Preparation:
This is an activity you should do yourself. Take one of the Pattern Block Train handouts and try it: collect the data, look for patterns, generate an equation and then see how general an equation you can derive.

Materials:
Set of Pattern Blocks for each group of four
Pattern Block Train handouts (2 pages)

Rationale
This activity is based on concrete materials to increase access and sensemaking, it involves working in groups and there are a variety of ways to solve the problem, it can be extended at a variety of levels, and it often leads students to ask further questions. In addition it puts students in a position where they need a variable to express the ideas they are discovering. It provides a concrete experiential introduction to the ideas of algebra.

Procedure
Have the students form into groups of four, give each group a set of pattern blocks.

Introducing the Problem:
Take a pattern block triangle and ask the students if we take the measurement of the side as one unit what will the perimeter be? Then tell them you are going to make a train and you want them to figure out the perimeter as you add each "car in the train." Put two blocks together and ask them what the perimeter is now. Informally record the data on a chart on the chalkboard. After you have done three ask them if they could tell you the perimeter of a 10 block train....(depending on time and pace you can interupt here and have them move to the first sheet, or you can have the class work on this problem. I usually interupt and tell them this is the kind of problem they will be doing, but rather than the triangle they will be working with the hexagon in small groups).
Tell them their job is to work with hexagons, to gather data, and see if they can find patterns, and then see if they can use the patterns they find to predict the perimeter of a train they have not actually built. (Pass out first problem sheet). Tell them if they finish the first problem you have a second one they can try. Also tell them to all work on the problem individually and then get together to share their approaches and see how many different ways the group came up with to think about it.

Lab time:
Have them work on the problem for a while. (assignment: if a group "finishes" the trapezoid (is able to write down a rule in english) you can give them the second sheet (pick another block)). In last five minutes have them get together and write up what they want to share.
(NOTE: it is important that the teacher find out what each group is doing so he/she can decide which group(s) to call on in what order. This problem tends to bring out a range of mathematical activity from students who only get to iterative rules (e.g. add 4 to get to the next perimeter) to the student who makes a formula. Our experience has been that starting with the least sophisticated is best.
Other issues: Keeping your eyes out for two groups that are doing the same things and encouraging them to talk with one another is useful strategy to model. It is also worthwhile to watch for groups that are exploring interesting tangents that you can later encourage to share their questions and findings.

Group discussion:
Have them share their results: (both verbal descriptions and equations)
This often takes a whole class. One option here is to go to another block, or have them choose a block and do the same thing. Once they have developed formulas or rules with numbers in them you will want to challenge them to make sense out of the rules they have found e.g. why is there a 4 in the hexagon rule and why is there a plus 2 in it? This can be given as homework and/or done in a lab context (See the folowing lab) (also see notes at end of this write up).

Lab time:
Challenge students to go back and try to explain the basis for the rule they got. Where did the numbers in their equation come from, what do they represent, why does the equation end with a 2?

Group Discussion: Share ideas and findings about what the numbers in the equations represent.

Extensions/Optional Homework:
Ask them, "Where do we go next, what other questions could we investigate here"?

Name:___________________________
In this activity you will build a train made out of one kind of pattern block and record its growing perimeter as you add each new block. The challenge is to look for patterns and rules.
E.G. For a triangle: a one triangle train would have a perimeter of 3, a two triangle train would have a perimeter of 4, etc.

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For this activity start with the yellow hexagon block.

Find the perimeter of a one block train.
What is the perimeter of a two block hexagon train?
Can you find and record the perimeter of a 3 block train, a 4 block train, a 5 block train?
Record your findings.

Based on your findings can you predict what the perimeter of a 10 block train will be?

How about a 50 or 100 block train?

Can you write down any rules that will help you calculate the perimeter of a train?

What patterns can you find? Use whatever tools will be helpful.

Record your data and your findings.

Name:__________________________

Once you have collected data for a hexagon choose a second shape and find the perimeter of a 1 block, a 2 block, and a 3 block train.....

Can you predict what the perimeter of a 10 block train will be?

Can you write down any rules that will help you calculate the perimeter of trains for this block?

What patterns can you find?

Can you predict the perimeter of a 50 or 100 block train?

Record your data and your findings.

Further Challenges:

Try finding perimeter patterns for another blockıs train. Can you make predictions?

Compare your findings to those of another group.

Can you make any generalizations?

Can you make predictions for trains made out of pentagons?

Record your data and your hypotheses/conjectures and your findings

What comes after students find a pattern in the Pattern Block Trains problem.

1. Exploring Students Ideas

In general discussions of rules for pattern block trains often bring up a lot of interesting mathematical issues and it is important to listen and be ready to respond to them.

For example. One seventh grade girl, Semhar, when working out a rule for hexagons, said that you could figure out the perimeter of a train by taking the number of blocks adding the next number and doubling the result. Neither the teacher nor any of the other students in the class had thought of this and most of them doubted that it worked at first. In fact, when they tried it out they found out it does work. Getting Semhar to explain her idea and asking others to try to understand it, apply it, and then apply it to other blocks (it only works for hexagons) was a good mathematical exercise.

[Note that algebraically Semharıs rule can be represented as follows: (N + (N+1)) 2 [using N for the number of blocks and (N+1) for the next number] Simplified this becomes 2(2N+1) and then 4N+2 the familiar equation for a hexagon train. It turned out that this algebraic representation was not ³sensible² for many of the students in this class.]

2. Mapping an equation back onto the blocks

Sometimes students generate rules for the pattern block trains directly from the tables of numbers they generate. If this is the case a next step is to ask them to explain where the numbers in the equations come from. e.g. the rule for hexagons can be expressed as the equation Perimeter = 4 N + 2 (four times the number of blocks plus 2).

The question is where does the 4 come from and where does the 2 come from. Challenge students to come up with an explanation for why there is a 4 in this equation and why there is a plus two in all the equations for the different blocks. The idea is to see if they can map the equation back onto the blocks and see that the equation is not magic but is sensible and that the numbers stand for something. Too often students experience algebra as a completely arbitrary set of rules invented by some nameless person that they have to memorize.

3. Comparing Different Rules

In making rules for the pattern block trains students may come up with rules that look or sound different (e.g. one group of students may articulate their rule in terms of figuring out the total perimeter of all the blocks in a train and then subtract the "covered up" sides while another group may count the perimeter of the first block, add the perimeters of all the middle blocks (which are different because "they donıt have an end showing") and then add the perimeter of the last block). The fact is that both these rules work and it is interesting to challenge students about why rules that sound or look so different produce the same results. Trying to get symbolic representations of these rules is difficult, but may be worth the effort. Trying to make sense out of how they are really the same is a fundamentally algebraic exercise.

4. Making a generalized equation

For really able students one can extend the challenge asking if they can make a generalized equation for all pattern blocks. To do this they will need to add the number of sides of a block as one of the variables in the equation. (e.g. can they write an equation that you could use to tell you the perimeter of a train of 8 12 sided shapes or 11 7 sided shapes).

Additional Pattern Block trains problem notes:

Some teachers like to use the red trapezoid blocks for the first pattern block train problem because of the confusion that often occurs as some students interpret the problem to mean the number of sides and others the total perimeter based on a unit of one (the shorter side of the trapezoid). With a hexagon this does not create a problem because all the sides are the same length, but the trapezoid has one side of length two. Student confusion about whether the problem is asking about number of sides or total perimeter (one side of the trapezoid has a length of two) is a good opportunity to deal with the question of what does perimeter mean.