begin the lesson, tell students you are interested in knowing the eye
colors of the students in the classroom. Explain that you want to know
the answer to this question:
- What colors are the eyes of the students in this class?
How can you find that out? Students may suggest that you count, that
you ask all the students whose eyes are the same color to stand
together in one part of the room, or that you make a graph. Listen
carefully to their suggestions in order to learn what the students know
about data collection and analysis. After the discussion, give students
colored paper matching their eye colors and have them hold the colored
paper in front of them as they form a human bar graph on a line in the
room. Record the number of students in each category.
- Ask the students to think about the relationships of the counts to
one another. How many more brown are there than green? Or alternately,
how many fewer green than brown? What fractional part of the class has
blue eyes? How many times as many blue are there as green? Green is
what fractional part of blue?
- Have the students point out these mathematical relationships
in their human graph. If there are 7 more brown than green, show that
on the graph. Move the emphasis toward multiplicative reasoning by
asking students to point out on the human graph that there are, for
example, twice as many blue as green, or half as many green as blue.
Explain to students that this data can be represented in a circle
graph as well as a bar graph. Ask them to stay with students who have
the same eye color, to hold their colored paper in front of them, and
to move into a circle. Ask a teacher's aide to stand in the center of
the circle. Give a string to any student who is standing at the point
where one eye color ends. Use these strings to form the radii (from the
outer rim to the teacher's aide in the center) bounding the eye-color
sectors of the circle. Which colors take up the most room in the
circle? The least? Are the fractional relationships the same as in the
bar graph? Be specific: Is green still half as much as blue, etc.?
Activity: Students formulate questions, gather and record data
When students are seated, tell them they have been working with
categorical data and that they have represented the data in two ways —
in a bar graph and in a circle graph. Explain that usually questions
about categorical data (the kind of data they were obtaining in the
eye-color activity) are answered with words. Give students some
examples and ask them to think of additional ones:
- What color are the eyes of students in this classroom?
- What is your favorite dessert?
- Whom do you want to be president?
Then tell students they are going to conduct a survey that will give
them a chance to think about categorical data. Remind them of the
question and explain the importance of phrasing the question carefully.
Ask them to speculate about whether responses in all classrooms of the
world would be similar to their responses, and point out the importance
of determining who will be asked the survey question.
Consider the following issues:
- What exact information do you want your classmates to give you?
- Do any words need to be defined more clearly?
- Do any questions need to be reworded?
- If the class wanted to group some of these responses into additional categories, would it help to ask a few more questions?
- Should we have everyone write down their answers, or should they just tell us the answers?
- Should we keep track of the names of people we survey?
Students can work in groups of two or three to select their survey question. Using the Survey Form, each group can collect its data during an appropriate period of time.
Alternatively, students can collect the data on individual index cards, which are set up as follows:
Student Survey Form
|Name || |
|Student Response |
Activity: Students represent and analyze data
Now that students have collected the data, they need to think about
organizing a bar graph of the raw data (names of books, leisure
activities, or TV shows). In graphing this raw data, students are
graphing categorical data, just as they have frequently done in primary
grades. Usually in these earlier grades, students identified the mode and were able to pick out their own data on the graph. In addition, students need to organize the data
in a variety of ways, analyze the data mathematically, and compare data sets.
Students should take a close look at the data to see if they can
think of other ways of organizing it on bar graphs. For example,
students might order the favorite books information according to
fiction/nonfiction; if most responses are fiction, students might
consider fictional subcategories. For free time activities, students
can organize information through the alone/with other, or
indoor/outdoor categories. Other categories may occur to students as
they work with the data.
Students can create bar grpahs of their survey results by using the NCTM Bar Grapher tool.
Remind the students of the mathematical relationships they
discovered when they analyzed the human graphs on eye color. Ask that
they make similar observations based on their current representations
of data. Using the total number of responses as the whole, they should count the number
of responses in each category. Then, they should record that number in
relation to the whole. For example, 15 out of 23 students chose
fiction; 8 out of 23 chose non-fiction. Then, give students time to
come up with further observations. Be sure to give them time to
generate ideas. If eventually students don't come up with observations,
you might offer some conjectures (true or false) and ask the students
to use their reasoning to refute or support them. The conversation
about the data will, of course, depend on the data collected by the
In conclusion, students should refer to the original survey
question. How do they think the question should be answered? Each
student might select one mathematical observation based on the data and
write a thorough explanation of it (as in the conversations above).
Students should also discuss any questions which the data suggest
should be pursued if the survey project were to be continued.