the text of the item specs, we learn that students are going to have to gather information from these types of graphical displays in order to find the mean, median, mode, or spread. Red: the first two (double bar graphs and pictographs) are mentioned in the item spec text but were “covered” in the previous grade level– this might be a need for remediation, if students have not yet demonstrated mastery of these graphical displays. The second two (select appropriate display and change intervals) are in the book but not mentioned in the standards or item specifications at this grade level. Do we spend our instructional time teaching these? If so, how much time?

FAIR GAME: Prerequisite Knowledge
MA.3.S.7.1: Construct and analyze frequency tables, bar graphs, pictographs, and line plots from data, including data collected through observations, surveys, and experiments. MA.5.S.7.1: Construct and analyze line graphs and double bar graphs. These are the fair-game standards students are expected to have mastered in the previous grade level.

FAIR GAME: Prerequisite Knowledge
These are the fair-game standards students are expected to have mastered in the previous grade level.

These are sample items from the grade 5 Go Math textbook
These are sample items from the grade 5 Go Math textbook. How might you “level up” the questions you ask to increase the level of cognitive demand on your 6th grade students?

What kinds of questions might you ask about these graphs that are tied to the standards?

Skills Trace mean median Add whole numbers, fractions, and decimals
mode range Add whole numbers, fractions, and decimals Divide whole numbers, fractions, and decimals Compare and order whole numbers, fractions, and decimals Add whole numbers, fractions, and decimals Divide whole numbers, fractions, and decimals Compare whole numbers, fractions, and decimals Subtract whole numbers, fractions, and decimals

Measures of Center mean median mode

Model: finding the median
Find the median of 2, 3, 4, 2, 6, 5. Usea strip of grid paper that has exactly as many boxes as data values. Place each ordered data value into a box. Fold the strip in half. The median is at the fold.

Model: finding the mean
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?

Model: finding the mean
You can use post-it notes to generate graphs with your students. Then to find the mean, you rearrange the post-its in such a way that every category (or “bar”) has the same number of observations.

Thinking about measures of center
The median of five numbers is 15. The mode is 6. The mean is 12. What are the five numbers? 6 6 15 n So, first thing: we know we have five numbers, so draw five “seats” We know the mean is 15… put it in the middle They also told us the mode is 6… put that in the two lowest seats We don’t know the last two numbers, let’s use the variable n…

The median of five numbers is 15. The mode is 6. The mean is 12. What are the five numbers? 15 6 n What does n=16.5 tell us about the solution? Can we use this decimal value twice as observations in our data set? Yes, because n and n appear twice in our data set, both variables should be replaced by the same value.

The median of five numbers is 15. The mode is 6. The mean is 12. What are the five numbers? 6 6 15 a b But what if we want two different values for a and b? (this time, the two numbers can be different, since a and b are different variables.) This affects the algebra slightly, as we shall see in the next slide.

The median of five numbers is 15. The mode is 6. The mean is 12. What are the five numbers? 15 6 a b NOW what are possible values for a and b? We know their sum is 33… so we could use 16 and 17….. Or 15 and 18… Could we use 14 and 19? Why or why not? (No, because if 14 were added to this data set, it becomes the median. The problem required the median to be 15.)

Missing Observations: Mean
Here are Jane’s scores on her first 4 math tests: What score will she need to earn on the fifth test for her test average (mean) to be an 80%? A x N = T Jane has 316 points. She needs 400 points. How many more does she need? Average Number Total before 79 4 316 after 80 5 400 84 points

Here are Jane’s scores on her first 4 math tests: What score will she need to earn on the fifth test for her test average (mean) to be an 80%?

Here are Jane’s scores on her first 4 math tests: There is one more test. Is there any way Jane can earn an A in this class? (Note: An “A” is 90% or above) What measure of center are we asking students to consider? No, there is no eway Jane can earn an A in the class. With only one test remaining, she would need to score a 134% in order to have her average be an A.

Here are Jane’s scores on her first 4 math tests: There is one more test. Is there any way Jane can earn an A in this class? (An “A” is 90% or above) No, there is no eway Jane can earn an A in the class. With only one test remaining, she would need to score a 134% in order to have her average be an A.

Missing Observations: Median
Here are Jane’s scores on her first 4 math tests: What score will she need to earn on the fifth test for the median of her scores to be an 80%? What is the first step? Order the data from least to greatest.

Missing Observations: Median
What score will she need to earn on the fifth test for the median of her scores to be an 80%? 70? 75? 79? 80? 81? 82? 83? 84?  

What is the fewest number of observations needed to accomplish this?
Think, Pair, Share Construct a collection of numbers that has the following properties. If this is not possible, explain why not. mean = 6 median = 4 mode = 4 What is the fewest number of observations needed to accomplish this? In this data set {2, 4, 4, 14} the mean is 6, the median is 4, and the mode is 4. Can I do it with three observations? I can! {4, 4, 10} {1, 3, 5, 16} this solution is close, since the median is 4 but the observation 4 does not show up in the data set. However, this data set has no mode and we want one with a mode of 4.

Think, Pair, Share Construct a collection of numbers that has the following properties. If this is not possible, explain why not. mean = 6 median = 6 mode = 4 What is the fewest number of observations needed to accomplish this? (close!) mean=6, median=6, but mode is not 4 (close! again) but you still have two modes Solution: mean=6, median=6, mode=4 Five observations are necessary.

Think, Pair, Share Construct a collection of 5 counting numbers that has the following properties. If this is not possible, explain why not. mean = 5 median = 5 mode = 10 What is the fewest number of observations needed to accomplish this? NOT POSSIBLE First, I know the mode is 10 and the median is 5, so this much is known: ____ ____ Even with the smallest positive values possible for the first two blanks, The mean is still only 5.4, and that is the smallest mean we can generate

Think, Pair, Share Construct a collection of 5 real numbers that has the following properties. If this is not possible, explain why not. mean = 5 median = 5 mode = 10 What is the fewest number of observations needed to accomplish this? THIS ONE IS POSSIBLE The word “counting” has been omitted so we can use zeroes in the data set First, I know the mode is 10 and the median is 5, so this much is known: ____ ____ Even with the smallest positive values possible for the first two blanks, The mean is still only 5.4, and that is the smallest mean we can generate

Think, Pair, Share mean = 6, mean > mode
Construct a collection of 4 numbers that has the following properties. If this is not possible, explain why not. mean = 6, mean > mode Solutions:

Think, Pair, Share mean = 6, mean > mode
Construct a collection of 5 numbers that has the following properties. If this is not possible, explain why not. mean = 6, mean > mode The previous slide used a 4-observation data set. Can you make a 5-observation data set that meets the same requirements? mean=6 mode=2

5 6 7 9 2 4 1 6 Adding a constant k mean = 5 median = 5.5 mode = 6
Suppose a constant k is added to each value in a data set. How will this affect the measures of center and spread? mean = 5 median = 5.5 mode = 6 range = 8 Ask participants to find the mean, median, mode, and range of this data set before showing them

Adding a constant k 5 6 7 9 2 4 1 5+2= 6+2= 7+2= 9+2= 2+2= 4+2= 1+2= 7
8 9 11 4 6 3 mean = 5 median = 5.5 mode = 6 range = 8 mean = 7 median = 7.5 mode = 8 range = 8 Adding a constant n to each term shifts the entire distribution, so the mean, median, and mode all increase by n. The range stays the same, since the min and the max were both shifted the same amount.

Multiplying by a constant k
Suppose a constant k is multiplied by each value in a data set. How will this affect the measures of center and spread? mean = 5 median = 5.5 mode = 6 range = 8 Multiplying a constant n by each term multiplies the mean, median, mode, and range by 2.

Multiplying by a constant k
5 6 7 9 2 4 1 5×2= 6×2= 7×2= 9×2= 2×2= 4×2= 1×2= 10 12 14 18 4 8 2 mean = 5 median = 5.5 mode = 6 range = 8 mean = 10 median = 11 mode = 12 range = 16 Multiplying a constant n by each term multiplies the mean, median, mode, and range by 2.

Watch out! Graphical Displays of Data and Measures of Center
Table Bar graphs Double bar graphs Line graphs Line plots Pictograph Frequency table Content limits specify that students will have to find the mean, median, and mode from data that is presented in these graphical forms. Understanding how the displays are made help students read them– and to gather data from the graphs students need to pay attention to -- units (labels) -- axes -- increments The most common mistake made seems to be trying to interpret the graph before really understanding the graph. Maybe it’s because students just want to get to the questions and get the work done, that make them skip the step of actively reading the graph. Graphs can be read just like you read the words in a story. Look at the title, the words describing the X and Y axis (the bottom and side of the graph), and the information that is presented. As you read the graph, think about whether the information makes sense. If it is a graph about the number of sunny days per month in each state, think about how much sun you usually get in your state in each month. Does the information seem to make sense with what you know? If it does, it is likely that you are reading the graph correctly. If it doesn’t make sense, try to figure out why it is not making sense before you try to answer the questions!

The most common mistake made seems to be trying to interpret the graph before really understanding the graph. Maybe it’s because students just want to get to the questions and get the work done, that make them skip the step of actively reading the graph. Graphs can be read just like you read the words in a story. Look at the title, the words describing the X and Y axis (the bottom and side of the graph), and the information that is presented. As you read the graph, think about whether the information makes sense. If it is a graph about the number of sunny days per month in each state, think about how much sun you usually get in your state in each month. Does the information seem to make sense with what you know? If it does, it is likely that you are reading the graph correctly. If it doesn’t make sense, try to figure out why it is not making sense before you try to answer the questions!

The FCAT test specifications make no mention of these bubbles, which identify the value represented by the bar. You want to ensure that your kids are FCAT-ready– that they can gather information from a bar chart without the “assist” bubbles

What is the mean number of fat grams in these 7 kinds of cheese? ( )/7 = What is the mode? 6 grams of fat What is the median number of fat grams? 6 grams In order to answer these questions, students must think about what is on the axes– labels, increments, units

Is this the best way to present this information (no, generally, we use a line graph to show change throughout time.) How will students find the mean, median, and mode on this bar graph? Is this inside or outside the content limits for this benchmark?

How is this graph different than the two before it? (this has numerical units on the dependent and independent axis– no categorical data… so when students go to gather the data they need to calculate a measure of center, they may not know which axis to read. What do you think: would a question like this appear on the FCAT? (outside the content limits– mean: 6 pieces of data median/mode: 10 pieces of data) also, dependent axis has units that are difficult to parse (break down into smaller units) i.e., what is the atomic radius of element with atomic number 8?) How does student background knowledge affect a student’s ability to understand and interpret data? If you don’t know what either of these quantities mean in the real world, can you effectively compare or interpret them relative to one another?

The graph below shows how many men and women were working out in the gym during the given time interval. What is the mean number of women who worked out between 1:00 pm and 3:00 pm? How many observations are there? We don’t know. They could have counted people every 10 minutes, every 30 minutes– who knows? How can we calculate a mean or a median if the graphical display of data does not indicate how many observations are in the data set? Discrete data has been displayed as continuous– but is it really? Teachers who make (or borrow) their own graphical displays of data need to be aware of this– you want to see DOTS (or coordinates) for the points that were actual observations.

So, tell me, What is the mode? Well, what did we measure? (book sales) What did the data set look like? (a list of how many books were sold in each year) 12,000 40,000 60,000 20,000 10,000 15,000 the distribution is bimodal: 10,000 and 20,000 (each with a frequency of 2) 25,000 17,000 VERY DIFFICUL FOR STUDENTS, right? They need to ask themselves: what did the data set look like? Working backwards really helps!

What would you title this graph?

What would you title this graph? What did the data set look like? (what observations did we collect?) How can we find the mean? How can we find the median? How can we find the mode?

Watch out! Line Graphs and Measures of Center
The Location  A sixth-grade teacher uses a secret location game to teach the class about statistics, connections, and reasoning. The Location Discussion guide at Video Overview Bill Stevenson explains to his sixth–grade class that each group will receive an envelope with a secret location. They're to decide, as a group, how many people will be present in that location at each hour of the day—starting at midnight and going until the next midnight. They record the information in a table. Once they have the table, each group creates a line graph that will show this information. Finally, Mr. Stevenson leads a wrap–up discussion in which students discuss the features of the graphs and try to guess what location the graphs might represent.  An Exploration for Teacher Workshops Materials: grid paper Imagine a supermarket, or wherever you buy your groceries. Think about what it's like at different times of the day. Work in pairs to make a table with 24 entries labeled by the 24 hours of the day, starting at midnight and ending the following night at 11 p.m. Record how many people you think will be in the market each hour. Think of it as a snapshot: how many are in the market at midnight? How many at 1:00 a.m.? Continue for the remaining hours. Finally, make a display that shows how the number of people in the market changes over time. Some Questions How did you and your partner decide how many people were there? How did you record your data? What kind of display did you make? Why did you choose that kind of display? What kinds of representations are appropriate for this kind of data? What makes a representation clear and communicative? These instructions were fairly open–ended. What additional structure (if any) would you give to students? What's the mathematical value of this activity?  Additional Discussion Topics Here are some additional ideas for discussion that arise in the video: What group skills did these students demonstrate? How experienced or inexperienced do they seem to be at working in groups? How do they compare with your classes? Discuss the connections this activity makes with places outside school. What purpose does this activity serve outside mathematics? Activities like this one occasionally raise sensitive issues such as family structure and lifestyle. How would you handle these issues if they arose in your math class? How would it have been different if students did the activity alone instead of in groups? What happened in the group conversations about the data? Teaching and Learning About Graphs This was the students' introduction to line graphs. Yet the teacher did not tell the students how to make one. Instead, he asked questions about what you need for a line graph and had the students answer. How did Mr. Stevenson orchestrate this introduction? What did students need to know to make this introduction to line graphs work\? Mr. Stevenson prompted students making presentations to describe what the features on the graphs meant in real life. For example, when someone said the graph "went down," the teacher remarked that people were leaving the location. Discuss the importance of this correspondence and difficulties students seem to have in understanding it.     Discussion Questions These questions appear at the end of the video. Here are some follow–up ideas and prompts to help get a discussion going.  What is the value of having students generate, record, and graph their own data?  What is the difference between the "made–up " data the students used and data generated by a simulation or collected from the real world? Discuss the use of "made–up " versus "real " data. Sometimes the data students generated were unrealistic, for example, the librarians getting in at 5:00 a.m. to set up. Mr. Stevenson did not intervene. In what circumstances, and how, would you intervene to correct unrealistic estimates?  Discuss Mr. Stevenson's decision not to supply grid paper. How might the lesson have gone differently if they had large grid paper to make their graphs? What are the advantages of using grid paper? Students spent a lot of time just scaling the axes correctly. What mathematics are students doing as they set up the graphs? When is that time worth spending, and when is it a waste? What kinds of graphs would grid paper inhibit?  Brainstorm the use of line graphs in other disciplines. A student mentioned using line graphs in the stock market. What other uses can you think of for line graphs?

Watch out! Line Graphs and Measures of Center
The Location What is the value of having students generate, record, and graph their own data? Discuss Mr. Stevenson’s decision not to supply grid paper. How does Mr. Stevenson stimulate discussion and statistical reasoning? How can background experiences affect a student’s ability to understand and generalize about data? ownership, investment, clarity about what the data means, discussion with classmates establishes relevance Mr. Stevenson forced students to think about units and measuring along the axes and not just plotting points. His decision not to supply graph paper required them to use rulers, talk about spacing, decide where to start numbering and how to choose increments– all of which are discussions that help students make meaning of data and prepare them to interpret these and other graphical displays. He “mirrors’ what students say (or repeats it back to them to verify and reinforce what they think), asks lots of open-ended questions, “what do you think”, “what kind of place/people/activities?”, hypothesizing encouraged, acknowledges very answer, Students who can’t make hypotheses about what is happening with the data cannot make predictions or interpret graphical displays.

Watch out! Frequency Tables and Measures of Center
MEDIAN: students must be cautious when finding the “middle” value– at least The position of the median can be found by the formula , where n is the number of observations in the data set.

MEAN: Students invariably want to add =17 and divide by 7… this is not appropriate!

Number Frequency 1 6 2 3 4 5 7 8 9 When reading frequency tables, students sometimes struggle to identify which data values should be added or ordered to find a mean or a median. Find the measures of center. Did you get a mean of 3.222? What did you do wrong? (Hint: What are the observations in the data set?) Here are the values in your data set: There are 29 observations in this data set. The mean is really

Watch out! Reviewing How Frequency Tables are Made

Choosing an appropriate measure of center
The student must: Distinguish between data sets that are symmetrical and those that are skewed Understand the effect of skewness on the mean Recognize outliers Understand why the median is outlier-resistant Remember that the mode is particularly helpful for categorical (vs. quantitative) data

Mean vs. Median Notice the use of line graphs to communicate statistical information

What is an outlier? An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs. Extreme observations In the real world, statisticians either discard them or use a robust (outlier-resistant) measure of center or spread.

What is an outlier?

How do we determine outliers?
1.5*IQR (interquartile range) 2, 5, 7, 9, 10, 12, 20 lower quartile: Q1= 5 median: 9 upper quartile: Q2=12 IQR = Q2-Q1= = 7 1.5*IQR= 10.5 Q1: lower quartile– median of the lower half of the data Q2: upper quartile-- median of the upper half of the data In order to be called a mild outlier, we say an observation has to be more than this distance below Q1 or above Q2. If an observation is 3 or more IQRs above/below Q1/Q3, we say an observation is an extreme outlier.

Outliers: What to do?

Describing Distributions
Students may at first not be comfortable thinking of a bar graph, line plot, or line graph as a smooth curve. It will help them notice skewness if they are able to see the trend in this way.

symmetric distribution
mean = median = mode skewed left distribution mean < median < mode skewed right distribution mean > median > mode

Visualizing how the outlier pulls the mean

mean or median Mean, Median or Mode? mean = 4.896 median = 5 mode = 1
Number Frequency 1 6 2 3 4 5 7 8 9 mean = 4.896 median = 5 mode = 1 Find the measures of center. Did you get a mean of 3.222? What did you do wrong? (Hint: What are the observations in the data set?) Here are the values in your data set: There are 29 observations in this data set. mean or median

mean or median Mean, Median or Mode? mean = 5 median = 5
Number Frequency 1 9 2 8 3 7 4 6 5 mean = 5 median = 5 modes = 1 and 9 If you got a mean of , remember: you don’t just average the numbers in the frequency column. These are the values in the data set: There are 81 values in this data set. mean or median

mode Mean, Median or Mode? mean = 4.58 median = 3 mode = 2 Number
Frequency 1 7 2 20 3 15 4 11 5 8 6 9 mean = 4.58 median = 3 mode = 2 If you got a mean of 9, remember: you don’t just average the numbers in the frequency column. These are the numbers in the data set: There are 81 observations in the data set. mode

median Mean, Median or Mode? mean = 8 median = 2 mode = 54 Number
Frequency 1 3 2 4 5 6 7 8 9 54 mean = 8 median = 2 mode = 54 median

symmetric distribution mean = median = mode
skewed left distribution mean < median < mode Remember, the mean is pulled in the direction of the tail. Right tailed, right skewed, mean is on the right. skewed right distribution mean > median > mode

Which measure of center is best for each data set?
Skew Which measure of center is best for each data set? Set A: symmetric; use mean or median Set B: symmetric; but mode would really highlight the fact that all data categories had the same frequency of observations in them Set C: bimodal (1 and 7), so we’d probably use the mode to describe this distribution

Using Boxplots to Show the Robustness of the Median
Why use this applet? It demonstrates to participants and students why we say the median is “robust” to outliers. Create a data set (or measure something to generate one). Using this applet, you can show students that one outlier on the right, if you move it further and further do the right, does not change the median. But the mean changes. The mean is pulled in the direction of the outlier.

Removing the Outlier, Recalculating the Mean
Fuel Economy (Miles per Gallon) for Two-Seater Cars Model City Highway Acura NSX 17 24 Audi TT Roadster 20 28 BMW Z4 Roadster Cadillac XLR 25 Chevrolet Corvette 18 Dodge Viper 12 Ferrari 360 Modena 11 16 Ferrari Maranello 10 Ford Thunderbird 23 Honda Insight 60 66 Lamborghini Gallardo 9 15 Lamborghini Murcielago 13 Lotus Esprit 22 Maserati Spyder Mazda Miata Mercedes-Benz SL500 Mercedes-Benz SL600 19 Nissan 350Z 26 Porsche Boxster 29 Porsche Carrera 911 Toyota MR2 32 With Outlier Without Outlier mean median mode range What is the mean, median, mode, and range of this fuel economy data for 2-seater cars on this list (city)? Are there any outliers? (Honda Insight– a hybrid vehicle) When we remove the outlier, what do you predict will be the new mean, median, and mode? (Calculate it.) How far were your predictions from the actual values?

Removing the Outlier, Recalculating the Mean

Encouraging Critical and Statistical Thinking
How can we encourage our kids to make inferences and draw conclusions?

What would you say to these students?
Gregory: "The boys are taller than the girls." Gregory does not quantify his statement and may only be looking at the upper extreme value. The teacher could ask Greg to determine "how much taller" the boys are than the girls.

Marie: "Some of the boys are taller than the girls, but not all of them." Marie is comparing the variation in the data and notices overlap in the values. The teacher might ask her to quantify her response

Arketa: "I think we should make box plots so it would be easier to compare the number of boys and girls." Arketa is considering other representations that might make certain patterns and relationships between the data sets more apparent. The teacher could ask the class to consider additional ways to represent the data that would make some comparisons more visible.

Michael: "The median for the girls is 63 and for the boys it's 65, so the boys are taller than the girls, but only by two inches." Michael correctly determines the median for each data set and quantifies "how much taller" the boys are than the girls by comparing the medians of the data sets. The teacher might ask the other students to react to Michael's statement and then consider why it can be useful to compare the medians of two data sets.

Paul [reacting to Michael's statement]: "I figured out that the boys are two inches taller than the girls, too, but I figured out that the median is 62 for the girls and 64 for the boys." Paul quantifies "how much taller" the boys are than the girls by comparing what he thinks are the medians of the data sets; what he found, though, was the middle of each range and not the middle of the data. This is an opportunity for the teacher to review the meaning of median as well as ways to find the median of a set of ordered data.

Kassie: "The mode for the girls is 62, but for the boys, there are three modes -- 61, 62, and so they are taller and shorter, but some are the same." Kassie believes that she is comparing the modes of the data sets, but when three or more values have the same number of data points, such as the boys, the data is considered not to have a mode. The teacher can review the meaning of mode and ask the students to speculate as to why statisticians say that a data set doesn't have a mode when three or more values have the same number of data points.

DeJuan: "But if you look at the means, the girls are only and the boys are 64.5, so the boys are taller." DeJuan correctly calculates the means and quantifies "how much taller" the boys are than the girls by comparing the means of the data sets. The teacher could now have the students compare the medians and means of the two data sets. What does each tell us about the data? In this situation, is one comparison more appropriate than the other one? Why or why not?

Carl: "Most of the girls are bunched together from 62 to 65 inches, but the boys are really spread out, all the way from 61 to 68." Carl is comparing intervals of the two data sets that contain the most data. The teacher could take this opportunity to focus further attention on the importance of examining intervals in considering how the data are spread out or bunched together.

Arketa: "There is a lot of overlap in heights between the boys and girls." Arketa is comparing the variation by looking at the range of each data set. The teacher might ask her to quantify her response.

Michael: "We can see that the median for the boys is higher than for the girls." Michael compares the data sets by looking at the medians. The teacher could ask Michael to point to the median on each box plot and then review that 50%, or half, of the data box plot is on each side of the median.

Monique: "It looks like just 12.5% of the boys are taller than all of the girls, and maybe about 10% of the girls are shorter than the shortest boy." Monique incorrectly reasons that one can further subdivide the lines (or boxes) and that a fractional part of a line reflects a fractional part of the data. The teacher should ask Monique how she arrived at those percentages and then show this same finding on the line pot to see if she recognizes the discrepancy.

Gregory: "The boys are taller than the girls, because 50% of the boys are taller than 75% of the girls." Gregory is correctly reasoning about the box plots with quartiles. The teacher might ask the rest of the class to evaluate Gregory's statement for its accuracy.

Morgan: "You can see that the middle 50% of the girls are more bunched together than the middle 50% of the boys, so the girls are more similar in height." Morgan is correctly reasoning about the spread of the middle 50% of the data on the box plots. The teacher might ask the rest of the class to evaluate the accuracy of Morgan's statement

Janet: "Why isn't the line in the box for the boys in the middle like it is for the girls? Isn't that supposed to be for the median, and the median is supposed to be in the middle? Janet does not understand how the box represents quartiles of the data. The teacher could go back to the line plots of the data and actually draw the box plot directly over the data so that Janet can see the distribution of the data within the quartiles of the box plot.

Discovering Math: Summary (3:45)
Remember, students have already been taught how to calculate mean, median, and mode in previous grades You should not be teaching how to calculate these– just reviewing Your are teaching how to gather observations from a graphical display– double bar chart, line graph, frequency table, etc.) The last part of this segment covers choosing appropriate measures of center.

Generating Meaningful Data
Make and fly paper airplanes—how far do they go? How long is a second? How many jumping jacks can you do in a minute? Handspan, arm span Food nutrition label analysis 3M Olympics: Peanut Flick, Cookie Roll, Marshmallow Toss

Instructional Resources

Read the article "What Do Children Understand About Average
Read the article "What Do Children Understand About Average?" by Susan Jo Russell and Jan Mokros from Teaching Children Mathematics. a. What further insights did you gain about children's understanding of average? b. What are some implications for your assessment of students' conceptions of average? c. What would be an example of a "construction" task and an "unpacking" task? d. Why might you want to include some "construction" and "unpacking" tasks into your instructional program?

To the tune of “Row, Row, Row Your Boat”
Mode, mode, mode– THE MOST Average is the mean Median, median, median, median The number in between

NLVM: Bar Chart

Another Representation of the Mean

Grade 6 Supporting Idea 6: Data Analysis

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