Download Pre-Algebra

Variables and Expressions
Lesson 1-1
Pre-Algebra
Objectives:
1. To identify variables, numerical expressions, and
variable expressions
2. To write expressions for word phrases
Variables and Expressions
Lesson 1-1
Pre-Algebra
Terms:
1. Variable – is a letter that stands for a number
2.
Variable Expression – a mathematical phrase that uses variables,
numerals, and operation symbols
Variables and Expressions
Lesson 1-1
Additional Examples
Pre-Algebra
Identify each expression as a numerical expression or a
variable expression. For a variable expression, name the variable.
a. 7 - 3
numerical expression
b. 4t
variable expression; t is the variable.
Variables and Expressions
Lesson 1-1
Pre-Algebra
Additional Examples
Write a variable expression for the cost of p pens
priced at 29¢ each.
Words
29¢
Let p
Expression
29
times
number of pens
= number of pens.
•
p
The variable expression 29 • p, or 29p, describes the cost
of p pens.
The Order of Operations
Lesson 1-2
Pre-Algebra
Objectives:
1. To use the order of operations
2. To use grouping symbols
The Order of Operations
Lesson 1-2
Pre-Algebra
Terms:
1.Order of Operations – the order in which you perform operations
2. Simplify – perform the order of operations until you get to the simplest value
Tips: make sure to enter an expression into the calculator correctly, otherwise,
you will get an incorrect solution because the calculator uses the order of
operations.
The Order of Operations
Lesson 1-2
Additional Examples
Simplify 8 – 2 • 2.
8–2•2
8–4
First multiply.
4
Then subtract.
Pre-Algebra
The Order of Operations
Lesson 1-2
Pre-Algebra
Additional Examples
Simplify 12 ÷ 3 – 1 • 2 + 1.
12 ÷ 3 – 1 • 2 + 1
4
–
2 + 1
2+1
3
Multiply and divide from left to right.
Add and subtract from left to right.
Add.
The Order of Operations
Lesson 1-2
Pre-Algebra
Additional Examples
Simplify 20 – 3[(5 + 2) – 1].
20 – 3[(5 + 2) – 1]
20 – 3[
7 – 1]
Add within parentheses.
20 –
3 [6]
Subtract within brackets.
20 – 18
2
Multiply from left to right.
Subtract.
Writing and Evaluating Expressions
Lesson 1-3
Pre-Algebra
Objectives:
1. To evaluate variable expressions
2. To solve problems by evaluating expressions
Writing and Evaluating Expressions
Lesson 1-3
Pre-Algebra
Terms:
1. Evaluate – first replace each variable with its number, the use the order of
operations to simplify
Writing and Evaluating Expressions
Lesson 1-3
Pre-Algebra
Additional Examples
Evaluate 18 + 2g for g = 3.
18 + 2g = 18 + 2(3)
Replace g with 3.
= 18 + 6
Multiply.
= 24
Add.
Writing and Evaluating Expressions
Lesson 1-3
Pre-Algebra
Additional Examples
Evaluate 2ab – c for a = 3, b = 4, and c = 9.
3
2ab – c = 2 • 3 • 4 – 9
3
3
Replace the variables.
=2•3•4–3
Work within grouping symbols.
=6•4–3
Multiply from left to right.
= 24 – 3
Multiply.
= 21
Subtract.
Writing and Evaluating Expressions
Lesson 1-3
Pre-Algebra
Additional Examples
The Omelet Café buys cartons of 36 eggs.
a. Write a variable expression for the number of
cartons the Café should buy for x eggs.
b. Evaluate the expression for 180 eggs.
a. x eggs
x
36
b. 180 eggs
x
180
=
36
36
= 5
Evaluate for x = 180.
Divide.
The Omelet Café should buy 5 cartons to get 180 eggs.
Writing and Evaluating Expressions
Lesson 1-3
Pre-Algebra
Additional Examples
The One Pizza restaurant makes only one kind of pizza,
which costs $16. The delivery charge is $2. Write a variable
expression for the cost of having pizzas delivered. Evaluate the
expression to find the cost of having two pizzas delivered.
Words
$16
Let p
Expression
16
for each
pizza
plus
$2 delivery charge
+
2
= number of pizzas.
•
p
Evaluate the expression for p = 2.
16 • p + 2 = 16 • 2 + 2
= 32 + 2
= 34
It costs $34 to have two pizzas delivered.
Integers and Absolute Value
Lesson 1-4
Pre-Algebra
Objectives:
1. To represent, graph, and order integers
2. To find opposite and absolute values
Integers and Absolute Value
Lesson 1-4
Pre-Algebra
Terms:
1. Opposites – numbers that are the same distance from zero on a number line
but in opposite directions
2.Integers – the whole numbers and their opposites
3. Absolute Value – a number’s distance from zero on a number line
Integers and Absolute Value
Lesson 1-4
Additional Examples
Pre-Algebra
Write a number to represent the temperature
shown by the thermometer.
The thermometer shows 2 degrees Celsius below zero, or –2°C.
Integers and Absolute Value
Lesson 1-4
Additional Examples
Graph 2, –2, and –3 on a number line. Order the
numbers from least to greatest.
The numbers from least to greatest are –3, –2, and 2.
Pre-Algebra
Integers and Absolute Value
Lesson 1-4
Pre-Algebra
Additional Examples
Use a number line to find |–5| and |5|.
|–5| = 5
|5| = 5
Adding Integers
Lesson 1-5
Pre-Algebra
Objectives:
1. To look at models to add integers
2. To use rules to add integers
Adding Integers
Lesson 1-5
Pre-Algebra
Terms:
1. When you add opposites the sum is zero. So opposites are also called
Additive Inverses
Adding Integers
Lesson 1-5
Pre-Algebra
Additional Examples
Use tiles to find (–7) + 3.
(–7) + 3
Model the sum.
–4
Group and remove zero pairs.
There are four negative tiles left.
(–7) + 3 = – 4
Adding Integers
Lesson 1-5
Additional Examples
Pre-Algebra
From the surface, a diver goes down 20 feet and then
comes back up 4 feet. Find –20 + 4 to find where the diver is.
Start at 0. To represent –20,
move left 20 units.
To add positive 4, move right 4
units to –16.
–20 + 4 = –16
The diver is 16 feet below the surface.
Adding Integers
Lesson 1-5
Pre-Algebra
Additional Examples
Find each sum.
a. –20 + (–15)
–20 + (–15) = –35
Since both integers are negative,
the sum is negative.
b. 13 + (–17)
|–17| – |13| = 17 – 13
=4
13 + (–17) = – 4
Find the difference of the absolute values.
Simplify.
Since –17 has the greater absolute
value, the sum is negative.
Adding Integers
Lesson 1-5
Pre-Algebra
Additional Examples
A player scores 22 points. He then gets a penalty of 30
points. What is the player’s score after the penalty?
22 + (–30)
Write an expression.
|–30| – |22| = 30 – 22
Find the difference of the absolute values.
=8
22 + (–30) = – 8
The player’s score is – 8.
Simplify.
Since –30 has the greater absolute value,
the sum is negative.
Adding Integers
Lesson 1-5
Pre-Algebra
Additional Examples
Find –7 + (– 4) + 13 + (–5).
–7 + (– 4) + 13 + (–5)
–11 + 13
+ (–5)
2 + (–5)
–3
–7 + (– 4) + 13 + (–5) = –3
Add from left to right.
The sum of the two negative
integers is negative.
|13| – |11| = 2. Since 13 has the greater
absolute value, the sum is positive.
|5| – |2| = 3. Since –5 has the greater
absolute value, the sum is negative.
Subtracting Integers
Lesson 1-6
Pre-Algebra
Objectives:
1. To look at models to subtract integers
2. To use a rule to subtract integers
Subtracting Integers
Lesson 1-6
Pre-Algebra
Additional Examples
Find –7 – (–5).
Start with 7 negative tiles.
Take away 5 negative tiles. There
are 2 negative tiles left.
–7 – (–5) = –2
Subtracting Integers
Lesson 1-6
Pre-Algebra
Additional Examples
Find 2 – 8.
Start with 2 positive tiles.
There are not enough positive tiles to take
away 8. Add 6 zero pairs.
Take away 8 positive tiles. There are
6 negative tiles left.
2 – 8 = –6
Subtracting Integers
Lesson 1-6
Pre-Algebra
Additional Examples
An airplane left Houston, Texas, where the
temperature was 42°F. When the airplane landed in
Anchorage, Alaska, the temperature was 50°F lower. What
was the temperature in Anchorage?
42 – 50
Write an expression.
42 – 50 = 42 + (–50)
To subtract 50, add its opposite.
= –8
Simplify.
The temperature in Anchorage was –8°F.
Inductive Reasoning
Lesson 1-7
Pre-Algebra
Objectives:
1. To write rules for patterns
2. To make predictions and test conjectures
Inductive Reasoning
Lesson 1-7
Pre-Algebra
Terms:
1.Inductive Reasoning – making conclusions based on patterns you observe
2. Conjecture – a conclusion you reach by inductive reasoning
3. Counterexample – an example that proves a statement false
Tips: all you need is one counter example to prove a conjecture is not true
Inductive Reasoning
Lesson 1-7
Additional Examples
Pre-Algebra
Use inductive reasoning. Make a conjecture about the
next figure in the pattern. Then draw the figure.
Observation: The circles are rotating counterclockwise
within the square.
Conjecture: The next figure will have a shaded circle at
the top right.
Inductive Reasoning
Lesson 1-7
Pre-Algebra
Additional Examples
Write a rule for each number pattern.
a. 0, – 4, – 8, –12, . . .
Start with 0 and subtract 4 repeatedly.
b. 4, – 4, 4, – 4, . . .
Alternate 4 and its opposite.
c. 1, 2, 4, 8, 10, . . .
Start with 1. Alternate multiplying by 2
and adding 2.
Inductive Reasoning
Lesson 1-7
Pre-Algebra
Additional Examples
Write a rule for the number pattern 110, 100, 90, 80, . . .
Find the next two numbers in the pattern.
110, 100, 90,
– 10 – 10 – 10
80
The first number is 110.
The next numbers are found by
subtracting 10.
The rule is Start with 110 and subtract 10 repeatedly.
The next two numbers in the pattern are 80 – 10 = 70 and
70 – 10 = 60.
Inductive Reasoning
Lesson 1-7
Additional Examples
A child grows an inch a year for three years in a
row. Is it a reasonable conjecture that this child will grow
an inch in the year 2015?
No; children grow at an uneven rate, and eventually they
stop growing.
Pre-Algebra
Inductive Reasoning
Lesson 1-7
Additional Examples
Is each conjecture correct or incorrect? If it is
incorrect, give a counterexample.
a. Every triangle has three sides of equal length.
The conjecture is incorrect. The figure below is a
triangle, but it does not have three equal sides.
b. The opposite of a number is negative.
The conjecture is incorrect. The opposite of –2 is 2.
Pre-Algebra
Inductive Reasoning
Lesson 1-7
Additional Examples
(continued)
c. The next figure in the pattern below has 16 dots.
The conjecture is correct. The diagram below
shows the next figure in the pattern.
Pre-Algebra
Problem Solving Strategy: Look for a Pattern
Lesson 1-8
Pre-Algebra
Objectives:
1. To find number patterns
Problem Solving Strategy: Look for a Pattern
Lesson 1-8
Pre-Algebra
Tips: there are many ways to find a solution to a problem that involves a
pattern, such as tables or a tree diagram
Problem Solving Strategy: Look for a Pattern
Lesson 1-8
Pre-Algebra
Additional Examples
Each student on a committee of five students
shakes hands with every other committee member. How
many handshakes will there be in all?
The pattern is to add the number of new handshakes to
the number of handshakes already made.
4
the number of handshakes by 1 student
4+3=7
the number of handshakes by 2 students
Problem Solving Strategy: Look for a Pattern
Lesson 1-8
Pre-Algebra
Additional Examples
(continued)
Make a table to extend the pattern to 5 students.
Student
1
2
3
4
5
Number of original
handshakes
4
3
2
1
0
Total number of
handshakes
4
4+3
=7
There will be 10 handshakes in all.
7+2
=9
9+1
= 10
10 + 0
= 10
Multiplying and Dividing Integers
Lesson 1-9
Pre-Algebra
Objectives:
1. To multiply integers using
repeated addition, patterns,
and rules
2. To divide integers using
rules
Multiplying and Dividing Integers
Lesson 1-9
Pre-Algebra
Tips: when only multiplying or dividing integers, you can count the number of
negative signs to see if your answer will be negative or positive. Even number
of negative signs the answer is positive and an odd number of negative signs
the answer will be negative.
Multiplying and Dividing Integers
Lesson 1-9
Additional Examples
A diver is descending from the surface of the water
at a rate of 5 ft/s. Write an expression with repeated
addition to show how far the diver is from the surface of
the water after four seconds.
Use a number line to show repeated addition.
4 (–5) = (–5) + (–5) + (–5) + (–5) = –20
The diver is 20 feet below the surface of the water.
Pre-Algebra
Multiplying and Dividing Integers
Lesson 1-9
Pre-Algebra
Additional Examples
Use a pattern to find each product.
a. –2(7)
2(7) = 14
Start with products you know.
1(7) = 7
0(7) = 0
–1(7) = –7
–2(7) = –14
Continue the pattern.
Multiplying and Dividing Integers
Lesson 1-9
Pre-Algebra
Additional Examples
(continued)
b. –2(–7)
2(–7) = –14
Start with products you know.
1(–7) = –7
0(–7) = 0
–1(–7) = 7
–2(–7) = 14
Continue the pattern.
Multiplying and Dividing Integers
Lesson 1-9
Pre-Algebra
Additional Examples
Multiply 6(–2)(–3).
6(–2)(–3) = (–12)(–3)
= 36
Multiply from left to right. The product of
a positive integer and a negative integer
is negative.
Multiply. The product of two negative integers
is positive.
Multiplying and Dividing Integers
Lesson 1-9
Pre-Algebra
Additional Examples
Use the table to find the average of the differences
in the values of a Canadian dollar and a U.S. dollar for
1999–2002.
–33 + (–33) + (–35) + (–36)
4
Write an expression for
the average.
Multiplying and Dividing Integers
Lesson 1-9
Pre-Algebra
Additional Examples
(continued)
=
–137
4
= –34.25
Use the order of operations.
The fraction bar acts as a
grouping symbol.
The quotient of a negative integer and a
positive integer is negative.
For 1999–2002, the average difference was –34¢.
The Coordinate Plane
Lesson 1-10
Pre-Algebra
Objectives:
1. To name coordinates and quadrants in the coordinate
plane
2. To graph points in the coordinate plane
The Coordinate Plane
Lesson 1-10
Pre-Algebra
Terms:
1.Coordinate Plane – formed by the intersection of two number lines
2. x-axis – the horizontal number line
3.y-axis – the vertical number line
4. Quadrants – the x and y axes divide the coordinate plane into 4 sections
5.Orgin – the point where the x and y axes intersect
6. Ordered Pair – gives the coordinates (x , y) and location of a point
7.x-coordinate – shows the position left or right of the y-axis
8. y-coordinate – shows the position above or below the x-axis
The Coordinate Plane
Lesson 1-10
Pre-Algebra
Additional Examples
Write the coordinates of point G. In which quadrant
is point G located?
Point G is located 2 units to the left of the y-axis.
So the x-coordinate is –2.
The point is 3 units below the x-axis.
So the y-coordinate is –3.
The coordinates of point G are (–2, –3). Point G is located in
Quadrant III.
The Coordinate Plane
Lesson 1-10
Additional Examples
Graph point M(–3, 3).
Pre-Algebra