Guided Practice Example 1

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1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
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Assignment #: 3.2.1
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
What’s wrong with this picture?
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Assignment #: 3.2.1
Write your answers for
the following numbers:
2.
3.
4.
5.
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Assignment #: 3.2.1 1-6
Write your answers for
the following numbers:
2. No real solution
3. +/- 3
4. x=-2 x=-4
5. No real solution
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Review
Ratio and Proportions
5
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Notes
Ratio:
Number of specific things
Total number of items
or
Part
Whole
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
What does a ratio mean?
In Orem, The male to female ratio
is 100:101
This means for every 100 males there are 101
females in Orem
At Mt. View High School the
student to teacher ratio in 28:1
This means for every 28 students there is 1
teacher at Mt. View
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
To change a ratio from a fraction to a percent,
divide then move the decimal two to the
right.
Move two to the right and then round to the tenth.
7/9 = .77777 = 77.8%
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Use the information to write a ratio in lowest
terms.
a. Hamburgers to veggie
burgers
The number of people
who prefer the
following:
b. Hot dogs to total
Hamburger: 32
Hot dogs: 26
c. Hot dogs to hamburgers
Veggie burger: 9
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
In a survey of 40 people the results are:
regular skittles: 24
sour skittles: 6
double flavor: 10
What is the proportion of sour to total?
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Notes
A proportion is an equation written in
the form
stating that two
ratios are equivalent.
To solve proportions, cross multiply.
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Solving Proportions
Cross Multiplying
7
x

12 36
7(36) = 12x
252 = 12x
21 = x
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Solve for x.
3 x

5 15
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Solve for x.
5
x

20 4
1
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Setting Up Proportions
If you are comparing apples to oranges,
you could set the equation up two ways:
Apples 1
Apples 2
=
Oranges 1 Oranges 2
Remember:
KEEP RATIOS
or
CONSISTANT!!
Apples 1
Oranges 1
=
Apples 2
Oranges 2
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
A dance number needs 4 guys for
every 3 girls. If there are 42 girls,
how many guys are there?
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
A dance number needs 4 guys for
every 3 girls. If there are 56
dancers, how many guys are there?
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
A dance number needs 4 guys for
every 7 girls. If there are 42 girls,
how many guys are there?
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
How to use the BLUE paper:
Example:
Word
Their Definition
My Definition
Variable
Represents a value or The letters in the
unknown quantity
equations
19
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Introduction
Algebraic expressions are mathematical statements
that include numbers, operations, and variables to
represent a number or quantity. We know that a variable
is a letter used to represent a value or unknown quantity
that can change or vary. We have seen several linear
expressions such as 2x + 1. In this example, the highest
power of the variable x is the first power. In this lesson,
we will look at expressions where the highest power of
the variable is 2.
20
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Notes
• A quadratic expression (ax2 + bx + c) is an
expression where the highest power of the variable is
the second power.
• The quadratic expression 4x2 + 6x – 2 is made up of
many component parts: terms, factors, coefficients,
and constants.
• A factor is one of two or more numbers or
expressions that when multiplied produce a given
product. In the given expression, the factors of 4x2 are
4 and x2 and the factors of 6x are 6 and x.
21
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Notes
• A coefficient is the number multiplied by (in front of)
a variable.
• Example: In the given expression, the coefficient
of the term 4x2 is 4 and the coefficient of the term
6x is 6.
• A constant term that does not contain a variable.
The value of the term does not change.
• Example: –2 is a constant.
22
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Pg. 253 in your Folder
In order to pay for upkeep of a local highway, the
transportation department has set up tollbooths at each
of the highway’s exits. Drivers are charged a toll of $1.20
for use of the highway, and are then charged an
additional $0.04 per mile driven.
1. Write an algebraic expression that can be used to
represent the total toll charged if m represents the
number of miles driven.
2. What is the toll charged to drive 14 miles on this
highway?
3.1.1: Identifying Terms, Factors, and Coefficients
23
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
2. What is the toll charged to drive 14 miles on this
highway?
• Substitute 14 for m and evaluate the expression.
1.20 + 0.04(14) = 1.20 + 0.56 = 1.76
• The toll charged for driving 14 miles on the
highway is $1.76.
24
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Notes
• A monomial is a number, a variable, or the product of
a number and variable(s). (Example: 5x2, 4, y)
• A polynomial is a monomial or the sum of
monomials. A polynomial can have any number of
terms.
• A binomial is a polynomial with two terms. (Example:
6x + 9)
• A trinomial is a polynomial with three terms.
(Example: 4x2 + 6x – 2)
• Like terms (same variables raised to the same
power) can be combined by adding.
25
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice
Example 1 – with steps
Identify each term, coefficient, and constant of
6(x – 1) – x(3 – 2x) + 12. Classify the expression as a
monomial, binomial, or trinomial. Determine whether it is
a quadratic expression.
26
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 1, continued
1. Simplify the expression.
The expression can be simplified by following the
order of operations and combining like terms.
6(x – 1) – x(3 – 2x) + 12
Original expression
6x – 6 – x(3 – 2x) + 12
Distribute 6 over x – 1.
6x – 6 – 3x + 2x2 + 12
Distribute –x over 3 – 2x.
3x + 6 + 2x2
Combine like terms: 6x and
–3x; –6 and 12.
2x2 + 3x + 6
Rearrange terms so the powers
are in descending order.
27
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 1, continued
2. Identify all terms.
There are three terms in the expression: 2x2, 3x,
and 6.
28
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 1, continued
3. Identify all coefficients.
The number multiplied by a variable in the term 2x2 is
2; the number multiplied by a variable in the term 3x
is 3; therefore, the coefficients are 2 and 3.
29
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 1, continued
4. Identify any constants.
The quantity that does not change (is not multiplied
by a variable) in the expression is 6; therefore, 6 is a
constant.
30
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 1, continued
5. Classify the expression as a monomial,
binomial, or trinomial.
The polynomial is a trinomial because it has three
terms.
31
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 1, continued
6. Determine whether the expression is a
quadratic expression.
It is a quadratic expression because it can be written
in the form ax2 + bx + c, where a = 2, b = 3, and c = 6.
✔
32
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Addition
Increased by
More than
Combined
Total of
Sum
Added to
Multiplication
Of
Times
Product of
Increased by a factor
of
Subtraction
Decreased by
Minus, less
Difference between
Less than, fewer
than
Division
per,
Out of
Ratio of, quotient of
Percent (divide by 100)
33
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice
Example 2
Translate the verbal expression ‘take triple the
difference of 12 and the square of x, then
increase the result by the sum of 3 and x” into
an algebraic expression. Identify the terms,
coefficients, and constants of the given
expression. Is the expression quadratic?
34
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice
Example 3 – with steps
A fence surrounds a park in the shape of a
pentagon. The side lengths of the park in feet
are given by the expressions 2x2, 3x + 1, 3x + 2,
4x, and 5x – 3. Find an expression for the
perimeter of the park. Identify the terms,
coefficients, and constant in your expression. Is
the expression quadratic?
35
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 3, continued
1. Find an expression for the perimeter of
the park.
Add like terms to find the perimeter, P.
P = 2x2 + (3x + 1) + (3x + 2) +
4x + (5x – 3)
Set up the equation
using the given
expressions.
P = 2x2 + 3x + 3x + 4x + 5x +
1+2–3
Reorder like terms.
P = 2x2 + 15x
Combine like terms.
The expression for the park’s perimeter is 2x2 + 15x.
36
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 3, continued
2. Identify all terms.
There are two terms in this expression: 2x2 and 15x.
37
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 3, continued
3. Identify all coefficients.
The number multiplied by a variable in the term 2x2 is
2; the number multiplied by a variable in the term 15x
is 15; therefore, 2 and 15 are coefficients.
38
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 3, continued
4. Identify any constants.
Every number in the expression is multiplied by a
variable; therefore, there is no constant.
39
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Guided Practice: Example 3, continued
5. Determine whether the expression is a
quadratic expression.
It is a quadratic expression because it can be written
in the form ax2 + bx + c, where a = 2, b = 15, and
c = 0.
✔
40
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Shanna wants to decorate the triangular deck behind
her house. The base of the triangle is 10 meters shorter
than the altitude. What are the terms, factors, and
coefficients of the quadratic expression that represents
the area of the deck to be decorated?
41
3.1.1: Identifying Terms, Factors, and Coefficients
1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1)
Assignment
Practice 3.1.1 Pg. 259-260 (odds)
R1-5 Ratio and Proportions (odds)
42
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