09-03-2009Addition and Subtraction of Whole NumbersWhole numbers answer the question “How many objects are there in a set?”Set Model of Addition for Whole Numbers• Recall that two sets are disjoint if they have no elements in common, i.e.A ∩ B = ∅.• addition Suppose a and b are whole numbers. If A and B are disjoint sets so that n(A) = a andn(B) = b then the sum of a and b, written as a + b is given bya + b = n(A ∪ B).• Example: Combining a box of ten crayons and a box of five crayons gives 15 crayons.• Physical Models: Blocks.Measurement Model of Addition for Whole Numbers• Number line representation: numbers are distances from zero.Properties of Addition for Whole Numbers• Closure property: If a and b are whole numbers then a + b is also a whole number.• Commutative property: a + b = b + a• Associative property: (a + b) + c = a + (b + c)• Additive identity property: a + 0 = a• Properties can be illustrated with any of the models from section 2.2.Subtraction for Whole Numbers• If a and b are whole numbers then the difference of a and b, written a − b, is the unique numberc such that a = b + c.• Take away model: Start with a objects take away b objects and determine how many are left.• Missing addend model: Start with a collection of b objects. How many more objects are needed toget to a objects?• Comparison model: Start with a collection of a objects and a collection of b objects. How manymore objects are in the collection of a objects?• Number line model. Draw a and b on the number line. Find the distance between a and b.(Remember to always write positive numbers with right pointing arrows!)More problems with Venn-diagrams. Use Venn diagrams to solve the following problems:• There are 150 senators on the student senate and there are three optional committees: Greek life(G), academic affairs (A), and social life (S). There are 30 senators on G, 50 on A, and 40 on S.Also, 10 senators are on A ∩ G, 10 are on A ∩ S, 15 are on S ∩ G. There are 5 senators on all threecommittees. How many senators did not volunteer for any committees? Answer: 60• There are 25 people at the ice cream social. 15 have chocolate; 15 have vanilla, 10 have strawberry;10 have both chocolate and vanilla; 5 have both chocolate and strawberry; 5 have both strawberryand vanilla; 3 people have all three. How many people have no ice-cream?
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