Activity
Ask the
students, “What are the various types of graphs
you have used in the past to represent data?” Hopefully
they will name bar graphs, histograms, line graphs, and pie-graphs
or some form of a similar list. Ask them, “Why are
there so many different types of graphs?” (There are
different types because each one has a fairly specific use).
Allow time for discussion.
Tell
them, “Today we’re going to look at bar
graphs and histograms since you have probably made these
graphs in the past. Let’s start with bar graphs. What
are bar graphs used to represent? What usually goes on the
horizontal axis and what goes on the vertical axis?” (See
teacher background) Allow time for discussion.
If you
have access to a computer laboratory or a computer with
a projection
screen, go to the following website http://nces.ed.gov/nceskids/graphing/bar.asp and use the education data from NCES to create a bar graph
for Popular Bachelor’s Degrees, 1999-2000. Have the
students discuss the structure of the bar graph and how they
would read the bar graph to determine the number of each
type of bachelor’s degrees awarded.
Another good website that uses bar graphs to represent the
probability of getting a given number when rolling a pair
of dice is http://nces.ed.gov/nceskids/probability/index.asp.
The students get to put in the number of times the dice are
rolled. Ask the students, “If we rolled the pair of
dice many times, what would expect the bar graph to look
like?” Allow time for discussion.
If in a laboratory,
have the students pair up and assign each pair of students
a specific number of rolls, going from
100 to 1000 by hundreds. Tell the students “Enter your
number of rolls into the box and then press the “Roll
Dice” button. Make a sketch of the bar graph that appears
on your screen. Does this bar graph agree with your expectations?” Have
them repeat the experiment with the same number of rolls.
Ask, “Why is the second bar graph different from the
first if you rolled the dice the same number of times in
each experiment?” Allow time for discussion.
Begin
a discussion about the differences between bar graphs and
histograms
by saying, “Histograms and bar graphs
look very similar. What's different about them?” Allow
time for the students to think about this and come up with
suggestions. (A good website discussing the differences is
http://bdaugherty.tripod.com/KeySkills/histograms.html )
In a
nutshell, histograms are closely related to bar charts,
but differ
in that they are used represent frequency distributions.
A frequency distribution is a tabular arrangement of data
to whereby the data is grouped into different intervals,
and then the number of observations that belong to each interval
is determined. Data that is presented in this manner are
known as grouped data. The smallest value that can belong
to a given interval is called the lower class limit, while
the largest value that can belong to the interval is called
the upper class limit. The difference between the upper class
limit and the lower class limit is defined to be the class
width. When designing the intervals to be used
in a frequency distribution, it is preferable that the class
widths of all
intervals be the same. (Source:http://library.thinkquest.org/10030/2sroffd.htm?tqskip1=1&tqtime=0624 )
Put up the Rocket Thrust Data overhead. The observations
have already been into classes of width 3 for the students.
Ask them, “How would we represent this data as a histogram?
What goes along the x-axis? Do we need to start at the origin
on the x-axis? What’s the lowest number we want to
use on the x-axis? How wide is each class width? What goes
on the y-axis? What’s the smallest number we want to
use on the y-axis?” Have the students take you through
the steps to create the histogram. (See http://bdaugherty.tripod.com/KeySkills/histograms.html for a picture of the histogram representing this data.)
Put up the Data overhead. Note that the values range from
0 to 10.0, therefore, it is easiest to create the following
10 classes, each with a class width of one unit.
| class
1: |
0
- 1.0 |
class
6: |
5.0
- 6.0 |
| class
2: |
1.0
- 2.0 |
class
7: |
6.0
- 7.0 |
| class
3: |
2.0
- 3.0 |
class
8: |
7.0
- 8.0 |
| class
4: |
3.0
- 4.0 |
class
9: |
8.0
- 9.0 |
| class
5: |
4.0
- 5.0 |
class
10: |
9.0
- 10.0 |
We
assume that a measurement that falls on the border between
two intervals belongs to the previous interval (e.g. the
value 4.0 belongs to class 4 instead of class 5).
By counting the number of observations that fall into each
class, we get the following frequency distribution:
Measurements Frequency
| 0.0
- 1.0 |
3 |
| 1.0
- 2.0 |
4 |
| 2.0
- 3.0 |
4 |
| 3.0
- 4.0 |
7 |
| 4.0
- 5.0 |
6 |
| 5.0
- 6.0 |
5 |
| 6.0
- 7.0 |
5 |
| 7.0
- 8.0 |
1 |
| 8.0
- 9.0 |
2 |
| 9.0
- 10.0 |
3 |
(Source:http://library.thinkquest.org/10030/2sroffd.htm?tqskip1=1&tqtime=0624 )
Homework
1. Have the students do the Activity Sheet where they must
decide if they would choose a bar graph or a histogram to
represent the data.
| Situation |
Bar Graph or
Histogram?
|
Reasoning |
We want to compare the
total revenues of five different companies.
|
Bar Graph |
No bins involved - you look at each company separately |
| We have measured revenues of several companies. We
want to compare numbers of companies that make from 0
to 10,000; from 10,000 to 20,000; from 20,000 to 30,000
and so on. |
Histogram |
You want to know the number of companies are in each
revenue bin |
| We want to compare heights of ten oak trees in a city
park |
Bar Graph |
No bins involved – you
look at each tree separately |
| We have measured several trees in a city park. We want
to compare numbers of trees that are from 0 to 5 meters
high; from 5 to 10; from 10 to 15 and so on. |
Histogram |
You want to know the number of trees are in each height
bin |
(Adapted
from: http://www.shodor.org/interactivate/discussions/sd4.html)
2. Have the students do problems 1 and 2 from the website:
http://www.shodor.org/interactivate/lessons/st5.html
|
Embedded
Assessment
Student
understanding of bar graphs can be assessed
through informal discussions with the
class about the structure and use of
bar graphs and informal observations
as the students make bar graphs to represent
the probabilities of getting a certain
number when rolling dice a given number
of times.
Student
understanding of histograms can be assessed
through informal discussions about the
structure of histograms and how they
represent the data of a grouped frequency
chart. As a given student leads the teacher
through the steps to draw a histogram
from a given frequency chart, other students
should be doing a self-evaluation to
see how well they understand the procedure.
Informal observations as the students
make a histogram from a given set of
data, without being given a frequency
chart, can be used to assess how well
the students understand the process of
making a histogram. |
