Chapter 2: Sets, Whole Numbers, and Functions

Chapter 2: Sets, Whole Numbers, and Functions

2.3 Addition and Subtraction of Whole Numbers

2.3.1. Vocabulary

2.3.1.1.  whole numbers – W = {w | w Î 0, 1, 2, 3, …}

2.3.1.2.  addends – terms being added together

2.3.1.3.  sum – the result of combining two or more addends

2.3.1.4.  greater than – larger than another quantity - >

2.3.1.5.  less than – smaller than another quantity - <

2.3.1.6.  greater than or equal to – larger than or the same as another quantity - ³

2.3.1.7.  less than or equal to – smaller than or the same as another quantity - £

2.3.1.8.  missing addend – the addend needed when only one addend and the sum are known

2.3.2. Addition of Whole Numbers

2.3.2.1.  Set Model

2.3.2.1.1.  one way to represent addition

2.3.2.1.2.  important to note that the sets being added are disjoint

2.3.2.1.3.  non-disjoint sets lead to faulty conclusions

2.3.2.2.  Definition of Addition of Whole Numbers: Let A and B be two disjoint finite sets. If n(A) = a and n(B) = b, then a + b = n(A È B).

2.3.2.3.  Now try this 2-11 p. 87: Work in your groups

2.3.2.4.  Number-Line Model

2.3.2.4.1.  See figure 2-17

2.3.2.4.2.  use directed arrows for each step – addend, addend, sum (separate arrow above addends)

2.3.2.4.3.  Definition of Less Than: For any whole numbers a and b, a is less than b, written a < b, if and only if there exists a natural number k such that a + k = b.

2.3.2.4.4.  Trichotomy Principle: One of the following three statements must be true for any a and b where a and b are any real numbers: a < b; a > b; a = b

2.3.3. Whole-Number Addition Properties

2.3.3.1.  Property: Commutative property of addition of whole numbers – If a and b are any whole numbers, then a + b = b + a.

2.3.3.1.1.  Counting on strategy can be problematic here

2.3.3.1.2.  Students need to know fact families

2.3.3.2.  Property: Associative property of addition of whole numbers – If a, b, and c are any whole numbers, then (a + b) + c = a + (b + c).

2.3.3.2.1.  order is not important in addition

2.3.3.2.2.  parentheses not usually used if more than 3 addends

2.3.3.3.  Property: Identity property of addition of whole numbers – There is a unique whole number 0, the additive identity, such that for any whole number a, a + 0 = a and 0 + a = a.

2.3.4. Mastering Basic Addition Facts

2.3.4.1.  Addition facts

2.3.4.1.1.  digit + digit = sum

2.3.4.1.2.  100 addition facts

2.3.4.1.3.  If know additive identity, how many more to learn?

2.3.4.1.4.  If know all +1, how many left to learn?

2.3.4.1.5.  If know additive identity and +1, how many left to learn?

2.3.4.1.6.  If understand commutative property of addition, how many facts left to learn?

2.3.4.1.7.  If know commutative and doubles, how many facts left to learn?

2.3.4.2.  Methods of learning the addition facts

2.3.4.2.1.  counting on

2.3.4.2.2.  doubles

2.3.4.2.3.  making 10

2.3.4.2.4.  doubles plus one

2.3.4.3.  Addition fact families

2.3.4.3.1.  0 0 + 0

2.3.4.3.2.  1 1 + 0 0 + 1

2.3.4.3.3.  2 2 + 0 1 + 1 0 + 2

2.3.4.3.4.  3 3 + 0 2 + 1 1 + 2 0 + 3

2.3.4.3.5.  etc.

2.3.5. Subtraction of Whole Numbers

2.3.5.1.  Subtraction facts

2.3.5.1.1.  sum – digit = digit

2.3.5.1.2.  How many subtraction facts are there?

2.3.5.2.  Take-Away Model

2.3.5.2.1.  Also called the set model

2.3.5.2.2.  remove one set from another set

2.3.5.2.3.  see figure 2-21

2.3.5.3.  Now try this 2-12 p. 92: Work in your groups

2.3.5.4.  Missing Addend Model

2.3.5.4.1.  if think of subtraction as missing addend, then if know addition already know subtraction

2.3.5.4.2.  sum – addend = missing addend

2.3.5.4.3.  re-write: addend + missing addend = sum

2.3.5.4.4.  using this method, if I can add I already know all of the subtraction facts

2.3.5.4.5.  cuts down on what needs to be memorized

2.3.5.5.  Definition of Subtraction of Whole Numbers: For any whole numbers a and b, such that a ³ b, a – b is a unique whole number c such that a = b + c.

2.3.5.6.  Comparison Model

2.3.5.6.1.  matches are removed

2.3.5.6.2.  see fig. 2-22

2.3.5.7.  Number-Line Model

2.3.5.7.1.  Most children find what your book has done in fig. 2-23 to be a mass of confusion

2.3.5.7.2.  recommend using missing addend model

2.3.6. Properties of Subtraction

2.3.6.1.  Now try this 2-13 p. 93: Work in your groups

2.3.6.2.  be careful, can’t subtract in whole numbers if a < b

2.3.6.3.  can subtract in integers

2.3.6.4.  try to not undermine future learning

2.3.7. Ongoing Assessment p. 95

2.3.7.1.  Home work: 2c, 2d, 4a, 4b, 4c, 5a, 5c, 9a, 9c, 9e, 10a, 10c, 12a, 15a, 20a