Algebra: Operations and Equations

Chapter 4 VocabularyAlgebraic expression – an expression that

includes at least one variableBase – a number used as a repeated factor bªDivision Property of Equality – the property

states that if you divide both sides of an equation by the same nonzero number, the sides remain equal

Evaluate – to find the value of a numerical or algebraic expression

Exponent – a number that tells you how many times the base is used as a factor bª

Vocabulary continuedIntegers – the set of whole numbers and their

opposites {… -3, -2, -1, 0, 1, 2, 3 …}

Multiplication Property of Equality – the property that states that if you multiply both sides of an equation by the same number, the sides remain equal

Order of Operations – the process for evaluating expressions (PEMDAS)

Subtraction Property of Equality – the property that states that if you subtract the same number from both sides of the equation, the sides remain equal

Chapter 4 Lesson 1Exponents

bª exponent

Base

The base is the number used as the repeated factorThe exponent is the boss & tells how many times the base is used as the factor.

Example: 10³ = 10 x 10 x 10 = 1000

Unlock the Problem pg. 125

Square NumbersThe product of a number and itself. A square

number can be represented using the repeated factor as the base and 2 as the exponent.

b²

Complete the activity on page 126

Explain how you can use multiplication to find the value of the number 21².

Problem Solving pg. 128 Use the table to solve 17 - 19

17. A frog egg has split. Several splits later the egg has 128 cells. How many splits have there been?

Problem Solving pg. 128 Use the table to solve 17 - 19

18. A frog cell splits. How many cells would a frog egg have after 9 splits? Write as the number of cells and as an expression in exponent form.

Write Math - JournalHow do you use an exponent?

Chapter 4 Lesson 2Order of OperationsConstruct a “cheat sheet”

Parentheses ()

Exponents bª

Multiply x ·

Divide ÷ n/d

Addition +

Subtraction -

Unlock the Problem pg. 129

Algebraic Expressions pg. 130

Unlock the Problem pg. 132

Write Math - JournalIn what order must operations be evaluated to find a correct solution to a problem?

Chapter 4 Lesson 3Balance EquationsWith a partner complete Investigate Materials – balance & cubes

Example pg. 134

Problem Solving pg. 136

What’s the error?

Describe the errorJill made.

Write Math – Journal How can you use a pan balance to solve an equation with a variable?

Chapter 4 Lesson 4Addition EquationsYara is going camping in the Everglades

National Park. Her backpack with camping gear weights 17 pounds. When she adds her camera gear, the total weight of her backpack is 25 pounds. How much does Yara’s gear weigh?

17 + c = 25Use a model to solve.

Another Way pg. 138

Subtraction Property of Equality - the property that states that if you subtract the same number from both sides of the equation, the sides remain equal

Example: x + 4 = 7 check your work: x + 4 = 7

- 4 -4 3 + 4 = 7 x = 3 7 = 7

Try This!Solve the equation. Check your answer

13 + d = 22

Can d in 13 + d = 22 have more than one value? Why or why not?

Problem Solving pg. 140Unlock the Problem 12 & 13-14

Write Math - JournalHow can an equation with addition be solved using subtraction?

Chapter 4 Lesson 5Subtraction EquationsAddition Property of Equality – states that if you add the

same number to both sides of an operation, the two sides remain equal.

Complete Activity on pg. 141 Materials: balance, cubes

Example: Addition Property of Equality Kent has a collection of CDs. He gives 5 CDs

to his brother. Kent then has 8 CDs left in his collection. How many CDs did Kent have before he gave some to his brother?

Solve: c – 5 = 8 Check: c – 5 = 8 + 5 +5 13 – 5 = 8

c = 13 8 = 8

Problem Solving pg. 144 Use the bar graph to solve for 22 - 24

22. Taking a shower uses 13 gallons less water than taking a bath. About how many gallons of water are used for taking a bath?

Use the equation b – 13 = 23, where b is equal to the number of gallons of water needed for a bath.

Problem Solving pg. 144Use the bar graph to solve for 22 - 24

23. Washing the dishes uses about 29 gallons less water than washing a load of laundry. How many gallons of water are used to wash a load of laundry?

Use the equation l – 29 = 15, where l is equal to the number of gallons of water needed for a load of laundry.

Write Math – Journal Use the bar graph to solve for 22 - 24

24. You use 2 gallons less water to brush your teeth than to wash your hands. Find how much water you use to wash your hands.

Write an equation you can use to solve the problem. Use h to represent the number of gallons of water used to wash your hands.

Then solve the equation. Don’t forget to check your work.

Chapter 4 Lesson 6Write and Solve EquationsWhen writing equations it’s important to

choose a variable and know what that variable represents. Choose the correct operation to solve the problem.

Underline what you are asked to find (in the word problem)

Circle the word that tells you which operation to use to write an equation.

Write and Solve EquationsThe Panthers won the basketball game with a

score of 73 points, which was 14 points greater than the score of the other team, the Bears. How many points did the Bears score?

What are you being asked to find?What word tells you the operation used in the

equation?

Example pg. 146

In the championship game, the Panthers scored 13 points fewer than the Dolphins scored. If the Panthers scored 54 points, how many points did the Dolphins score?

What are you being asked to find?What word tells you the operation used in the

equation?

Problem Solving pg. 148Use the table to solve 8 – 10. Complete the table.

8. The Knights’ score was 15 points less than the Bulls’ score. What was the Bulls’ score?

Write & solve the equation.

9. The Tigers’ score was 17 points greater than the Cubs’ score. What was the Cubs’ score?

Write & solve the equation.

Write Math – Journal The Cougars scored 14 points in the first half

of the game. In the second half, the Cougars scored enough points to beat the Hawks by 5 points. How many points did the Cougars score in all? In the second half of the game?

Write and solve the equations.

Chapter 4 Lesson 7 Solve a Simpler Problem – Function TablesWhen creating and solving a function table you

must create a possible rule for the table to work correctly.

Example: Samantha is making a scarf using a pattern of equilateral triangles. Each triangle has a perimeter of 6 inches. If Samantha adds triangles from left to right, what is the perimeter of a scarf made from 15 triangles?

Continue the function table: what is the rule?

# of ‘s

1 2 4 6 8 10 12 15

Perimeter

6 8 12

Unlock the Problem pg. 149

On an archaeological dig. Gabriel divides his dig site into square sections that are 1 meter on each side. He uses 4 meters of rope to rope of the first section. He only needs 3 meters of rope for each additional section. How many meters of rope will Gabriel need for 10 sections?

Finish the table – what is the rule?

Number of sections

1 2 3

Amount of rope (m)

4 7 10

On Your Own pg. 152

4. Jane works as a limousine driver. Her base fee is $50, and she makes $25 for every hour that she drives. How much does Jane make if she works for 8 hours? Complete the table.Possible rule: ________________________

h 1 2 3 4 8

m $75 $100 ? ? ?

Write Math - JournalHow can you solve a problem by solving a simpler problem?

Chapter 4 Lesson 8Multiplication EquationsUse a model.At the movies, Jake bought 3 bags of popcorn

for himself and his friends Larry and Sal. Each bag of popcorn was the same price. Jake paid $12 for the 3 bags. How much did each bag of popcorn cost?

Complete the model on pg. 155

So, 1 bag of popcorn cost $___.

Another Way pg. 156 Division Property of Equality – states that when

you divide both sides of an equation by the same non-zero number, the two sides remain the same.

Solve: 3 x p = 12

3 3

p = 4

Check: 3 x p = 12

3 x 4 = 12

12 = 12

Try This! Solve the equationWhen solving an equation with two operations, use

the properties of equally twice to get the variable by itself on one side of the equation. To get the variable by itself, the order of operations is reversed so addition & subtraction is undone first before multiplication & division.

6 x n + 3 = 27 - 3 - 3

6 x n = 24 6 6 n = 4

Problem Solving pg. 158 Use the table to solve 15 - 17

15. The drama club sees a movie. Each member of the club buys one fruit snack. The club spends a total of $76 on fruit snacks. How many members of the club went to the movies? Solve the equation 4f = 76, where f represents the number of fruit snacks bought.

Problem Solving pg. 158Use the table to solve 15 - 17

On Friday, the snack bar made $992 selling buckets of large popcorn. How many buckets of large popcorn did the snack bar sell in Friday? Solve the equation 8p = 992, where p represents the number of buckets of large popcorn sold.

Write Math - Journal18. Michael solves the equation 8y = 2 and finds that y is equal to 16. Explain how you know Michael’s solution is not correct.

Chapter 4 Lesson 9 Division EquationsMultiplication Property of Equality – states

that if you multiply both sides of the equation by the same non-zero number, the two sides remain equal.

Solve the equation & check your solution.Unlock the Problem pg. 159

Problem Solving pg. 162

20.

Problem Solving pg. 162

22. Asher wants to buy a handheld video game console. In order to save the money needed to buy it, he divides the cost to see how much he needs to save every month for 5 months. He finds that he must save $37 a month. Write and solve a division equation with a variable that describes the problem.

***Don’t forget to check your solution.***

Write Math - Journal23. Jasmine says that x = 348 is the solution to

x ÷ 12 = 29. Explain how you can justify Jasmine’s solution.

Chapter 4 Lesson 10Use Substitution Underline what you are asked to find.Circle the information you will use.Use a Model.

Erik knows that 1 cube weighs 2 ounces and that 9 cubes weigh the same as 3 bouncy balls. Use this information to find the weight of 1 bouncy ball.

Another Way pg. 164Use the Substitution Property of Equality Substitution Property of Equality – states that if you know that one quantity is equal to

another, you can substitute that quantity for the other in an equation.

Erik remembers that 1 cube weighs 2 ounces and that 9 cubes weigh the same as 3 bouncy balls. How can Erik use this information in another way to recall the weight of 1 bouncy ball?Step 1: Write an equation for the information given in the problem.

1 cube = 2 ounces 1c = 2 or (c=2)9 cubes -= 3 bouncy balls 9c = 3b

Step 2: Use the Substitution Property of EqualitySubstitute c=2 9(2) = 3bMultiply 18 = 3b

Step 3: Solve the equationUse the Division Property of Equality 18 = 3b

3 3 6 = b

Problem Solving pg. 166Use the pan balances to solve 12 - 14

12. The weight of 6 blocks is shown on the first pan balance. What is the weight of one block? If one green cylinder on the second pan balance has the same weight as 4 blocks what is the weight of the cylinder?

Step 1: Write an equationStep 2: Use the Substitution Property of EqualityStep 3: Solve the equationStep 4: Check your work

Problem Solving pg. 166Use the pan balances to solve 12 - 14

14. Using the two pan balances shown, what if the weight on the right side of the first pan balance weighed 90 ounces, and a green cylinder weighed the same as 3 blocks? What is the weight of the block? What is the weight of one green cylinder?

Step 1: write an equationStep 2: use the substitution property of equalityStep 3: solve the equationStep 4: check your work

Write Math - Journal13. Using the information from Problem 12,

what is the weight of one triangle? Explain how you know.

Don’t forget to follow the steps to solve a substitution problem.

Chapter 4 Lesson 11Understand IntegersIntegers – the set of whole numbers and their

opposites. For example: +8 and -8 are opposites

Positive Integers – any integer greater than 0. For example: +19 is read positive 19

Negative Integers – any integer less than 0.

For example: -47 is read negative 47

Name the integer for each situationThe highest point in Florida, Britton Hill, is

345 feet above sea level.Larry withdraws $30 from his bank account.The lowest recorded temperature in Florida

was 2 degrees below zero in Tallahassee in 1899.

A team loses 10 yards in a football game.Larry deposits $300 into a bank account.Tiger Woods hit 7 under par.

Opposite Numbers & GraphingOpposite integers are the same distance from

zero on a number line in opposite directions.

Complete Try This!

Example pg. 168Use a vertical number line

Problem Solving pg. 170

17. Miriam goes scuba diving. She dives to a depth of 25 meters below sea level. What integer represents her dive?

18. Neil earns $17. He owes his brother $23. What integers represent the amount Neil earns and the amount he owes?

20. Which integer represents 7 days before now if today is Day 0?

Connect to Science pg. 170

Write Math - JournalWhat real-world situations can be described using positive and negative numbers?

Chapter 4 Lesson 12Compare & Order Integers

One Way & Another Way pg. 172

Try This! Pg. 172

Problem Solving pg. 174Use the table for 15 - 17

15. Which is greater, the average temperature of Earth or the average temperature of Mars?Explain how you know.

16.The average temperature of the planetoid Pluto is -393°F. Is that greater than or less than the average temperature of Mercury? Is it greater than or less than the average temperature of Neptune?

19. At 7:00 A.M. the temperature was -4°C. At 10:00 A.M. the temperature was +6°C. By how many degrees Celsius did the temperature change?

Write Math – Journal How do you compare and order integers?

Chapter 4 Review