Getting Started 
Misconception/Error The student does not understand how frequencies of data are presented in a histogram, a dot plot, or both. 
Examples of Student Work at this Level The student indicates that one or both dot plots do not correspond to the histogram because:
 The dots donâ€™t display what the bars tell us.
 The frequencies donâ€™t match.
 The frequency of the first bar is two but just one in the dot plot.
 The shapes of the dot plots are significantly different.

Questions Eliciting Thinking What does the histogram indicate about the data? How do you read a histogram?
What does the scale along the vertical axis of the histogram represent? How many trees have heights less than five meters? How many trees are there altogether?
What do the dot plots indicate about the data? How do you read a dot plot? 
Instructional Implications Provide direct instruction on the structure of a histogram. Emphasize that histograms are used to summarize the frequency of quantitative data (as opposed to categorical data) that has been placed in intervals or classes of uniform width. Explain that the intervals are shown on the horizontal axis and the heights of the bars indicate the frequency of data in each interval they span. Pose questions that address interval widths, frequencies of data within intervals, the interval(s) of greatest or least frequency, the overall shape of the histogram, and an interpretation of the distribution in context.
Have the student extract the raw data (e.g., tree heights) from the first dot plot and write each tree height inside the appropriate bar on the histogram. Then have the student consider whether the histogram could correspond to the first dot plot. Ask the student to repeat the exercise for the second dot plot but encourage the student to take a â€śshort cutâ€ť (e.g., circling the dots in the dot plot that correspond to each interval in the histogram) if he or she can discover one.
Have the student construct histograms for each dot plot using the same interval widths.
Provide the student with a set of test scores and ask the student to construct both a dot plot and a histogram with intervals that correspond to a letter grade scale. Then challenge the student to change some scores so that the dot plot changes but the histogram stays the same. 
Making Progress 
Misconception/Error The student provides a correct response but with insufficient reasoning. 
Examples of Student Work at this Level The student indicates that both dot plots correspond to the histogram. However, the student is unable to clearly and completely explain how he or she determined the correspondence between the pairs of graphs. For example, the student says both dot plots correspond to the histogram because:
 They display the same data.
 The totals add up.
 The points on the dot plots are on the histogram.

Questions Eliciting Thinking Can you explain how you compared the dot plots to the histogram? What did you actually do?
How can the histogram correspond to more than one dot plot? 
Instructional Implications Review terminology associated with histograms and dot plots (e.g., quantitative data, intervals, classes, frequencies, and bar height). Ask the student to explain how he or she determined that the dots plots correspond to the histogram. Assist the student in using correct terminology. Then ask the student to revise his or her response. Expose the student to wellconstructed explanations and justifications posed by classmates. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student counts groups of dots corresponding to each of the histogram intervals and checks for consistency. The student indicates that both dot plots are consistent with the histogram because the frequencies of data within the intervals (given by the histogram) are the same as those in the histogram.

Questions Eliciting Thinking Could there be other dot plots that correspond to this histogram? Why is that?
Can you determine the mean of the tree heights using the dot plots? Can you determine the mean of the tree heights using the histogram? 
Instructional Implications Provide the student with more examples of histograms and encourage the student to describe the shape, center, and spread of each using conventional terminology correctly.
Ask the student to compare bar graphs, circle graphs, box plots, dot plots, and histograms in terms of the kind of data that is displayed (categorical or quantitative) and whether the graph can display frequencies of raw data, frequencies of data in intervals, relative frequencies, and/or percents. 