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Friday Letters - MS Math, Mr. Gritzan

                 May 2, 2008

Dear Middle School Parents,

Mathematics, arguably, revolves about the manipulation of symbols. The very numbers themselves are abstract ideas that stand for the idea of that many elements in a group. Of all the symbols employed,  the equal sign is probably used the most often. This symbol was first used in 1557 by Robert Recorde in his book the ‘The Whetstone of Wit’.

“To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lengthe, thus: =====, bicause noe .2. thynges, can be moare equalle.”

It is the anchor of all those math facts learned in the 3rd grade where 2 plus 3 is equal to 5 and 7 times 8 is equal to 56. In middle school, it is found in the center of equations with variables hovering on either side. To most students, it stands primarily as a giant signpost that indicates the answer   which   immediately follows.

 Often lost in this view of the equal sign is the fact that it also represents   a conditional   statement and represents a relationship between what is on the left and what is on the right.   What number when doubled and increased by 5 is equal to 36?  Today we use symbolic notation and would write the question like this:  2n + 5 = 36. We are looking for the value of n that will make the left side of the statement equal to the right. Any number other than ’15 and a half’ will make this statement false. In algebra, we find solutions to such problems by manipulating the question until it assumes a simpler form. As long as we maintain the equality of the statement by performing each and every adjustment to both sides of the balance we can generate an infinite set of different questions with the same answer.   2n = 31 is a gentler question,  n = 31/2 is simpler yet. Students that do not appreciate this aspect of the equal sign struggle when it comes to solving equations.

It gets even more complicated when inequalities are introduced and we are forced to think in terms of more than a single answer.  What numbers when doubled and 5 is added to them are bigger than or 36? Well, again we would employ modern symbolic notation and express this as       2n + 5 > 36. This time, there are infinite possibilities for making this statement true and the singular answer of  15.5 from the earlier equality is the lower boundary of these values. 15.6 works, so does 15.50006.

All the grades have been working on inequalities. Sometimes this gets left out of the curriculum. I’ve made a point of making sure that this gets included as much as the simple equal sign denoting a single answer in order to strengthen our student’s ability to think in terms of changing mathematical relationships. At the end of the day, the answers are transitory and fleeting. The true purpose is to be able to think proportionally.

Sincerely,  Thorsten Gritzan                                                                   tgritzan@ringmountain.org