Overview for Exam 3
Included is a list of main topics you should know for the exam. To see how these might be asked, study your class notes and review your HW problems.
· Be able to solve basic equations using the allowed properties (i.e., the ones you have used all through algebra – Theorems 4.1 – 4.6, pp. 207-209).
· Be able to demonstrate simple algebra properties in the integers using the balanced scales approach.
· Concentrate (as always) on HW-type problems, such as being able to interpret word problems and translate them into equations that can be solved. Make sure you can check whether you have the right one or not.
· Know about relations (any collection of ordered pairs or graph) and how to check whether a relation on a set is reflexive, symmetric and/or transitive:
o Reflexive: (a,a) is in the relation for any a from the underlying set.
o Symmetric: If (a,b) is in the relation, so is (b,a).
o Transitive: If (a,b) and (b,c) are in the relation, so is (a,c).
· Know the definition of a function (especially uniqueness) and the ways of interpreting whether a relation is a function: arrow diagrams, ordered pairs, and graphs.
· Know that a sequence is a function with values plugged in from the natural numbers. Be able to understand sequences and put them together as in your HW problems and in-class examples.
· Understand what is meant by the set of integers.
· Know and be able to carry out the following approaches for representing addition of integers: chip/charged field, number-line, and pattern.
· Know the definition (p. 255, blue box) of absolute value and how to calculate absolute values.
· Understand the properties of integer addition: closure in the set of integers, commutative, associative, and the additive identity element (that is, 0).
· Know and be able to carry out the following approaches for representing subtraction of integers: chip/charged field, number-line, pattern, and missing addend.
· Always use parentheses when adding, subtracting or multiplying by negative integers. Order of operations is also important in the integers.
· Know and be able to carry out the following approaches for representing multiplication of integers: pattern, chip/charged field, number-line (using starting direction and possible change of direction)
· Understand the properties of integer multiplication: closure in the set of integers, commutative, associative, multiplicative identity element (that is, 1), distributive property of multiplication over addition/subtraction. Ours (and the standard) zero multiplication property is that ab = 0 means either a = 0 or b = 0.
· Remember the difference of squares formula (top of p. 275) and its use in mental math (i.e., something like 39*41 = (40-1)(40+1) = 40*40 – 1) and algebra as in your HW.
· Know the definition of division in the integers – that is, a/b means there is a unique integer c such that bc = a. This relates division to a multiplication property.
· Understand and be able to use the definition of b divides a (or equivalently a is a multiple of b) and the symbol b|a (remember that this means there is a unique integer c such that bc = a). Remember that this means either a positive or negative integer may divide a positive or negative integer as well.
· Know the basic concepts of divisibility as per the theorems we studied, the T/F examples and your HW. For example, everything except 0 divides 0.
· Know the divisibility rules and be able to apply them for 2, 3, 4, 5, 6, 8, 9 and 10.