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Transcript
PreAlgebra Concepts
Important Concepts
exponent
In a power, the number of times a
base number is used as a factor
order of operations
The rules which tell which
operation to perform first when
more than one operation is used.
1. Simplify expressions
inside parentheses.
2. Find the value of all
numbers with exponents.
3. Multiply OR divide in
order from left to right.
4. Add OR subtract in order
from left to right.
Examples
In 53, the exponent is 3. So, 53 is equal to 5 x 5 x 5 or 125.
Find the value of the expressions.
3 x 62 + 4 = 3 x 36 + 4
= 108 + 4
= 112
60 ÷ (12 + 23) ∙ 9 =
=
=
=
60 ÷ (12 + 8) ∙ 9
60 ÷ 20 ∙ 9
3∙ 9
27
evaluating algebraic expressions
Substitute values for the variables Find the value of the expression if x = 3, y = 5, and z = 20.
and evaluate using the order of
operations above.
2x + z ÷ y = 2 ∙ 3 + 20 ÷ 5
= 6 + 20 ÷ 5
= 6 + 4
= 10
Evaluate each algebraic expression for x= 12, y = 8, and z = 2. Write your work on the lines provided.
1.
40
–
y2 ÷
z
2.
( x + y ) ∙ 32
______________________________
______________________________
______________________________
______________________________
______________________________
______________________________
3.
( y + 1 )2 – x
4.
5x – z3
______________________________
______________________________
______________________________
______________________________
______________________________
______________________________
Data Analysis
Important Concepts
mean
The sum of the values divided by
the number of values in the set.
median
The number that marks the
middle of an ordered set of data.
mode
The mode is the value that occurs
with greatest frequency in a set of
data.
range
The difference between the
greatest and least values in a data
set.
Examples
{1, 2, 3, 4}
1 + 2 + 3 + 6 = 12 and 12 ÷ 4 = 3 so the mean is 3
{4, 5, 6, 7, 8}
The median is 6 because it marks the middle value when the
numbers are ordered from least to greatest.
{1, 2, 2, 2, 3, 4}
The mode is 2 because it is the value that occurs most often.
{1, 2, 3, 4, 5}
The range is 4 because 5 – 1 = 4.
Use the data table to answer each question.
Calories of Selected
Vegetables (per serving)
15
35
50
31
5
25
85
25
20
1.
What is the mean of the calories? Round to the nearest tenth, if necessary.
2.
Order the set of data from least to greatest. What is the median of the calories?
3.
What is the mode of the calories? How do you know?
4.
What is the range of the calories?
Basic Integer Concepts
Important Concepts
integer
The set of numbers that includes
all the whole numbers and their
opposites, including 0.
negative integers
All integers less than zero and
found to the left of zero on the
number line.
positive integers
All integers greater than zero and
found to the right of zero on the
number line.
opposites
Two numbers the same distance
from zero but in opposite
directions on the number line.
*The sum of two opposites is 0.
absolute value
The distance a number is from 0
on the number line.
Absolute value is never negative.
Examples
{. . .-4, -3, -2, -1, 0, 1, 2, 3, 4. . .}
-8
-7
-6
-5
-4
-3
-2
-1
0
0
1
2
3
4
5
6
7
8
-4 and 4
-9 and 9
|-9| = 9
-36 and 36
|8| = 8
|-13| = 13
-17 and 17
|100| = 100
Identify numbers that meet the given requirements.
1.
2 positive numbers that are not integers
___________
__________
2.
2 negative numbers that are not integers
___________
__________
Graph the given numbers on the number line.
3.
6, 0, -5, 1, -3
Insert <, >, or = to make a true statement.
4.
-9
-8
5.
-3
-6
6.
-2
-4
Name the opposite of each integer given.
7.
-12
__________
8.
9
__________
__________
10.
8 + 7 x 6  14
__________
Identify the absolute value.
9.
-10
Ratio and Proportion
Important Concepts
Examples
ratio
A comparison of two numbers by
division. Does NOT have to be a
PART to WHOLE comparison.
Can be written in 3 forms
a
, a to b, a:b
b
3
, 3 to 7, 3:7
7
7
, 7 to 6, 7:6
6
ratio of triangles to circles
ratio of nonpolygons to polygons
Corresponding sides of similar shapes are in proportion. All
measures shown are in units.
proportion
An equation that shows that two
ratios are equivalent.
3
3 5
=
9 15
9
5
15
Use the numbers to write ratios for #1-3. Write each ratio in 3 forms.
-14
35
1
2
24
100
-4
-8
9
16
1.
ratio of square numbers to nonsquare numbers
_____ _____ _____
2.
ratio of negative integers to positive integers
_____ _____ _____
3.
ratio of abundant numbers to total numbers
_____ _____ _____
Identify the missing value in the proportion.
4.
5 20
=
8
x
5.
y
7
=
60 10
6.
72
9
=
a
20
Probability
Important Concept
probability
Probability is the likelihood that an
event will occur. Probability can be
expressed as a fraction, decimal, or
percent.
P(event) = number of favorable outcomes
number of possible outcomes
The probability of 2 events occurring can be
found by multiplying the probability of each
independent event.
Examples
You are rolling a standard number cube {1, 2, 3, 4, 5, 6}.
Find the probability of each event.
1
P(5) =
6
3 1
P(even number) =
=
6 2
P(composite number) =
2 1
=
6 3
P(4, 6) =
1
1
1
x
=
6
6
36
P(1, prime number) =
1
1
1
x
=
6
2
12
The letters T, H, E, O, R, E, T, I, C, A, and L are put in a bag. Find each probability.
1.
P (A, I, or O) =
________
2.
P (not O or T) =
________
3.
P (U) =
________
You choose 1 letter from the bag and then put in back before choosing a second letter.
Find each probability.
4.
P (T, vowel) =
________
5.
P (E, T) =
________
Operations with Fractions and Mixed Numbers
Addition Algorithm
1.
2.
3.
4.
1. Find a common denominator.
2. Write equivalent fractions.
3. Add numerators and keep
denominator.
4. Add whole numbers, if necessary.
5. Simplify your answer.
5.
6.
1.
2.
3.
4.
5.
Multiplication Algorithm
1. Change all whole or mixed
numbers to improper fractions.
2. Multiply numerators.
3. Multiply denominators.
4. Simplify your answer.
Look to simplify before you multiply by
canceling a numerator with a
denominator that has a common factor.
Subtraction Algorithm
Find a common denominator.
Write equivalent fractions.
Borrow if necessary.
Subtract numerators and keep
denominator.
Subtract whole numbers, if
necessary.
Simplify your answer.
OR
Find a common denominator.
Write equivalent fractions.
Change to improper fractions so
no borrowing is necessary.
Subtract numerators.
Simplify your answer.
Division Algorithm
1. Change all whole or mixed
numbers to improper fractions.
2. Keep the dividend and change the
divisor to its reciprocal.
3. Then follow the rules for
multiplication.
OR
1. Change all whole or mixed
numbers to improper fractions.
2. Find a common denominator for
both fractions.
3. Write equivalent fractions.
4. Divide the numerators only.
5. Simplify your answer.
Solve each problem. Use separate paper if necessary. NO CALCULATORS!
2
5
+ 1
5
6
2.
5
÷ 3
7
2
1
- 1
6
5
6.
2
1.
2
5.
10
1
x 3
2
3.
7
9
7.
10
x 1
1
2
÷
1
2
7
22
3.
3
2
9
- 1
2
3
8.
3
2
9
+ 1
2
3
Operations with Decimals
Addition Algorithm
Subtraction Algorithm
1. Line up equal place values so you
are adding equal sized pieces.
2. Put in zeros as place holders, if
necessary.
3. Add beginning with the smallest
place value.
4. Bring down the decimal point into
the sum.
1. Line up equal place values so you
are adding equal sized pieces.
2. Put in zeros as place holders, if
necessary.
3. Borrow and rename when
necessary.
4. Subtract beginning with the
smallest place value.
5. Bring down the decimal point into
the difference.
Multiplication Algorithm
Division Algorithm
1. Multiply as you would with whole
numbers.
2. Find the sum of the decimal places
in both factors.
3. Count the same number of decimal
places going right to left in the
product and place the decimal.
example:
tenths x tenths = hundredths
1 place + 1 place = 2 places
1. If the divisor is a whole number,
bring the decimal straight up into
the quotient. Follow your division
algorithm for whole numbers,
adding zeros to the dividend as
necessary.
2. If the divisor is a decimal number,
multiply divisor and dividend by a
power of ten that will make the
divisor a whole number. Then
follow your division algorithm for
whole numbers, adding zeros to
the dividend as necessary.
Solve each problem. Use separate paper if necessary. NO CALCULATORS!
1.
5.5( 9.9)
2.
0.6 + 5.8
3.
3.4 – 0.972
4.
25.041 – 8.3
5.
1.009 + 12 + 0.87
6.
936 ÷ 0.12
7.
0.492 ÷ 4
8.
12.2 ∙
9.
43.59 x 0.1
1
2
Solve for n
Making a fact family may help you.
2
1
= 3
4
5
2
5
+ n = 5
3
6
2.
n–4
2 n = 20
1
2
4.
n
1
= 1
5
3
5.
4.3 + n = 13.42
6.
n – 12.03 = 8.2
7.
0.7 ∙ n = 0.084
8.
n ÷ 0.5 = 6.4
1.
1
3.
Mixed Problem Solving
Tips for Problem Solving






Read the question first so you know what you need to find.
Underline the important facts as you read the entire problem.
Look for key words as you read.
Plan how to solve the problem.
Find the solution and check your solution for reasonableness/accuracy.
Remember to label your solution.
Solve each problem. Show work to support your answer. You may use a calculator on
this section.
1.
Laura has 121 carpet tiles. Each tile is a 1-foot by 1-foot square. Can she use all
of the carpet tiles to make a square pattern without cutting any of the tiles?
2.
A florist has 28 white roses, 35 red roses, 14 pink roses, and 70 yellow roses to use
in making floral arrangements. He wants to use all the flowers and place an equal
number of each color rose in each arrangement. Find the greatest number of floral
arrangements he can make.
3.
According to a survey, 9 in 10 teens volunteer at least once a year. Of these, onethird help clean up their communities. What fraction of teen volunteers help by
cleaning up their communities?
4.
Andrew is going to plant three new types of vegetables in his garden. The garden
store sells packages of tomatillo seeds for $1.67, chili pepper seeds for $0.89, and
pumpkin seeds for $2.32. Andrew buys one of each package. How much change
will he receive from a ten-dollar bill?
5.
Find the perimeter and area of the figure below.
7 cm
5 cm
6.
Claire and Chad want to design a rectangular pen for their new puppy. They want
the pen to have an area of 48 square feet. Fencing costs $0.85 per foot. What
would be the dimensions and the cost of the least expensive pen Claire and Chad
could build, assuming the side lengths are whole numbers?
7.
A triangle has side lengths measuring 3 and 7. The measurement of the longest
side is missing. Ted says that one possibility for the unknown side length is 11. Is
Ted correct? Why or why not?
8.
The figure below shows the dimensions of Adam’s bedroom. Adam wants to put
new carpet in the room. How much carpet will Adam need?
6
3
2
3
4
9.
The Make A Difference Club raised 30% of their fundraising goal of $200. What
percent do they have left to earn? How much money do they have left to earn?
10.
Alexa wants to use ready-made 6-foot long fence sections for her yard. The yard is
a rectangle with dimensions 30 feet by 36 feet. How many fence sections will she
need to enclose her entire yard?
11.
The number of registered voters in the town of Cedarville is 8,900. In the last
election, Mayor Burgis won reelection with approximately 72% of the vote.
a.
If approximately 52% of registered voters voted in the election, how many
people voted?
b.
Based on your answer to part a, how many voters voted for Mayor Burgis?
12.
The Midtown Middle School cheerleaders earned $175 at a car wash. If this
amount is 25% of the cost of a new set of uniforms, what is the total cost for a set
of uniforms?
13.
Skateboards are on sale at Susan’s Skateshop for 30% off. If the regular price of a
skateboard is $89, what is the discounted price with a 7% sales tax?