From: Gail (for Teacher2Teacher Service)
Date: Apr 18, 2010 at 08:21:21
Subject: Re: Difference between ordered lists, and a tree diagram

Hi Teresa,   

I think it would be great for your students to see the relationship between
the two. In the end, you probably want them to realize that if they can
create a tree, they don't need to do the ordered list. An ordered list gives
them all the possible outcomes, and from it they can determine the
probability of a certain type of outcome occurring. They can really do the
same thing with a tree.

For example, if you are looking at the different sundaes that can be made 
from three types of ice cream, two types of sauce and two types of sprinkles,
you could start your lists with one of those categories and then 
keep listing until you had them all:

Chocolate  - hot fudge - nuts
Chocolate  - hot fudge - coconut
Chocolate  - butterscotch- nuts
Chocolate  - butterscotch- coconut

Vanilla    - hot fudge - nuts
Vanilla    - hot fudge - coconut
Vanilla    - butterscotch- nuts
Vanilla    - butterscotch- coconut

Strawberry - hot fudge - nuts
Strawberry - hot fudge - coconut
Strawberry - butterscotch- nuts
Strawberry - butterscotch- coconut

Or you could make the tree (I can't figure out how to make the connecting 
lines, here...

                          - nuts
           - hot fudge 
                          - coconut

Chocolate
   
                          - nuts
           - butterscotch
                          - coconut
           

                  

                          - nuts
           - hot fudge 
                          - coconut

vanilla

                          - nuts
           - butterscotch
                          - coconut
 



                          - nuts
           - hot fudge 
                          - coconut

strawberry

                          - nuts
           - butterscotch
                          - coconut


If you wanted to know what the chance was of getting a sundae with 
butterscotch topping if you just ordered randomly, you just count the times 
it shows up on the ordered list: 6 times out of 12 (or half the time, 50% 
chance)

To do that with the tree, your students should notice that you end up with 
the same number of outcomes (12), and there are three branches out of 6 
naming butterscotch as the topping. This could be a great way to point out 
how useful it is that they know about equivalent fractions, too!

One thing I would be sure to do is to have them find the outcomes of the 
ordered list "hidden" in the tree. For example, if they are looking for 
"Vanilla - butterscotch- nuts" in the tree, they start with Vanilla, then 
take the bottom branch to butterscotch, and the top branch that comes next 
for nuts. They won't readily see this, unless you guide them to "discover" 
it.

One thing I have done is bring in little vanilla wafers, oatmeal and 
chocolate cookies, some different types of frosting, and some different 
sprinkles.  First we determined all the possible ways we could decorate the 
cookies, and then had a cookie decorating session.

Eventually you will want your students to figure out that they can find the 
total number of outcomes by multiplying, but if they are actually looking 
for certain outcomes, knowing that won't get them the answer they are 
looking for. A tree or ordered list is a necessary skill, especially when 
they begin looking at dependent and independent events in probability later 
on.  

Can you tell I love desserts?  Often this is done with outfits, or
sandwiches. I hope I have given you something to start with.

 -Gail, for the T2T service

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