Download Reasoning in Algebra

2-4
Reasoning in Algebra
Objectives: 1) To develop an awareness of the structure of a mathematical
system, connecting postulates, logical reasoning, and theorems.
2) To connect reasoning in algebra and geometry.
2-4
Reasoning in Algebra
Objectives: 1) To develop an awareness of the structure of a mathematical
system, connecting postulates, logical reasoning, and theorems.
2) To connect reasoning in algebra and geometry.
Properties of Equality
c
c
c
c
c
c
c
c
a
a
c
a
ab
ac
Both properties can be named APE,
short for Addition Prop. of Equality.
Subtraction is actually adding the
negative of the number.
Both properties can be named MPE,
short for Multiplication Prop. of
Equality. Division is actually
multiplying by the reciprocal of the
number.
2-4
Reasoning in Algebra
Objectives: 1) To develop an awareness of the structure of a mathematical
system, connecting postulates, logical reasoning, and theorems.
2) To connect reasoning in algebra and geometry.
Properties of Congruence
AB
A
AB
A
EF
C
2-4
Reasoning in Algebra
Objectives: 1) To develop an awareness of the structure of a mathematical
system, connecting postulates, logical reasoning, and theorems.
2) To connect reasoning in algebra and geometry.
There are three types of properties in mathematics:
assumed,
POSTULATES
defining,
and deduced.
DEFINITIONS
THEOREMS
Corollaries
PROPERTIES
Simplifying expressions and solving equations both represent a series of justified steps.
A proof with a given and only one justified conclusion is often called a one-step proof.
Common Types
of Proofs:
two-column proof
flow proof
paragraph proof
Indirect proof
2-4
Reasoning in Algebra
Objectives: 1) To develop an awareness of the structure of a mathematical
system, connecting postulates, logical reasoning, and theorems.
2) To connect reasoning in algebra and geometry.
Justify each step used to solve for 5x  12 = 32 + x .
APE (Addition Property of Equality)
APE
MPE (Multiplication Property of Equality)
Suppose that points A, B, and C are collinear with point B between points A and C.
Solve for x if AC = 21, AB =15  x , and BC = 4 + 2x . Justify each step.
15  x
A
4 + 2x
B
Segment Addition Postulate
C
15  x
4 + 2x
19
21
Substitution Property
21
2
APE
2-4
Reasoning in Algebra
Objectives: 1) To develop an awareness of the structure of a mathematical
system, connecting postulates, logical reasoning, and theorems.
2) To connect reasoning in algebra and geometry.
Given
Angle
Substitution Property
APE
MPE
Recall that the length of segment AC is 21.
15  x
A
4 + 2x
B
C
Find the lengths of AB and BC by substituting x = 2 into the expressions in
The diagram. Check that AB + BC = 21.
AB = 13
BC = 8
AB + BC = 13 + 8 =
21
2-4
Reasoning in Algebra
Objectives: 1) To develop an awareness of the structure of a mathematical
system, connecting postulates, logical reasoning, and theorems.
2) To connect reasoning in algebra and geometry.
Name the property that justifies each statement.
a. If x = y and y + 4 = 3x, then x + 4 = 3x.
The conclusion of the conditional statement is the same as the equation y + 4 = 3x (given) after x
Substitution Property
has been substituted for y (given).The property used is the ____________________.
b. If x + 4 = 3x, then 4 = 2x.
The conclusion of the conditional statement shows the result after x is subtracted
APE
from each side of the equation in the hypothesis. The property used is the ____________________.
2-4
Reasoning in Algebra
Objectives: 1) To develop an awareness of the structure of a mathematical
system, connecting postulates, logical reasoning, and theorems.
2) To connect reasoning in algebra and geometry.
c. If P  Q, Q  R, and R  S, then P  S.
Transitive Property of Congruence
P  R
Transitive Property of Congruence
P  S.
Transitive Property of Congruence
2-4
Reasoning in Algebra
Objectives: 1) To develop an awareness of the structure of a mathematical
system, connecting postulates, logical reasoning, and theorems.
2) To connect reasoning in algebra and geometry.
Name the property the property of equality or congruence illustrated.
Reflexive Property of Congruence
Transitive Property of Congruence
2-4
Reasoning in Algebra
*Connect reasoning in algebra and geometry.
Angle Addition Postulate
Substitution Property
Simplify
APE
MPE
Segment Addition Postulate
Substitution Property
Distributive Property
Simplify
APE
MPE
textbook page 105
2-4
Reasoning in Algebra
*Connect reasoning in algebra and geometry.
textbook page 106
MPE
Distributive Property
Distributive Property
APE
APE
MPE
Distributive Prop.
Reflexive Prop. of =
Symmetric Prop. of ≅
MPE
MPE
Reflexive Prop. of Equality ( or = )
Symmetric Prop. of =
APE
Substitution Prop.
APE
Transitive Property of Congruence ( or ≅ )

2-4
Reasoning in Algebra
*Connect reasoning in algebra and geometry.
15
YU=AB
PQR
5x
K
3
WYT
EF + 7
2-4
Reasoning in Algebra
*Connect reasoning in algebra and geometry.
2-4
Reasoning in Algebra
*Connect reasoning in algebra and geometry.
2-4
Reasoning in Algebra
*Connect reasoning in algebra and geometry.
2-4
Reasoning in Algebra
*Connect reasoning in algebra and geometry.
End of Notes and Exercises for Lesson 2-4.
You may proceed to work on Practice 2-4 Worksheet, DNG page 281.
Please make an effort to print copies of the additional exercises for each lesson
as they will be good reference when you make your homework.
You use the library services to print these pages. I saved them all in PDF file and
You may choose to print 2-3 pages for each page of printing paper. Mr. Chapa,
The librarian or Ms. A. Rivera can help you on this matter.
Once again thank you for all your cooperation and work.