Addition Property of Equality - Add (+) the same thing to both
sides of the equation Subtraction Property of Equality - Subtract
(-)the same thing to both sides of the equation Multiplication
Property of Equality - Multiply (x) the same thing to both sides of
the equation Division Property of Equality - Divide (/)the same
thing to both sides of the equation Distributive Property of
Equality - Multiplying the number outside of the parenthesis to
each individual number in the group (in the parenthesis)
Substitution Property of Equality - Replacing a variable with a
value/number Reflexive Property of Equality - A value that is
equal/congruent to itself; a = a Symmetric Property of Equality -
The values being compared are equal to each other and order doesnt
matter; a=b, b=a Transitive Property of Equality - If a=b, and b=c,
then a=c
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Equation - a mathematical statement that proves that two
expressions are equal to one another (ex: 20+4=24 Expression - The
side of the equation that is grouped with numbers, operators,
and/or variables (the side that needs to be solved) (ex: 20x+40-30
= 90) Coefficient - a number that is used to multiply a variable
(ex: 4 x) Variable - a symbol for a number that is unknown (needs
to be solved) (ex: 2m) Sum - the answer for addition (ex: 6+4= 10
Difference - the answer for subtraction (ex: 20-18= 2) Product -
the answer for multiplication (14x10= 140) Quotient - the answer
for division (ex: 20/4= 5) Operation - a mathematical process used
to solve equations using symbols (addition +, subtraction -,
multiplication x, and division / )
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Add the same thing to both sides Subtract the same thing to
both sides Addition Property of Equality If a = b, Then a + c= b +
c The same thing is being added to both sides of the equation.
Subtraction Property of Equality If a = b, Then a c = b c The same
thing is being subtracted from both sides of the equation.
Slide 6
Multiply the same thing to both sides Divide the same thing to
both sides Multiplication Property of Equality Multiply both sides
by the same thing. If a = b, Then a c = b c Division Property of
Equality Divide both sides by the same thing If a = b AND c is NOT
0, Then a/c = b/c
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Distributive Property of Equality Multiplying the number
outside of the parenthesis to each individual number in the group
(in the parenthesis) When distributing, keep in mind that the signs
also matter! For example: 5(x-4) 4*x= 4x 5+(-4)= -20 (notice how
the -4 in the original equation makes it a negative) Substitution
Property of Equality Replacing a variable with a value/number This
property allows us to use any given information to solve an
equation. For example: Find the value of x using the given
information: Y=8 6+y=x 6+8=x X=14
Slide 8
Reflexive Property of Equality A value that is equal/congruent
to itself; Ex: a = a *One way to remember this is by remembering
that the word Reflexive is a little similar to the word reflect
(like reflecting from a mirror) Mirror images are the same, so you
can remember that a = a (equals itself). Symmetric Property of
Equality The values being compared are equal to each other and
order doesnt matter; a=b, b=a *Check to make sure that if the
solution is flipped around, that is actually still equals the same
thing. For example: K=9 9=K Even though it is flipped, they still
equal the same thing.
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Transitive Property of Equality If a=b, and b=c, then a=c The
Transitive Property of Equality is similar to one of the laws of
logic- The Law of Syllogism If the measure of angle PIG equals the
measure of angle DOG, and the measure of angle DOG equals the
measure of angle COW, then the measure of angle PIG equals the
measure of angle COW I G D O G C OW P......... Congruent
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1)W=64; The Division Property of Equality was used to find the
value of w 2)1. The variable k randomly appears, and has nothing to
do with the statements above; Correct Answer: Then, the measure of
angle b equals the measure of angle h 2. There are no errors,
therefore everything is correct 3. The order of the letters of the
angles are mixed up, so the angles are not the same; Correct
Answer: Angle BYE, is congruent to angle YUM 3)The measure of angle
z is congruent to the measure of angle z, therefore, the measure of
the reflexive property of equality is also 36 degrees. 4)Line
Segment AB is equal to Line Segment YZ and Line Segment YZ is equal
to Line Segment AB 5)H= 1039; Properties Used: The Substitution
Property of Equality and the Distributive Property of Equality
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Conditional Statement - If (Hypothesis), then (Conclusion); If,
then statement Hypothesis - The If of a conditional statement (p)
Conclusion - The Then of a conditional statement (q) Law of
Detachment -If a conditional statement is true and the hypothesis
is true, then the conclusion is also true; If p then q, p is true,
therefore q is true. Law of Syllogism - The conclusion becomes
another conditional statement; If p then q, if q then r, therefore
if p then r. Venn Diagram - A diagram to show the relationship
between the hypothesis and conclusion - Therefore
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If p, then q. P is true. Therefore, q is also true. The
conclusion for the Law of Detachment is just a con. If it is a dog,
then it barks. Venn Diagram: Marshmallow is a dog. Therefore,
Marshmallow barks. Hypothesis: If it is a dog Conclusion: Then it
barks Barks Dogs. Marshmallow
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If p, then q. If q, then r. Therefore, if p then r. For the Law
of Syllogism, the conclusion is another conditional statement. If
you cry, then you will have wet eyes. If you have wet eyes, then
you need a tissue. Therefore, if you cry, then you will need a
tissue.
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1) The Law of Syllogism 2) The Law of Detachment 3) Therefore,
Santa will come 4) 1. The therefore statement is incorrect because
crying never showed up until the end. It did not relate to the
conclusion or hypothesis. The correct statement is: Therefore, if
you do not eat, your tummy will growl. 2. In this case, both the
hypothesis and conclusion in the therefore statement did not relate
to the conditional statement. The information that is given is what
the therefore statement will be based on. Correct Statement:
Therefore, you will get a cavity. 5) If it is snowing, then it is
cold. If it is cold, then you need a jacket. Therefore, If it is
snowing, you need a jacket.
Slide 28
Thank You for reviewing: Algebraic Reasoning and the Laws of
Logic
