Getting Started 
Misconception/Error The student does not understand how to construct a histogram. 
Examples of Student Work at this Level The student:
 Indicates he or she does not know how to construct a histogram or attempts to construct another type of graph.
 Constructs a histogramlike graph that includes bars whose heights correspond to the value of each data point.

Questions Eliciting Thinking What does a histogram look like? How is it constructed?
How must the data first be organized before constructing the histogram?
What should be graphed on the horizontal axis? What should be graphed on the vertical axis?
How are the widths and the heights of the bars on a histogram determined? 
Instructional Implications Explain the distinction between categorical and quantitative data and be sure the student understands that histograms are used to display quantitative data while bar graphs are used to display categorical data. Provide clear instruction on the structure of a histogram. Emphasize that histograms are used to summarize the frequency of quantitative data that has been placed in intervals or classes of uniform width. Using an interval width of 10 (e.g., 30 â€“ 39, 40 â€“ 49, 50 â€“ 59, 60 â€“ 69, 70 â€“ 79, and 80 â€“ 89), ask the student to determine the frequency of data in each interval. Help the student to appropriately scale the frequency axis. Then guide the student through the construction of the histogram for the data. Remind the student to always label both axes and title the graph.
Show the student histograms that summarize data in context. 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.
Give the student sets of data and ask the student to construct histograms. Provide a checklist of features the histogram must contain: precisely defined intervals of uniform width shown along the horizontal axis, a scale on the vertical axis that is appropriate for graphing the frequencies of data within each interval, bars that span the interval width and whose heights reflect the frequencies within the intervals, axes labels, and a title. 
Moving Forward 
Misconception/Error The student does not appropriately represent the data. 
Examples of Student Work at this Level The student places each data point in an interval and counts the frequency of data within each interval. However, when constructing the graph the student:
 Does not scale the vertical axis appropriately. The student scales it to values that are much too large so that the histogram is very small and differences in frequencies are hard to detect.
 Constructs bars whose heights do not reflect the frequency of data within each interval.
 Does not use a linear scale on one or both axes.
 Does not label axes or labels them incorrectly.
 Does not indicate the intervals that were used or does not use a uniform width for the intervals.
 Uses interval widths that are too small or too large which makes it difficult to characterize the shape of the distribution.

Questions Eliciting Thinking What is the frequency of data in each interval? How should you scale the vertical axis to accommodate these frequencies?
How are the bar heights determined? Should there be spaces between the bars?
Does your axis look like a number line? Are the coordinates equally spaced?
What are the widths of your intervals? How did you decide how wide to make them?
Ideally, about how many intervals should a histogram contain? 
Instructional Implications Remind the student that a histogram should ideally contain about five bars or intervals. Emphasize that the intervals or classes must be of uniform width. Using an interval width of 10 (e.g., 30 â€“ 39, 40 â€“ 49, 50 â€“ 59, 60 â€“ 69, 70 â€“ 79, and 80 â€“ 89), ask the student to determine the frequency of data in each interval. Then guide the student to appropriately scale the frequency axis. Assist the student in determining labels for the axes in terms of the context. Ask the student to construct the histogram using the given intervals.
Give the student sets of data and ask the student to construct histograms. Provide a checklist of features the histogram must contain: precisely defined intervals of uniform width shown along the horizontal axis, a scale on the vertical axis that is appropriate for graphing the frequencies of data within each interval, bars that span the interval width and whose heights reflect the frequencies within the intervals, axes labels, and a title. 
Almost There 
Misconception/Error The student omits a minor component of the graph or makes a small error. 
Examples of Student Work at this Level The student:
 Does not title the graph or label the axes.
 Does not show the scale on one or both axes.
 Displays the wrong frequency of data withinÂ oneÂ interval.
 Leaves spaces between the bars as if it were a bar graph.

Questions Eliciting Thinking How should your axes be labeled? What might be a good title for your graph?
Why is it important that the scales on the axes be shown? 
Instructional Implications Review any issue with the studentâ€™s histogram and ask the student to revise his or her graph. Provide the student with additional opportunities to construct histograms. 
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 produces a histogram with precisely defined intervals of uniform width shown along the horizontal axis, a scale on the vertical axis that is appropriate for graphing the frequencies of data within each interval, bars that span the interval width and whose heights reflect the frequencies within the intervals, axes labels, and a title.For example, the student produces a histogram with interval endpoints labeled 30, 40, 50, 60, 70, 80 and corresponding heights equal to 1, 2, 4, 2, 1.
Note: The intervals were definedÂ to include the left endpoint value shown on the horizontal axis but not the right endpoint value. For example, the first interval isÂ . 
Questions Eliciting Thinking How would the histogram change if the intervals are changed?
How do you know how to determine the width of the intervals? What would happen if the intervals had been only two units wide? What would happen if the intervals had been 25 units wide?
What useful information does a histogram provide?
How would you describe the shape of your histogram?
Could you construct the original data set from the histogram? 
Instructional Implications Ask the student:
 To indicate on the histogram how it would change if one includes 50 wins in 2010 and 60 wins in 2011.
 IfÂ one could construct a histogram with intervals of width five wins given a histogram with intervals of width ten wins.
 To use another reasonable interval width and reconstruct the histogram. Then ask the student to compare the shapes of the two histograms.
