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Project Grant Team
John S. Pazdar Peter A. Wursthorn Project Director Principal Investigator Capital Community College Capital Community College Hartford, Connecticut Hartford, Connecticut
This project was supported, in part, by the Patricia L. Hirschy National Science Foundation Principal Investigator Opinions expressed are those of the authors Asnuntuck Community College and not necessarily those of the Foundation Enfield, Connecticut
SPINOFFS
Spinoffs are relatively short learning modules inspired by the LTAs. They can be easily implemented to support student learning in courses ranging from prealgebra through calculus. The Spinoffs typically give students an opportunity to use mathematics in a real world context.
LTA - SPINOFF 18A Space Exclusion Versus Inclusion
LTA - SPINOFF 18B Creating a Scaled Room Design
Karen Gaines - AMATYC Writing Team Member St Louis Community College - Meramec, Kirkwood , Missouri
Johanna Halsey - AMATYC Writing Team Member Dutchess Community College, New York
Kristine Kennedy - NASA Scientist/Engineer Kennedy Space Center, Florida (Currently at The Johnson Space Center, Houston, Texas)
NASA - AMATYC - NSF
SPINOFF 18B
Creating a Scaled Room Design
When thinking about various placements of furniture in a room, it is easiest to work with scaled, two dimensional models of the room and the furniture. The purpose of this lesson is to help you determine how to create a meaningful scale to use for your model, and how to convert the original dimensions of the room and furniture to the scaled dimensions.
When appropriate, we will use the common abbreviations for units: Inch or Inches - in Foot or Feet - ft Centimeter or centimeters - cm
We will create appropriate scales for two mediums:
- a piece of standard size poster board (22 inches by 28 inches) which contains no previous grids
- and a piece of 8 1/2 inch by 11 inch, quad ruled graph paper (4 squares per inch)
Let’s start by working with a living room that measures 16 feet by 12 feet.
Considerations
- Use the majority of the space available for your model.
- Try to use grid sizes that will be easy to work with. Sides of 1 unit, 1.5 units, 2 units, 2.5 units, etc., often work well.
Using the Poster board
Clearly, we want to align the longer side of the living room with the longer side of the poster board. That means we want to determine a scale with which we can reflect 16 feet using most of the 28 inches available on the poster. We will start out by deciding on a scaling equation (an equation which shows what measure in the scaled model will equal a unit measure of the original object). It would be easiest to try using a scale so that 1 foot = 1 inch. However, using this scale would mean that we would use only 16 inches of the poster. We should try to see if we can find an equation so that more of the poster board is used. Let’s try the equation, 1.5 inch = 1 foot.
Is the poster long enough to accommodate this scale? To determine this, we want to convert 16 feet into inches using this scale. We would multiply 16 feet by a scaled unit fraction (a fraction equivalent to 1 which incorporates the scale). A scaled unit fraction will often be referred to as a scaled unit factor or simply a scale factor. In all the work we do, it is crucial to keep track of appropriate units.
NASA - AMATYC - NSF
Method I
Changing original measurement in inches to a measurement in feet
and then applying the scaled unit factor of
1 5 in 1 ft
From common knowledge, we know that 1 foot = 12 inches. We can divide both sides by 12
inches to create the unit fraction:
1 ft 12 in
If we multiply our original measure of 20 inches by this unit fraction and cancel units where
possible, we end up with:(20 in) 1 ft 12 in
ft
FHG IKJ = = feet
Since we now have our measurement in feet, we can multiply this by our scaled unit factor: 5 3
ft
1.5 in 1 ft
FHG IKJFHG IKJ =2.5 inches
Likewise, the other original dimension would be:
(15 in)
1 ft 12 in
ft =
F feet
HG^
I
KJ^
Then: 5 4
ft
1.5 in 1 ft
F 1.875 inches
HG^
I
KJ
F
HG^
I
KJ^
=. To make our measuring easier, but still fairly accurate,
we would probably choose to use 1.9 inches.
So our rectangle representing the coffee table should be 2.5 inches by 1.9 inches.
Method II
Creating a new scaled unit factor which can be used to convert original inches to scaled inches
We recognize that both sides of our scaled equation need to involve inches. Therefore, we again use the common knowledge that 1 foot = 12 inches, and write the scaling equation of 1. inches = 1 foot as 1.5 inches = 12 inches. Dividing both sides by 12 inches produces the scaled
unit factor of 1. 12
. (Notice that the units cancel out here, so our scaled unit factor will not cause
unit cancellation when multiplied by an original dimension in inches.)
To convert an original measurement in inches to a scaled measurement in inches, we will
multiply the original measurement by 1. 12
NASA - AMATYC - NSF
For the end table:
(
in) F = 2.5 inches
HG^
I
KJ
in) F = 1.875 inches
HG^
I
KJ^
(exactly what we got using Method I!)
You can use whichever of these methods makes the most sense to you.
Practice Exercises
Using the scaling equation of 1.5 inches = 1 foot, and either Method I or Method II, determine the scaled dimensions of:
- An entertainment unit which is 4 feet long and 22 inches wide.
- A coffee table which is 3 feet long and 20 inches wide.
- A chair which is 32 inches long and 32 inches wide.
- A bookcase which is 2.5 feet long and 1 foot deep.
Make sure you show all your work clearly and completely, so that you will be able to explain exactly what you did to another classmate.
Using a piece of grid paper (Quad ruled - 4 squares per inch)
Now we have a grid of squares already on the paper, where we know that each side of a square is 1/ inch long.
Again, we will want to line the longer side of the living room up with the longer side of the paper. Realistically, we would want to avoid using any incomplete squares along the border of the paper. If you count the number of complete squares along the long edge of the paper, you should find at least 42 complete squares. Doing the same along the shorter edge, you should have 32 complete squares. Therefore, if we use only this number of complete squares, we will be utilizing a region that measures 8 inches by 10 1/2 inches.
Again, using our living room dimensions of 12 feet by 16 feet, we want to determine an appropriate scaling equation, which will help us develop a scaled unit factor for conversion purposes.
Try using 1/2 inch = 1 foot. Following the same general steps as we did for the poster board, divide
both sides of this equation by 1 foot, creating a scaled unit factor of 0 5.^ in 1 ft
