For a science project, Jean wanted to make a scale model of a Space Shuttle. The length of a real shuttle is 37.24 meters, and the height is 17.25 meters.

1. Jean decided to make her shuttle half a meter long.
   1. Fill as much of the following table as you can from just the information given above. (Don’t do any calculations yet! You’ll have to leave one entry blank.)

      	Length	Height
      Real shuttle
      Jean’s model

   2. When you set up a proportion, you use three values you know. In the table, you filled three of the four entries. You can now write a proportion from the table, like this:
      

      

      Use this idea to write and solve a proportion to find the height of Jean’s model.

2. What if you had arranged the original table in a different, but sensible, way? For example, what if it had been set up like either of these?
   

   	Length	Height
   Jean’s model
   Real shuttle

   	Real shuttle	Jean’s model
   Length
   Height

   1. Write and solve the proportion statements that come from these tables.
   2. Do these give the correct height for Jean’s model?
3. Consider this proportion:
   
   1. Does this proportion have a corresponding table? If so, create it. Be sure to write the row and column headers!
   2. Solve the proportion. Does it give the correct height for Jean’s model?
   3. Arrange the values 37.24, 17.25, 0.5, and x into a proportion that has not been used so far. Does it have a corresponding table? Does the solution give the correct height for Jean’s model?
4. For the proportions you’ve looked at that did not have a corresponding table, what about them made it impossible to create the table?
5. Two brothers are reading on a long trip in a car. In the time it takes Rex to read 10 pages, Carlos can read 17 pages. Carlos’s book has 184 pages, and they both start on page 1 at the same time.
   1. Set up and solve a proportion to find the approximate number of pages Rex can read by the time Carlos finishes his book.
   2. Without solving, decide which of the following proportions will give the correct answer:
      1. =
      2. =
      3. =
      4. =
      5. =
      6. =

Hint to problem 4. The structure of a table is very similar to the structure of a proportion. Describe how you decide where in the table a particular value must go.

1. 1. Here is the completed table:
      

      	Length	Height
      Real shuttle	37.24	17.25
      Jean’s model	0.5	x

   2. = ; x 0.2316
2. 1. First table: =
      Second table: =
      The solutions are both about 0.2316.
   2. Yes
3. Consider this proportion:
   
   1. No
   2. x 1284.78; no, this isn’t the correct height.
   3. Possible proportions are as follows. Students may also switch the left-hand and right-hand sides of each equation. (For example, give = instead of = .)

      i.	=		ii.	=		iii.	=
      iv.	=		v.	=		vi.	=
      vii.	=		viii.	=		ix.	=

      Of these, only proportion vi has a corresponding table, and that is also the only proportion that gives the correct height for Jean’s model.

4. Explanations may vary. When a proportion doesn’t have a corresponding table, the four values are arranged so that the numerators, the denominators, or the parts of either fraction don’t have a characteristic in common.

   In other words, to have a correct proportion, there must be some correspondence between parts. For example, the first fraction might be a ratio of lengths. Then the other fraction must be a ratio for the height--but it also must be the corresponding ratio. If one is real/model, then the other must also be real/model and not model/real. The table follows the same rules--each entry is a row has something in common, such as being measurements for the real shuttle, or for the model. Each column also has something in common, such as being the length measurement or the height.
   Teacher’s Note: The fact that a proportion must have corresponding information (i.e. corresponding labels) for numerators (and denominators) and at the same time have corresponding information for each ratio (fraction) is often where students make their mistakes. A class discussion of this problem--just what one does when building a table, and how it relates to the building of a correct proportion--would be well worth the time.
   

5. 1. There are four correct proportions (ignoring which fraction is on which side of the equation): = ; = ; = ; and = . Rex can read about 108 pages (about 108.235) by the time Carlos finishes his book.
   2. ii and iii