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Chapter 1Number Sense, Algebra, and Functions

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Chapter 1Number Sense, Algebra, and Functions

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http://ca.gr5math.com/

Lesson 1-1 Prime Factors

Lesson 1-2 Powers and Exponents

Lesson 1-3 Order of Operations

Lesson 1-4 Problem-Solving Investigation: Use the Four-Step Plan

Lesson 1-5 Algebra: Variables and Expressions

Lesson 1-6 Algebra: Functions

Lesson 1-7 Problem-Solving Strategy: Guess and Check

Lesson 1-8 Algebra: Equations

Lesson 1-9 Algebra: Area Formulas

Lesson 1-10 Algebra: The Distributive Property

11Number Sense, Algebra, and Functions



Five-Minute Check

Main Idea and Vocabulary

California Standards

Key Concept: Prime and Composite

Example 1

Example 2

Example 3

1-11-1 Prime Factors



• I will find the prime factorization of a composite number.

• factor

• prime number

• composite number

• prime factorization



Standard 5NS1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 × 2 × 2 × 3 = 23 × 3).


Factors of 13: 1, 13

Tell whether the number 13 is prime, composite, or neither.


Answer: So, 13 is a prime number.

A whole number that has exactly two unique factors, 1 and the number itself, is a prime number.



Which of the following numbers is prime?

A. 4

B. 7

C. 6

D. 8




Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Answer: So, 30 is a composite number.

A number greater than 1 with more than two factors is a composite number.



Which of the following is a composite number?

A. 3

B. 5

C. 17

D. 9


Write the prime factorization of 48.


48

4 × 12

2 × 2 2 × 6×

2 2 2 × 32× × ×

48

2 × 24

2 6 × 4×

2 × 3 × 2 2 × 2×



What is the prime factorization of 64?

A. 2 × 4 × 2 × 4

B. 8 × 8

C. 2 × 32

D. 2 × 2 × 2 × 2 × 2 × 2


Five-Minute Check (over Lesson 1-1)



Example 1

Example 2

Example 3

Example 4

1-21-2 Powers and Exponents

Example 5

Example 6

Example 7



• I will use powers and exponents in expressions.

• base

• exponent

• power

• squared

• cubed



Standard 5NS1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor.

Standard 5NS1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication.


Write 7 × 7 × 7 × 7 × 7 using an exponent.


The base is 7. Since 7 is used as a factor five times, the exponent is 5.

7 × 7 × 7 × 7 × 7 = 75

Answer: So, the answer is 75.

Write as a power.



Which of the following is 4 × 4 × 4 × 4 × 4 × 4 written using an exponent?

A. 64

B. 45

C. 46

D. 44


Write 93 as a product of the same factor. Then find the value.


The base is 9. The exponent is 3. So, 9 is used as a factor 3 times.

93 = 9 × 9 × 9 Write 93 using repeated multiplication.

= 729 Multiply.



Which of the following is 27 written as the product of the same factor?

A. 2 × 2 × 2 × 2 × 2 × 2 = 64

B. 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128

C. 2 × 64 = 128

D. 2 × 2 × 32 = 128


King’s Peak is the highest point in Utah. It stands just a bit more than 46 meters. What is the height of the peak?


The base is 4. The exponent is 6. So, 4 is used as a factor 6 times.

46 = 4 × 4 × 4 × 4 × 4 × 4 Write 46 as a product.

= 4,096 Multiply.

Answer: So, Pike’s Peak is 4,096 meters.



The Taipei 101 building in Taiwan is just under 37 feet tall. What is the height of the building?

A. 2,187 feet

B. 3,000 feet

C. 2,000 feet

D. 1,678 feet


The hottest temperature at Salton Sea State Park in California can reach 53 degrees. What is the temperature?


53 = 5 × 5 × 5 Write 53 as a product.

= 125 Multiply.

Answer: So, the highest temperature is 125 degrees.



The coastal shelf of the Pacific Ocean is 36 feet deep. How deep is it?

A. 700 feet

B. 650 feet

C. 828 feet

D. 729 feet


Write the prime factorization of 28 using exponents.


28 = 2 × 2 × 7 Write the prime factorization.

= 22 × 7 Write products of identical factors using exponents.



Which of the following is the prime factorization of 36?

A. 33 × 22

B. 22 × 33

C. 22 × 32

D. 34 × 2




45 = 3 × 3 × 5 Write the prime factorization.





A. 33 × 52

B. 24 × 33

C. 22 × 33

D. 52 × 2




34 = 2 × 17 Write the prime factorization.





A. 33 × 2

B. 2 × 3

C. 2 × 13

D. 17 × 2





Key Concept: Order of Operations

Example 1

Example 2

Example 3

Example 4

Example 5

1-31-3 Order of Operations



• I will find the value of expressions using the order of operations.

• numerical expression

• order of operations



Reinforcement of Standard 4AF1.2 Interpret and evaluate mathematical expressions that now use parentheses.


Find the value 7 + 4 × 8.

7 + 4 × 8 = 7 + 32


= 39

Multiply 4 and 8 first.

Add 7 and 32.



Find the value of 6 + 2 × 7.

A. 84

B. 44

C. 20

D. 56



Find the value of 6 + 12 – 8.

6 + 12 – 8 = 18 – 8

= 10

Add 6 and 12 first.

Subtract 8 from 18.



Find the value of 4 + 7 – 3.

A. 8

B. 7

C. 9

D. 2


Find the value of 80 ÷ 4 + (8 – 5) – 10.


80 ÷ 4 + (8 – 5) – 10 = 80 ÷ 4 + 3 – 10

= 20 + 3 – 10

Subtract 5 from 8.

Divide 80 by 4.

= 23 – 10 Add 20 and 3.

= 13 Subtract 10 from 23.



Find the value of 50 ÷ 5 + (6 – 2) + 3.

A. 15

B. 13

C. 17

D. 12


Find the value of 24 + 6 × 3 – 5.


24 + 6 × 3 – 5 = 24 + 18 – 5

= 42 – 5

Multiply 6 and 3 first.

Add 24 and 18.




Find the value of 15 + 2 × 8 – 12.

A. 19

B. 124

C. 21

D. 18


To find the total cost, write an expression and then find its value.


Maria and 2 friends buy school supplies. Each person buys a pen, a folder, and a book. Write an expression for the total cost of their school supplies. Then find the cost.


3 × $1 + 3 × $2 + 3 × $8


= $3 + 3 × $2 + 3 × $8 Multiply 3 and 1.

= $3 + $6 + 3 × $8 Multiply 3 and 2.

= $3 + $6 + $24 Multiply 3 and 8.

= $33 Add 3, 6 and 24.

Answer: So, the total cost for school supplies was $33.



Kelly and her 2 sisters buy a new outfit. Each girl buys a shirt for $5, a skirt for $6, and a hair ribbon for $2. Choose the correct expression and total cost of their clothes.

A. 2 × $5 + 2 × $6 + 2 × $2 = $26

B. 3 × $5 + 3 × $6 + 3 × $2 = $39

C. 2 × $5 + 2 × $6 + 2 × $2 = $39

D. 3 × $5 + 3 × $6 + 3 × $2 = $26



Main Idea


Example 1: Problem-Solving Investigation

1-41-4 Problem-Solving Investigation: Use the Four-Step Plan



• I will use the four-step plan to solve a problem.



Standard 5MR1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.

Reinforcement of Standard 4NS3.4 Solve problems involving division of multi-digit numbers by one-digit numbers.


DAVID: Today, I learned that there are 5,280 feet in one mile. I wonder how many pennies would be in one mile if I lined the pennies up side by side?

YOUR MISSION: Find how many pennies are in a row that is one mile long.



Understand

What facts do you know?

• There are 5,280 feet in one mile.

What do you need to find?

• You need to find how many pennies are in a row that is one mile long.



Plan

Plan a strategy for solving the problem.

Find how many pennies are in one foot. Then multiply by 5,280 to find how many pennies are in one mile.



SolveUse your plan to solve the problem.


Line up pennies along a ruler. There are 16 pennies in one foot, and 16 × 5,280 = 84,480.


SolveWhat is the solution?


Answer: So, a row of pennies one mile long will contain 84,480 pennies.


Check

Does the answer make sense?

Look back at the problem. Use estimation to check.

15 × 5,000 = 75,000






Example 1

Example 2

Example 3

Example 4

Example 5

1-51-5 Algebra: Variables and Expressions



• I will evaluate algebraic expressions.

• algebra

• variable

• algebraic expression

• evaluate



Standard 5AF1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution.


Evaluate 30 + d if d = 6.


30 + d = 30 + 6

= 36

Replace d with 6.

Add 30 and 6.



Evaluate 42 – c if c = 12.

A. 54

B. 40

C. 30

D. 10


Evaluate e – f if e = 12 and f = 9.


e – f = 12 – 9

= 3

Replace e with 12 and f with 9.

Subtract 9 from 12.



Evaluate a + b if a = 6 and b = 7.

A. 12

B. 13

C. 14

D. 15


Evaluate 4y – 5 if y = 3.


4y – 5 = 4 • 3 – 5

= 12 – 5

Replace y with 3.

Multiply 4 and 3.




Evaluate 3x + 7 if x = 5.

A. 50

B. 21

C. 22

D. 30


The expression 6x – 3 represents the amount of money Dhara will need to pay for 6 binders with a $3 off coupon where x is the cost of each binder. How much will she pay if each binder is $5?


6x – 3 = 6 • 5 – 3

= 30 – 3

Replace x with 5.

Multiply 6 and 5.


Answer: So, Dhara will pay $27 for the binders.



The expression 5x + 8 represents the amount of money Myra will need to pay for 5 candles with an $8 off coupon where x is the cost of each candle. How much will she pay if each candle is $8?

A. $48

B. $69

C. $104

D. $55


An expression for finding the area of a triangle whose height is 5 units longer than its base is (b + 5) • b ÷ 2, where b is the measure of the base. Find the area of a triangle with a base of 6.


(b + 5) • b ÷ 2 = (6 + 5) • 6 ÷ 2

= 11 • 6 ÷ 2

Replace b with 6.

Add 6 and 5.

= 66 ÷ 2 Multiply 11 and 6.

Answer: So, the area of the triangle is 33 square units.

= 33 Divide 66 and 2.



An expression for finding the area of a triangle whose height is 7 units longer than its base is (b + 7) • b ÷ 2, where b is the measure of the base. Find the area of a triangle with a base of 3.

A. 12 square units

B. 20 square units

C. 15 square units

D. 10 square units





Example 1

Example 2

Example 3

Example 4

1-61-6 Algebra: Functions



• I will complete function tables and find function rules.

• function

• function table

• function rule

• defining the variable



Standard 5AF1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.


Complete the function table below.


The function rule is x + 5. Add 5 to each input.

4 + 5

10 + 5

11 + 5

9

15

16



Which answers complete the function table?

A. 10, 14, 19

B. 11, 14, 19

C. 10, 13, 19

D. 10, 14, 20


Find the rule for the function table.


Study the relationship between each input and output. Each output is three more than the input.

Answer: So, the function rule is n + 3.



Find the function rule for the function table.

A. n + 5

B. n + 7

C. 3n

D. 2n




Study the relationship between each input and output. Each output is one-ninth the input.

Answer: So, the function rule is t ÷ 9.




A. t – 6

B. t – 10

C. t ÷ 3

D. t ÷ 4


$20 for each yard

Taylor earns $20 for each yard she mows. Define a variable. Then write a function rule that relates the money she earns to the number of yards she mows.


Words

Variable

Expression

Let y represent the number of yards Taylor mows.

20 • y



Shane earns $2 for every item he sells during the fundraiser. Define a variable. Then write a function rule that relates the money he earns to the number of items sold.

A. 2f

B. f – 2

C. f + 2

D. 2 ÷ f



Main Idea


Example 1: Problem-Solving Strategy

1-71-7 Problem-Solving Strategy: Guess and Check



• I will solve problems by using the guess and check strategy.



Standard 5MR2.6 Make precise calculations and check the validity of the results from the context of the problem.

Reinforcement of Standard 4NS2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to two places.


A comic book store sells used comic books in packages of 5 and new comic books in packages of 3. Keisha buys a total of 16 comic books for her brother Trent for his birthday. How many packages of new and used comic books did Keisha buy for Trent?



UnderstandWhat facts do you know?

• The comic book store sells 3-book packages and 5-book packages.

• The used comic books come in packages of 5. The new comic books come in packages of 3.

• 16 books were bought.

What do you need to find?

• How many packages of new and used comic books did Keisha buy for Trent?



Plan

Plan a strategy for solving the problem.

Make a guess until you find an answer that makes sense for the problem.



Solve

Use your plan to solve the problem.

Answer: So, Keisha bought two 3-book packages and two 5-book packages.


1

12

2

1

2

12

1(3) + 1(5) = 8

1(3) + 2(5) = 13

2(3) + 1(5) = 112(3) + 2(5) = 16


Check

Does the answer make sense?

Look back at the problem. Two 3-book packages result in 6 books.

Two 5-book packages result in 10 books. Since 6 + 10 is 16, the answer is correct.






Example 1

Example 2

Example 3

1-81-8 Algebra: Equations



• I will solve equations by using mental math and the guess and check strategy.

• equation

• equals sign

• solve

• solution




Standard 5AF1.1 Use information taken from a graph or equation to answer questions about a problem situation.


4 + 9 = 1513 ≠ 15

Is 4, 5, or 6 the solution of the equation f + 9 = 15?


4

5

6

no

no

yes

5 + 9 = 1514 ≠ 15

6 + 9 = 1515 = 15

Answer: The solution is 6 since replacing f with 6 results in a true sentence.



Is 5, 6, or 7 the solution of the equation n + 8 = 14?

A. 5

B. 6

C. 7

D. 8


Solve 24 = 8w mentally.

24 = 8w


THINK 24 equals 8 times what number?

24 = 8 • 3 You know that 24 = 8 • 3.

Answer: So, w = 3. The solution is 3.



Solve 45 = 5x mentally.

A. 8

B. 10

C. 9

D. 7


The highest temperature recorded in Saskatchewan, Canada was 113 degrees. This is 21 degrees fewer than the highest temperature recorded in Death Valley, California. Solve the equation t – 21 = 113 to find the Death Valley temperature.

Use the guess and check strategy.



Try 130.


t – 21 = 113

109 ≠ 113

130 – 21 = 113?

NOT TRUE

Try 132.

t – 21 = 113

111 ≠ 113

132 – 21 = 113?

NOT TRUE


Try 134.


t – 21 = 113

113 = 113

134 – 21 = 113?

TRUE

Answer: So, the highest temperature recorded in Death Valley, California is 134 degrees.



The average depth of Lake Erie is 65 feet. This is 130 feet shallower than Lake Huron. Solve the equation d – 65 = 130 to find the depth of Lake Huron.

A. 200 feet

B. 195 feet

C. 155 feet

D. 175 feet





Key Concept: Area of a Rectangle

Key Concept: Area of a Square

Example 1

Example 2

Example 3

Example 4

1-91-9 Algebra: Area Formulas

Area of Rectangles and Squares



• I will find the areas of rectangles and squares.

• area

• formula




Standard 5MG1.4 Differentiate between, and use appropriate units of measures for, two- and three dimensional objects.


Find the area of the rectangle with length 13 feet and width 10 feet.


A = w

A = 13 • 10 Replace with 13 and w with 10.

A = 130 Multiply.

Answer: The area is 130 square feet.



Find the area of a rectangle with a length 8 feet and a width 5 feet.

A. 40 square feet

B. 35 square feet

C. 45 square feet

D. 50 square feet


Find the area of a square with side length of 5 meters.

A = s2


A = 52 Replace s with 5.

Area of a square

A = 25 Multiply.

Answer: The area is 25 square meters.



Find the area of a square with a side length of 6 inches.

A. 25

B. 30

C. 32

D. 36


An adult regulation soccer field is 100 yards long and 60 yards wide. What is the area of the field?


A = w

A = 100 • 60 Replace with 100 and w with 60.

A = 6,000 Multiply.

Answer: The area is 6,000 square yards.

Area of a rectangle



A professional basketball court is 90 feet long by 50 feet wide. What is the area of the court?

A. 450 square feet

B. 4,500 square feet

C. 9,000 square feet

D. 5,000 square feet


The gymnasium is 30 square. What is the area of the gymnasium?


A = s2

A = 302 Replace s with 30.

Area of a square

A = 900 Multiply.

Answer: The area is 900 square feet.



The dance floor is 20 feet square. What is the area of the dance floor?

A. 400 square feet

B. 600 square feet

C. 40 square feet

D. 300 square feet





Key Concept: Distributive Property

Example 1

Example 2

1-101-10 Algebra: The Distributive Property



• I will use the Distributive Property in equations and expressions.

• Distributive Property



Standard 5AF1.3 Know and use the distributive property in equations and expressions with variables.


7 × 84 = 7 × (80 + 4)

Find 7 × 84 mentally using the Distributive Property.


Write 84 as 80 + 4.

= (7 × 80) + (7 × 4) Distributive Property

= 560 + 28 Find each product mentally.

= 588 Add 560 and 28 mentally.



Find 6 × 37 mentally using the Distributive Property.

A. 6 × (30 + 7) = (6 × 30) + (6 × 7) = 180 + 42 = 322

B. 6 × (30 + 7) = 322

C. 6 × (30 + 7) = 222

D. 6 × (30 + 7) = (6 × 30) + (6 × 7) = 180 + 42 = 222


Suppose it costs students $3 to bowl a game and $2 to rent shoes. What is the cost for 25 students?


25 × (3 + 2) = (25 × 3) + (25 × 2) Distributive Property

= 75 + 50 Multiply.

= 125 Add.

Answer: The total cost is $125 for 25 students.



Suppose it costs students $7 to see a movie at a theater and $4 for popcorn. What is the cost for 20 students?

A. $200

B. $250

C. $220

D. $180



Five-Minute Checks

Area of Rectangles and Squares


Lesson 1-1

Lesson 1-2 (over Lesson 1-1)










Divide.

A. 20

B. 18

C. 22

D. 11

44 ÷ 2


Multiply.

A. 182,080

B. 1,842,080

C. 1,202,202

D. 18,828

92,104 × 20


Estimate. Tell whether the estimate is more or less than the actual product.

$55× 60

A. $3,400; less

B. $3,000; more

C. $3,600; more

D. $3,100; less


(over Lesson 1-1)


A. prime

B. composite

C. neither


(over Lesson 1-1)

Find the prime factorization of 48.

A. 2 × 2 × 3 × 4

B. 4 × 4 × 3

C. 2 × 2 × 2 × 2 × 3

D. 2 × 4 × 6


(over Lesson 1-1)

Find the prime factorization of 102.

A. 2 × 3 × 17

B. 6 × 7 × 10

C. 2 × 2 × 2 × 17

D. 4 × 5 × 5 + 2


(over Lesson 1-2)

Write 5 × 5 × 5 × 5 × 5 × 5 using an exponent.

A. 55

B. 253

C. 56

D. 1252


Write 13 as a product of the same factor. Then find the value.

A. 1 × 3; 3

B. 1 × 1 × 1; 3

C. 1 + 1 + 1; 3

D. 1 × 1 × 1; 1

(over Lesson 1-2)


Write the prime factorization of 100.

(over Lesson 1-2)

A. 22 × 52

B. 4 × 5 × 5

C. 10 + 20 + 20 + 20 + 30

D. 10 × 10


Carlos has 92 CDs. What whole number is this?

A. 18

B. 72

C. 11

D. 81

(over Lesson 1-2)


(over Lesson 1-3)

Find the value of the following expression.

A. 6

B. 15

C. 24

D. 42

27 – 18 ÷ 3 × 2


(over Lesson 1-3)


A. 7

B. 10

C. 13

D. 19.5

4 + 62 ÷ 4


(over Lesson 1-3)


A. 45

B. 10

C. 57

D. 17

5 × (6 – 4) + 7


(over Lesson 1-3)


A. 35

B. 28

C. 14

D. 2

12 ÷ (4 – 1) × 7


(over Lesson 1-4)

Use the Four-Step Plan to solve the problem. Halima runs one mile in 9 minutes. If she continues at this rate, how long will it take for her to run 6 miles?

A. 52 minutes

B. 14 minutes

C. 54 minutes

D. 15 minutes


(over Lesson 1-4)

Use the Four-Step Plan to solve the problem. Reynaldo stencils the following pattern on his bedroom wall: circle, circle, triangle, square, circle, circle. If the pattern continues, what will the 14th shape be?

A. triangle

B. circle

C. rectangle

D. square


(over Lesson 1-5)

Evaluate the following expression if a = 4 and b = 2.

A. 4

B. 6

C. 10

D. 14

4a – b


(over Lesson 1-5)


A. 9

B. 12

C. 7

D. 6

3a


(over Lesson 1-5)


A. 7

B. 8

C. 10

D. 28

14 ÷ b


(over Lesson 1-5)


A. 0

B. 2

C. 4

D. 6

a + b – 4


(over Lesson 1-6)


A. 2x

B. x + 12

C. 3x

D. x + 20


Osvaldo hit 7 more home runs than Josh. Define a variable. Write a function rule that relates the number of home runs hit by Osvaldo to the number hit by Josh.

A. Let J represent Josh’s hits and O represent Osvaldo’s hits; J + O = 7

B. Let h represent Josh’s hits; 7 – h

C. Let J represent Josh’s hits; J × 7

D. Let h represent Josh’s hits; h + 7

(over Lesson 1-6)


(over Lesson 1-7)

Solve. Use the Guess and Check strategy. Aja has 10 coins in his pocket that total $2.17. What are the coins?

A. 8 quarters, 1 dime, 1 nickel, 2 pennies

B. 2 half-dollars, 4 quarters, 1 dime, 1 nickel, 2 pennies

C. 6 quarters, 1 dime, 1 nickel, 2 pennies

D. 1 half-dollar, 6 quarters, 3 nickels, 2 pennies


(over Lesson 1-8)

Identify the solution.

A. 5

B. 6

C. 7

D. 8

19 – x = 12


(over Lesson 1-8)

Identify the solution.

A. 3

B. 5

C. 7

D. 9

35 = 7y


(over Lesson 1-8)

Solve the following equation mentally.

A. 7

B. 9

C. 6

D. 8

56 ÷ w = 7


(over Lesson 1-8)

Solve the following equation mentally.

A. 26

B. 18

C. 16

D. 6

x + 22 = 38


(over Lesson 1-9)

Find the area of a rectangle with a length of 13 ft and a width of 9 ft.

A. 117 ft2

B. 111 ft2

C. 108 ft2

D. 44 ft2

13 ft

9 ft


(over Lesson 1-9)

Find the area of a rectangle with a length of 20 m and a width of 11 m.

A. 40 m2

B. 220 m2

C. 62 m2

D. 231 m2

20 m

11 m


(over Lesson 1-9)

Find the area of a square with a side length of 18 yd.

A. 72 yd2

B. 162 yd2

C. 316 yd2

D. 324 yd218 yd

18 yd


(over Lesson 1-9)

Find the area of a square with a side length of 7 cm.

A. 42 cm2

B. 28 cm2

C. 49 cm2

D. 32 cm2

7 cm

7 cm

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