If you do not have access to Cuisenaire Rods, access the Materials section to download paper versions. It is recommended that you print these on sturdy paper.
Explain to students that there are 5 different lengths (1,2,3,4,5) of train cars:
Show them the train below and explain that it is a train of length 4 made from a car of length 3 and a car of length 1.
Show the trains below and explain that even though they use the same cars, they are two different trains.
|Train of length 4 made|
from 2 cars of length 1
and 1 car of length 2.
| ||Train of length 4 made|
from 1 car of length 1,
1 car of length 2 and
another car of length 1.
Explain that the train below is a train of length 5 made from 1 car of length 5.
Give students the opportunity to ask questions and allow them to hold
the trains if it helps them to explain their thinking. Emphasizing the
difference between train length and car length at the beginning helps
students to talk about the different types of trains. It gives them a
common language. To assess their understanding, hold up a train and ask
students to describe it to you. Repeat this until all students
understand. Have students work in groups of 2 to 4 to build the trains.
This will decrease the amount of train combinations you will have to
Pass out the trains and the Cusinaire Trains Activity Sheet.
Cuisinaire Trains Activity Sheet
Give students colored pencils, markers or crayons and grid paper (to
accommodate students who can’t see certain colors, students can put a
number in each car: ).
Explain that they should use the grid paper and the colors to record
their trains. In order to make all the trains simultaneously, students
will need 50 cars of length 1, 20 cars of length 2, 15 cars of length
3, 10 cars of length 4 and 5 cars of length 5. Remind students that
they must build the trains together, and record their trains on
their individual Activity Sheets. It might help to ask, “If I were to
take your trains away, would you still be able to tell me all
the combinations you built by reading what is on your paper?” Suggest
that they use black to represent the white trains to avoid problems
with recording white trains.
Make sure that students have built and recorded all the different trains before proceeding to the table, graph and rule.
- When a group of students thinks they have all the trains, be sure to ask how they know they aren't missing any.
- If students are missing some trains, you can scaffold by
You might need to point out another train that has the same cars or
point to another train of a different length that looks similar. When
all else fails, give them the pieces they need and let them build the
If some groups finish building all the trains sooner than others, it is
okay for them to move on to the table, graph, and rule. They can also
do the extension activities.
Lead a whole-group discussion to allow students to present how they
figured out the number of trains of length 6. Ask them to explain or
show how to get from the 4 trains of length 3 to the 8 trains of length
4. Some may add a car of length 1 to each of the trains of length 3 and
extend the last car in each of the trains of length 3 by 1. Allow
students to share their strategies.
Ask students if the same method works for getting from the trains of length 4 to the trains of length 5.
Ask a student to present or explain their table and graph. Have
students discuss any difficulties they ran into in making the table.
You may want to point out the train length should be along the x-axis
since it is the independent variable (save this discussion if this is
not part of your curriculum). Have students present any conjectures
they are making regarding rules that they think will work to determine
the number of trains possible if they know the length of the train.
- Driscoll, Mark. 1999. Fostering Algebraic Thinking: A guide for teachers grades 6-10. Heinemann, Portsmouth, NH.
- Benjamin, A. T., and J. J. Quinn. 2003. Proofs That Really Count: The Art of Combinatorial Proof.
- Dolciani Mathematical Expositions, Volume 27. Mathematical Association of America.
By the end of this lesson, students will:
- Represent data using tables, graphs and rules.
- Investigate patterns and make conjectures.
- Explain their reasoning when making conjectures.
NCTM Standards and Expectations
- Relate and compare different forms of representation for a relationship.
- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
Common Core State Standards – Practice
Look for and express regularity in repeated reasoning.