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.
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| 
|
train of length 4 made
from 2 cars of length 1
and 1 car of length 2.
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| 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 check.
Pass out the trains and the 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 listed below.
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.
and
only count as one way.
