Thinking MathematicallyThinking Mathematically

A review and summary of A review and summary of Algebra 1Algebra 1

By: Bryan McCoy and Mike PelantBy: Bryan McCoy and Mike Pelant

In the next slides you will review:In the next slides you will review:

All of the Properties and Equations All of the Properties and Equations needed to succeed in the upcoming needed to succeed in the upcoming

ExamExam

In the next slides you will In the next slides you will review:review:

Review all the Properties and Review all the Properties and then take a Quiz on identifying then take a Quiz on identifying

the Property Namesthe Property Names

Addition Property (of Equality)Addition Property (of Equality)

Multiplication Property (of Equality)Multiplication Property (of Equality)

Example: Example:

If a=b then a+c=b+c If a=b then a+c=b+c

Example: Example: If a=b then a(c)=b(c) If a=b then a(c)=b(c)

Reflexive Property (of Equality)Reflexive Property (of Equality)

Symmetric Property (of Equality)Symmetric Property (of Equality)

Transitive Property (of Equality)Transitive Property (of Equality)

Example:Example:If a=b then b=aIf a=b then b=a

Example:Example:

If a=b then b=aIf a=b then b=a(order does not matter)(order does not matter)

Example:Example: If a=b and b=c then a=c If a=b and b=c then a=c

Associative Property of AdditionAssociative Property of Addition

Associative Property of Associative Property of MultiplicationMultiplication

Example: Example: (1+2)+3=6 1+(2+3)=6(1+2)+3=6 1+(2+3)=6

(Does not matter where you put the parenthesis)(Does not matter where you put the parenthesis)

Example:Example:

(1 2) 3 6 1 (2 3) 6

Commutative Property of AdditionCommutative Property of Addition

Commutative Property of Commutative Property of MultiplicationMultiplication

Example: Example: 5+3+2=3+5=25+3+2=3+5=2

Example: Example: 2 4 4 2

Distributive Property (of Distributive Property (of Multiplication over AdditionMultiplication over Addition

Example: Example: 3(2+7-5)=3(2)+3(7)+(3)(-5)3(2+7-5)=3(2)+3(7)+(3)(-5)

Prop of Opposites or Inverse Prop of Opposites or Inverse Property of Addition Property of Addition

Prop of Reciprocals or Inverse Prop of Reciprocals or Inverse Prop. of Multiplication Prop. of Multiplication

Example: Example: +8-8=0+8-8=0

Example: Example:

Identity Property of Addition Identity Property of Addition

Identity Property of Multiplication Identity Property of Multiplication

Example: Example: 4+0=44+0=4

Any number plus 0 equals the original numberAny number plus 0 equals the original number

Example: Example: Any number times 1 will equal itselfAny number times 1 will equal itself

4 1 4

Multiplicative Property of Zero Multiplicative Property of Zero

Closure Property of Addition Closure Property of Addition

Closure Property of Multiplication Closure Property of Multiplication

Example: Example: A number times 0 equals 0A number times 0 equals 0

Example: Example: ..

Example: Example:

0 0a

Product of 2 real numbers = a real number 5 7 35

Product of Powers PropertyProduct of Powers Property

Power of a Product Property Power of a Product Property

Power of a Power Property Power of a Power Property

Example: Example:

7722 × 7× 766 = = 7788

Example: Example: 3322 · 4 · 422 = = 121222

Example: Example:

Quotient of Powers PropertyQuotient of Powers Property

Power of a Quotient Property Power of a Quotient Property

Example: Example:

Example: Example:

Zero Power Property Zero Power Property

Negative Power Property Negative Power Property

Example: Example: (-3)(-3)00 = 1 = 1

Example: Example:

Zero Product Property Zero Product Property

Example: Example:

if if abab = 0, then either = 0, then either aa = 0 or = 0 or bb = 0 (or both). = 0 (or both).

Product of Roots Property Product of Roots Property

Quotient of Roots Property Quotient of Roots Property

The product of the square roots is the square root of the product.

For any non-negative (positive or 0) real number a and any positive real number b: =√a -- √b

Root of a Power Property Root of a Power Property

Power of a Root Property Power of a Root Property

Example: Example:

Example: Example:

Now you will take a quiz!Now you will take a quiz!Look at the sample problem and Look at the sample problem and

give the name of the property give the name of the property illustrated. illustrated.

1. a + b = b + a 1. a + b = b + a

Click when you’re ready to see the answer.Click when you’re ready to see the answer.

Answer: Answer: Commutative Property (of Addition)Commutative Property (of Addition)

Now you will take a quiz!Now you will take a quiz!Look at the sample problem and Look at the sample problem and

give the name of the property give the name of the property illustrated. illustrated.

2. 2.

Click when you’re ready to see the answer.Click when you’re ready to see the answer.

Answer: Answer:

If a=b then a(c)=b(c) If a=b then a(c)=b(c)

Multiplication Property (of Equality)Multiplication Property (of Equality)

In the next slides you will review:In the next slides you will review:

Solving inequalities Solving inequalities

Solution Set: {x: x > -2}Solution Set: {x: x > -2}

Sample Problem:Sample Problem:

- Remember the Multiplication Property of Inequality! If you multiply or divide by a negative, you must reverse the inequality sign.

-2

-5x < 10 x > -2

Linear Equations in 2 VariablesLinear Equations in 2 Variables

Here’s a sample problem: can you graph Here’s a sample problem: can you graph this: y=x-5?this: y=x-5?

Linear SystemsLinear Systems

Can you solve this? y = 3x – 2 y = –x – 6

Y=-x-6

3x-2=-x-6

4x=8

X=2

Now solve for Y

Y=3(2)-2

Y=6-2

Y=4

The answer is (2,4)

In the next slides you will In the next slides you will review:review:

All of the Factoring All of the Factoring MethodsMethods

Find GCF!Find GCF!

Finding the GCF will make the problem Finding the GCF will make the problem simpler greatly.simpler greatly.

Ex: Ex: – 2x-4y=82x-4y=8– GCF = (2)GCF = (2)– = (2)(x-y=4)= (2)(x-y=4)

The following is a list of the rest The following is a list of the rest of the properties:of the properties:

Difference of SquaresDifference of Squares

Sum/Difference of CubesSum/Difference of Cubes

PSTPST

Reverse FoilReverse Foil

Factor by Grouping(4 or more terms)Factor by Grouping(4 or more terms)

Rational ExpressionsRational ExpressionsTry this Problem:

FunctionsFunctions

f(x)= is another way to write y=f(x)= is another way to write y=

Functions are relations only when every input has a Functions are relations only when every input has a distinct output, so not all relations are functions but all distinct output, so not all relations are functions but all functions are relations. functions are relations.

Let’s say you had the points (2,3) and (3,4) and you needed to find a Let’s say you had the points (2,3) and (3,4) and you needed to find a linear function that contained them. This is how you would do that.linear function that contained them. This is how you would do that.

3-4 over (divided by) 2-3 (rise over run, Y is rise, X is run)3-4 over (divided by) 2-3 (rise over run, Y is rise, X is run)

you would get -1 over -1. This equals 1, which will be the slope. To you would get -1 over -1. This equals 1, which will be the slope. To find y-intercept, substitute: 2=1(3)+bfind y-intercept, substitute: 2=1(3)+b

2=3+b 2=3+b -1=b -1=b

So your final equation is: Y=X-1. You can now graph this.So your final equation is: Y=X-1. You can now graph this.

ParabolasParabolas

See if you can graph this one: See if you can graph this one: xx22-6x+5-6x+5

The x-intercepts are (5,0) and (1,0).The x-intercepts are (5,0) and (1,0).  

y-intercept:

Vertex: and

So the vertex is (3, -4).

Now just graph it.

Simplifying Expressions With Simplifying Expressions With ExponentsExponents

Simplify this:Simplify this:

The "minus" on the 2 says to move the variable; the "minus" on the 6 says that the 6 is negative. Warning: These two "minus" signs mean entirely different things, and should not be confused. I have to move the variable; I should not move the 6.

Your answer is:

Simplifying expressions with Simplifying expressions with radicalsradicals

Try this one:Try this one:

Word problemsWord problemsYou need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a 30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you 30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you decide to mix 10% solution with 30% solution, to make your own 15% solution. You need 10 liters decide to mix 10% solution with 30% solution, to make your own 15% solution. You need 10 liters of the 15% acid solution. How many liters of 10% solution and 30% solution should you use? of the 15% acid solution. How many liters of 10% solution and 30% solution should you use?

Let Let xx stand for the number of liters of 10% solution, and let stand for the number of liters of 10% solution, and let yy stand for the number of liters of 30% stand for the number of liters of 30% solution. (The labeling of variables is, in this case, very important, because "solution. (The labeling of variables is, in this case, very important, because "xx" and "" and "yy" are not at " are not at all suggestive of what they stand for. If we don't label, we won't be able to interpret our answer in all suggestive of what they stand for. If we don't label, we won't be able to interpret our answer in the end.) For mixture problems, it is often very helpful to do a grid: the end.) For mixture problems, it is often very helpful to do a grid:

Word Problems ContinuedWord Problems Continued

A collection of 33 coins, consisting of nickels, dimes, and quarters, A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. If there are three times as many nickels as has a value of $3.30. If there are three times as many nickels as quarters, and one-half as many dimes as nickels, how many coins of quarters, and one-half as many dimes as nickels, how many coins of each kind are there?each kind are there? I'll start by picking and defining a variable, and then I'll use translation to convert this exercise into I'll start by picking and defining a variable, and then I'll use translation to convert this exercise into mathematical expressions.mathematical expressions.

Nickels are defined in terms of quarters, and dimes are defined in terms of nickels, so I'll pick a Nickels are defined in terms of quarters, and dimes are defined in terms of nickels, so I'll pick a variable to stand for the number of quarters, and then work from there:variable to stand for the number of quarters, and then work from there:

number of quarters: number of quarters: qq number of nickels: 3number of nickels: 3qq number of dimes: (½)(3number of dimes: (½)(3qq) = (3/2)) = (3/2)qq

There is a total of 33 coins, so:There is a total of 33 coins, so:

qq + 3 + 3qq + (3/2) + (3/2)qq = 33 = 33 44qq + (3/2) + (3/2)qq = 33 = 33 88qq + 3 + 3qq = 66 = 66 1111qq = 66 = 66 qq = 6 = 6

Then Then there are six quartersthere are six quarters, and I can work backwards to figure out that, and I can work backwards to figure out that there are 9 dimes and there are 9 dimes and 18 nickels18 nickels..