1-1 Variables and Expressions PRE-ALGEBRA LESSON 1-1 How many weeks in 363 days? 51 weeks 6767.

1-1

Variables and ExpressionsVariables and ExpressionsPRE-ALGEBRA LESSON 1-1PRE-ALGEBRA LESSON 1-1

How many weeks in 363 days?

51 weeks67


Complete each equation.

1. 1 week = days 2. 1 foot = inches

3. 1 nickel = cents 4. 1 gallon = quarts

5. 1 yard = feet

Check Skills You’ll Need

1-1

Solutions

1. 1 week = days 2. 1 foot = inches

3. 1 nickel = cents 4. 1 gallon = quarts

5. 1 yard = feet


7 12

5 4

3

1-1

Identify each expression as a numerical expression or a

variable expression. For a variable expression, name the variable.


numerical expression

variable expression; t is the variable.

a. 7 3

b. 4t

Quick Check

1-1


Write a variable expression for the cost of p pens

priced at 29¢ each.

= number of pens.

Words times

Let

29¢ number of pens

p

The variable expression 29 • p, or 29p, describes the cost of p pens.

Expression 29 p•

Quick Check

1-1


Write a variable expression for each word phrase.

1. the total of h and 56 2. three less than d

3. p decreased by three 4. a divided by 7

p – 3

h + 56 d – 3

a ÷ 7

1-1

1-2

The Order of OperationsThe Order of OperationsPRE-ALGEBRA LESSON 1-2PRE-ALGEBRA LESSON 1-2

Divide. Round answers to the nearest tenth.

a. 75 ÷ 8 b. 740 ÷ 7

9.4 105.7


(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


1-2


Solutions

1. Estimate: 2. Estimate: 3. Estimate:160 ÷ 2 = 80 360 ÷ 8 = 45 280 ÷ 4 = 70 82 43 712 164 8 344 4 284 –16 –32 –28 4 24 4 – 4 – 24 – 4 0 0 0

4. Estimate: 5. Estimate: 6. Estimate:140 ÷ 7 = 20 195 ÷ 13 = 15 625 ÷ 25 = 25 19 14 267 133 13 182 25 650 –7 –13 –50 63 52 150 –63 – 52 –150 0 0 0

1-2


Simplify 8 – 2 • 2.

8 – 2 • 2

8 – 4 First multiply.

Then subtract.4

Quick Check

1-2


Simplify 12 ÷ 3 – 1 • 2 + 1.

12 ÷ 3 – 1 • 2 + 1

Add and subtract from left to right.2 + 1

4 – 2 + 1 Multiply and divide from left to right.

3 Add.

Quick Check

1-2


Simplify 20 – 3[(5 + 2) – 1].

20 – 3[(5 + 2) – 1]

20 – 3[ 7 – 1] Add within parentheses.

Subtract within brackets.20 – 3 [6]

20 – 18 Multiply from left to right.

Subtract.2

Quick Check

1-2


Simplify each expression.

1. 7(3) – 2 • 4 2. 6 ÷ 2 + 1 • 5

3. 10 ÷ (4 + 1) 4. 3[9 • 2 ÷ (10 – 4)]

2

13 8

9

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1-3

Writing and Evaluating Expressions Writing and Evaluating Expressions PRE-ALGEBRA LESSON 1-3PRE-ALGEBRA LESSON 1-3

Evaluate each expression.

a. 8 + 6 – 2 b. 16 – 8 + 2 c. 12 + 9 • 2

12 10 30


(For help, go to Lesson 1-2.)


1. 6(9 + 1) 2. 17 – 2 + 3

3. 9 + 8 • 2 + 4 4. [3(5) + 1] • 2


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


Evaluate 18 + 2g for g = 3.

18 + 2g = 18 + 2(3) Replace g with 3.

= 18 + 6 Multiply.

= 24 Add.

Quick Check

1-3


Evaluate 2ab – for a = 3, b = 4, and c = 9.

= 2 • 3 • 4 – 3 Work within grouping symbols.

= 6 • 4 – 3 Multiply from left to right.

= 24 – 3 Multiply.

= 21 Subtract.

c3

2ab – = 2 • 3 • 4 – Replace the variables.c3

93

1-3

Quick Check


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

b. 180 eggs

x36 = Evaluate for x = 180.180

36

x36

= 5 Divide.

The Omelet Café should buy 5 cartons to get 180 eggs.

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Quick Check


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.

$2 delivery charge

= number of pizzas.

Words for each

Let

$16

p

pizza plus

Evaluate the expression for p = 2.

16 • p + 2 = 16 • 2 + 2 = 32 + 2 = 34

It costs $34 to have two pizzas delivered.

Expression 16 p• 2+

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Quick Check


Evaluate.

1. 7(b) – 4 for b = 3 2. h ÷ 2 + 1 for h = 12

3. 3c + 4 ÷ d for c = 8, d = 2 4. fg – g for f = 5, g = 7

26

17 7

28

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1-4

Integers and Absolute ValueIntegers and Absolute ValuePRE-ALGEBRA LESSON 1-4PRE-ALGEBRA LESSON 1-4

Order from least to greatest: , –1, 0.5, – 0.2, 23

18

– 1, – 0.2, , 0.5,18

23



Write an integer for each situation.

1. lose $7 2. find $9

3. 8 steps forward 4. 3 yards gained

5. 5 floors down


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Solutions

1. lose $7 2. find $9

–7 9

3. 8 steps forward 4. 3 yards gained

8 3

5. 5 floors down

–5

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


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.

1-4

Quick Check


Use a number line to find |–5| and |5|.

|–5| = 5 |5| = 5

1-4

Quick Check


Write an integer to represent each situation.

1. A debt of $50

2. A dive of 23 feet below the surface

Simplify.

3. | –12 | 4. | 8 |

12

–50

–23

8

1-4

1-5

Adding IntegersAdding IntegersPRE-ALGEBRA LESSON 1-5PRE-ALGEBRA LESSON 1-5

List each set of fractions in order from greatest to least.

a. , , b. , , c. , , 56

910

35

16

23

27

14

13

15

27

13

14

15

27

23

16

, , , , 9 10

56

35

, ,



Compare. Use >, <, or = to complete each statement.

1. –6 –3 2. 2 –15

3. –5 | 5 | 4. | 10 | | –10 |

5. | 9 | | –2 | 6. | –8 | | 0 |


1-5

PRE-ALGEBRA LESSON 1-5PRE-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

Adding IntegersAdding Integers

(–7) + 3 Model the sum.

– 4 Group and remove zero pairs. There are four negative tiles left.


Use tiles to find (–7) + 3.

(–7) + 3 = – 4

Quick Check

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.

The diver is 16 feet below the surface.

–20 + 4 = –16

Start at 0. To represent –20, move left 20 units. To add positive 4, move right 4 units to –16.

1-5

Quick Check


Find each sum.

b. 13 + (–17)

a. –20 + (–15)

Since –17 has the greater absolute value, the sum is negative.

13 + (–17) = – 4

Since both integers are negative, the sum is negative.

–20 + (–15) = –35

Find the difference of the absolute values.|–17| – |13| = 17 – 13

= 4 Simplify.

1-5

Quick Check


A player scores 22 points. He then gets a penalty of 30

points. What is the player’s score after the penalty?

Write an expression.22 + (–30)

Find the difference of the absolute values.|–30| – |22| = 30 – 22

Since –30 has the greater absolute value, the sum is negative.

22 + (–30) = – 8

The player’s score is – 8.

= 8 Simplify.

1-5

Quick Check


Find –7 + (– 4) + 13 + (–5).

Add from left to right.–7 + (– 4) + 13 + (–5)

–7 + (– 4) + 13 + (–5) = –3

–3 |5| – |2| = 3. Since –5 has the greater absolute value, the sum is negative.

The sum of the two negative integers is negative.

–11 + 13 + (–5)

|13| – |11| = 2. Since 13 has the greater absolute value, the sum is positive.

2 + (–5)

1-5

Quick Check

Find each sum.

1. –37 + (–5) 2. 14 + (–4)

3. –100 + 5 + (–3) 4. Evaluate 33 + t for t = –11.


–98

–42 10

22

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1-6

Subtracting IntegersSubtracting IntegersPRE-ALGEBRA LESSON 1-6PRE-ALGEBRA LESSON 1-6

Find all possible number pairs for which the sum of two wholenumbers is 63 and the difference between the numbers is lessthan 10.

31 and 32; 30 and 33; 29 and 34; 28 and 35; 27 and 36



Find each sum.

1. 8 + (–9) 2. –11 + (–18) 3. –4 + (–6)

4. 14 + (–3) 5. 6 + (–6) 6. –13 + (–10)


1-6


Solutions

1. 8 + (–9) 4. 14 + (–3)

| –9 | – | 8 | = 9 – 8 | 14 | – | –3 | = 14 – 3

= 1 = 11

8 + (–9) = –1 14 + (–3) = 11

2. –11 + (–18) 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

Start with 7 negative tiles.

Take away 5 negative tiles. There are 2 negative tiles left.


Find –7 – (–5).

–7 – (–5) = –2

Quick Check

1-6

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.


Find 2 – 8.

2 – 8 = –6

1-6

Quick Check


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?

The temperature in Anchorage was –8°F.

Write an expression.42 – 50

To subtract 50, add its opposite.42 – 50 = 42 + (–50)

= –8 Simplify.

1-6

Quick Check

Find each difference.

1. –24 – (–5) 2. 19 – (–4) 3. –33 – 11

4. 14 – 46 5. –200 – 50 – (–10)


–44–19 23

–32 –240

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1-7

Inductive ReasoningInductive ReasoningPRE-ALGEBRA LESSON 1-7PRE-ALGEBRA LESSON 1-7

Find the difference between and . 312

18

324

18

=


(For help, go to Skills Handbook.)

Find each difference.

1. –3 – 4 2. –7 – 4

3. –11 – 4 4. –15 – 4


1-7


Solutions

1. –3 – 4 2. –7 – 4

–3 – 4 = –3 + (–4) –7 – 4 = –7 + (–4)

= –7 = –11

3. –11 – 4 4. –15 – 4

–11 – 4 = –11 + (–4) –15 – 4 = –15 + (–4)

= –15 = –19

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


Write a rule for each number pattern.

a. 0, – 4, – 8, –12, . . .

b. 4, – 4, 4, – 4, . . .

c. 1, 2, 4, 8, 10, . . .

Start with 0 and subtract 4 repeatedly.

Alternate 4 and its opposite.

Start with 1. Alternate multiplying by 2 and adding 2.

1-7

Quick Check


Write a rule for the number pattern 110, 100, 90, 80, . . .

Find the next two numbers in the pattern.

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.

The next numbers are found by subtracting 10. – 10 – 10 – 10

The first number is 110. 110, 100, 90, 80

1-7

Quick Check


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.

1-7

Quick Check


Is each conjecture correct or incorrect? If it is incorrect, give a counterexample.

a. Every triangle has three sides of equal length.

b. The opposite of a number is negative.

The conjecture is incorrect. The figure below is a triangle, but it does not have three equal sides.

The conjecture is incorrect. The opposite of –2 is 2.

1-7

(continued)


c. The next figure in the pattern below has 16 dots.

The conjecture is correct. The diagram belowshows the next figure in the pattern.

1-7

Quick Check


Find the next three numbers in each pattern.

1. 1, –1, 2, –2, 3, . . . 2. 1, 3, 7, 15, 31, . . .

3. –11, –8, –5, –2, . . .

1, 4, 7

–3, 4, –4 63, 127, 255

1-7

1-8

Problem Solving Strategy: Look for a Pattern Problem Solving Strategy: Look for a Pattern PRE-ALGEBRA LESSON 1-8PRE-ALGEBRA LESSON 1-8

Describe the pattern in this sequence of numbers and find the nexttwo numbers. 5, 8, 4, 9, 3, . . .

description: + 3, – 4, + 5, – 6; next two numbers: 10, 2



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, . . .


1-8


Solutions

1. 8 11 14 17 20 23 26

+3 +3 +3 +3 +3 +3Start with 8 and add 3 repeatedly.

2. 1 5 4 8 7 11 10 14

+4 –1 +4 –1 +4 –1 +4Start with 1. Alternate adding 4 and subtracting 1.

3. 3 5 10 12 24 26 52 54

+2 2 +2 2 +2 2 +2Start with 3. Alternate adding 2 and multiplying by 2.

4. 1 4 7 10 13 16 19

+3 +3 +3 +3 +3 +3Start with 1 and add 3 repeatedly.

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 the number of handshakes already made.

4 the number of handshakes by 1 student

4 + 3 = 7 the number of handshakes by 2 students

1-8


(continued)

There will be 10 handshakes in all.

Make a table to extend the pattern to 5 students.

Student

Number of originalhandshakes

Total number ofhandshakes

1

4

4

2

3

4 + 3

3

2

7 + 2

4

1

9 + 1

5

0

10 + 0= 7 = 9 = 10 = 10

Quick Check

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

1-9

Multiplying and Dividing IntegersMultiplying and Dividing IntegersPRE-ALGEBRA LESSON 1-9PRE-ALGEBRA LESSON 1-9

For 3 weeks, Jessica worked 2 hours per day, 4 days per week babysitting. If she earned $3.50 per hour, how much did she earn in all?

$84.00




1. 5 4 2. 3 8 3. 5 5

4. 14 2 5. 6 5 6. 20 7


1-9


Solutions

1. 5 2. 3 3. 5 4 8 5 20 24 25

4. 14 5. 6 6. 20 2 5 7 28 30 140

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.

4 (–5) = (–5) + (–5) + (–5) + (–5) = –20

The diver is 20 feet below the surface of the water.

Use a number line to show repeated addition.

Quick Check

1-9


a. –2(7)

Use a pattern to find each product.

2(7) = 14 Start with products you know.

1(7) = 7

0(7) = 0

–1(7) = –7 Continue the pattern.

–2(7) = –14

1-9


b. –2(–7)

(continued)

2(–7) = –14 Start with products you know.

1(–7) = –7

0(–7) = 0

–1(–7) = 7 Continue the pattern.

–2(–7) = 14

1-9

Quick Check


Multiply 6(–2)(–3).

6(–2)(–3) = (–12)(–3) Multiply from left to right. The product of a positive integer and a negative integer is negative.

= 36 Multiply. The product of two negative integers is positive.

1-9

Quick Check


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.

Write an expression for the average.

–27 + (–27) + (–26) + (–28)4

1-9


(continued)

The quotient of a negative integer and a positive integer is negative.

= –27

For 1994–1997, the average difference was –27¢.

Use the order of operations. The fraction bar acts as a grouping symbol.

–1084=

1-9

Quick Check

Find each product or quotient.

1. –7(–3) 2. –36 ÷ (–9)

3. –12 • 2 4. 7(–3)

5. –6 • (–2) • (–1)


–24

21 4

–21

–12

1-9

1-10

The Coordinate PlaneThe Coordinate PlanePRE-ALGEBRA LESSON 1-10PRE-ALGEBRA LESSON 1-10

Order from greatest to least: –1, 3, 0, – 2, – 4

3, 0, – 1, – 2, – 4



Graph numbers on a number line.

1. –2, 1, –5 2. 0, 2, –4

3. –3, 3, –2 4. –1, –5, –8


1-10


Solutions

1. –2, 1, –5 2. 0, 2, –4

3. –3, 3, –2 4. –1, –5, –8

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 Quadrant III. Quick Check

1-10


Graph point M(–3, 3).

1-10

Quick Check


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

1-1 Variables and Expressions PRE-ALGEBRA LESSON 1-1 How many weeks in 363 days? 51 weeks 6767.

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