Download 1-1 A Plan for Problem Solving Understand Read the problem

1-1 A Plan for Problem Solving
 Understand Read the problem carefully.
What information is given?
What do you need to find out?
Is enough information given?
Is there any extra information?
 Plan
How do the facts relate to each other?
Select a strategy for solving the problem. There may
be several that you can use.
Estimate the answer.
 Solve
Use your plan to solve the problem.
If your plan does not work, revise it or make a new
plan.
What is the solution?
 Check
Does you’re answer fit the facts given in the problem?
Is your answer reasonable compared to your
estimate?
If not, make a new plan and start again.
Example 1
There were about 268 million TV’s in the U.S. in 2007. This amount
increases by 4 million each year after 2007. In what year will there be at
least 300 million TVs?
Example 2
A diagonal connects two
nonconsecutive
vertices in a figure, as shown at
the right.
Find how many diagonals a figure with
7 sides would have.
1-2 Powers and Exponents
35
Base
Exponent
= 3 · 3 · 3 · 3 · 3 = 243
common factors
The exponent tells you how many times the
base is used as a factor.
Example 1
Write 63 as a product of the same factor.
Example 2
Evaluate 54.
Example 3
Write 4 · 4 · 4 · 4 in exponential form.
Write each power as a product of the same factor.
1. 73
2. 27
3. 92
4. 154
Evaluate each expression.
5. 35
6. 73
7. 84
8. 53
Write each product in exponential form.
9. 2 · 2 · 2 · 2
10. 7 · 7 · 7 · 7 · 7 · 7
1-3 Squares and Square Roots
The area of the square at the
right is 5 · 5 or 25 square units.
The product of a number and
5 units
itself is the square of that number.
5 units
25
units2
Numbers like 9, 16, and 225 are called square
numbers or perfect squares because they are
squares of whole numbers.
The factors multiplied to form perfect squares are
called square roots. A radical sign, , is the
symbol used to indicate a square root of a number.
Square Root
Words: A square root of a number is one of its two
equal factors.
Examples:
Numbers
Algebra
4 · 4 = 16, so 16 = 4 If x · x or x2 = y, then y = x
Examples:


Find the square of each number.
1) 3
2) 9
3) 12
4) 23
Find each square root.
5)
6)
7)
8)
25
100
400
49
Sports The infield of a baseball field is a square with an
area of 8,100square feet. What
are the dimensions
of



the infield?
1-3 Squares and Square Roots
The area of the square at the
right is 5 · 5 or 25 square units.
The product of a number and
5 units
itself is the square of that number.
5 units
25
units2
Numbers like 9, 16, and 225 are called square
numbers or perfect squares because they are
squares of whole numbers.
The factors multiplied to form perfect squares are
called square roots. A radical sign, , is the
symbol used to indicate a square root of a number.
Square Root
Words: A square root of a number is one of its two
equal factors.
Examples:
Numbers
Algebra
4 · 4 = 16, so 16 = 4 If x · x or x2 = y, then y = x
Examples: Answers are in red!


Find the square of each number.
1) 3 9 2) 9 81 3) 12 144 4)
Find each square root.
5)
6)
25 5
100 10 7)
400 20 8)
Sports The infield of a baseball field is a
square with an
area of 8,100
square feet.



What are the dimensions of the infield?
23 529
49
7
1-4 Order of Operations
A numerical expression is a combination of
numbers and operations, such as
5+3x4
To evaluate expressions, use the order of
operations. These rules ensure that numerical
expressions have only one value.
Order of Operations
1. Evaluate all expressions inside grouping symbols.
2. Evaluate all powers.
3. Multiply and divide in order from left to right.
4. Add and subtract in order from left to right.
Besides using the symbols x and ·, multiplication
can be indicated by using parentheses. For
example, 2(3 + 5) means 2 x (3 + 5).
Examples:
Evaluate.
1) 5 + (12 – 3)
2) 8 – 3 · 2 + 7
3) 5 · 32 – 7
4) 14 + 3(7 – 2)
1-4 Order of Operations
A numerical expression is a combination of
numbers and operations, such as
5+3x4
To evaluate expressions, use the order of
operations. These rules ensure that numerical
expressions have only one value.
Order of Operations
1. Evaluate all expressions inside grouping symbols.
2. Evaluate all powers.
3. Multiply and divide in order from left to right.
4. Add and subtract in order from left to right.
Besides using the symbols x and ·, multiplication
can be indicated by using parentheses. For
example, 2(3 + 5) means 2 x (3 + 5).
Examples:
Evaluate. Answers in red.
1) 5 + (12 – 3)
2) 8 – 3 · 2 + 7
5+9
8–6+7
14
2+7=9
3) 5 · 32 – 7
4) 14 + 3(7 – 2)
5 · 9 – 7 = 45 – 7 = 38
14 + 3 · 5 = 14 + 15 = 29
Review for Quiz
Vocabulary terms:
A)base
B)cubed
C)evaluate
D)exponent
E)exponential form
F)factors
G)numerical expression
H)order of operations
I)perfect square
J)powers
K)radical sign
L)square
M)squared
N)square root
O)standard form
1) To _____ a number means to multiply that number
by itself.
2) A(n) _____ is the symbol used to indicate the
positive square root of a number.
3) A(n) _____ tells how many times a base is used as a
factor.
4) Mathematicians agreed on a(n) _____ so that numerical expressions would have only one value.
Topics on Quiz
 Four-step plan for solving problems
 Powers and exponents – write a power as
the product of the same factor – evaluate
powers – write products of the same factor
in exponential form
 Find squares and square roots
 Evaluate numerical expressions using order
of operations
1-5 Problem-Solving Investigation:
Guess and Check
When solving problems, one strategy that is helpful to use
is guess and check. Based on the information given:
 make a guess of the solution.
 use computations to check your guess.
 repeat until you get the correct solution.
You can use guess and check along with the 4-step
problem solving plan to solve a problem.
Understand – read carefully; understand problem
Plan – make a plan and estimate the solution
Solve – use your plan to solve the problem
Check – check the reasonableness of your solution
Example 1
Dr. Miller saw 40 birds and cats in one day. All together the pets he saw had 100 legs. How many of
each type of animal did Dr. Miller see in one day?
Example 2
In a math class of 26 students, each girl drew a
triangle and each boy drew a square. If there were
89 sides in all, how many girls and how many boys
were in the class?
1-5 Problem-Solving Investigation:
Guess and Check
When solving problems, one strategy that is helpful to use is
guess and check. Based on the information given:
 make a guess of the solution.
 use computations to check your guess.
 repeat until you get the correct solution.
You can use guess and check along with the 4-step problem
solving plan to solve a problem.
Understand – read carefully; understand problem
Plan – make a plan and estimate the solution
Solve – use your plan to solve the problem
Check – check the reasonableness of your solution
Example 1
Dr. Miller saw 40 birds and cats in one day. All together the pets he saw had 100 legs. How many of
each type of animal did Dr. Miller see in one day?
Understand
Plan
You know that Dr. Miller saw 40 birds and cats
total. You also know that there were 110 legs
in all. You need to find out how many of each
type of animal he saw in one day.
Make a guess and check it. Adjust the guess
until you get the right answer.
Solve
Check
Number of birds Number of cats
20
30
25
20
10
15
Total number of feet
2(20) + 4(20) = 120
2(30) + 4(20) = 100
2(25) + 4(15) = 110
25 birds have 50 feet. 15 cats have 60 feet.
Since 50 + 60 is 100, the answer is right.
Example 2
In a math class of 26 students, each girl drew a
triangle and each boy drew a square. If there were
89 sides in all, how many girls and how many boys
were in the class?
Understand You know that there are 26 boys and girls total.
Plan
Solve
Girls drew triangles, which have 3 sides and
boys drew squares, which have 4 sides. There
were 89 sides in all. You need to find out how
many boys and girls were in the class.
Make a guess and check it. Adjust the guess
until you get the right answer.
Number of girls Number of boys
13
16
15
Check
13
10
11
Total number of sides
3(13) + 4(13) = 91
3(16) + 4(10) = 88
3(15) + 4(11) = 89
15 girls have 45 sides. 11 boys have 44 sides.
Since 45 + 44 is 89, the answer is right.
1-6 Algebra: Variables and Expressions
1. Draw the next three figures in the pattern.
2. Find the number of squares in each figure and record
your data in a table like the one shown below.
Figure
1
2
3
4
5
6
Number of Squares 3
4
5
?
?
?
3. Without drawing the figure, determine how many squares
would be in the 10th figure. Check by making a drawing.
4. Find a relationship between the figure and its number of
squares.
A variable is a symbol that represents an unknown
quantity.
Figure number  n + 2
number of squares
The branch of mathematics that involves expressions
with variables is called algebra. The expression n + 2 is
called an algebraic expression because it contains
variables, numbers, and at least one operation.
To evaluate an algebraic expression you replace each
variable with its numerical value, then use the order
of operations to simplify.
In algebra, the multiplication sign is often omitted.
6d

6 times d
9st

9 times s times t
mn

m times n
The numerical factor of a multiplication expression that
contains a variable is called a coefficient. So, 6 is the
coefficient of 6d.
Examples:
Evaluate
1) 8w – 2v if w = 5 and v = 3.
2) 5m – 3n if m = 6 and n = 5.
3)

ab
3
if a = 7 and b = 6.
4) Evaluate x3 + 4 if x = 3.
In algebra, the multiplication sign is often omitted.
6d

6 times d
9st

9 times s times t
mn

m times n
The numerical factor of a multiplication expression that
contains a variable is called a coefficient. So, 6 is the
coefficient of 6d.
Examples:
Evaluate
1) 8w – 2v if w = 5 and v = 3.
8(5) – 2(3)
40 – 6 = 34
2) 5m – 3n if m = 6 and n = 5.
5(6) – 3(5)
30 – 15 = 15
3)

ab
if a = 7 and b
3
(7)(6)
= 42 = 14
3
3
= 6.
4) Evaluate x3 + 4 if x = 3.


(3)3 + 4
27 + 4 = 31
1-7 Algebra: Equations
An equation is a sentence that contains two
expressions separated by an equals sign, =. The
equals sign tells you that the expression on the left
is equivalent to the expression on the right.
5=7–2
5(3) = 15
23 = 15 + 5 + 3
An equation that contains a variable is neither
true nor false until the variable is replaced with a
number. A solution of an equation is a number for
the variable that makes the sentence true.
The process of finding a solution is called solving
an equation.
Choosing a variable to represent an unknown
quantity is called defining the variable.
Examples:
1) Solve 23 + y = 29 mentally.
2) On their annual family vacation, the Whites
travel 790 miles in two days. If on the first
day they travel 490 miles, how many miles
must they drive on the second day to reach
their destination?
1-7 Algebra: Equations
An equation is a sentence that contains two
expressions separated by an equals sign, =. The
equals sign tells you that the expression on the left
is equivalent to the expression on the right.
5=7–2
5(3) = 15
23 = 15 + 5 + 3
An equation that contains a variable is neither
true nor false until the variable is replaced with a
number. A solution of an equation is a number for
the variable that makes the sentence true.
The process of finding a solution is called solving
an equation.
Choosing a variable to represent an unknown
quantity is called defining the variable.
Examples:
1) Solve 23 + y = 29 mentally. y = 6
2) On their annual family vacation, the Whites
travel 790 miles in two days. If on the first
day they travel 490 miles, how many miles
must they drive on the second day to reach
their destination? Distance traveled the
second day = d. d = 300 miles
1-8 Algebra: Properties
Distributive Property
To multiply a sum by a number, multiply each addend
of the sum by the number outside the parentheses.
Numbers
Algebra
3(4 + 6) = 3(4) + 3(6) a(b + c) = a(b) + a(c)
5(7) + 5(3) = 5(7 + 3) a(b) + a(c) = a(b + c)
Properties are statements that are true for all numbers.
Real Number Properties
Commutative The order in which two numbers are
Properties
added or multiplied does not change
their sum or product.
a+b=b+a
Associative
Properties
The way in which three numbers are
grouped when they are added or
multiplied does not change their sum
or product.
a + (b + c) = (a + b) + c
Identity
Properties
ab=ba
a  (b  c) = (a  b)  c
The sum of an addend and 0 is the
addend. The product of a factor and 1
is the factor.
a+0=a
a1=a
Examples
1) Use the Distributive Property to evaluate
5(8 + 9).
5(8) + 5(9) = 40 + 45 = 85
2) Name the property shown by each
statement.
6 + (1 + 4) = (6 + 1) + 4
Use one or more properties to rewrite each
expression as an equivalent expression that does
not use parentheses.
3) (y + 1) + 4
= y + (1 + 4) = y + 5
4) 7(5x)
= (7  5) x = 35x
1-9 Algebra: Arithmetic Sequences
A sequence is an ordered list of numbers.
Each number in a sequence is called a term.
In an arithmetic sequence, each term is found by
adding the same number to the previous term.
8, 11, 14, 17, 20, . . .
each term is found by adding 3
to the previous term
Example 1
Describe the relationship between terms in the
arithmetic sequence 17, 23, 29, 35, . . . Then write
the next three terms in the sequence.
17, 23, 29, 35, . . . Each term is found by adding 6
to the previous term.
35 + 6 = 41, 41 + 6 = 47, 47 + 6 = 53
The next three terms are 41, 47, and 53.
Example 2
Brian’s parents have decided to start giving him a
monthly allowance for one year. Each month they
will increase his allowance by $10. Suppose this
pattern continues. What algebraic expression can
be used to find Brian’s allowance after any given
number of months? How much money will Brian
receive for allowance for the tenth month?
Make a table to display the sequence.
Position
1
2
3
n
Operation
1  10
2  10
3  10
n  10
Value of Term
10
20
30
10n
Each term is 10 times its position number. So,
the expression is 10n. How much money will
Brian earn after 10 months?
1-10 Algebra: Equations & Functions
A relationship that assigns exactly one output
value for each input value is called a function.
In a function, you start with an input number,
perform one or more operations on it, and get
an output number. The operation(s) performed
on the input is given by the function rule.
Input FUNCTION RULE  Output
You can organize the input numbers, output
numbers, and the function rule in a function
table. The set of input values is called the
domain, and set of output values is called the
range.
Make a function table.
James saves $20 each month. Make a function
table to show his savings after 1, 2, 3, and 4
months. Then identify the domain and range.
Input

Number
Of Months
1
2
3
4
Function Rule

Multiply
by 20
20 x 1
20 x 2
20 x 3
20 x 4
Output

Total
Savings
20
40
60
80
The domain is {1, 2, 3, 4}, and the range is {20,
40, 60, 80}.
Functions are often written as equations with
two variables – one to represent the input and
one to represent the output. Here’s an equation
for the situation in Example 1:
Function rule: multiply by 20
20x = y
Input (x): number of months
Output (y): total savings