AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have created as part of our AIM Project. Thanks for looking! Kathy Monaghan & Pat Peterson MENU- click a link 1.1 Introduction to Algebra 3.3a The Slope of a Line 7.2 Solving Systems by Substitution SAMPLE KM & PP

Click here to go back to the menu. Thanks for Looking! Click here to go back to the menu. SAMPLE KM & PP

Getting Started with Algebra What is Algebra ? Algebra is a branch of mathematics in which symbols, usually letters, are used to represent quantities that can be replaced by a number or an expression. ( ) 4 X 3 2 SAMPLE KM & PP

Getting Started with Algebra Who invented Algebra ? Algebra is a reasoning skill and language that developed and evolved along with civilization. No one person invented Algebra SAMPLE KM & PP

Getting Started with Algebra Where did the word Algebra originate ? The word is from Kitab al-Jabr wa-l-Muqabala which was a book written in approximately 820 A.D. by a Persian mathematician. Algebra SAMPLE KM & PP

a letter used to represent various numbers. Variables variable A is a letter used to represent various numbers. X “x” is frequently used as the variable, but many other letters can be used. SAMPLE KM & PP

leg inseam measurement L Variables For example, jean sizes are often given by waist and leg inseam measurements. waist measurement W leg inseam measurement L SAMPLE KM & PP

Here are three examples: Define each Variable We must always define what quantity or measurement the letter represents. Here are three examples: X = unknown number W = waist measurement = leg inseam measurement L SAMPLE KM & PP

= the speed of light in Einstein’s equation E = mc2 c Constants A is constant a letter used to represent a number that doesn’t change its value in the problem. For example: = “pi” = 3.1416…. p = the speed of light in Einstein’s equation E = mc2 c SAMPLE KM & PP

Algebraic Expressions An algebraic expression is a mathematical phrase using variables, constants, numerals, & operation signs. algebraic expression An will NOT have any of the following symbols: = > < > < SAMPLE KM & PP

Algebraic Expressions: Examples x is the variable. + is the operation 5 is a numeral and a constant. is the algebraic expression SAMPLE KM & PP

Algebraic Expressions: Examples p is the variable. . is the indicated operation 3 is a numeral and a constant. is the algebraic expression SAMPLE KM & PP

Algebraic Expressions: Examples z is the variable. - is the operation 9 is a numeral and a constant. is the algebraic expression SAMPLE KM & PP

Algebraic Expressions: Examples y is the variable. ÷ is the operation 5 is a numeral and a constant. is the algebraic expression SAMPLE KM & PP

Algebraic Expressions: Examples x and y are variables.  and + are operations 2 and 5 are numerals and constants. is the algebraic expression SAMPLE KM & PP

Algebraic Expressions: Examples x and y are variables. + ÷  are operations 5 and 2 are numerals and constants. is the algebraic expression SAMPLE KM & PP

Substitution When a variable is replaced with a numerical value, that is called substitution. Sometimes, in higher mathematics, a variable is replaced with an expression. That is also called substitution. SAMPLE KM & PP

Evaluate an Algebraic Expression When a numerical value is substituted into an algebraic expression and then simplified, that is called evaluating the expression. Evaluating means you will compute a numerical value. SAMPLE KM & PP

Evaluate an Expression: Example 1a when SAMPLE KM & PP

Evaluate an Expression: Example 1b when SAMPLE KM & PP

Evaluate an Expression: Example 1c when SAMPLE KM & PP

Evaluate an Expression: Example 1d when SAMPLE KM & PP

Evaluate an Expression: Example 2 a) when b) when SAMPLE KM & PP

Evaluate an Expression: Example 3 when SAMPLE KM & PP

Evaluate an Expression: Example 4a when SAMPLE KM & PP

Evaluate an Expression: Example 4b when SAMPLE KM & PP

Evaluate an Expression: Example 5 When and SAMPLE KM & PP

Evaluate an Expression: Example 6a When and SAMPLE KM & PP

Evaluate an Expression: Example 6b When and SAMPLE KM & PP

Evaluate an Expression: Example 6c When and SAMPLE KM & PP

Evaluate an Expression: Example 6d When and SAMPLE KM & PP

Application: Area of a Rectangle The AREA of a Rectangle Area = length x width A=l.w SAMPLE KM & PP

Translating: English into Algebra In order to solve problems, English phrases must be translated into the language of algebra. The following slides list keywords which can help us translate. SAMPLE KM & PP

English & Algebra ADDITION The following words translate as ADDITION: Plus Sum Add Added to Total More than Increased by SAMPLE KM & PP

The following phrases would translate to : X + 7 The following phrases would translate to : A number plus seven The sum of a number and seven Add a number and seven Seven added to a number The total of seven and a number Seven more than a number A number increased by seven SAMPLE KM & PP

English & Algebra SUBTRACTION The following words translate as SUBTRACTION: Minus Difference Subtract Subtracted From Take away Less Than Decreased by SAMPLE KM & PP

The following phrases would translate to : X - 7 The following phrases would translate to : A number minus seven The difference of a number and seven Subtract a number and seven Seven subtracted from a number Seven take away a number Seven less than a number A number decreased by seven SAMPLE KM & PP

English & Algebra MULTIPLICATION The following words translate as MULTIPLICATION: Multiplied by Multiply Product Times Of SAMPLE KM & PP

The following phrases would translate to : 7x The following phrases would translate to : A number multiplied by seven Multiply seven and a number The product of a number and seven The product of seven and a number Seven times a number SAMPLE KM & PP

English & Algebra DIVISION The following words translate as DIVISION: Divided by Divide Quotient SAMPLE KM & PP

The following phrases would translate to : x/7 The following phrases would translate to : A number divided by seven Divide a number by seven The quotient of a number and seven SAMPLE KM & PP

The following phrases would translate to : 7/x The following phrases would translate to : Seven divided by a number Divide a seven by a number The quotient of seven and a number SAMPLE KM & PP

“Half of a number” would be or or SAMPLE KM & PP

“Thirty percent of a number” is: or SAMPLE KM & PP

“Twice” or “Double” “Twice a number” is: “Double a number” is: SAMPLE KM & PP

Translate a Phrase “Seven more than twice a number” Seven more than SAMPLE KM & PP

Translate a Phrase “Seven less than twice a number” Seven less than SAMPLE KM & PP

Translate a Phrase (watch for the comma) “the quotient of seven and a number increased by two” “the quotient of seven and a number, increased by two” comma SAMPLE KM & PP

Salary Increase? Suppose you will get a salary increase of 3%. Let s represent your old salary. The increase is 3% of your current salary, so 0.03s is the increase. Your new salary will be the sum of the old salary and the increase. So, s + 0.03s is your new salary. SAMPLE KM & PP

Discount? Suppose the bookstore has all merchandise on sale for 15% off. Let p represent the regular price. The discount is 15% of the regular price, so 0.15p is the discount. The sale price will be the difference of the regular price and the discount So, p - 0.15p is the sale price. SAMPLE KM & PP

Click here to go back to the menu. That’s All for Now! Click here to go back to the menu. SAMPLE KM & PP

Slope Slope of a Straight Line SAMPLE KM & PP

Definition: Slope The slope of the line containing points P1(x1, y1) and P2(x2, y2) is given by The denominator can’t be zero. SAMPLE KM & PP

A Slope Triangle RISE y2 - y1 x2 - x1 RUN P2(x2, y2) RISE y2 - y1 P1(x1, y1) RUN x2 - x1 SAMPLE KM & PP

m Calculus SAMPLE KM & PP

The line is rising (going upwards) Compute the Slope The line is rising (going upwards) The slope is positive. (4, -1) (-2, -3) SAMPLE KM & PP

Another Slope (0, 5) (-4, -3) SAMPLE KM & PP

The line is falling (going downwards). Compute this Slope The line is falling (going downwards). The slope is negative. (-5, 2) (4, -1) SAMPLE KM & PP

HORIZONTAL Compute the Slope The line is horizontal. The slope is zero, 0. HORIZONTAL (-5, -1) (4, -1) SAMPLE KM & PP

VERTICAL Compute the Slope The line is vertical. The slope is undefined. vertical california VERTICAL (-4, 0) (-4, -5) SAMPLE KM & PP

SLOPE BASICS POSITIVE SLOPE The line rises from left to right. ZERO SLOPE The line is HORIZONTAL (zero rise) NEGATIVE SLOPE The line falls from left to right. UNDEFINED SLOPE The line is VERTICAL (zero run) SAMPLE KM & PP

Determine the slope of the line containing points P1 and P2. Practice Problem 1a Determine the slope of the line containing points P1 and P2. RISE = 2 RUN = 3 The line rises from left to right. SAMPLE KM & PP

Determine the slope of the line containing points P1 and P2. Practice Problem 1b Determine the slope of the line containing points P1 and P2. The slope is undefined. Vertical Line SAMPLE KM & PP

Determine the slope of the line containing points P1 and P2. Practice Problem 1c Determine the slope of the line containing points P1 and P2. RUN = 2 RISE = -1 The line falls from left to right. SAMPLE KM & PP

Determine the slope of the line containing points P1 and P2. Practice Problem 1d Determine the slope of the line containing points P1 and P2. RISE = 0 RUN = 6 The line is Horizontal. SAMPLE KM & PP

Click here to go back to the menu. That’s All for Now! Click here to go back to the menu. SAMPLE KM & PP

Solving Systems of Linear Equations Substitution Method Systems of Equations Solving Systems of Linear Equations Substitution Method SAMPLE KM & PP

In order to solve a system of equations by substitution: Systems of Equations In order to solve a system of equations by substitution: Solve one equation for x or for y (pick the easiest one). Substitute for that letter in the OTHER equation. Solve this equation for the variable. Substitute the answer back into the original equation. SAMPLE KM & PP

No Graph? WHAT LUCK! Just Substitute for x! SAMPLE KM & PP

Let’s Check the Answer Replace x with 2 & y with -4 SAMPLE KM & PP

Another Substitution 4 U ! Substitute for x in the second equation! Solve this equation for y and then determine x. is the solution set. SAMPLE KM & PP

Be Sure to Check the Answer! Replace x with 16 & y with 5 SAMPLE KM & PP

Solve this equation for x to get: Would I Try to Fool You? Substitute for y in the second equation! Solve this equation for x to get: YIKES! That is FALSE! NO SOLUTION! is the solution set. SAMPLE KM & PP

Substitute for x in the second equation! is the solution set. Guess what? Substitute for x in the second equation! is the solution set. SAMPLE KM & PP

Check To Be Sure! Replace x with 0 & y with 0 SAMPLE KM & PP

Solve the second equation for x. OK…last one for now! Solve the second equation for x. Substitute for x in the the first equation. Simplify. continued on next page... SAMPLE KM & PP

Always TRUE means they are the same line. Problem continued! TRUE! Always TRUE means they are the same line. Write the answer using the slope intercept form of the equation. {(x, mx + b)} SAMPLE KM & PP

Click here to go back to the menu. That’s All for Now! Click here to go back to the menu. SAMPLE KM & PP