Download Pearson Square

Pearson’s Square: A Different Approach from Simultaneous Equations
While looking at some of the mathematics used in agriculture I came across a problem
such as the following:
Mix 100 pounds of an 11% protein feed made up of corn (9% protein) and
soybeans (38% protein). How many pounds will you need of corn and
how many pounds will you need of soybeans?
Upon reading this problem, my mind automatically went to simultaneous equations. I knew I
could create one equation to represent the percentage of protein and one equation to
represent the proportions needed of each feed type to get the final feed mixture. This gave
me,
where x = pounds of corn and y = pounds of soybean,
1. .09x + .38y = 11
2. x + y = 100
Solving equation 2 for x,
2. x = 100 – y
Then substituting back into equation 1 to get y,
1. .09(100 – y) + .38y = 11
9 - .09y + .38y = 11
.29y = 11 – 9
.29y = 2
y = 6.9 pounds
So x = 100 – 6.9 = 93.1 pounds.
I thought this was a pretty simple (and painless) way to approach the problem.
Apparently, in the eyes of someone who does not enjoy algebra it is not. I looked at the
Cynthia A. Thomas
EMAT 6690
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solution for the problem and found that simultaneous equations were not even mentioned!
Instead, many people in the agriculture world use what is called Pearson’s Square (or the
Pearson Square) to solve problems involving rations.
Brief research on this mysterious “Pearson Square” was inconclusive on finding out
where the method originally came from. All I could determine is that it is used in agriculture
(for feed mixtures), winemaking, and sausage making. It is a simplified way of figuring out the
number of “parts” of each product is needed to make the correct percentage for the final
product. The attached Geometer’s Sketchpad file explains the process of Pearson’s Square for
determining rations.
See: Pearson’s Square (GSP).
I found Pearson’s Square interesting mainly because simultaneous equations are one of
my favorite topics to teach in the classroom. I enjoy every aspect of solving simultaneous
equations, from graphing to substitution and elimination. I think it’s an interesting subject
because it is fairly easy for students to “see” where the equations come from and it is
convenient for the teacher to come up with real world problems where simultaneous equations
are applicable.
Pearson’s Square is great because the problems it solves, involving rations and parts and
percentages, can often be difficult to understand with simultaneous equations. Such as above,
where I worked the problem using simultaneous equations, equation 1 is .09x + .38y = 15. Why
fifteen? Why not .15? Everything else in the equation is in decimal form so it does not make
“sense” firsthand why I used fifteen. The explanation is that you have to find the total pounds
Cynthia A. Thomas
EMAT 6690
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of protein wanted in the final mixture, hence .15 * 100 = 15 pounds. This was not obvious to
me at first so I immediately know that some students may not completely understand it either.
Pearson’s Square gives a method for working out what could be confusing problems in a
straightforward method that is actually used by people all over the country in the real world
(always a plus in any classroom). It works because you are determining how many “parts” of
the final mixture come from each of the beginning feeds (or wines, sausage products, etc.).
These parts are used to find the ratio of one beginning product to the other.
This method is probably not ideal to be used on its own in a mathematics classroom.
Obviously it is not found in a normal mathematics curriculum but that does not mean it is not
worthwhile. It is a great method for students that may be involved in agriculture education
courses or even for some students that may struggle with typical simultaneous equations
problems. It is always nice to see a different side of mathematics and a different way to work
out a problem. I hope to use Pearson’s Square in my own classroom both to generate interest
and to compare with simultaneous equations.
Cynthia A. Thomas
EMAT 6690
Page 3
Resources
http://www.northtexaswinemakers.org/Wine/using_pearsons_square.pdf
http://www.gactaern.org/ms/curriculum_ag_ms_8th.html
Cynthia A. Thomas
EMAT 6690
Page 4