Getting Started |
| Misconception/Error The student makes significant errors in scaling the horizontal axis. |
| Examples of Student Work at this Level The student understands the basic structure of a histogram but does not appropriately scale the horizontal axis. The student:
- Uses intervals that are not uniform in width.
- Uses too many or too few intervals.
- Does not show the scale.
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| Questions Eliciting Thinking Can the intervals be any width? Does it matter if they are not all the same width?
How did you decide how many intervals to use?
What is actually graphed on your horizontal axis? Can you provide a scale for this axis? |
| Instructional Implications Review the concept of a histogram and emphasize that a histogram summarizes the frequency of data in intervals of equal width which are shown on the horizontal axis. Help the student organize the data in a frequency table. Assist the student in devising a strategy for finding the width of the interval and the number of intervals. Explain there is no right number of intervals for a given data set but generally a minimum of five is used. Caution the student against using too many as this produces a very flat, uniform looking histogram which makes it difficult to determine the shape of the distribution. Caution the student against using intervals that overlap (e.g., 30 – 35, 35 – 40, 40 – 45 …); otherwise, it is possible for a data point to fall into more than one interval. Model selecting intervals for this set of data (e.g., 30 – 39, 40 – 49, 50 – 59, 60 – 69, and 70 – 79). Instruct the student to scale the horizontal axis using the endpoints of the intervals (e.g., 30, 40, 50, 60, 70 and 80) and explain that it is acceptable to show a break in the scale after zero if there would otherwise be a number of empty intervals. Be sure the student understands that the height of each bar corresponds to the frequency of data in the intervals and the width of each bar is equal to the interval width. Guide the student to scale the vertical axis so that it does not extend well beyond the highest frequency. Remind the student to title the graph and label the axes.
Provide additional opportunities to create histograms for sets of data. Engage the student in a discussion of the differences between bar graphs and histograms in order to eliminate confusion. |
Moving Forward |
| Misconception/Error The student does not correctly draw bars whose heights reflect the frequency of the data. |
| Examples of Student Work at this Level The student appropriately scales each axis but the bars drawn do not reflect the frequency of the data.
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| Questions Eliciting Thinking Did you organize your data before drawing your histogram?
What do the bars on a histogram signify? How do you determine the heights of the bars?
What determines the width of the bars? |
| Instructional Implications Guide the student to organize the data in a frequency table, clearly describe the intervals, and tally the number of data points in each interval. Be sure the student understands that the height of each bar corresponds to the frequency of data in its interval and the width of each bar is equal to the interval width. Guide the student to scale the vertical axis so that it does not extend well beyond the highest frequency. Remind the student to title the graph and label the axes.
Provide a checklist of features that 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.
Provide additional opportunities to create histograms for sets of data. |
Almost There |
| Misconception/Error The student makes a minor error in some component of the graph. |
| Examples of Student Work at this Level The student makes one or more of the following errors:
- Does not show a break on the x-axis from the origin to the first interval.
- Omits a title or labels on one or both axes.
- Uses intervals that overlap (e.g., 35 – 40, 40 – 45, 45 – 50…).
- Does not show the scale on the horizontal axis appropriately.
- Includes significant portions of the horizontal or vertical axis for which there is no data.
- Makes an error in counting the frequency of the data within an interval.
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| Questions Eliciting Thinking Where is zero on the horizontal axis?
Does your graph contain all necessary titles and labels?
What if a data point had been 45? In which interval would you have placed it?
Can you scale your axes so that the histogram fills the graph?
I think you may have made an error in counting the frequency of your data. Can you check your counts and see if you can find it? |
| Instructional Implications Review any issue with the student’s histogram and ask the student to revise his or her graph. Provide a checklist of features that 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.
Provide the student with additional opportunities to construct histograms.
Consider implementing the MFAS task Shark Attack Data and Chores Data (6.SP.2.4). |
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 intervals labeled 30-39, 40-49, 50-59, 60-69, and 70-79 and corresponding heights equal to 1, 2, 4, 2, 1.
*Each interval was precisely defined to include its lower limit but not its upper limit. For example the interval 30 – 40 corresponds to the interval  where w represents the number of wins per year. |
| Questions Eliciting Thinking How did you determine the width of your intervals?
How would the histogram change if the intervals are changed?
How would you describe the shape of your histogram?
Could you construct the original data set from the histogram? |
| Instructional Implications Review terms used to describe the shapes of distributions, for example, uniform, symmetric, bimodal, skewed left, and skewed right. Provide a number of histograms of various shapes and ask the student to describe each distribution’s shape and explain what the shape indicates about the data in context.
Ask the student to use another reasonable interval width and reconstruct the histogram. Then ask the student to compare the shapes of the two histograms. |
