9 If-Then Statements:
Relations Involving Addition, Subtraction, Multiplication,
Division, and Equality
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This chapter does not provide a model activity for elementary students
Discuss basic properties that underlie arithmetic and beginning algebra
Together with previously discussed topics this chapter explores the basis for most of arithmetic and
beginning algebra
Relating Subtraction to Addition and Division to Multiplication
Addition and Subtraction
 9 boys and 12 girls, how many children in the class?
 21 children, 9 are boys, how many girls are in the class?
o Both problems describe the same situation
o Different problems can be made from this information
o Different problems require different methods for solving
 Addition
 Subtraction
 Missing addend
o Demonstrate the relation between addition and subtraction
o Addition or subtraction can be thought of as positive numbers where a whole is composed of
two parts for the set of whole numbers
 This is commonly thought of as the missing addend model for subtraction
 Addend + addend = sum for addition of whole numbers
 Addend + missing addend = sum for subtraction of whole numbers (Sum – addend = missing addend)
The Relation between Addition and Subtraction
 If a – b = c, then a = c + b
 If d + e = f, then d = f – e and e = f – d
 Might use this relation in several commonly encountered contexts
o Number facts
 Fact families
 Can make learning facts easier and more robust
o Solving word problems
 Open number sentences
 Provides for alternative solutions to problems
 Provides context for discussing the relation between addition and subtraction
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Teacher Commentary 9.1
 3rd grade students
 Some subtracted to find answer, but realized addition could also be used
 Wrote a conjecture to see if subtraction would always work for this type of problem
 Students encouraged to find big ideas in their conjectures
 Teacher provided necessary mathematical language to help students focus thinking
 Students came up with generalized statement for this conjecture using symbols
 If  +  = , then  -  = 
Multiplication and Division
 Related in essentially as addition and subtraction
 Commonly referred to in division as the missing factor model
 Factor x factor = product for multiplication of whole numbers
 Factor x missing factor = product for division of whole numbers (Product  factor = missing factor)
The Relation between Multiplication and Division
 If a  b = c, then a = c x b
 If d x e = f, then d = f  e for e  0, and e = f  d for d  0
 Critical for learning division number facts
Operating on Both Sides of the Equal Sign
 3rd grade students using relational thinking
 345 + 576 = 342 + 574 + d
Operating on Both Sides of the Equal Sign
 If a = b, then a + c = b + c
 If a = b, then a x c = b x c
 If a = b, then a - c = b - c
 If a = b, then a  c = b  c, c  0
 Play a central role in solving algebraic equations
 See 9.1 p. 127
Proving Conjectures about Subtraction and Division
 Prove by relating to corresponding conjectures for addition and multiplication
 Prove d – d = 0 for all numbers
 Prove p  1 = p for all numbers
 Show why we cannot divide by zero: If r  0 = , then r =  x 0. Thus r could only be zero, not any
number
 Show why we cannot divide zero by zero: If 0  0 = , then 0 =  x 0. Thus any number could replace ,
which contradicts the Fundamental Theorem of Arithmetic which says the prime factorization of a
number is unique
 Can also consider fractions or the set of rational numbers for each of these ideas
Inverses
1
1
 For every number r except zero, there is a unique number such that r   1
r
r
 Multiplicative inverse of r or the reciprocal of r
 Any division problem can be recast as a multiplication problem by multiplying by the inverse
1
 p q  p
q
 Any subtraction problem can be recast as an addition problem by adding the opposite
 p  q  p   q 
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Important for students to distinguish vocabulary between inverse and opposite
Addition and multiplication have simpler rules than subtraction and division
Does not mean we do away with subtraction and division, just want to understand the relation between
them
The Basic Properties Revisited
 Some properties more critical than others
 Some properties derived from others
 See table 9.1 p. 130
 Properties of subtraction and division are missing – WHY?
 Zero property of multiplication not included – WHY?