MATH_0382_LINFUN5_LES_02.docx382 LessonLINFUN5 (2 instructional days)Simultaneous Equations (Systems of Linear Equations in y = mx + b form)

8(9)  Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to:(A) identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations. Supporting

The student will know… The student will be able to… the solution to simultaneous linear

equations (system of equations) represents the values that make both equations true.

the solution to simultaneous linear equations (system of equations) can be identified in a graph.

that simultaneous linear equations are a system of equations.

graph a system of equations on the graphing calculator, in the form y = mx + b, and determine the solution.

determine if an ordered pair is the solution to a system of linear equations.

explain the meaning of the intersection point’s values in terms of a given situation.

Technology TEKS:8(3)  Research and information fluency. The student acquires, analyzes, and manages content from digital resources. The student is expected to:

(D)  process data and communicate results.8(4)  Critical thinking, problem solving, and decision making. The student makes informed decisions by applying critical-thinking and problem-solving skills. The student is expected to:

(C)  collect and analyze data to identify solutions and make informed decisions; (E)  make informed decisions and support reasoning;

ELPS:

(2) Cross-curricular second language acquisition/listening. The student is expected to:(E) use visual, contextual, and linguistic support to enhance and confirm understanding of increasinglycomplex and elaborated spoken language;

(3) Cross-curricular second language acquisition/speaking. The student is expected to:(E) share information in cooperative learning interactions;

(4) Cross-curricular second language acquisition/reading. The student is expected to:(F) use visual and contextual support and support from peers and teachers to read grade-appropriatecontent area text, enhance and confirm understanding, and develop vocabulary, grasp of languagestructures, and background knowledge needed to comprehend increasingly challenging language;

(5) Cross-curricular second language acquisition/writing. The student is expected to:(B) write using newly acquired basic vocabulary and content-based grade-level vocabulary;

MATH_0382_LINFUN5_LES_02.docxMaterials

Graphing Calculator Dry-erase Boards and Markers (Day 1) 1 copy of Watch Out for That Intersection! Practice Problems per student (Day 1) 1 copy of Spin to Win! Practice Problems per student (Day 2) 1 copy of Jet Ski Rental Alternative Assessment (optional)

Graphing Calculator Notes: 8th grade students should have graphing calculators in hand throughout the entire class period

every day. Teachers should frequently model graphing calculator strategies with TI-SmartView or with a

graphing calculator under the HoverCam as students press keys on their handheld graphing calculator at the same time.

We will NOT use the linear regression feature on the graphing calculator in the LINFUN unit. Linear regression will be used as a strategy in the Math 8 STAAR Strategy Booklet and during STAAR review.

MATH_0382_LINFUN5_LES_02.docxProcedure:Day 1Use the MATH_0382_LINFUN5_MAT_1SIMULTEQUPRES_02 notebook file to guide this lesson.

Slide 1: Title Slide

Use this column to write your own notes:

Slide 2: Have students locate the intersection point and name it as an ordered pair. Click on the yellow “Click to Reveal” box to check the solution which is (2, 1).

Slide 3:Have students verify algebraically on dry-erase boards that the ordered pair (2, 1) makes both equations true. Then pull the screen shade to check student work.

Slide 4:Have students also verify on the graphing calculator that the ordered pair (2, 1) makes both equations true. Students can use the graph on the calculator to verify, but point out that the table feature (2nd, Graph) is much more exact. Pull the screen shade to check student work.

MATH_0382_LINFUN5_LES_02.docxSlide 5:On this slide, the term “system of linear equations” is used as another name for two equations which are solved simultaneously on a graph. Have students locate the intersection point and name it as an ordered pair. Click on the yellow “Click to Reveal” box to check the answer which is (6, -2).

Slide 6:Have students verify algebraically on dry-erase boards that the ordered pair (6, -2) makes both equations true. Then pull the screen shade to check student work.

Slide 7:Have students also verify on the graphing calculator with the graph and table that the ordered pair (6, -2) makes both equations true. Pull the screen to check student work.

Slide 8:Guide students through this application problem which appears to be very similar to the types of problems worked when students were solving equations with variables on both sides. Pull the screen shade to ask the questions.The solution is (4, 20). At 4 weeks, both Plant A and Plant B will be 20 inches tall.Also ask students what other ordered pairs mean on either of the graphs of the equations in relationship to the scenario. For example, at 2 weeks, which plant is taller? (Plant B) Shorter? (Plant A) At 6 weeks, which plant is taller? (Plant A) Shorter? (Plant B) This helps students understand that all ordered pairs have meaning on either of the graphs of the equations in terms of the scenario.

MATH_0382_LINFUN5_LES_02.docxSlide 9:Have students verify algebraically on dry-erase boards that the ordered pair (4, 20) makes both equations true. Then pull the screen shade to check student work.

Slide 10:Have students also verify on the graphing calculator with the graph and table that the ordered pair (4, 20) makes both equations true. Pull the screen to check student work.

Slide 11:Guide students through this application problem. Pull the screen shade to ask the questions.The graphs intersect at (3, 28).For 3 pizzas, both Pizza Planet and Eatza Pizza will charge $28.Also ask students what other ordered pairs on either of the graphs of the equations mean in relationship to the scenario. For example, for 2 pizzas, which restaurant is cheaper? (Pizza Planet) More expensive? (Eatza Pizza) For 4 pizzas, which restaurant is cheaper? (Eatza Pizza) More expensive? (Pizza Planet) This helps students understand that all ordered pairs have meaning for the graphs of these equations in terms of the scenario.Slide 12:Have students verify algebraically on dry-erase boards that the ordered pair (3, 28) makes both equations true. Then pull the screen to check student work.

MATH_0382_LINFUN5_LES_02.docxSlide 14:Transition slide for partner or independent practice

Assign students Watch Out for That Intersection! Practice Problems as individual practice/homework. Encourage students to verify solutions algebraically and/or with the graphing calculator when equations are given.

MATH_0382_LINFUN5_LES_02.docxProcedure:Day 2Use the MATH_0382_LINFUN5_MAT_2SIMULTEQUPRES_02 notebook file to guide this lesson. Give each student a copy of the Spin to Win Practice Problems. Teachers may choose to have students work the problems before or during the game. Encourage students to verify solutions algebraically and/or with the graphing calculator when equations are given. An answer key is provided and answers are on each slide of the game beneath a “click to reveal” graphic.

Slide 1: This slide provides rules for the game. The number on the problem corresponds to the number on the student copy of the Spin to Win Practice Problems.

If neither team answers correctly, use the problem as a teaching moment.

Use this column to write your own notes:

Slide 2:This is the game board slide. Be sure to place a check mark over problems once they are completed. Keep score in the lower left-hand corner.

Slides 3 – 22:Problems for the game are located on these slides and are linked to the rectangles on the left-hand side of slide 2.

Slide 3 is pictured here.

Click the wheel in the lower right-hand corner of each slide to return to slide 2 once a problem is completed.

Students most likely will not be able to finish all 20 practice problems in class. The remaining problems can be assigned for homework or practice on another day in class.

MATH_0382_LINFUN5_LES_02.docxOptional Activity:

Jet Ski Rental is provided as an optional alternative assessment/activity and should be completed with a partner or small group. Graphing calculator is required. Points are assigned to each question on the student copy and a key is provided.

MATH_0382_LINFUN5_LES_02.docx

Watch out for that intersection! Practice ProblemsBe sure to verify solutions algebraically and/or with the graphing calculator when equations are given.1. The two lines graphed on the coordinate grid each represent an equation.

Which ordered pair represents a solution to both equations?

A (0, 5)B (-3, -1)C (-1, -3) D (0, -2)

_____________________________________________________________________________________________________________________________________________2. Moriah is verifying the x and y values that simultaneously satisfy the two linear equations graphed below. First Equation Second Equation

y = - x + 20 y = 1/2x + 10

15 = - (15) + 20 15 = (10) + 10 15 = -7.5 + 20 15 = 5 + 10

15 = 12.5 15 = 15

What mistake did she make?

A She used 10 for the value of x in the second equation.

B She multiplied and 10 and got 5 in the second equation.C She used 15 for the value of x in the first equation.D She added -7.5 and 20 and got 12.5 in the first equation.

_____________________________________________________________________________________________________________________________________________3. Darryl is trying to find the values of x and y that simultaneously satisfy both of the equations

graphed.

Which of the following ordered pairsis the best estimate for the solution?

A (4, 7.5)B (0, 5)C (-8, 0)D (7.5, 4)

4. Which two equations have (-6, -2) as their solution?

MATH_0382_LINFUN5_LES_02.docx

I. y = -x – 8

II. y = x + 2

III. y = x – 1A I and IIB I and IIIC II and IIID none of these

_____________________________________________________________________________________________________________________________________________5. Which pair of linear equations has the ordered pair (-6, 2) as its solution?

A y = x C y = x + 4 y = x + 8 y = -x – 4

B y = -2x + 10 D y = x + 6

y = x + 5 y = x – 4 _____________________________________________________________________________________________________________________________________________6. Jose and Maris work as sales people for different car dealerships. Jose earns a monthly salary of

$3,500 plus a 6% commission on his sales, x. Maris earns a monthly salary of $4,000 plus a 4% commission on her sales, x. The equations that represent the total monthly earnings, y, based on the amount of sales, x, are shown below.

Jose y = 3,500 + 0.06xMaris y = 4,000 + 0.04x

When the equations are graphed, the lines intersect at the point (25,000, 5,000). What does this point mean in relationship to the scenario above?

A Both sales persons will earn a monthly salary of $5,000 for $25,000 in sales.B Both sales persons will earn a monthly salary of $25,000 for $5,000 in sales.C Both sales persons will earn a monthly salary of $25,000 for selling 5,000 vehicles.D Both sales persons will earn a monthly salary of $5,000 for selling 25,000 vehicles.

_____________________________________________________________________________________________________________________________________________

7. In the graph below, which two lines have (-3, -5) as the x and y values that simultaneously satisfy both linear equations?

A lines a and cB lines b and dC lines b and cD lines a and d

bc

d

a

MATH_0382_LINFUN5_LES_02.docx

Watch out for that intersection! Practice ProblemsAnswer Key1. B2. C3. A4. A5. C6. A7. D

MATH_0382_LINFUN5_LES_02.docx

Practice ProblemsUse this worksheet to answer questions as they are presented on the SMART Board during the Spin to Win class game. Be sure to verify solutions algebraically and/or with the graphing calculator when equations are given.1. Which of the following ordered pairs is the solution to the system of linear equations shown below?

A (4, 1)B (1, 4)C (0, 2.5)D (0, 5)

_____________________________________________________________________________________________________________________________________________2. Kate is trying to decide between two health clubs. Company A charges a one-time fee of $70 and $35 a month. Company B charges a one-time fee of $10 and $40 a month.

Kate wrote two equations to represent the cost of each plan, C, after m months as shown below.

C = 35m + 70C = 40m + 10

When Kate graphed the equations on her graphing calculator, the lines intersected at the point (12, 490). What does this point mean in relationship to the scenario above?

A After 12 months, Kate will have 490 points at the club.B After 12 people buy memberships, the club will earn $490.C After 12 months, both plans will cost $490.D After Kate spends $490, she will get 12 months free.

_____________________________________________________________________________________________________________________________________________3. Which of the following ordered pairs is the solution to the system of linear equations graphed

below?

A (3, -4) B (4, -3)C (7, 0)D (0, 6)

MATH_0382_LINFUN5_LES_02.docx4. Marine biologists studied the lengths of two species of shark for several years. The initial length

and growth rate of each species are shown in the table.

Greenland Shark Spiny Dogfish SharkInitial Length 37 cm 22 cm

Rate of Growth 0.75 cm per year 1.5 cm per year

Mayra wrote two equations to represent the length of a shark, y, after x years of growth as shown below.

y = 37 + 0.75xy = 22 + 1.5x

When she graphed the equations on her graphing calculator, the lines intersected at the point (20, 52). What does this point mean in relationship to the scenario above?

A After 20 years, the rates of growth are the same for both sharks.B After 20 years, the sharks have increased in length by 52 cm.C After 52 years, the sharks have both grown by 20 cm.D After 20 years, the sharks are both 52 cm in length.

_____________________________________________________________________________________________________________________________________________5. The graphs of two linear equations are shown at right.

What are the values of x and y that simultaneously satisfy these two linear equations?

A (4, 3)B (3, 4)C (2, 0)D (5, 0)

_____________________________________________________________________________________________________________________________________________6. Which of the following ordered pairs is the best estimate for the solution of the system of linear

equations shown in the graph below?

A (2, 0)B (-5, 0)C (4, -3)D (4, 3)

MATH_0382_LINFUN5_LES_02.docx7. Which two equations have (2, 5) as their solution?

I. y = 4x – 3 II. y = x + 5

III. y = 2x + 1A I and IIB I and IIIC II and IIID none of these

_____________________________________________________________________________________________________________________________________________8. Which pair of linear equations have the ordered pair (-6, -4) as the x and y values that

simultaneously satisfy both equations?

A C

B D

_____________________________________________________________________________________________________________________________________________9. In the graph below, which two lines have (1, 6) as their solution?

A lines a and dB lines b and dC lines c and aD lines d and c

_____________________________________________________________________________________________________________________________________________10. Two hot-air balloons are set to launch on opposite sides of a field. Balloon A is 6 feet off the ground

and starts to rise at a rate of 1.8 feet per second. Balloon B is 3 feet off the ground and starts to rise at a rate of 2.4 feet per second.

Allison wrote 2 equations to represent the height of the balloons, h, after t seconds as shown below.

Balloon A h = 6 + 1.8tBalloon B h = 3 + 2.4t

When Allison graphed the equations on her graphing calculator, the lines intersected at the point (5, 15). What does this point mean in relationship to the scenario above?

A After 5 balloons are launched, the balloons are 15 feet high.B After 5 seconds, 15 balloons are in the air.C After 5 seconds, the balloons are 15 feet in the air.D After 15 seconds, the balloons are 5 feet apart.

d

c

b

a

MATH_0382_LINFUN5_LES_02.docx11. The graphs of two linear equations are shown at right.

What are the values of x and y that simultaneously satisfy the two linear equations?

A (-2, -3)B (-3, -2)C (0, 1.5)D (0, -5)

_____________________________________________________________________________________________________________________________________________12. Which two linear equations have (4, -1) as their solution?

I. y = 2x – 7 II. y = -3x + 11III. y = x – 5

A I and IIB I and IIIC II and IIID I, II, and III

_____________________________________________________________________________________________________________________________________________13. The school band will be selling poinsettias for the holidays. Holiday Happenings sells poinsettias for

$5 each plus an order processing fee of $50. Fancy Plants sells poinsettias for $5.50 apiece with no order processing fee. The equations shown below can be used to represent this situation where x represents the number of poinsettias purchased and y represents the total cost of the order in dollars.

Holiday Happenings y = 50 + 5xFancy Plants y = 5.5x

When the equations are graphed on the graphing calculator, the lines intersect at the point (100, 550). What does this point mean in relationship to the scenario?

A For $100, the band can buy 550 poinsettias at both Holiday Happenings and Fancy Plants.B The difference in the processing fees for Holiday Happenings and Fancy Plants is $100 for 550

poinsettias.C If 100 poinsettias are purchased, the band can make $550 in profit.D The total cost of 100 poinsettias will be $550 at both Holiday Happenings and Fancy Plants.

MATH_0382_LINFUN5_LES_02.docx14. Two linear equations are shown below.

y = -x + 8y = x – 2

What are the values of x and y that simultaneously satisfy these two linear equations?

A (5, 3)B (3, 5)C (0, 8)D (2, 0)

_____________________________________________________________________________________________________________________________________________15. Danny and Andrew both own landscaping companies. Danny charges a fuel fee of $25 and $20 a lawn. Andrew charges a $75 fuel fee and $10 a lawn.

Maddie is trying to decide which company to use. She wrote the equations to represent the cost of each company, C, to mow x lawns. The equations are shown below.

C = 20x + 25C = 10x + 75

When Maddie graphed the equations on her graphing calculator, the lines intersected at the point (5, 125). What does this point mean in relationship to the scenario above?

A For 5 hours of work, both companies need $125 in fuel.B For 125 hours of work, both companies will need 5 people to help.C For 5 lawns mowed, both companies will charge $125.D For 5 lawns mowed, both companies will use 125 gallons of fuel.

_____________________________________________________________________________________________________________________________________________16. In the graph below, which two lines have (0, -4) as their solution?

A lines p and qB lines s and qC lines p and rD lines r and s

s

p

r

q

MATH_0382_LINFUN5_LES_02.docx17. Keyonna is considering two different cable plans. Plan A charges $5.50 per movie viewed. Plan B

requires a one-time fee of $24 plus $4 per movie viewed. Keyonna wrote two equations to represent the cost of each plan, P, for the number of movies viewed, m.

P = 5.50mP = 4m + 24

When Keyonna graphed the equations on her graphing calculator with Plan A in y1 and Plan B in y2, she saw the following table of values.

Which of the following statements is NOT true about this scenario based on the information above?

A If Keyonna views 10 movies, Plan A is less expensive.B If Keyonna views 16 movies, the cost of both plans will be $88.C If Keyonna views 20 movies, Plan B is less expensive.D If Keyonna views 50 movies, Plan A is less expensive.

_____________________________________________________________________________________________________________________________________________18. Ken and Kayla are reading the same book for their language arts class. Ken has already read 8

pages of the book. He reads at a rate of 2 pages every minute. Kayla has not started reading the book, but she reads at a rate of 5 pages in 2 minutes. The graph below shows the relationship between the number of minutes, x, and the total number of pages read, y, for both students.

Which of the following statements is NOT true about this relationship?

A Kayla will have read more pages than Ken after 18 minutes.B Ken and Kayla will have read the same number of pages after 16 minutes.C Kayla will have read more pages than Ken after 14 minutes.D Ken will have read more pages than Kayla after 10 minutes.

MATH_0382_LINFUN5_LES_02.docx19. Roberto is saving money to buy an antique car. The price of the car is $4300, but the price

decreases by $100 every month. Roberto currently has $700 in his savings account. He is able to save $300 every month. Roberto wrote two equations to model this situation where x represents the number of months and y represents the total amount of money.

y = 4,300 – 100xy = 700 + 300x

When Roberto graphed the equations on his graphing calculator, the lines intersected at the point (9, 3,400). What does this point mean in relationship to the scenario above?

A After 9 months, Roberto will have saved $3,400 which is the price of the antique car.B After 9 months, the amount of money in Roberto’s savings account will have increased by

$3,400.C After 9 months, the cost of the car will have decreased by $3,400.D After 9 months, Roberto will have saved $3,400 more than the initial amount of money he had.

_____________________________________________________________________________________________________________________________________________20. Jamal is looking for a summer job. Job 1 pays $10 per hour. Job 2 pays $14 per hour but requires a

uniform that costs $50. Jamal wrote 2 equations to represent the total pay he would receive, p, for h hours of work as shown below.

Job 1 p = 10hJob 2 p = 14h – 50

When Jamal graphed the equations, the lines intersected at the point (12.5, 125). What does this point mean in relationship to the scenario above?

A At $12.50 per hour, Jamal will earn $125 at Job 2.B For 12.5 hours of work, both jobs will pay $125.C For 12.5 hours of work, Job 1 pays $125 more than Job 2.D At $12.50 per hour, Jamal will earn $125 at Job 1.

MATH_0382_LINFUN5_LES_02.docx

Practice Problems – Answer KeyNote: Answers to these questions are also included on each slide of the Spin to Win SMART Board, (MATH_0382_LINFUN5_MAT_SIMULTANEOUSEQPRESENT2_02) beneath a “Click to Reveal” graphic.

1. B2. C3. A4. D5. B6. C7. B8. D9. A10. C11. B12. C13. D14. A15. C16. B17. D18. C19. A20. B

MATH_0382_LINFUN5_LES_02.docx

Alternative Assessment for Math 382You are opening a new jet ski rental business at Lake Conroe. There are two other jet ski rental businesses already in operation, but you feel that your rental business can be competitive. The rental plans offered by the other two businesses are shown in the graph below.

Line A represents Alfie’s Jet Ski Rental Company. Alfie charges a deposit of $50 and $10 per hour. His rental plan can be represented by the equation y = 50 + 10x where x represents the number of hours and y represents the total cost.

Line B represents Bubba’s Jet Ski RentalCompany. Bubba charges a deposit of $70 and $7.50 per hour. His rental plan can be represented by the equation y = 70 + 7.5x where x represents the number of hours and y represents the total cost.

1. What values of x and y simultaneously satisfy these two linear equations? Remember you can use your graphing calculator to verify this solution with the table of values for each equation. (10 pts.)

(______, ______)2. What does the intersection point mean in relationship to the scenario above? (15 pts.)

3. If a customer wants to rent a jet ski for 2 hours, which company offers the better deal? How do you know? (10 pts.)

4. If a customer wants to rent a jet ski for 14 hours, which company offers the better deal? How do you know? (10 pts.)

MATH_0382_LINFUN5_LES_02.docx5. On the graph at right, graph a line to represent your company’s rental plan so that your jet ski rental business has the same break-even point as your competitors. Label the graph of your linear equation C. (10 pts.)

6. Choose 2 or 3 ordered pairs on the graph of your rental plan. Use a table to help you write an equation for your rental plan where x represents the number of hours a jet ski is rented and y represents the total cost. (10 pts.)

The equation for my company is:

_______________________________

7. What values of x and y simultaneously satisfy the three linear equations graphed above? Remember you can use your graphing calculator to verify this solution with the table of values for each equation. (10 pts.)

(______, ______)8. What does the intersection point mean in relationship to the entire scenario? (15 pts.)

9. If a customer wants to rent a jet ski for only 5 hours, which of the three businesses offers the best deal? How do you know? (10 pts.)

x Process y

MATH_0382_LINFUN5_LES_02.docx

Answer Key

1. (8, 130)

2. If a customer rents a jet ski for 8 hours, the cost of $130 will be the same at both rental companies.

3. Alfie’s Jet Ski Rental Company; At 2 hours, Alfie’s company charges a total of $70 while Bubba’s Jet Ski Rental Company charges a total of $85.

4. Bubba’s Jet Ski Rental Company; At 14 hours, Bubba’s company charges a total of $175 while Alfie’s Jet Ski Rental Company charges a total of $190.

5. Answers vary, but the graph must have (8, 130) as the intersection point with the other two companies. y = 16.25x is one possibility.

6. Answers vary, but the equation must have (8, 130) in its table of values. Use the graphing calculator to verify student answers.

7. (8, 130)

8. If a customer rents a jet ski for 8 hours, the cost of $130 will be the same at all three rental companies.

9. Answers vary. Use the graph to determine if the student’s linear equation is a better deal than Alfie’s Jet Ski Rental Company.