Transcript CH I: Connections to Algebra

CH I:
Connections to Algebra
1.1) Variables in Algebra
1.2) Exponents and Powers
1.3) Order of Operations
1.4) Equations and Inequalities
1.5) Translating Words into Mathematical Symbols
1.6) A Problem Solving Plan Using Models
1.7) Tables and Graphs
1.8) An Introduction to Functions
1.1) Variables in Algebra
Let’s take a look at a problem:
You are driving to LA at 80 miles per hour. How many
miles have you traveled after 2 hours? How about
after 5 hours?
The problem can be written this way:
d = 80x ; x = {2, 5}; d = ?
Here, d and x are called variables.
The answer, d, will vary depending on the
number plugged in for x which in this case
are 2 and 5.
1.1) Variable in Algebra (cont.)
Let’s solve the problem:
d = 80x ; x = {2, 5}; d = ?
d = 80(2) = 80 x 2 = 160
d = 80(5) = 80 x 5 = 400
The car will travel 160 miles after traveling
for 2 hours at 80 miles/hr.
The car will travel 400 miles after traveling
for 5 hours at 80 miles/hr.
1.2) Exponents and Powers
Look at this:
2³ = 2 x 2 x 2
2³ means you multiply 2 three times.
x ³¹=x x x x x x x x x x x x x x x x x x x x x x x x x x x x
x xx xx xx xx xx xx xx xx xx xx xx xx xx xx
xx xx
x ³¹
means you multiply x thirty one times.
1.2) Exponents and Powers (cont.)
BE AWARE!
2x ³ DOES NOT EQUAL (2x )³.
2x ³
=2xx xx xx
(2x )³ = 2x x 2x x 2x
1.3) Order of Operations
1/ First do operations that occur within
grouping symbols (parentheses or
brackets)
2/ Then evaluate powers
3/ Then do multiplications and divisions from
left to right
4/ Finally, do additions and subtractions
from left to right.
1.3) Order of Operation (cont.)
Left-to-right rule- when there are same
types of signs, you operate the left one
first.
Ex.1) in 4+2+6, you do (4+2) first. Then
move on to +6.
Ex.2) in 4–2+5, you do the – first. Then
move on to +.
Ex.3) in 4x2÷3, you do x first. Then move
on to ÷.
1.4) Equations and Inequalities
Equation: Any statement that has equal sign.
Take a look at the following problem:
4x +1 = 9
When you see a statement, “Find the solution
to the equation,” it just means to find the right
values for the variable
(in this case x).
1.4) equations and Inequalities (cont.)
So, we got
4x +1 = 9
If you subtract 1 from both sides,
4x +1-1 = 9-1
4x = 8
Then divide both sides by 4.
4x /4 = 8/4
x=2
Yay~!
1.4) equations and inequalities (cont.)
There are 4 inequality symbols:
>
is greater than
<
is less than
 is greater than or equal to

is less than or equal to
Is the following statement true?
4x +1  9 when x ={2,3}
1.4) equations and inequalities (cont.)
4x +1  9 when x ={2,3}
Step1) Replace x with the given value, 2 and 3.
4(2)+1  9
4(3)+1  9
Step2) Do the calculation
8+1  9
12+1  9
99
13  9
Step3) answer the question.
Is the following statement true?
4x +1  9 when x ={2,3}
{9,13} is greater than or equal to 9. TRUE.
1.5)Translating words into
mathematical symbols
It’s not that hard, just MEMORIZE these:
Sum, more than, plus, increased => +
Difference, minus, less than, decreased => Product, times, multiplied by => x
Division is worded a little differently.
One fourth of 6 => 6 x ¼
The quotient of 6 and 4 => 6/4
6 divided by 4 => 6/4
Practice problems on pg.30-32. :)
1.6) A Problem Solving Plan Using Models
A problem solving plan using models
#1) Verbal model-Ask yourself what you need to know to solve the
problem. The write a verbal model that will give you what you need
to know.
#2) Labels-Assign labels to each part of your verbal model.
#3) Algebraic Model-Use the labels to write an algebraic model based
on your verbal model.
#4) Solve-Solve the algebraic model and answer the original question.
#5) Check that your answer is reasonable.
1.6) A Problem Solving Plan Using Models
Pg.36) You and some friends are at a Chinese restaurant.
You order several $2 plates of wontons, egg rolls, and
dumplings. Your bill is $25.20, which includes tax of
$1.20. Use modeling to find how many plates you
ordered.
Step1) cost/plate x number of plates = amount of bill – tax.
Step2)
cost/plate = 2
Number of plates = p
Amount of bill = 25.20
Tax = 1.20
Step3)
2p = 25.20 – 1.20
Step4)
2p = 24
p = 12
Step5) We ordered 12 plates
1.7) Tables and Graphs
Table
time
types
1990
1995
2000
2005
SAT Average
English Score.
800
700
600
500
SAT Average
Math Score
200
300
400
500
1.7) Tables and Graphs (cont.)
Bar Graph
Line Graph
800
900
700
800
700
600
600
500
400
English
500
English
300
Math
400
Math
300
200
200
100
100
0
1990
1995
2000
2005
0
1990
1995
2000
2005
1.8) An Introduction to Functions
A function is a rule that establishes a
relationship between two quantities, called
the input and the output. For each input,
there is exactly one output—even though
two different inputs may give the same
output.
An input-output table (p.48) describes a
function.
1 input => 1 output (Function? O)
1 input => multiple outputs (Function? X)
1.8) An Introduction to Functions (cont.)
Take a look at the following equation:
h = 250+20t
For every value of t (input), there is exactly
one value of h (output). Therefore, it is a
function.
Domain: a group of inputs
Range: a group of outputs
Terms
Variable
Variable expression
Value
Numerical expression
Evaluate
Power
Exponent
Base
Grouping symbols
Order of operations
Left-to-right rule
Equation
solution
Inequality
Modeling
Verbal model
Algebraic model
Data
Bar graph
Line graph
Function
Input
Output
Input-output table
Domain
range