Ineed to to plan an investigation to answer the the following material is best at keeping a beaker of water hot: foil, paper or foam

Foil would be the best option. Paper does not insulate and is very thing, while foam is better at retaining cold than heat. The foil with heat up with the water and give energy back to the system.

 
-- Get three beakers, all with identical shape and size.

-- Get a lab thermometer if possible ... the kind that's made of
glass and is like a foot long.  If you can get 2 or 3 of them to
work with, that would be even better.
If you do get to use more than 1 thermometer, then whatever kind
they are, they should all be the same thing.

-- Get a timer or a clock that's easy to read.

-- Get a fresh pad of paper and a few pens.

-- Draw three long columns from the top of the paper to the bottom.
At the top of one column, write "Foil".  At the top of another column,
write "Paper, and at the top of the third column, write "Foam".

-- On the lines down one side of the paper, write  "0",  "10",  "20,  "30",
"40",  "50", ... and keep going all the way to, I'd say", at least  "150".

Now you're ready to investigate.


-- Wrap one beaker with foil, one with paper, and the third one
with foam.  Put a rubber band or a strip of tape around each one
to keep the wrapping in place. 
This is not easy.  In order to get any meaningful conclusion from your
investigation, all three wraps should have the same thickness !
Any material that's 6 inches thick will certainly be a better insulator
than any other material that's only as thick as a sheet of paper.
You want the results of your investigation to be caused only by
the kind of material ...NOT by how thick it is.

-- Set up your beakers in a place where they can stay for a few hours,
and with some space between them.  It should be a place where there's
not too much wind, and where the temperature in the room won't make
any sudden changes.

-- Boil a teapot of water.  When the water boils, quickly pour the same
amount of water into all 3 beakers, and start the timer or the clock.

-- Take the temperature of the water in each beaker, and write it down
on the top line of your paper ... the one labeled " 0 " . 
Do this very carefully.  Don't move the beakers. Don't take anything out
of them, and don't put anything into them except the thermometer. 
If you were able to find three lab thermometers to use, then leave one in each
beaker, and just read them.  You don't need to move or touch anything
as long as your investigation lasts.

-- Take the temperature of the water in each beaker every 10 minutes.
Write each one down on the paper, in the space under the proper wrap
material for that beaker, and on the line for the proper time of the reading.

-- Keep reading and writing every 10 minutes.
If your paper is numbered up to "150", then it'll take you 2 and 1/2 hours,
and you'll have lots of delicious numbers to play with.

When you're done, before you take the data into your office to analyze it,
unwrap the beakers and clean and dry them.  Also, any other equipment
that you borrowed for your investigation, clean it and return it to whoever
lent it to you.  Say "Thank you" and give them a piece of chocolate.

Now, you can draw beautiful graphs with your numbers.  The graph will
have three lines on it ... one for each material.  The lines will all start out
at the same point, and then one will drop fast, one will drop slow, and one
will drop in between the other two.  I don't know which one will drop the
slowest  That's the whole point of your investigation.  The slowest one
is the winner ... it's the best insulator, no matter whether you're trying
to keep something hot OR cold.

One thing that might surprise you when you graph your numbers . . . the
lines of temperature are NOT straight.  The lower each temperature gets,
the slower it cools more.  Each line begins dropping fast, but drops slower
and slower as time goes on, and heads for a smooth landing at the bottom
as it approaches the temperature in the room.

You're welcome, and good luck.
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So what is the question