COOPERATIVE INVESTIGATION Counting Cars and Trains Materials Needed The order in which the cars appear is taken A set of Cuisenaire rods (or colored number strips) for each into account, so we consider the red-white and cooperative group of three or four students. The rods, which have white-red trains as two different trains. (Imagine lengths from 1 to 10 centimeters, are color coded as follows: that the engine is the rightmost car and the caboose is the leftmost car.) White N Red Questions to Consider 3 Light Green 1. How many trains can you form that have the same length as the purple rod? + Purple 2. How many trains can you form that have the same length Yellow as the yellow rod? Dark Green 3. What pattern do you observe in the number of trains of length 3, 4, and 5? On the basis of the pattern, predict the Black number of trains of length 8. 00 Brown 4. How many trains can be formed of length n, where n is any whole number and you use cars of lengths I up to n? Can Blue you justify your conjecture? Can you give more than one justification? 10 Orange 5. A train made up of only red and white cars is called an Directions RW-train. Answer questions 1 through 4 for RW-trains. Form a train by placing one or more rods end to end; each rod 6. Call a train with no white cars a W-train. Answer questions in a train is a car. For example, in the figure to the right column 1 through 4 for W-trains. there are four ways to form trains that have the same overall 7. (Writing) Write two additional questions that could be length as the light-green (LG) rod. asked and then give solutions to them