This lesson will help students understand the concepts of range, median, mode, and mean through the use of graphs drawn from models.
- Chart paper
- Markers/colored pencils
- Connecting colored
- Flat squares of paper
the same size as the
cubes (This will be used
to represent 0)
- Math journals (lined
- Vocabulary Matching
- Navigating through Data Analysis and Probability in
Grades 3-5, by
Peggy A. House; ISBN 0-87353-521-9
Background for Teachers
Many students can calculate the range, median, mode, and mean of a
set of numbers. But if asked what that looks like, students have difficulty
expressing these concepts. This lesson will help students understand the
concepts of range, median, mode, and mean through the use of graphs
drawn from models. Students will use manipulatives to represent data
that they have collected within groups. By using models, students can
compare the differences between mode, median, and mean. Students
should understand that the mean, median, and mode are just different
ways of calculating what the middle is in a set of data. The range will
be represented visually as students arrange their models from smallest to
greatest. As students show a representation of the mean, the teacher can
lead a discussion on the meaning of what the remainder represents.
Depending on the students background knowledge, the teacher can either
have the students disregard the remainder or model how the remainder
can be represented in fraction or decimal form.
Intended Learning Outcomes
2. Become mathematical problem solvers.
4. Communicate mathematically.
5. Make mathematical connections.
6. Represent mathematical situations.
Invitation to Learn
Display a set of objects such as various sizes of crayons on the
overhead. Ask the students if they can tell what the average length is in
the set of crayons. Who can draw a picture on the board of what an
average looks like? When do we use averages?
- Tell students that they
are going to learn four key math words
today (range, median, mode, and mean) and that they will draw
pictures about these words.
- Hold up the Vocabulary Matching Cards. Read
the words and
their definitions to students. Explain that these terms will be used
during the days activities.
- Discuss how graphs and charts represent data
that was collected
to answer a question (purpose). Explain that you need a question
to answer before you can begin to collect data.
- As a class, come up with
a simple question for students to
answer with a number that is between 0-10.
- How many TVs are in your home?
- How many pairs of shoes do you have?
- How many times have you been
to Disney Land?
- How many hours a day do you read?
- Display the question on the board.
- Put the students into groups of three,
five, or seven. This will
make it easier to calculate the median.
- Pass out 10 like-colored cubes
per student and assign each
student in the group a different color. Pass out 1 flat square to
each student to use as a 0 if needed.
- Tell students that they can use
the cubes to make a tower that
shows their answer to the class question. Have students in each
group compare their answer (tower) with the others in their group.
Have each group line up their towers in order from least to
- Teach the concept of range. Tell the students that the highest
tower is the maximum number in their set of data and that the
shortest tower is the minimum number in their set of data. To
find the range, they need to subtract the minimum from the
maximum. Have each group calculate the range for their set of
data. Refer back to the vocabulary definitions used at the
beginning of the lesson. Have each student draw a picture of their
groups towers (this should look like a bar graph) in their math
journals. Have them write the equation that represents the range
and an explanation of how they calculated it.
- Ask students how many in
their group have the same size tower.
Explain that if they do, this is he mode. (Refer back to the
vocabulary cards.) Have students draw a picture of the mode on
their journals and give an explanation of how they calculated it.
(The teacher should explain that groups could have more than one
- Have the students line their towers up from least to greatest again.
Explain that the middle tower represents the median. Have
students draw and explain this in their journals. (This part of the
activity is optional because this is not an assessed concept in the
- To teach the mean, ask students if all of them have the same
tower. Ask them how they would find the average height of
their towers. (Many will be able to explain how to calculate it
using paper pencil or a calculator.) Demonstrate how to show the
average height of their towers by moving blocks until all the
towers are the same height. Have them set the leftover blocks
aside until you are ready to discuss remainders. (Give them time
to manipulate their blocks.)
- Explain that the remainder would have to
be divided up evenly
among the towers to make them truly even. This requires
splitting wholes into parts. Have students estimate how much of
each remaining block would need to be added to each tower to
make them even. Demonstrate on the board using either decimals
or fractions what that looks like.
- Explain that an average is a way to
even things out, and that all
the towers should be the same size in their groups. Have students
draw and explain this in their journals as before.
- Have each group come
up with their own questions to use to
calculate the range, mean, mode, and median. Let them draw
charts representing their data in their journals and share their
results with the class.
- Students who have a difficult
time writing may dictate their
explanations to the teacher or another person and have them
record it in students journal next to the illustrations.
- Using the scientific
method, have students create experiments
where the data can be graphed using bar graphs, and the mean,
median, range, and mode can be calculated.
Have the students measure each
member in their family and
record the results. Use the information in a daily activity where
students calculate the mean, mode, median, and range for the
heights. Ask the students if they can tell by the data the age of
the family members as it applies to range.
- Students will be able to show understanding of the
mode, median, and range by:
- Using their cubes to model the data they have
- Drawing their models in their journals using markers or
colored pencils. They should have four illustrationsone for
each vocabulary word/concept.
- Having a written explanation next to each
explaining how they calculated the data.
- Median is not a concept that
needs to be assessed in the 5th grade;
however, it relates directly to understanding mean, mode, and
range in a set of data.
Lappan, G., Fey, J., Fitzgerald, W. Friel, S. & Phillips, E. (1996).
Data about us. Connected
Mathematics Project, Palo Alto, CA.
The mode, median, and mean are kinds of
averages that are a part of
representations and statistics used to analyze data. Students need to
understand each of these measures and how they are applied and
calculated. This article examines two ways in which the concept of
mean can be demonstrated.
Hitch, C. & Armstrong, G. (1994). Daily activities for data analysis.
Children develop mathematical concepts by seeing them in a variety
of settings. For students to understand statistics and graphs, they need
exposure to the process of collecting, organizing, and describing data.
This article describes useful activities that help students understand and