Integrated Math I Linear Equations and Inequalities Packet...

1

Integrated Math I

Linear Equations and Inequalities Packet 1

This week we will begin by going over class expectations. Then, we will

go over both numeric and graphic representations of data. We will

review writing expressions and equations from scenarios, before

solving equations and learning about the two-column proof. We’ll

take a break from lessons to do our beginning of year benchmark

testing. Finally, we will review none and infinitely many solutions. You

will have a quiz on this section of the unit on Tuesday next week.

Day Activity Pages Topic Assignment

Monday

8/21

0.5 3-7 Class Intro Practice 1.1

Pg. 47-48 1.1 8-12 Numeric and Graphic

Representations of Data

Tuesday

8/22

1.2 13-19 Writing Expressions Practice 1.2-1.3

Pg. 45-46 1.3 20-22 Writing and Interpreting

Linear Equations

Wednesday

8/23

2.1 23-28 Solving Linear Equations Practice 2.1-2.2

Pg. 43-44 2.2 29-32 Solving More Complex

Equations

Thursday

8/24 Testing

None Pro-Core Benchmark Testing None

Friday 8/25 2.3 33-35 Modeling with Equations Practice 2.3-2.4

Pg. 41-42 2.4 36-40 Equations with No Solution

or Infinitely Many Solutions

Looking Forward to Next Week

Monday: No School (HAPPY LABOR DAY!)

Tuesday: Assessment 1 (over activities 1-2)

Wednesday-Friday: Activities 3 and 4

3

AIM(S):

✓ WWBAT understand and follow classroom procedures

DO NOW Directions: Complete the following questions.

Do you know what problem-based learning is? If so, explain! If not, give a reasonable guess.

Answers will vary

What is your history with math? Have you had trouble or has it seemed easy?

Answers will vary

Integrated

Math I

Linear Equations and Inequalities 0.5

Classroom Expectations

8/21/2017

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Integrated Mathematics I Description Course Overview

Welcome to eSTEM Academy! This semester I will challenge you to explore and inquire. I will

guide you through difficult problems that demand perseverance and creativity. We will cover a

variety of mathematical topics in algebra, geometry, and probability. While exploring these topics,

we will focus on the themes of strategic problem solving and creative thinking. My hope for this

course is that you will learn both valuable mathematics and valuable skills which will prove useful

to you in many areas of your life. If you put in your hardest work and believe that you are capable

of learning, you will leave my classroom not only as a better mathematician, but also as a better

thinker.

Habits of Mind

During a typical class session, you will work collaboratively in groups with three or four of your

classmates. It will be important that you clearly communicate your ideas to your group. You will

work through challenging problems that will require you to be persistent and flexible--you will

often need to try many different approaches and not give up even when you feel stuck. Many

problems will also require creative problem solving and risk taking--you will need to come up

with and try new ways of solving a problem, even if you are not sure if they will work.

Classroom Expectations

• Present: You are in your seat when the class begins, with all necessary materials (pencil,

calculator, necessary notes, etc.). Focus on learning, which means no distractions are on/out. Note

on cell phones: you are expected to have electronics stowed and off during class. Students caught

on their phones on a first offense will have 30 seconds to put them away, and if caught a second

time must put them in distractions jail on my desk for the remainder of the class.

• Kind: Remember that all of your classmates and your teacher are fellow humans, make

sure to treat them with respect and kindness. Don’t interrupt the learning of others. Follow

instructions.

• Honest: Present your own work. Be willing to express respectfully when you’re confused.

Ask questions in an effort to understand. Give a genuine effort.

Major Units of Study

1. Linear Equations and Inequalities

2. Linear Functions

3. Line Segments and Angles

4. Linear and Exponential Relationships

5. Transformations and Congruence

6. Triangles and Quadrilaterals

7. Statistics

Instructor

Ms. Chelsea Huber

Room

205A

Phone

(614) 859-0019

Email

[email protected]

Office Hours

Tuesdays and Thursdays 2-3

*Will vary due to meetings

Website:

MsHuberMath.weebly.com

How to Contact

Best Way:

I prefer text! I will respond to

calls, texts, and emails sent

before 7:30pm.

mailto:[email protected]

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Homework Policy

You will receive homework most days, and is considered incomplete if work is not shown (on paper assignments). Homework is

10% of your grade, and completing it is strongly correlated with higher test grades and overall grades.

▪ Completed homework turned in by the due date can receive up to full credit and will be returned with time to review before the

next quiz. Missing or incomplete homework will result in a 0%

▪ Late or redone homework can be turned in within one week of due date. The last day I will accept homework is labeled on the

paper assignments.

▪ Turn in homework at the bin for your block in the back of the room.

▪ If you are absent or lose your homework, assignments can be found at mshubermath.weebly.com – either print the homework

page or do the assignment on loose leaf paper, making sure to write the assignment title on the paper.

Grading System

• • Assigned • % of Quarter Grade

• Homework/Practice Assignments • ~3 a week • 10

• Check for Understanding • ~4 a week • 0 - Graded to check understanding only

• Unit Tests

• Topic Quizzes

• ~3-4 a quarter

• ~1 a week

• 90

•

• Total • • 100

Midterms and Finals

Your midterm and final test will each account for 10% of your course grade. The other 80% of your grade for the course comes from

your two quarter grades.

Retakes

Every student has the option to retake tests and quizzes. If you would like to do a retake, you must turn the test back in within two

weeks of receiving it graded, with the appropriate section filled in to schedule the retake time. Up to half of the points missed can be

made up by retaking. Flex or Office Hours are the two times you will have to retake assignments.

Weighted vs. Unweighted

If you choose to take this as a weighted course, you will be given a list of interesting math related theorems and topics. You will be

required to create two assignments and presentations surrounding your two selections from these topics.

Tardy Policy

Students who enter the room after class begins will follow the following sequence:

• 1st offense: verbal warning to student, logged in system

• 2nd offense: verbal warning to student, logged in system, parent call

• 3rd offense: parent call, logged in system, one after school detention

• 4th offense: parent call, logged it system, two after school detentions

• 5th offense: logged in system, consequence determined by administrative team

6

Before you enter Class • Stow (put in pocket, backpack, etc) your smart device. If it is seen out in class and

I give you “the look” put it away. If I have to do this twice, you’ll need to turn in

your smart device in the blue distractions bin. You can grab it as you leave class.

• Other things that go in the blue distractions bin after a second correction:

anything that will cause others or yourself to be distracted:

Examples: fidget spinners, rubber bands, etc.

• Consequence for not putting distractions in the bin: parent call and after-school

detention assigned.

• Make sure you have your materials: pencil, calculator ( TI-84 and edition),

KEEP SHEETS. Occasionally packets will be used for multiple days. In those cases,

you need to have those packets with you as well.

Starting class • As you enter, grab any materials sitting on the edge of the kidney table

• Once seated, begin working on the Do Now. I will have a timer and once it beeps

we will read the AIM, review the agenda, and go over the Do Now.

• WWBAT stands for we will be able to

Daily Class Set-Up

• Our school promotes problem-based learning

• This means, our class will mostly start with a real-world situation which we will

analyze. In my class, often this will include working in pairs and think/pair/share and

group work in threes or fours.

• After working through the problem in these groups, we will come back together to

synthesize what we’ve learned.

• Many days we will end with a check for understanding. I will not put these in the

gradebook the majority of the time, because they are formative (vs. summative)

assessments to see how we are doing. If they are entered in the gradebook, they

count as an assessment grade

7

Practice Assignments • Practice assignments (homework) are worth 10% of your grade.

• Even though practice assignments are only 10%, having them done and checked is

strongly correlated with higher assessment scores.

• Turn in homework to the correct bin for your block

• I will always grade the practice I assign: I will never assign something that is just “busy

work”

• I will grade any assignment until the “accepted until” date, which is always one week

after the original due date.

• When graded, you will receive them back the next class. If you would like to resubmit,

just check the “is this a resubmit?” line and turn back in after making corrections. You

may do this once on most assignments.

• Homeworks will always be the last page(s) of the packet so you can tear them off

Website • MsHuberMath.weebly.com

• We reviewed the website last week, but remember that this website will have all

assignments and notes under the Math 1 tab

• If you are doing an assignment from the website (and you can’t print it), you MUST

write the title of the assignment at the top of the page. I cannot grade it otherwise

Cheating • Any assignment in which I have clear evidence of cheating will result in an after-school

detention, receiving a zero on the assignment and no possibility of retaking the test or

redoing the practice

• Cheating is obvious: showing your work and both making the same obvious mistake,

one student copying something wrong which makes no sense, trying to make a small

change in wording and thinking that is no longer cheating. I assigned 40 zeros and

detentions last year due to students cheating on practice. If you cheat or let a

classmate borrow your assignment to cheat, you will both have a parent call and be

assigned a detention.

• Multiple cheating offenses can result in suspension

• Only exception to both parties receiving the consequence: if one student is clearly

unaware of the cheating (someone looking at their paper during a test without their

knowledge), or if one student comes forth before they are caught.

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AIM(S):

✓ WWBAT identify patterns in data

✓ WWBAT use tables, graphs, and expressions to model data

✓ WWBAT use expressions to make predictions

Identifying patterns at Yellowstone

Ms. Huber loves to travel, and a couple in 2016 she

went on a road trip west to visit various National Parks,

including Yellowstone. Yellowstone is famous for its

geysers, especially one commonly referred to as Old

Faithful. A geyser is a spring that erupts intermittently,

forcing a fountain of water and steam from a hole in

the ground.

Old Faithful can have

particularly long and fairly

predictable eruptions. As a

matter of fact, park rangers

have observed the geyser over many years and have developed

patterns they use to predict the timing of the next eruption. Park

rangers have recorded the information in the table below.

1. Describe any patterns you see in the table.

Answers will vary. EX: for every one minute added to the eruption length, there are 12 minutes added

To the time until next eruption.

2. Why might it be important for park rangers to be able to predict the timing of Old

Faithful’s eruptions?

Answers will vary. EX: to control the crowd, for visitor safety.

Integrated

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Numeric and Graphic Representations Of Data

8/21/2017

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3. If an eruption lasts 8 minutes, about how long must park visitors wait to see the next

eruption? Explain your reasoning using the patterns you identified in the table.

130 minutes. Reasoning will vary, EX: 46+12(7)

4. Graph the data below.

5. Reason quantitatively. Ms. Huber arrived at Old Faithful to find a sign indicating they had

just missed an eruption and that it would be approximately 2 hours before the next one.

How long was the eruption they missed? Explain how you determined your answer.

About 7 minutes; 2 hours is 120 minutes; 46 + 12 + 12 + 12 + 12 + 12 + 12 = 118 minutes. 118 minutes is very close to 120

minutes. Students may also extend the table from Item 1.

Patterns can be written as sequences.

6. Using the table or graph above, write the approximate times

until the next Old Faithful eruption as a sequence.

46, 58, 70, 82, …

10

7. How would you describe this sequence of numbers?

Answers will vary. It starts at 46 and then each subsequent term increases by 12.

In the table below, 5 and 8 are consecutive terms. Some sequences have a common

difference between consecutive terms. The common difference between the terms in the

table below is 3.

Sequence: 5, 8, 11, 14…

8. Identify two consecutive terms in the sequence of next eruption time that you created in

Item 6.

Answers will vary. 70 and 82

9. The sequence of next eruption times has a common difference. Identify the common

difference.

12 minutes

11

10. Each term in the sequence above can be written using the first term and repeated

addition of the common difference. For example, the first term is 5, the second term is

5 + 3, and the third term can be expressed as 5 + 3 + 3 or 5 + 2(3). Similarly, the terms

in the sequence of next eruption times can also be written using repeated addition of

the common difference.

a. Write the approximate waiting time for the next eruption after eruptions lasting 4

and 5 minutes using repeated addition of the common difference.

4 minutes: 46 + 12 + 12 + 12 or 46 + 12(3)

5 minutes: 46 + 12 + 12 + 12 + 12 or 46 + 12(4)

b. Model with mathematics. Let 𝑛 represent the number

of minutes an eruption lasts. Write an expression using

the variable 𝑛 that could be used to determine the

waiting time until the next eruption.

46 + 12(n − 1)

c. Check the accuracy of your expression by evaluating it when 𝑛 = 2.

58

d. Use your expression to determine the number of minutes a visitor to the park must

wait to see another eruption of Old Faithful after a 12-minute eruption.

178 minutes

12

Check Your Understanding 1.1 SB-Mobile charges $20 for each gigabyte of data used on any of its smartphone plans.

11. Complete the tableshowing the charges for data based on the number of gigabytes

used.

12. Graph the data from the table. Be sure to label your axes.

13. Write a sequence to represent the total price of a data plan.

20, 40, 60, 80, …

14. The sequence you wrote in Item 13 has a common difference. Identify the common

difference.

$20

15. Let 𝑔 represent the number of gigabytes used. Write an expression that can be used to

determine the total data charge for the phone plan.

𝟐𝟎𝒈 or 𝟐𝟎 + 𝟐𝟎(𝒈 − 𝟏)

16. Use your expression to calculate the total data charge if 10 gigabytes of data are used.

17. How many gigabytes are used if the total data charge is

a. $160?

b. $320?

c. $10?

13

AIM(S):

✓ WWBAT Use patterns to write expressions.

✓ WWBAT Use tables, graphs, and expressions to model situations.


Determine the relationship between the number of chairs in the classroom and the

number of chair legs.

In words: In an expression:

There are 2 legs for every chair, 𝟐𝒍 + 𝟐

plus 2 chairs with one leg

In a table: In a graph:

Number of chairs Number of legs

3 4

4 6

5 8

6 10

Predict how the number of chair legs would change if 8 more chairs were brought into the

room.

16 more legs

Integrated

Math I


Writing Expressions

8/22/2017

14

Patterns at Mesa Verde

Ms. Huber also visited Mesa Verde National

Park in Colorado. As she investigated the

artifacts on display from the ancestral

Pueblo people who once called the area

home, she began to notice that the

patterns used to decorate pottery, baskets,

and textiles were geometric. She found a

pattern similar to

the one below

particularly

interesting.

1. Reason abstractly. Draw the next two figures in the pattern.

2. Create a table showing the relationship between the figure number and

the number of small squares in each figure.

15

3. Use the variable 𝑛 to represent the figure number. Write an expression that could be

used to determine the number of small squares in any figure number.

4. Use your expression to determine the number of small squares in the 12th figure.

Ms. Huber noticed that many times the centers of the figures in the pattern were filled in

with small squares of the same size as the outer squares but in a different color.

5. Fill in the centers of the diagrams with small colored squares

6. Draw the next two figures in the pattern. Be sure to include the inner colored squares.

7. Copy the first two columns of the table you created in Item 2 and add a column to

show the relationship between the figure number and the number of inner colored

squares.

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8. Describe any numerical patterns you see in the table.

9. Write the numbers of inner colored squares as a sequence.

10. Does the sequence of numbers of inner colored squares have a common difference? If

so, identify it. If not, explain.

11. Model with mathematics. Graph the data from the table on the appropriate grid. Be

sure to label an appropriate scale on the 𝑦-axis.

12. Compare the graphs

13. Reason quantitatively. Use the patterns you have described to predict the number of

inner colored squares in the 10th figure of the pattern.

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14. How is the number of inner squares related to the figure number?

15. Use the variable 𝑛 to represent the figure number. Write an expression that could be

used to determine the number of inner colored squares in any figure number.

16. Use your expression to determine the number of inner colored squares in the 17th

figure.

17. In what figure will the number of inner colored squares be 400?

Ms. Huber discovered another pattern in the artifacts. She noticed that when triangles were

used, the triangles were all equilateral and often multicolored.

18. Attend to precision.

a. Determine the perimeter of each figure in the pattern if each side of one triangle

measures 1 cm.

b. Use the variable 𝑛 to represent the figure number. Write an expression that could

be used to determine the perimeter of any figure number.

𝟐𝒏 + 𝟏

18

19. Ms. Huber found that she could determine the perimeter of any figure in the pattern

using the expression 2𝑛 + 1. Use Ms. Huber’s expression to calculate the perimeters of

the next three figures in the pattern. Use the table below to record your calculations.

20. Create a sequence to represent the perimeters of the figures in the pattern. Does the

sequence have a common difference? If so, identify it. If not, explain.

21. Represent the relationship between the figures in the pattern and their perimeters as a

graph. Label the axes and the scale on your 𝑦-axis.

22. Is it possible to have a figure with a perimeter of 28? Why or why not?

23. Is it possible for the perimeter to be an even number? Why or why not?

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Check Your Understanding 1.2 A pattern of small squares is shown below. Use the pattern to respond to the following

questions.

24. Create a table to show the number of small squares in the first through the fifth figures,

assuming the pattern continues.

Figure Number Number of Small Squares

25. Write the number of small squares in each figure as a sequence. Does the sequence

have a common difference? If so, identify it. If not, explain.

26. How many small squares would be in the 10th figure? Justify your response using the

sequence or the table.

27. Use the variable 𝑛 to write an expression that could be used to determine the number

of small squares in any figure in the pattern.

28. Use your expression to determine the number of small squares in the 20th figure.

20

AIM(S):

✓ WWBAT Write a linear equation to model a real-world situation.

✓ WWBAT Interpret parts of a linear equation.

Equations in Central Park Mizing and his family visited New York City. During their stay, they decided to take a walk

around the entire edge of Central Park. The park is a large rectangle.

1. Let 𝑤 represent the width of the park and let 𝑙 represent the length.

a. Use 𝑙 and 𝑤 to write two different expressions for the perimeter of the park.

b. Mizing learns that the perimeter of Central

Park is 9600 meters. Write an equation for the

perimeter of Central Park in terms of 𝑙 and 𝑤.

2. Mizing learns that the length of the park is 5 times as its width. Use this fact to write an

expression for the length of the park in terms of 𝑤 only.

3. Model with Mathematics. Draw a rectangle to represent Central Park. Label each

side length with expressions written in terms of 𝑤.

4. Write an expression for the perimeter of the park in terms of 𝑤. Simplify the expression.

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Writing and Interpreting Linear Equations

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5. Write an equation that relates the perimeter of Central Park to its width 𝑤 only. Explain

your thinking.

6. Based on the equation you wrote in Item 5, how is the perimeter of Central Park

related to the width of Central Park?

7. Write an expression for the area of Central Park in terms of its length 𝑙 and its width 𝑤.

8. Given that Central Park’s length is five times its width, write an expression for the area

of Central Park in terms of 𝑤 only.

9. The area of Central Park is about 3.4 square kilometers. Write an equation that relates

this area to the width 𝑤 only. How is this equation different from the one you wrote in

Item 5?

22

Check Your Understanding 1.3 10. The width of a rectangle is half the length of the rectangle.

a. Write an expression for the rectangle’s perimeter in terms of its length 𝑙.

b. The perimeter of the rectangle is 15 inches. Write an equation that relates this

perimeter to the length of the rectangle.

11. The height ℎ of a rectangular mirror is four times the width of the mirror.

a. Write an expression for the width of the mirror in terms of the height. Then write

an expression to represent the perimeter of the mirror in terms of the height.

b. The perimeter of the mirror is 60 inches. Write an equation that relates this

perimeter to the height of the mirror.

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AIM(S):

✓ WWBAT Use the algebraic method to solve an equation.

✓ WWBAT Write and solve an equation to model a real-world situation.


Solve the following one-step equations.

1. 15𝑥 = 60

𝑥 = 4

2. 𝑡 − 23 = 6

𝑡 = 29

3. 𝑛 + 12 = −18

𝑛 = −30

Integrated

Math I


Solving Linear Equations

8/23/2017

24

Solving Linear Equations 1. Make sense of problems. Determine the number, and explain how you came up with

the solution.

2. You could have used an equation to answer Item 1. Write an equation that could be

used to represent the problem in Item 1.

3. For each equation, tell whether the given value of x is a solution. Explain.

a. 2 + 4(𝑥 − 1) = 22; 𝑥 = 6

b. 12 −𝑥

4= 8; 𝑥 = 24

c. 3.8 + 6𝑥 = 8.6; 𝑥 = 0.9

One way to solve an equation containing a variable is to use the algebraic method. This

method is also called the symbolic method or solving equations using symbols.

25

Example A

Solve the equation 𝟑𝒙 + 𝟗𝟎 + 𝟐𝒙 = 𝟑𝟔𝟎 using the algebraic

method, showing each step. List a property or provide an

explanation for each step. Check your solution.

Try These A

a. Solve the “What’s My Number?” problem using the equation

you wrote in Item 2 and the algebraic method.

Solve each equation using the algebraic method, showing

each step. List a property or provide an explanation for each step.

b. −5𝑥 − 6 = 1

d. 12𝑑 + 2 − 3𝑑 = 5

26

e. 7.4𝑝 − 9.2𝑝 = −2.6 + 5.3

f. 𝑥

4+ 7 = 12

g. 20𝑥 − 3 + 5𝑥 = 22

h. 8 =3

2𝑤 − 12 + 8

Check Your Understanding 2.1a 4. Is 𝑥 = 4 a solution of 2(𝑥 − 3) + 7 = 23? How do you know?

5. Which property of equality would you use to solve the equation 𝑥

5= 13? Explain your

answer.

27

Julio has 5 more dollars than Dan. Altogether, Julio and Dan have 19 dollars. How much

money does each young man have?

6. Let 𝑑 represent the amount of money, in dollars, that Dan has. Use 𝑑 to write an

expression that represents the amount of money that Julio has.

7. Write an equation to represent the problem situation. Then

solve your equation, showing each step. State a property or

provide an explanation for each step. Check your solution.

8. Interpret your solution to Item 7 within the context of the problem.

9. Verify the reasonableness of your solution by checking that your answer to Item 9

matches the information given in the original problem situation at the top of the

page.

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Check Your Understanding 2.1b 10. In Item 6, the variable 𝑗 could have been defined as the amount of money, in dollars,

that Julio has.

a. Use 𝑗 to write an expression to represent the amount of money, in dollars, that

Dan has. Describe the similarities and differences between this expression and

the one you wrote in Item 6.

b. Write and solve an equation using 𝑗 to represent the problem situation. Interpret

the solution within the context of the problem.

c. Does the definition of the variable in a problem situation change the solution to

the problem? Explain your reasoning.

11. Eva is 8 years younger than Leo. The sum of their ages is 34. Define a variable. Then

write and solve an equation to find Eva’s and Leo’s ages.

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AIM(S):

✓ WWBAT Solve complex equations with variables on both sides and justify each step in

the solution process.


Two-Column Proof The expressions on each side of an equation are assumed equal. To solve a complex

equation, isolate the variable by performing the same operations on each side. Justify each

step by writing the operations or properties you use.

1. The equation 3𝑥 − 2(𝑥 + 3) = 5 − 2𝑥 is solved in the table below. Complete the

table by stating a property or providing an explanation for each step.

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2. Solve the following equations. State a property of equality or provide an explanation

for each step.

a. 5𝑥 + 8 = 3𝑥 − 3

b. 2(4𝑦 + 3) = 16

c.

2

3𝑝 +

1

5=

4

5 d.

3

4𝑎 −

1

6=

2

3𝑎 +

1

4

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3. Model with mathematics. Bags of maple granola cost $2 more than bags of apple

granola. The owner of a restaurant ordered 6 bags of maple granola and 5 bags of

apple granola. The total cost of the order was $56.

a. Let 𝑚 represent the cost of a bag of maple granola. Write an expression for the

cost of 6 bags of maple granola.

b. Use the variable 𝑚 to write an expression for the cost of a bag of apple

granola.

c. Write an expression for the cost of 5 bags of apple granola.

d. Write an equation to show that the cost of 6 bags of maple granola and 5 bags

of apple granola was $56.

e. Solve your equation to find the cost per bag of each type of granola.

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Check for Understanding 2.2

4. Suppose you are asked to solve the equation 3

4𝑥 −

2

3=

1

6𝑥

a. What number could you multiply both sides of the equation by so that the

numbers in the problem are integers and not fractions?

b. What property allows you to do this?

5. Explain how the Commutative Property of Addition could help you solve the equation

−6𝑥 + 10 + 8𝑥 = 12 − 4𝑥.

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AIM(S):


✓ WWBAT Interpret parts of an expression in terms of its context.

Modeling with Equations The Future Engineers of America Club (FEA) wants to raise money for

a field trip to the science museum. The club members will hold an

engineering contest to raise money. They are deciding between two

different contests, the Straw Bridge contest and the Card Tower

contest. The Straw Bridge contest will cost the club $5.50 per

competitor plus $34.60 in extra expenses. The Card Tower contest will

cost $4.25 per competitor plus $64.60 in extra expenses. To help

decide which contest to host, club members want to determine

how many competitors they would need for the costs of the two contests to be the same.

1. Define a variable, explaining what it represents. Write an equation that sets the costs

of the two contests equal.

2. Solve the equation from Item 1 by using the algebraic method, showing each step.

List a property of equality or provide an explanation for each step.

Integrated

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Modeling with Equations

Date:

34

3. Model with mathematics. Interpret the meaning of the solution in Item 2 in the context

of the problem.

4. The FEA club estimates they will have more than 30 competitors in their contest. Make

a recommendation to the club explaining which contest would be the better choice

to sponsor and why.

5. The FEA club will charge each competitor $10 to enter the engineering contest. Write

an expression for the club’s revenue if 𝑥 competitors enter the contest.

6. a. Write an equation to find the break-even point for the fundraiser using the contest

you recommended to the FEA club.

b. Solve the equation. State a property of equality or provide an explanation for

each step. How many competitors does the club need to break even?

7. How much profit will the FEA club earn from 32 competitors if they use the contest you

recommended?

35

8. The Future Engineers of America Club treasurer was going back through the

fundraising records. On Monday, the club made

revenue of $140 selling contest tickets at $10 each. One

person sold 8 tickets, but the other person selling that

day forgot to write down how many she sold. Write and

solve an equation to determine the number of tickets

the other person sold.

Check for Understanding 2.3 9. How can you use the Multiplication Property of Equality to rewrite the equation

0.6𝑥 + 4.8 = 7.2 so that the numbers in the problem are integers and not decimals?

10. When writing an expression or equation to represent a real-world situation, why is it

important to be able to describe what each part of the expression or equation

represents?

36

AIM(S):

✓ WWBAT Identify equations that have no solution.

✓ WWBAT Identify equations that have infinitely many solutions.

How many solutions? Remember that a solution of an equation with one variable is a value of the variable that

makes the equation true.

1. The set of numbers { 1

2, 3, 6, 17, 0, 11 } contains possible solutions to the following

equations. Determine which of these numbers are solutions to each of the following

equations.

a. 9𝑥 + 5 = 4(𝑥 + 2) + 5𝑥

b. 7𝑥 − 10 = 3𝑥 + 14

c. 3𝑥 − 12 = 3(𝑥 + 1) − 15

An equation has no solution if there is no value of the variable that will create a true

mathematical statement. An equation has infinitely many solutions if there are an unlimited

number of values of the variable that will create a true mathematical statement.

2. Laura, Nia, and Leo solved the following three equations as shown. Identify each of

the equations as having one solution, no solution, or infinitely many solutions. Justify

your responses.

Integrated

Math I


Equations with No Solution or Infinitely Many Solutions

Date:

37

Check Your Understanding Determine solutions to each of the following equations.

3. 3(2𝑧 + 4) = 6(5𝑧 + 2)

4. 3(𝑥 + 1) + 1 + 2𝑥 = 2(2𝑥 + 2) + 𝑥

5. 8𝑏 + 3 − 10𝑏 = −2(𝑏 − 2) + 3

38

Some equations are true for all values of the variable. This type of equation has all real

numbers as solutions. This means the equation has infinitely many solutions. Other equations

are false for all values of the variable. This type of equation has no solutions.

6. Which of the following equations has no solutions and which has all real numbers as

solutions? Explain your reasoning.

a. 3𝑥 + 5 = 3𝑥

b. 4𝑟 − 2 = 4𝑟 – 2

7. Critique the reasoning of others. A student claims that the equation 2𝑥 + 6 = 4𝑥 + 4

has no solutions because when you substitute 0 for 𝑥, the left side has a value of 6 and

the right side has a value of 4. Is the student’s reasoning correct? Explain.

39

8. Explain why the equation 𝑥 + 3 = 𝑥 + 2 has no solution.

9. Reason quantitatively. For what value of a does the equation 3𝑥 + 5 = 𝑎𝑥 + 5 have

infinitely many solutions? Explain.

10. Consider the equation 𝑛𝑥 – 4 = 6. a. What are the solutions if the value of 𝑛 is 0? Explain.

b. What if the value of 𝑛 is 2? What is the solution of the equation? How do you

know?

40

Check Your Understanding 2.4 Make use of structure. Create an equation that will have each of the following as its

solution.

11. One solution

12. No solution

13. Infinitely many solutions

14. A solution of zero

41

Name:

_________________ ______

Date: August 25th, 2017

Block: ___________________

Practice

2.3 and 2.4 Linear Equations and Inequalities

Modeling with Equations

Equations with No Solution or

Infinitely Many Solutions

Ms. Huber

614-859-0019 Mshubermath.weebly.com

/ 5

A B C D F Due Date:

08/29/17

Accepted Until:

09/05/17

Directions: Complete all of the below problems (FRONT AND BACK). If you have questions, first check the

examples in your packet. Then, check the class website or ask a classmate for help. Then, you can meet with

Ms. Huber at office hours of by texting if you still have questions.

2.3

On-the-Go Phone Company has two monthly plans for their customers.

The EZ Pay Plan costs $0.15 per minute. The Base Rate Plan charges customers $40 per month

plus $0.05 per minute.

1. a. Write an expression that represents the monthly bill for 𝑥 minutes on the EZ Pay Plan.

b. Write an expression that represents the monthly bill for 𝑥 minutes on the Base

Rate Plan.

c. Write an equation to represent the point at which the monthly bills for the two

plans are equal.

2. Solve the equation, showing each step. List a property of equality or provide an

explanation for each step.

3. Interpret the solution of the equation within the context of the problem.

FLIP OVER

Is this a re-submit?

42

2.4

Solve each equation. If an equation has no solutions or if an equation has infinitely many

solutions, explain how you know.

4. 3𝑥 − 𝑥 − 5 = 2(𝑥 + 2) – 9

5. 7𝑥 − 3𝑥 + 7 = 3(𝑥 − 4) + 20

6. −2(𝑥 − 2) − 4𝑥 = 3(𝑥 + 1) − 9𝑥

7. 5(𝑥 + 2) − 3 = 3𝑥 − 8𝑥 + 7

8. 4(𝑥 + 3) − 4 = 8𝑥 + 10 − 4𝑥

43

Name:

_________________ ______

Date: August 23rd, 2017

Block: ___________________

Practice

2.1 and 2.2 Linear Equations and Inequalities

Solving Linear Equations

Solving More Complex Lin. Eq.

Ms. Huber


/ 9

A B C D F Due Date:

08/24/17

Accepted Until:

08/31/17




2.1

1. Attend to precision. Justify each step in the solution of 5𝑥 + 15 = 0 below by stating a

property or providing an explanation for each step.

2. Solve the equations below using the algebraic method. State a property or provide an

explanation for each step. Check your solutions.

a. 3𝑥 − 24 = −6 b. 4𝑥+1

3= 11

Define a variable for each problem. Then write and solve an equation to answer the

question. Check your solutions.

3. Last week, Donnell practiced the piano 3 hours longer than Marcus. Together, Marcus

and Donnell practiced the piano for 11 hours. For how many hours did each young

man practice the piano?

4. Olivia ordered 24 cupcakes and a layer cake. The layer cake cost $16, and the total

cost of the order was $52. What was the price of each cupcake?

FLIP OVER


44

2.2 Solve the following equations, and explain each step.

5. 6𝑥 + 3 = 5𝑥 + 10

6. 6 + 0.10𝑥 = 0.15𝑥 + 8

7. 5 − 4𝑥 = 6 + 2𝑥

8. 9 − 2𝑥 = 7𝑥

9. 2(𝑥 − 4) + 2𝑥 = −6𝑥 – 2

45

Name:

_________________ ______

Date: August 22nd, 2017

Block: ___________________

Practice

1.2-1.3 Linear Equations and Inequalities

Writing Expressions

Writing and Int. Equations

Ms. Huber


/ 9

A B C D F Due Date:

08/23/17

Accepted Until:

08/30/17




1.2 A toothpick pattern is shown below. Use the pattern for Items 1-5.

1. Create a table to show the number of toothpicks in the first through the fifth figures,

assuming the pattern continues.

Figure Number Number of toothpicks

2. Write the number of toothpicks in each figure as a sequence. Does the sequence

have a common difference? If so, identify it. If not, explain.

3. Express regularity in repeated reasoning. How many toothpicks would be in the 15th

figure? Justify your response using the sequence of the table.

4. Use the variable 𝑡 to write an expression that could be used the determine the

number of toothpicks in any figure in the pattern.

5. What figure number will use 51 toothpicks?

FLIP OVER


46

1.3

6. The length of Central Park is five times its width. How is the width of Central Park

related to its length? Write an expression for the width in terms of the length 𝑙.

7. The perimeter of Central Park is 9600 meters. Write an equation that relates this

perimeter to the length 𝑙 of the park.

8. Write an equation for the area 𝐴 of a square in terms of its side length 𝑠. Then write an

equation for the perimeter 𝑃 of the square in terms of 𝑠.

9. The floor of a room is a rectangle. The length of the floor is 3.5 times the width, and the

perimeter of the room is 81 feet.

a. Let 𝑤 represent the width of the room. Write an equation to relate the perimeter

of this room to its width.

b. Make sense of problems. Describe the relationship between the perimeter of the

room and the width of the room.

47

Name:

_________________ ______

Date: August 21st, 2017

Block: ___________________

Practice

1.1 Linear Equations and Inequalities

Numeric and Graphic

Representations of Data

Ms. Huber


/6

A B C D F Due Date:

8/22/2017

Accepted Until:

8/29/2017




1.1 Luisa owns stock in the SBO Company. After the first year of ownership the

stock is worth $45 per share. Luisa estimates that the value of a share will

increase by $2.80 per year.

1. Complete the table showing the value of the stock over the course of several years.

2. Write a sequence to show the increase in the stock value over the course of several

years.

3. Make use of structure. The sequence you wrote in Item 18 has a common difference.

Identify the common difference.

4. Let 𝑛 represent the number of years that have passed. Write an expression that can be

used to determine the value of one share of SBO stock.

FLIP OVER


48

5. Use your expression to calculate the value of one share of stock after 20 years.

6. After how many years will the share value be greater than $60?

Integrated Math I Linear Equations and Inequalities Packet...

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