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Transcript
```Variables and Expressions
PRE-ALGEBRA LESSON 1-1
How many weeks in 363 days?
51 6 weeks
7
1-1
Variables and Expressions
PRE-ALGEBRA LESSON 1-1
Complete each equation.
1. 1 week =
days
2. 1 foot =
3. 1 nickel =
cents
4. 1 gallon =
5. 1 yard =
inches
quarts
feet
Check Skills You’ll Need
1-1
Variables and Expressions
PRE-ALGEBRA LESSON 1-1
Solutions
1. 1 week =
7
days
2. 1 foot =
3. 1 nickel =
5
cents
4. 1 gallon =
5. 1 yard =
3
feet
1-1
12
inches
4
quarts
Variables and Expressions
PRE-ALGEBRA LESSON 1-1
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.
Quick Check
1-1
Variables and Expressions
PRE-ALGEBRA LESSON 1-1
Write a variable expression for the cost of p pens
priced at 29¢ each.
Words
29¢
Let p
Expression
29
number of pens
times
= number of pens.
•
p
The variable expression 29 • p, or 29p, describes the cost
of p pens.
Quick Check
1-1
Variables and Expressions
PRE-ALGEBRA LESSON 1-1
Write a variable expression for each word phrase.
1. the total of h and 56
2. three less than d
d–3
h + 56
3. p decreased by three
4. a divided by 7
p–3
a÷7
1-1
The Order of Operations
PRE-ALGEBRA LESSON 1-2
Divide. Round answers to the nearest tenth.
a. 75 ÷ 8
b. 740 ÷ 7
9.4
105.7
1-2
The Order of Operations
PRE-ALGEBRA LESSON 1-2
(For help, go to Skills Handbook p. 760.)
Find each quotient.
1. 164 ÷ 2
2. 344 ÷ 8
3. 284 ÷ 4
4. 133 ÷ 7
5. 182 ÷ 13
6. 650 ÷ 25
Check Skills You’ll Need
1-2
The Order of Operations
PRE-ALGEBRA LESSON 1-2
Solutions
1. Estimate:
160 ÷ 2 = 80
82
2 164
–16
4
– 4
0
2. Estimate:
360 ÷ 8 = 45
43
8 344
–32
24
– 24
0
3. Estimate:
280 ÷ 4 = 70
71
4 284
–28
4
– 4
0
4. Estimate:
140 ÷ 7 = 20
19
7 133
–7
63
–63
0
5. Estimate:
195 ÷ 13 = 15
14
13 182
–13
52
– 52
0
6. Estimate:
625 ÷ 25 = 25
26
25 650
–50
150
–150
0
1-2
The Order of Operations
PRE-ALGEBRA LESSON 1-2
Simplify 8 – 2 • 2.
8–2•2
8–4
First multiply.
4
Then subtract.
Quick Check
1-2
The Order of Operations
PRE-ALGEBRA LESSON 1-2
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.
Quick Check
1-2
The Order of Operations
PRE-ALGEBRA LESSON 1-2
Simplify 20 – 3[(5 + 2) – 1].
20 – 3[(5 + 2) – 1]
20 – 3[
7 – 1]
20 –
3 
Subtract within brackets.
20 – 18
2
Multiply from left to right.
Subtract.
Quick Check
1-2
The Order of Operations
PRE-ALGEBRA LESSON 1-2
Simplify each expression.
1. 7(3) – 2 • 4
2. 6 ÷ 2 + 1 • 5
13
8
4. 3[9 • 2 ÷ (10 – 4)]
3. 10 ÷ (4 + 1)
2
9
1-2
Writing and Evaluating Expressions
PRE-ALGEBRA LESSON 1-3
Evaluate each expression.
a. 8 + 6 – 2
12
b. 16 – 8 + 2
c. 12 + 9 • 2
10
30
1-3
Writing and Evaluating Expressions
PRE-ALGEBRA LESSON 1-3
(For help, go to Lesson 1-2.)
Simplify each expression.
1. 6(9 + 1)
2. 17 – 2 + 3
3. 9 + 8 • 2 + 4
4. [3(5) + 1] • 2
Check Skills You’ll Need
1-3
Writing and Evaluating Expressions
PRE-ALGEBRA LESSON 1-3
Solutions
1. 6(9 + 1)
2. 17 – 2 + 3
6(10)
15 + 3
60
18
3. 9 + 8 • 2 + 4
4. [3(5) + 1] • 2
9 + 16 + 4
[15 + 1] • 2
29
16 • 2
32
1-3
Writing and Evaluating Expressions
PRE-ALGEBRA LESSON 1-3
Evaluate 18 + 2g for g = 3.
18 + 2g = 18 + 2(3)
Replace g with 3.
= 18 + 6
Multiply.
= 24
Quick Check
1-3
Writing and Evaluating Expressions
PRE-ALGEBRA LESSON 1-3
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.
Quick Check
1-3
Writing and Evaluating Expressions
PRE-ALGEBRA LESSON 1-3
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.
1-3
Quick Check
Writing and Evaluating Expressions
PRE-ALGEBRA LESSON 1-3
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.
1-3
Quick Check
Writing and Evaluating Expressions
PRE-ALGEBRA LESSON 1-3
Evaluate.
1. 7(b) – 4 for b = 3
2. h ÷ 2 + 1 for h = 12
17
7
4. fg – g for f = 5, g = 7
3. 3c + 4 ÷ d for c = 8, d = 2
26
28
1-3
Integers and Absolute Value
PRE-ALGEBRA LESSON 1-4
Order from least to greatest: 2 , –1, 0.5, – 0.2, 1
3
8
– 1, – 0.2, 1 , 0.5, 2
8
3
1-4
Integers and Absolute Value
PRE-ALGEBRA LESSON 1-4
(For help, go to Skills Handbook p. 775.)
Write an integer for each situation.
1. lose \$7
2. find \$9
3. 8 steps forward
4. 3 yards gained
5. 5 floors down
Check Skills You’ll Need
1-4
Integers and Absolute Value
PRE-ALGEBRA LESSON 1-4
Solutions
1. lose \$7
2. find \$9
–7
3. 8 steps forward
9
4. 3 yards gained
8
3
5. 5 floors down
–5
1-4
Integers and Absolute Value
PRE-ALGEBRA LESSON 1-4
Write a number to represent the temperature
shown by the thermometer.
The thermometer shows 2 degrees Celsius below zero, or –2°C.
Quick Check
1-4
Integers and Absolute Value
PRE-ALGEBRA LESSON 1-4
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.
Quick Check
1-4
Integers and Absolute Value
PRE-ALGEBRA LESSON 1-4
Use a number line to find |–5| and |5|.
|–5| = 5
|5| = 5
Quick Check
1-4
Integers and Absolute Value
PRE-ALGEBRA LESSON 1-4
Write an integer to represent each situation.
1. A debt of \$50
–50
2. A dive of 23 feet below the surface
–23
Simplify.
3. | –12 |
4. | 8 |
12
8
1-4
PRE-ALGEBRA LESSON 1-5
List each set of fractions in order from greatest to least.
a. 5 , 9 , 3
6 10
5
9 , 5, 3
10 6 5
b. 1 , 2 , 2
6
3
7
2 ,2,1
3 7 6
c. 1 , 1 , 1
4
3
5
1 , 1, 1
3 4 5
1-5
PRE-ALGEBRA LESSON 1-5
(For help, go to Lesson 1-4.)
Compare. Use >, <, or = to complete each statement.
1. –6
–3
2. 2
3. –5
|5|
4. | 10 |
| –10 |
6. | –8 |
|0|
5. | 9 |
| –2 |
–15
Check Skills You’ll Need
1-5
PRE-ALGEBRA LESSON 1-5
Solutions
1. –6 < –3
2. 2 > –15
3. –5 < | 5 |
4. | 10 | = | –10 |
5. | 9 | > | –2 |
6. | –8 | > | 0 |
1-5
PRE-ALGEBRA LESSON 1-5
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
Quick Check
1-5
PRE-ALGEBRA LESSON 1-5
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.
Quick Check
1-5
PRE-ALGEBRA LESSON 1-5
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.
Quick Check
1-5
PRE-ALGEBRA LESSON 1-5
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
Simplify.
Since –30 has the greater absolute value,
the sum is negative.
The player’s score is – 8.
Quick Check
1-5
PRE-ALGEBRA LESSON 1-5
Find –7 + (– 4) + 13 + (–5).
–7 + (– 4) + 13 + (–5)
–11 + 13
+ (–5)
2 + (–5)
–3
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.
–7 + (– 4) + 13 + (–5) = –3
Quick Check
1-5
PRE-ALGEBRA LESSON 1-5
Find each sum.
1. –37 + (–5)
2. 14 + (–4)
–42
3. –100 + 5 + (–3)
10
4. Evaluate 33 + t for t = –11.
–98
22
1-5
Subtracting Integers
PRE-ALGEBRA LESSON 1-6
Find all possible number pairs for which the sum of two whole
numbers is 63 and the difference between the numbers is less
than 10.
31 and 32; 30 and 33; 29 and 34; 28 and 35; 27 and 36
1-6
Subtracting Integers
PRE-ALGEBRA LESSON 1-6
(For help, go to Lesson 1-5.)
Find each sum.
1. 8 + (–9)
2. –11 + (–18)
3. –4 + (–6)
4. 14 + (–3)
5. 6 + (–6)
6. –13 + (–10)
Check Skills You’ll Need
1-6
Subtracting Integers
PRE-ALGEBRA LESSON 1-6
Solutions
1. 8 + (–9)
4. 14 + (–3)
| –9 | – | 8 | = 9 – 8
| 14 | – | –3 | = 14 – 3
=1
= 11
8 + (–9) = –1
2. –11 + (–18)
14 + (–3) = 11
5. 6 + (–6)
–11 + (–18) = –29
| –6 | – | 6 | = 6 – 6
=0
6 + (–6) = 0
3. –4 + (–6)
6. –13 + (–10)
–4 + (–6) = –10
–13 + (–10) = –23
1-6
Subtracting Integers
PRE-ALGEBRA LESSON 1-6
Find –7 – (–5).
Take away 5 negative tiles. There
are 2 negative tiles left.
–7 – (–5) = –2
Quick Check
1-6
Subtracting Integers
PRE-ALGEBRA LESSON 1-6
Find 2 – 8.
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
Quick Check
1-6
Subtracting Integers
PRE-ALGEBRA LESSON 1-6
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.
Quick Check
1-6
Subtracting Integers
PRE-ALGEBRA LESSON 1-6
Find each difference.
1. –24 – (–5)
–19
4. 14 – 46
–32
2. 19 – (–4)
23
5. –200 – 50 – (–10)
–240
1-6
3. –33 – 11
–44
Inductive Reasoning
PRE-ALGEBRA LESSON 1-7
Find the difference between 3 and 1 .
12
8
3
1
=
24
8
1-7
Inductive Reasoning
PRE-ALGEBRA LESSON 1-7
(For help, go to Skills Handbook.)
Find each difference.
1. –3 – 4
2. –7 – 4
3. –11 – 4
4. –15 – 4
Check Skills You’ll Need
1-7
Inductive Reasoning
PRE-ALGEBRA LESSON 1-7
Solutions
1. –3 – 4
2. –7 – 4
–3 – 4 = –3 + (–4)
–7 – 4 = –7 + (–4)
= –7
3. –11 – 4
= –11
4. –15 – 4
–11 – 4 = –11 + (–4)
–15 – 4 = –15 + (–4)
= –15
= –19
1-7
Inductive Reasoning
PRE-ALGEBRA LESSON 1-7
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.
Quick Check
1-7
Inductive Reasoning
PRE-ALGEBRA LESSON 1-7
Write a rule for each number pattern.
a. 0, – 4, – 8, –12, . . .
b. 4, – 4, 4, – 4, . . .
Alternate 4 and its opposite.
c. 1, 2, 4, 8, 10, . . .
Quick Check
1-7
Inductive Reasoning
PRE-ALGEBRA LESSON 1-7
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 next two numbers in the pattern are 80 – 10 = 70 and
70 – 10 = 60.
Quick Check
1-7
Inductive Reasoning
PRE-ALGEBRA LESSON 1-7
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.
Quick Check
1-7
Inductive Reasoning
PRE-ALGEBRA LESSON 1-7
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.
1-7
Inductive Reasoning
PRE-ALGEBRA LESSON 1-7
(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.
Quick Check
1-7
Inductive Reasoning
PRE-ALGEBRA LESSON 1-7
Find the next three numbers in each pattern.
1. 1, –1, 2, –2, 3, . . .
2. 1, 3, 7, 15, 31, . . .
–3, 4, –4
63, 127, 255
3. –11, –8, –5, –2, . . .
1, 4, 7
1-7
Problem Solving Strategy: Look for a Pattern
PRE-ALGEBRA LESSON 1-8
Describe the pattern in this sequence of numbers and find the next
two numbers. 5, 8, 4, 9, 3, . . .
description: + 3, – 4, + 5, – 6; next two numbers: 10, 2
1-8
Problem Solving Strategy: Look for a Pattern
PRE-ALGEBRA LESSON 1-8
(For help, go to Lesson 1-7.)
Write a rule for each pattern. Find the next three numbers.
1. 8, 11, 14, 17, . . .
2. 1, 5, 4, 8, 7, . . .
3. 3, 5, 10, 12, 24, . . .
4. 1, 4, 7, 10, . . .
Check Skills You’ll Need
1-8
Problem Solving Strategy: Look for a Pattern
PRE-ALGEBRA LESSON 1-8
Solutions
1. 8
11
14
17
20
23
26
+3
+3 +3 +3 +3 +3
2. 1
5
4
8
7
11 10
14
+4 –1
+4 –1 +4 –1 +4
3. 3
5
10 12 24 26 52 54
+2 2
+2 2 +2 2 +2
4. 1
4
7
10 13 16 19
+3 +3 +3
+3 +3 +3
1-8
Problem Solving Strategy: Look for a Pattern
PRE-ALGEBRA LESSON 1-8
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
4
the number of handshakes by 1 student
4+3=7
the number of handshakes by 2 students
1-8
Problem Solving Strategy: Look for a Pattern
PRE-ALGEBRA LESSON 1-8
(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
7+2
=9
9+1
= 10
10 + 0
= 10
There will be 10 handshakes in all.
Quick Check
1-8
Problem Solving Strategy: Look for a Pattern
PRE-ALGEBRA LESSON 1-8
Solve using any strategy.
1. You have a penny, a nickel, a dime, and a quarter. You give
away three coins. How many different amounts of money can
you give away? Name the values.
4; 16¢, 31¢, 36¢, 40¢
1-8
Multiplying and Dividing Integers
PRE-ALGEBRA LESSON 1-9
For 3 weeks, Jessica worked 2 hours per day, 4 days per week baby
sitting. If she earned \$3.50 per hour, how much did she earn in all?
\$84.00
1-9
Multiplying and Dividing Integers
PRE-ALGEBRA LESSON 1-9
(For help, go to Skills Handbook p. 759.)
Simplify each expression.
1. 5  4
2. 3  8
3. 5  5
4. 14  2
5. 6  5
6. 20  7
Check Skills You’ll Need
1-9
Multiplying and Dividing Integers
PRE-ALGEBRA LESSON 1-9
Solutions
1.
5
4
20
2.
3
8
24
3.
5
5
25
4.
14
2
28
5.
6
5
30
6.
20
7
140
1-9
Multiplying and Dividing Integers
PRE-ALGEBRA LESSON 1-9
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.
1-9
Quick Check
Multiplying and Dividing Integers
PRE-ALGEBRA LESSON 1-9
Use a pattern to find each product.
a. –2(7)
2(7) = 14
1(7) = 7
0(7) = 0
–1(7) = –7
Continue the pattern.
–2(7) = –14
1-9
Multiplying and Dividing Integers
PRE-ALGEBRA LESSON 1-9
(continued)
b. –2(–7)
2(–7) = –14
1(–7) = –7
0(–7) = 0
–1(–7) = 7
Continue the pattern.
–2(–7) = 14
Quick Check
1-9
Multiplying and Dividing Integers
PRE-ALGEBRA LESSON 1-9
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.
Quick Check
1-9
Multiplying and Dividing Integers
PRE-ALGEBRA LESSON 1-9
Use the table to find the average of the differences
in the values of a Canadian dollar and a U.S. dollar for
1994–1997.
–27 + (–27) + (–26) + (–28)
4
Write an expression for
the average.
1-9
Multiplying and Dividing Integers
PRE-ALGEBRA LESSON 1-9
(continued)
=
–108
4
= –27
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 1994–1997, the average difference was –27¢.
Quick Check
1-9
Multiplying and Dividing Integers
PRE-ALGEBRA LESSON 1-9
Find each product or quotient.
1. –7(–3)
2. –36 ÷ (–9)
21
4
3. –12 • 2
4. 7(–3)
–24
–21
5. –6 • (–2) • (–1)
–12
1-9
The Coordinate Plane
PRE-ALGEBRA LESSON 1-10
Order from greatest to least: –1, 3, 0, – 2, – 4
3, 0, – 1, – 2, – 4
1-10
The Coordinate Plane
PRE-ALGEBRA LESSON 1-10
(For help, go to Lesson 1-4.)
Graph numbers on a number line.
1. –2, 1, –5
2. 0, 2, –4
3. –3, 3, –2
4. –1, –5, –8
Check Skills You’ll Need
1-10
The Coordinate Plane
PRE-ALGEBRA LESSON 1-10
Solutions
1. –2, 1, –5
2. 0, 2, –4
3. –3, 3, –2
4. –1, –5, –8
1-10
The Coordinate Plane
PRE-ALGEBRA LESSON 1-10
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
1-10
Quick Check
The Coordinate Plane
PRE-ALGEBRA LESSON 1-10
Graph point M(–3, 3).
Quick Check
1-10
The Coordinate Plane
PRE-ALGEBRA LESSON 1-10
Draw a coordinate grid. Graph each point.
1. S(2, 3)
2. T(2, –3)
3. U(–2, 3)
4. K(0, –3)
5. L(3, 0)
1-10
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