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## Fibonacci Trains

• Lesson
6-8
1

In this lesson, students use Cuisenaire Rods to build trains of different lengths and investigate patterns.  Students make algebraic connections by writing rules and representing data in tables and graphs.

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 madefrom 2 cars of length 1and 1 car of length 2. Train of length 4 madefrom 1 car of length 1,1 car of length 2 andanother 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 check.

Pass out the trains and the Cusinaire 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 offering hints. 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 missing combinations.

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.

### References

• 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.
http://www2.edc.org/makingmath/mathprojects/trains/trains.asp

Assessment Options

1. Ask students to build all the trains of length 1, 2, 3, 4, 5, and 6 using only cars of length one and/or two. Then, repeat the 5 questions above in the Questions for Students section.
 Train Length 1 2 3 4 5 6 Number of Ways 1 2 3 5 8 13

This assessment also provides another connection to the Fibonacci numbers.

2. Use the Cuisinaire Trains Activity Sheet as another form of assessment.

Extensions

1. Ask students, How many cars of length 3 would you expect to occur in all the trains of length 3? 4? 5? 6?

Students can fill in the following table of occurrences and make predictions for the next row(s).

Train Length # of Cars of Length 1 # of Cars of Length 2 # of Cars of Length 3 # of Cars of Length 4 # of Cars of Length 5 # of Cars of Length 6

1

2
3
4
5
6
They can record any patterns they find.
2. Ask students to repeat the task from the activity sheet, except that now:    and   only count as one way.

Questions for Students

1. How do you know you have all of the trains?
2. Explain why you think the number of possible trains doubles each time the length of the train is increased by one.
3. In your scatter plot, should you connect the points? If you connect the points, what do the segments that connect the points represent?
4. Compare your experience with drawing the trains of length 4 with the trains of length 3. Which was more difficult and why?
5. What were your variables when you made your scatter plot? How did you decide which variable was independent and which variable was dependent? Explain your reasoning.

Teacher Reflection

• How did you challenge the achievers?
• How does this lesson relate to permutations and combinations?
• In what ways did students exceed or not meet your expectations?
• What were the misconceptions exhibited by students? What did you do to address those misconceptions?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what were the adjustments, and were they effective?

### Learning Objectives

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

• CCSS.Math.Practice.MP8
Look for and express regularity in repeated reasoning.