# Rod Trains

In the problem Rod Trains, students use mathematical concepts of combinatorics, number theory, and discrete mathematics. The mathematical topics that underlie this problem are knowledge of number sense, number patterns, counting principles, systematic charting, and closed form equations. The mathematics that includes counting principles and systematic charting is often referred to as discrete mathematics. In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

PRE-K
In this task, students compare the length of rods. They use the length of one rod to determine how many it takes to make the length of the other rods.

LEVEL A
In this level, students compare the length of rods to determine a numerical measurement of each of the rods.

This level addresses Common Core standard 1.MD.A.2 by having students express the length of an object by laying multiple copies of a shorter object end to end.

LEVEL B
In this level, students analyze problems to determine the number of trains that are created by arranging the rods in different orders and different lengths.

This level can be used as an extension of Common Core standards 1.MD.A.2 and 1.OA.C.6. Students can use what they learned from 1.MD.A.2 to help determine 2 different-length rods that can be combined to equal the length of a third rod. They will then use what they know about the value of each rod (determined in Level A) and what they know about addition and subtraction strategies from 1.OA.C.6 to write addition sentences for each of the combinations of rods they found.

LEVEL C
In this level, students are presented with situations that require them to use counting principles and organized lists to determine the number of ways trains can be assembled.

Students make an organized list or other method such as tables or tree diagrams to make an exhaustive list of combinations (7.SP.C.8b).

LEVEL D
In this level, students are presented with situations that require them to use counting principles and organized lists to determine the number of ways trains can be assembled.

Students represent sample spaces for events using multiple approaches (7.SP.C.8b).

LEVEL E
In this level, students are asked to determine a formula that will give the number of rod train combinations needed for a train length of n units. Before working this level, it may benefit students to work level D to lay the groundwork for this level.

This level address Common Core standard F-BF.A.1.a by having students determine an expression that gives the number of rod arrangements for a train length of n.

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