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Unit 6 Cycle 1 – Rewriting Expressions

Lesson 6.1.1 – Equivalent Algebraic Expressions

Vocabulary

Algebraic Expression Term Constant Coefficient Variable Order of Operations

Active Instruction

ta(1) What are the different parts of the following algebraic expression?

ta(2) How can you rewrite the following

algebraic expression?

th(1) For a-d, tell whether the expression is

equivalent to

a) b) c)

d)

Team Mastery

(5) Write the following expression two different ways:

Lesson Quick Look

Team Name

Team

Complete?

Team Did Not Agree On

Questions…

Write the vocabulary introduced in this lesson:

Today you learned to write equivalent algebraic expressions using the order of operations and simplifying the expressions. You also used properties like the distributive property and the commutative property to help you. Rewriting algebraic expressions in different forms helps you make connections and see the processes you can use to evaluate them later on.

Rewriting expressions can also turn complex algebraic expressions into simpler ones. Use the following techniques to successfully rewrite algebraic expressions:

Order of operations: The sequence in which mathematical operations are done.

Parentheses (and other grouping symbols)

Exponents

Multiplication and Division from left to right

Addition and Subtraction from left to right

Here is an example!

6.2(x + 2.8) – 5x + 2x2 + 3.1 • |

–3 +

–2|

6.2x + 17.36 – 5x + 2x2 + 3.1 • |

–5|

6.2x + 17.36 – 5x + 2x2 + 15.5

2x2 + 6.2x – 5x + 17.36 + 15.5

2x2 + 1.2x + 32.86

The final step shows the expression in simplified form because there are no more like terms to be combined

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Lesson 6.1.1 - Homework

1) For a–d, tell whether the expression is equivalent to

6.1pq – q2 + 5.9pq + 2.1q2 – pq + p2

a. 13p2q2

b. 11pq – 1.1q2 + p2 c. –2.1q2 – 11pq – p2 d. 6.1pq3 – q6 + p2

2) Rewrite the expression two ways and explain your thinking.

22d3 + 2(17.5d2) – d + (–5)

3) Rewrite the expression two ways.

3d + 3

6 d2 + 2 • |–2 • 3| – d2

4) For a–d, tell whether the expression is equivalent to

4

1 xy + 2

1 x(y2) +

xy

2

11– – 2y(xy)

a. 4

1 xy + 2

1 x2

1 y2 +

xy

2

11– – 2y2xy

b. –24

1 xy3

c. –12

1 xy2 – 14

1 xy

d. –1.5xy2 – 1.25xy

5) Rewrite the expression two ways.

3.9b3 + 2.1b(b2) – 4.5b2(b)

6) For a–d, tell whether the expression is equivalent to

–49(r – t) + 32t – 15r + 750

a. –64r + 81t +750

b. –64r – 17t + 750 c. 49r – 49t – 32t – 15r + 750 d. 17rt +750

Mixed Practice

7) A coat that costs $248.00 is on sale for 33% off. What is the sale price of the coat?

8) Evaluate.

5

2 •

4

1– •

2

1

9) Evaluate.

(–3)3

10) Evaluate.

2.2(–8.5 + 4.5) • (3 – 0.7)

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Unit 6 Cycle 1 – Rewriting Expressions

Lesson 6.1.2 – Evaluate Algebraic Expressions

Vocabulary

Evaluate

Active Instruction

ta(1) If we know the values of variables, we can evaluate algebraic expressions. What is the temperature in Fahrenheit when it is 37° Celsius? The following expression can be used to convert temperatures in Celsius (C) to Fahrenheit:

th(1) Evaluate. Show your work. ; when and

Team Mastery

(3) Evaluate. Show your work.

; when

and

Lesson Quick Look

Team Name

Team

Complete?

Team Did Not Agree On

Questions…

Write the vocabulary introduced in this lesson:

Today you learned to evaluate algebraic expressions when values are given for the variables. In an expression, you can substitute the numeric value for the variable and then simplify the expression using mathematical properties and the Order of Operations. There may be more than one variable in an expression; we substitute different values for each variable.

Here’s an example! The following is an expression for the area of a trapezoid.

2

1 (b1 + b2)h

If b1 = 1, b2 = 3, and h = 2, we can evaluate the

expression to find the area of the trapezoid:

2

1 (1 + 3)2

2

1 (4)(2) = 4

The area of this trapezoid is 4 square units.

Lesson 6.1.2 - Homework

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Directions for questions 1–5: Evaluate. Show your work.

1) Explain your thinking.

y2

1

+ y2 – (–3 + y); when y = 2

2) 5x – x2 + 2x + 3x3 – x; when x = 3

3) –4

2–

4

1 x – 2x(x – 1); when x = 4

4) 5.3y – 11.4 + 0.4y; when y = 83.4

5) 5

2 (–a)2 • 6.3a + (4a – 1.2a);

when a = 5

Mixed Review

6) What is 39.4% of $120.50?

7) What is another way to write t + t + t + s + s ?

8) What is the missing value in this

proportion comparing the number of miles

traveled to the number of gallons of gas

consumed?

gal 9

mi 180= gal 10

mi x

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Unit 6 Cycle 1 – Rewriting Expressions

Lesson 6.1.3 – Writing Expressions with Percents

Active Instruction

ta(1) At XYZ Shoes, all sports shoes are on sale at 40% off. What expression can we use to describe this situation?

th(1) Mr. Gomez wants to know how much simple interest he will earn on his savings of $5,500. Write an expression he could use to determine how much interest he would earn. Define your variables(s). Evaluate the expression if Mr. Gomez keeps his money in the account for 2 years at 1.3% interest.

Team Mastery

(4) Andrea wants to increase her grade average by 15%. Write an expression Andrea can use to

determine her new grade. Define your variable(s).

Lesson Quick Look

Team Name Team Complete?

Team Did Not Agree On

Questions…

Write the vocabulary introduced in this lesson:

Today we wrote expressions involving percents. The expressions involved one or more unknowns, and it is important to identify what the unknown represents. To include a percent in an expression, it is necessary to covert the percent to its decimal or fractional equivalent. To do this, simply divide the percent number by 100 (or move the decimal point two places to the LEFT); for example, 1% = 0.01.

Remember, an expression is a mathematical representation of information.

Here’s an example!

During an end-of-season sale, winter coats are 20% off and gloves are 15% off. We’ll let c = the cost of a winter coat and g = the cost of a pair of gloves. We can use the following expression to describe the purchase: (c – 0.2c) + 2(g – 0.15g).

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Lesson 6.1.3 - Homework

1) At XYZ jewelry store, a salesperson’s weekly income is a salary plus 5% of the salesperson’s total sales. Write an expression to describe this situation. Define any variable(s) you use. Evaluate the expression when the salary is $626.50 and the total sales are $340.

2) The Dust Bowl was a 10 year drought that hit the high plains, including Colorado, Nebraska, and the Texas panhandle. Average precipitation dropped about 20% from the previous normal levels during this time. Write an expression to describe this situation. Define any variable(s) you use. Evaluate the expression when the average precipitation in this area was formerly 23.08 inches. What was the amount of loss of precipitation in inches during the period of the drought? Explain your thinking.

3) Write an expression that describes the new cost of gasoline after an increase of 4.7%. Define your variable(s).

4) For an investment in the stock market, Lorenzo received a gain of 4.1% after the first year, but lost 3.9% at the end of the second year. Write an expression to describe this situation. Define any variable(s) you use. Evaluate the expression when Lorenzo’s initial investment was $500.

Mixed Review

5) Maxine borrowed $155 to buy a new bike. At the end of the year, she paid back the principal and $3.10 in interest. What was her simple interest rate?

6) 13 is what percent of 35?

7) Simplify the expression.

b

7

14 – 3(3 + b)

8) Evaluate the expression for the given variables.

1.2x + 3.56y + 4.5(x + y)

when x = –2 and y = 7.9

9) In the local July 4th parade, 30% of the floats’ sponsors are new to the parade. If there are 40 floats in the parade, what is the number of new sponsors floats? Write and evaluate an expression that describes this situation. Define any variable(s) you use.

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Unit 6 Cycle 1 – Rewriting Expressions

Lesson 6.1.4 – Writing Expressions Multiple Ways

Active Instruction

ta(1) Quincy and Andie are both saving money each week for new bikes. Quincy saves

times more

than Andie. Quincy uses the expression 4(

to describe the total amount saved by both Quincy

and Andie in 4 weeks. Andie uses the expression 4(

to describe the situation. Who's right?

th(1) The heart rate (beats per minute) of a mouse is 17.9 times faster than the heart rate of an elephant.

Write two different expressions to describe how many more heart beats a mouse has than an elephant in 1 hour. Define any variable(s) you use.

Team Mastery

(4) Greg charges the same price per bagel for the 4 bagel flavors he sells: sesame, chocolate chip, plain, and everything. Write two different expressions to describe the cost of buying 1 dozen plain bagels, 1.5 dozen sesame bagels, 8 chocolate chip bagels, and 2 everything bagels. Define any variable(s) you use.

Lesson Quick Look

Team Name Team Complete?

Team Did Not Agree On

Questions…

Write the vocabulary introduced in this lesson:

Today you learned to write expressions for contextual situations in more than one way. Seeing situations in more than one way may give you insights on how to solve problems. Writing expressions in several ways improves your flexibility in math.

Remember, to prove that different expressions are equivalent, you can substitute a value for the variable or variables in the expression.

Here’s an example!

Jane has 1.75 times more red marbles as blue ones. Harriet has 3 times more blue marbles as red ones.

The following expressions could be used to describe how many marbles the girls have in all. When j = the number of blue marbles Jane has and h = the number of red marbles Harriet has: j + 1.75j + h + 3h or 2.75j + 4h

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Lesson 6.1.4 - Homework

Directions for questions 1–4: Write each expression in two different ways.

1) A movie ticket at the local theater is $11. The price of a bucket of popcorn is the 90% the cost of the ticket, and the price of a soda is half the cost of the popcorn. Write two expressions that could describe the cost of 2 tickets to the movies, 1 popcorn, and 2 sodas. Define any variable(s) you use.

2) There is an admission fee for the county fair. At the fair a meal of a hot dog, fries, and a lemonade is 50% of the admission fee. Write two expressions that could describe what a family of 4 would pay to attend the fair and get the hot dog meal. Define any variable(s) you use.

3) Sides a and b are twice as long as side c. Write two different expressions to describe the perimeter of a triangle one-half as large as this one.

4) At the Happy Tails Groomer, there is a flat rate for a dog bath. A hair cut adds 35% to the price of the bath. Mrs. Peffer brings in three dogs. Two dogs are getting baths, and one dog is getting a bath and a haircut. Write two expressions that describe what Mrs. Peffer will pay to groom her dogs.

Mixed Review

5) Angela lives 9 blocks from the store. She walked a third of the way there to buy some milk when she remembered she left her wallet at home. So she went home to get it. How far is Angela from her home now? Write a number sentence to show Angela’s distance from home.

6) (–73.57) + 8.62 + 3.65

7) The original price of a computer is $783.99. The computer is on sale for 15% off. What is the sale price of the computer?

8) What is the constant of proportionality between x and y?

x y

40.2 20.1

52.182 26.091

63.54 31.77

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Unit 6 Cycle 1 – Rewriting Expressions

Lesson 6.1.5 – Write and Evaluate Expressions

Active Instruction

ta(1) Mr. Chang is ordering furniture for his new eat-in pizza shop. He is choosing between two sets of

tables and chairs. He needs to buy 8 tables, each table with 4 chairs. In one furniture set, padded chairs cost $25.50 each, while the chairs for the second set are plain and cost $24.95 each. For both sets, the tables cost 1.5 times as much as the chairs. How much will Mr. Chang pay for the new furniture he needs?

th(1) The nearby amusement park uses this chart to help them set their yearly prices for admission for

adults, children, and senior citizens.

Fri-Sun Mon-Th

Adults a a - 5

Children 5 - 16 c c - 6.50

Senior Citizens s s - 10

The Montoya family (2 adults, 2 children, and 1 senior citizen) is considering going to the amusement park on either a Wednesday or a Sunday.

Write an expression to find the difference in the cost between the two days. Evaluate the expression when this year, a = $35, c = $40, and s = $25. Show your work.

Team Mastery

(3) The following chart summarizes the relationships of the average temperatures on Mars, Jupiter,

Saturn, Uranus, and Neptune.

Planet Mars Jupiter Saturn Uranus Neptune

Temperature (°C)

m 3.6 times that of Mars 110°C colder

than Mars 85°C colder than Mars

1.8 times that of Jupiter

Write an expression for the temperature of each planet in terms of the temperature on Mars and evaluate the expressions when the average Mars temperature is -30°C.

Lesson Quick Look

Team Name Team Complete?

Team Did Not Agree On

Questions…

Write the vocabulary introduced in this lesson:

Today we practiced writing and evaluating expressions to answer every-day, real-world questions. Translating information into mathematical language can help solve problems. Remember to identify:

1) what the question is asking, 2) what the known values are, 3) what the unknown(s) values are, 4) the variable(s) you will use, and what operations you will use.

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Here is an example!

A store advertised a special sale in the shoe department—all shoes will be discounted 10.5% on Thursday. An expression that describes the cost of a pair of shoes during the sale (when s is the cost of a pair of shoes) is: s – 0.105s. So for a pair of shoes originally costing $16.99 the expression is: 16.99 – 0.105(16.99) 16.99 – 1.78 = 15.21, this pair of shoes will cost $15.21 What expression could you write to show the savings a customer would get from buying two pairs of shoes on sale?

Lesson 6.1.5 - Homework

Directions for questions 1–3: Define any variable(s) you use.

1) The following chart summarizes the relationships of the deepest parts of the Great Lakes.

Lake Erie Huron Michigan Ontario Superior

Maximum distance

from surface in

feet

d 3.5 times as deep as Lake Erie

Lake Huron +

–175 feet

Lake Michigan +

–123 feet

1.7 times as deep as

Lake Ontario

Write an expression for the depth of each lake in terms of the maximum distance from the surface of Lake Erie and evaluate the expressions when the maximum distance from the surface for Lake Erie is –210 feet.

2) At Chantal’s Pizza Shop, a bottle of

iced tea is 12

1 the cost of a medium size

pizza. Write and evaluate an expression when Pablo buys 3 bottles of ice tea and a medium pizza when the pizza costs $15.00.

3) A historic house is redoing the windows in one of the rooms with new curtains, insulated shades, and curtain rods. The decorator uses the chart below to help him determine the costs:

Item Cost

curtains per pair c

insulated shade 9

1 c

curtain rods 9

1 c – 7

Write an expression the decorator can use to determine the cost of buying 2 curtains, 2 shades, and 2 curtain rods for the front room of the house. Evaluate the expression when a pair of curtains costs $382.50. Explain your thinking.

Mixed Review

4) What is 16% of 39.40?

5) Simplify the expression.

7.3(x + 3.9) – 6x + 3x3 + 4.2 • |–2 + 1|

6) Evaluate the expression.

2.9(–8.65 + 4.34) • (5 – 0.63).

7) Evaluate the expression when

d = –42

1 .

438 + 239d

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Unit 6 Cycle 1 – Rewriting Expressions

Lesson 6.1.6 - Think Like a Mathematician: Find the Patterns and Structure II

Active Instruction

ta(1) Andre purchased a pack of erasers. He put half of them in his pencil case. Then he divided the

other half equally between his friends, Dez, Tanya, Sabine, and Carl. Tanya ended up with 14 erasers. How many erasers did Andre purchase?

th(1) The Ramirez family comes home from vacation to find their mailbox overflowing with mail. There

are 6 more magazines than there are letters. There are twice as many bills as letters. There are

of the

amount of bills as pieces of junk mail. There are 18 pieces of junk mail. How many magazines are there?

Team Mastery

(3) Braylon, Dimitri, and Myung are going camping. They each brought the same amount of money, and

they put it all together for the trip. When they got to the campground, they spent $50 to pay for the campsite. Over the next 4 days, they spent what was left of their money. They spent $30 each day. How much money did they each start with?

Lesson Quick Look

Team Name Team Complete?

Team Did Not Agree On

Questions…

Write the vocabulary introduced in this lesson:

Today we used the Work Backwards strategy to solve word problems. Here’s an example.

Larry has a cup of paper clips. He took out 10 and gave them to his friend. Then he put half of what was left in his desk. After that, 7 were left in the cup. How many paper clips were in the cup when he started?

First, organize the data:

Then work backwards:

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So there were 24 paper clips to begin with.

Don’t forget to check your work! Just use your answer and work forward through the steps of the problem:

Lesson 6.1.6 - Homework

Directions for questions 1–5: Solve each problem.

1) Karina won money on a game show. She spent $110 of the winnings on clothes. She spent 1.75 that amount on groceries. Then, outside the grocery store, she found $30! With that $30 plus her remaining winnings, Karina had $1,315 left. How much did Karina win?

2) Gervas is a huge sports fan. He has 7 more football jerseys than baseball jerseys. He has one and a half as many soccer jerseys as baseball jerseys. He has 2 more basketball jerseys than soccer jerseys. Gervas has 5 basketball jerseys. How many jerseys does Jimmy have in all?

3) At the start of a concert, the venue was not filled up. Five minutes later, 400 more people showed up. Half the people at the end stayed around to get an autograph. If 1,675 people stayed around trying to get an autograph, then how many people were there at the start of the concert?

4) Hayden has a jar of pushpins. She used half of the pushpins to decorate her cork board. Then she took the rest and divided them into 6 equal groups. Each pile had 17 pushpins. How many pushpins were in the jar when Hayden started?

5) Nevena walked from her house to Levi’s house in 45 minutes. They played whiffle ball for 1.5 hours and then had dinner. Dinner took twice as long as the walk to Levi’s. Dinner was over at 7:30 p.m. What time did Nevena leave for Levi’s house?

Mixed Review

6) Is 11a • 8 • a equivalent to 88a?

7) Rewrite the expression

56x + 42

8) Evaluate the expression when k = 6.4.

(7k + 18) + 9k + 0.9?

9) It took Rika 39 minutes to read 4 chapters.

a) What is Rika’s reading rate? b) Write a unit rate to describe how much

Rika reads in 1 hour.

10) Shakil put his nickels in two piles. The first pile had 165 nickels, and he put that pile in his coin bank. He put the nickels from the second pile into paper rolls. It took 7 rolls to hold all those nickels. Each roll holds 40 nickels. How many nickels did Shakil have in all? Explain your thinking.