Whether you're a teacher learning to use manipulatives or a student learning to use signed numbers, GO GET SOME BOTTLE TOP or COINS so you can complete the operations described here.

    If you just wish WRITTEN INSTRUCTIONS on how to compute with signed numbers, use the last page.

    PRACTICE is found on other pages linked through the QUESTION buttons.

    For a CALCULATOR demonstration, press the CALAULATOR button.

    Begin your work here with REPRESENTATION. It is the most important part of the page.

    Though four operations are listed here, there are really only three sets of instructions. This page justifies the algorithms for signed number computation listed on the last page. It also provides practice briefly and links to more practice on other pages.

• Representation
• Addition
• Subtraction
• Multiplication or Division is not presented manipulatively.

    Use bottle tops as token for numbers.

    Use heads-up tops for positives.

    Use bottoms-up tops for negatives.

• Zero is represented by an equal number of positives and negatives.
  
• A positive number is represented by more positives than negatives.
  
• A negative number is represented by more negatives than positives.
  
• The "take the opposite," flip every top. To see this done, use the RESTATE button, then your browser to return to this page.

• Respresent each addend with tops.
• Simplify by pairing a positive with a negative to make a zero.
• What's left is the sum.

ADD:
+ 4 + 3	- 4 - 3	- 4 + 3	+ 4 - 3

    Two methods for the manipulative operation subtraction are presented. The author prefers the first method because it parallels the terminology used in algebra classes. Some students prefer the second definition of subtraction because it is the definition they learned in first grade.

    A teacher should investigate both methods. A student should with practice be able to complete this subtraction menatlly, without the use of manipulatives.

    Two sets of RESTATE buttons permit examples of both computation methods.

SUBTRACTION means ADD THE OPPOSITE.

1st: Represent the first number, the minuend.

 

2nd: Represent the opposite of the second number,

the subtrahend.

 

3rd: Simplify by pairing a positive with a negative

to make a zero.

 

4th: What's left is the remainder.

SUBTRACTION means TAKE AWAY.

1st: Represent the first number, the minuend.

 

2nd: Remove, TAKE AWAY, the second number,

the subtrahend, adding zeros -- pairs of

positives and negatives -- as needed.

 

3rd: Simplify by removing zeros.

 

4th: What's left is the remainder.

    Manipulatives are not the best format for the presentation of signed number multiplication or division. The "rules say:"

• An ODD number of NEGATIVE SIGNS is NEGATIVE.
• An EVEN number of NEGATIVE SIGNS is POSITIVE.

    And an explanation based on factors and opposites is shown on the last page and on the Opposites on Your Computer's Calculator page.

    The explanation here involves money (the manipulative or token) and a win/loss (signed) monetray meaning for.

• (3)(-2)= -6

  Three times, you lost 2 dollars. Result: you lost 6 dollars.

   

• (3)(+2)= 6

  Three times, you won 2 dollars. Result: you won 6 dollars.

   

• (-3)(+2)= -6

  This equals (-1)(3)(+2) or the opposite of (3)(+2). Result: you lost 6 dollars.

   

• (-3)(-2)= 6

  This equals (-1)(3)(-2) or the opposite of (3)(-2). Result: you won 6 dollars.