how much lasagna

Question

there are two thirds of a pan of lasagna in the refrigerator. Bill and his friends eat half of what was left. write a number sentence and draw a model to represent the problem. How much of the pan did they eat

Answer 1

In this problem, we are asked to visualize the action of removing half of the amount of something and reason about how much is left.  Since removing half will leave exactly one half remaining, we can think about this in terms of dividing what's left of the pan of lasagna into two even portions-- one which Bill and his friends will eat, and one which will be saved in the pan for later.  Since the problem states that the pan is two thirds full to start out, splitting this amount into two even portions, each of size one third pan, reveals the answer to the problem.

 

Bill and his friends eat one of the two thirds that were in the pan, leaving one third left over.  Essentially there will be one portion left for every two portions there were to begin with.

 

We can express this numerically:

(2/3 pan lasagna to begin) * (1 portion left / 2 portions to begin) = (1/3 pan lasagna left)

Notice how the units work out cleanly to relate the amount of lasagna before Bill and his friends ate some to the amount afterwards.

 

We can also visualize this graphically:

 

Before:

- - - - - - - - - - - - - - - - - - - - - - -

|   1 / 3    |   1 / 3   |   1 / 3   |

|    pan    |     pan   |  empty |

- - - - - - - - - - - - - - - - - - - - - - -

After:

- - - - - - - - - - - - - - - - - - - - - - -
|   1 / 3    |   1 / 3   |   1 / 3   |
|    pan    |  empty  |  empty |
- - - - - - - - - - - - - - - - - - - - - - -

Notice how in the "after" diagram, exactly half of the portions present in the "before" diagram have been emptied out, leaving exactly one third of the pan left.  (As a side note, the 1/3 portions of lasagna may be easier to reason about if you use a round pan.)

 

I think I'll have seconds!