## Teaching Notes for Raw Recruits

This project can be useful to introduce two important topics, alternate representations of a problem and invariants.  It is relatively short if you stick with only the original questions posed at the beginning of the project, but can be much longer if you consider some of the extensions.

Depending on the time you have in each class and how quickly the original problem is solved, these notes could be broken up differently to take three or more days.  If you plan for more days and finish early, there are several different extensions that can be discussed.

#### Day 1

Describe the problem to your class.  You might want to try acting out a "live" situation by telling some of your students.  Here are a few tips from Michelle Manes (a member of the Making Mathematics staff) who's tried this:

When I did this in class, it took some real work to have kids doing it right and not just spinning wildly.  Here's what I did:
As a whole class, do a demo:
- Line up six kids.
- Do a trial "left face" so eveyrone knows the right and wrong way.
- Tell one kid to face the wrong way when the order is given.
- Say "left face"  -- eveyone faces left (or right) and stays there.
- Say "If your facing someone, turn around."   Make sure everyone who should, does.
- Say "If your facing someone, switch."
- "Switch" ... "Switch" ...   etc., each time making sure kids stop and check if they're facing someone and switch together.
Ask all of them to face forward again.  Have someone in class who isn't in the lineup pick who faces the wrong way.  Then that student can call "left face" and "switch," making sure everyone follows the rules.
When you split into groups, make sure one person acts as sargeant - picking the starting position, calling "left face" and calling the times to switch.

Have each team try to create initial configurations that take a long time to settle down.  After they all understand the basic problem, suggest that they work out more examples on paper.  At this point, you might want to break the class into groups of two or three students.

After the teams have been working on the original questions for a while longer, see if it might be appropriate to introduce the idea of alternate representations.  You might want to see what representations the teams are using and integrate it into the discussion.  Again, some advice from Michelle Manes:

Teachers should know that this is not a trivial transition to make.  In my class, only one kid figured out a way to write stuff down.  The rest stopped working on the problem in frustration until I gave a way to write it.  What I would do if I were teaching it again: pose the problem of how to represent it on paper and have them work together. Say "To record and analyze data we can't just act it out - we need to write it down."  Have them record all of the steps for a particular case, like the one given in the project statement.   Then ask groups to share the representations they came up with.  The teacher can suggest others, or not, depending on how it goes.

Let teams the groups work until the end of class on the original problems with 6 or fewer recruits.  For homework, choose one or two of the the warmup problems, which will help kids move on to the next stage of the investigation.

#### Day 2

Discuss the work so far.  Have they noticed anything?  Do they have questions?  Allow enough time for for each group to present their observations and/or pose follow-up questions.  Ask the students to consider whether their observations so far would apply to lines of recruits with more than six soldiers.

If none of the teams have noticed the invariant in this problem, you might want to introduce the idea of an invariant and lead the class in a discussion of how invariants can help solve problems.

Time permitting, ask the class if they can think of an extension to "Raw Recruits."  You might want to pose this question as "Can you think of a way to change this project so that it is different in at least one way and might lead to interesting questions?"

#### Ideas for a final report

To assess the success of this activity and to give students a chance to communicate their results you might want to ask them to turn in a report with some of these components.
-  A statement of the problem, in their own words.
-  A representation of the process.
-  Some data.  For example they might take all of the initial configurations of 4 (or 3, or 5)  recruits, record how long they took to reach the final state, and tabulate the results in some way.
-  Conclusions, either from the data for a specific number of recruits; or in more general terms - whatever the student is able to observe.
-  A "new"  but related problem - an extension- that they can design.

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