Algebraic Reasoning

 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 doesn’t matter; a=b, b=a  Transitive Property of Equality - If a=b, and b=c, then a=c

 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 = 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 / )

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.

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

 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

 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 doesn’t 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.

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

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

 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 ”

 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

 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.

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.

Thank You for reviewing: Algebraic Reasoning and the Laws of Logic