What is the operation? What is the sign of the integer?

There is so much within the current, common teaching of mathematics that is wrong, yet just accepted that it is correct.  Below, is a link to a letter that I emailed to Cathy Williams, Coordinator, Curriculum and Instruction Unit at the San Diego County Office of Education.  I sent the letter and July 23, 2009 and did not recieve a reply from Cathy.  I will follow up on this some time, soon.

The point of the letter is that students are often (almost always) taught operations with integers without having to distinguish between the operation and the sign of the integer.  Often, the symbol for subtraction is magically attached to the integer, making the integer negative.  This is not mathematically correct.  I believe that the lack of attention to these details creates confusion for students, many of whom never are able to work their way through that confusion.

So, take a look at the link and let me know what you think.  Thanks.

What's the operation?  What's the sign of the integer?

An Introduction to Equations
There are several distinct concepts either introduced or reviewed during this introduction to equations. These are:
1.     balance – expressions on each side of the equal sign are “balanced” or equal
2.     factors and a little bit of multiples, but mostly factors
3.     Fact Families; both for multiplication/division and addition/subtraction
4.     Four operations
        related in pairs of inverse operations
5.     the fraction bar used as division
6.     the answer to a division problem will tell how much per 1 group (in each group)
7.     the identity property of 1 for multiplication
8.     a number divided by itself equals 1
9.     expressions can be switched across the equal sign:
        3 = x is the same as x = 3
10.   An expression as one side of an equation
 
There are several physical ideas of balance that can be used to show the concept:
1.     a pan balance
2.     a teeter totter or see-saw
 
Seesaw-aa.jpg
 
3.     a hanger -- tie cups to each end
4.     tug of war
5.     a meter stick balanced across a second meter stick on its edge
        can use centimeter cubes to balance each side
6.     There is a nice tetter-totter game at the website:
        http://www.meddybemps.com/teetertotter/index.html
7.     Two good activity books with activities on balance (I have both of these):
        In The Balance; Algebra Logic Puzzles by Lou Kroner ISBN 0-7622-0551-2
        Algebra Puzzles & Problems; Grade 6 by Greenes and Findell
        ISBN 0-7622-0558-X (now titled Groundworks; Creative Publications)
Using multiplication as an example:

  Think of the Fact Family for 18


To solve a multiplication equation, think of "calculating forward" -- what is the OPERATION.
Now, think of "calculating backward" -- the INVERSE operation; UNDO the operation.
 
  
As you move more into algebra, you will use the division symbol less and the fraction bar more to show the operation, division.  The figure, below, include the identity property for multiplication (1x = 3; x = 3) and simplifying fractions by canceling common factors in the numerator and denominator (6/6 = 1)
























 

California 6th grade math standard Algebra and Functions (AF) 1.1

AF 1.1:  write and solve one-step linear equations in one variable.

The main idea of the Holt (course 1: Number to Algebra; 2008) lessons:  lsn 1-7 through 1-11 is that an unknown amount can be calculated by “undoing” an operation.  So, we should talk about the 4 basic operations and the inverse of each operation.  For multiplication and division, we can also show multiplying by a unit fraction is the inverse of multiplying by the denominator, e.g., ½ of 8 = 4; 2 x 4 = 8; 3 x 5 = 15; 1/3 of 15 = 5.  This is also a nice way to make the point that 2/2 = 1; 3/3 = 1; etc. because ½ of 2 = 1 = 2/2; 1/3 of 3 = 1 = 3/3; ¼ of 4 = 1 = 4/4; etc.

So, the idea is to put together several problems and ask, 1) what is the operation; 2) what is the inverse operation; calculate “forwards”, and, then, undo the calculation by calculating “backwards”.

Examples:
1.        3 x 7 = 21               operation is multiplication; the inverse is division
          Think of this as 3 groups of 7 things,
          so, 21 things divided into 3 groups = 7 per group.
          21 divided by 3 = 7
          1/3 of 21 = 7
          3/3 x 7 = 21/3     (this is the algebra: divide both sides by 3)
          1 x 7 = 21/3
          7 = 7
 
2.       5 + 3 = 8                  operation is addition; the inverse is subtraction
          How do you get back to 5 from the 8?
          8 – 3 = 5
          Addition and subtraction problems work well on a number line,
          particularly when adding and subtracting integers.
          The Algebra:  5 + 3 - 3 = 8 - 3     (subtract 3 from both sides of the equation)
          5 = 5
 
3.       multiplication and division with integers:
          4 x -8 = -32            operation is multiplication; the inverse is division
          think of this as 4 groups of -8; -32 divided into 4 groups = -8 per group
          -32 divided by 4 = -8
          ¼ of -32 = -8
          4/4 x -8 = -32/4     (this is the algebra: divide both sides by 4)
          1 x -8 = -32/4
          -8 = -8

summer planning - the beginning

I've spent the last month mulling over some things to emphasize during this next school year (2010-2011) both in the math content as well as behaviorally. First the behavior: 1.  students must work neatly and carefully:  "neatness and completeness" 2.  be a responsible student:      supplies      behavior 3.  A few phrases to say often      "You must be RESPONSIBLE"      "It's on YOU!"      "It is what you make of it." The content: This is a list of four concepts that I think are ... (read more)