Lesson 16: Polygraph Linear Systems - MR....

Hart Interactive – Algebra 1 M2 Lesson 16 ALGEBRA I

Lesson 16: Polygraph Linear Systems

Opening Exploration

You will need: a Chrome book and the Desmos call code

1. Go to student.desmos.com and type in your class code: ___________________ to play Polygraph: Linear Systems. First, you’ll play a practice round again on the computer to learn how the game works. Next, you’ll be randomly paired up with someone from your class to play polygraph. One person chooses a graph and their partner asks yes/no questions in order to narrow a field of suspects down to one. Between rounds, you’ll answer questions about the strategies you used and the vocabulary that helped you find the correct graph.

2. What are some of the words you or your partner used to determine the correct graph?

3. Draw a picture to represent each type of solution you could get with a linear system. Then describe how you would get that type of solution.

No Solution One Solution Infinitely Many Solutions

Lesson 16: Polygraph Linear Systems Unit 4: Systems of Equations and Inequalities

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Exploration 1 – Evaluating Equations

4. The Math Club is having a Math Mystery Party. You are given the following clues about the location of the party.

A. Circle ALL the possible restaurants for the party based on the hint.

B. How did you decide whether or not an ordered pair was a solution to the equation?

The Math Mystery Party is at a restaurant on the line 4x – y = 10.

New clues are on the way!

Alamo is at (3, 4). Foster’s Freeze is at (5, 10).

Baja Fresh is at (2, 3). Grand Panda is at (0, -10).

Chi-Chi’s Pizza is at (-1, -14). The Habit is at (3, 2).

Dippin’ Dots is at (0, 0). Islands is at (6, 0).

El Taco Llama is at (1, -6). Jazmine Thai Barbeque is at (4, -1).


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5. A. Discover as many additional solutions to the equation 4𝑥𝑥 − 𝑦𝑦 = 10 as possible. Consider the best way to organize all the solutions you have found. Be prepared to share the strategies you used to find your solutions.

B. Now, find five more solutions where one or more variables are negative numbers or non-integer values. Be prepared to share the strategies you used to find your solutions.

C. How many ordered pairs (𝑥𝑥,𝑦𝑦) will be in the solution set of the equation 4𝑥𝑥 − 𝑦𝑦 = 10? D. Create a visual representation of the solution set

by plotting each solution as a point (𝑥𝑥,𝑦𝑦) in the coordinate plane.

E. Why does it make sense to represent the solution

to the equation 4𝑥𝑥 − 𝑦𝑦 = 10 as a line in the coordinate plane?


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6. A. What is the slope of the line in Exercise 5D? What is the y-intercept?

Slope: ________________________ y-intercept: ________________________

B. Rewrite the equation, 4x – y = 10, in slope-intercept form.

C. What is the slope and y-intercept based on the equation?

Slope: ________________________ y-intercept: ________________________

D. If your answers to Parts A and D did not agree, what error did you make?

7. Plot all the points of the circled restaurants from Exercise 4 onto the grid in Exercise 5D.

Exploration 2 – Vertical & Horizontal Lines

8. Read the clue below and determine the location of the Math Club Party.

The point (3, -6) has a horizontal line going through it.

The equation of this horizontal line is y = -6.

What is the equation of a vertical line going through that point?

Which other point of the five you found, does the vertical line go through?

That is the location of the Math Club Party!


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Exploration 3 – Continuous and Discrete Data Points 8. Richard thinks of two numbers. The only clue he gives is

“The sum of two numbers is 10.” What are the numbers Richard could be thinking of? A. Create an equation using two variables to represent

this situation. Be sure to explain the meaning of each variable.

B. List at least six solutions to the equation you created in part (A).

C. Create a graph that represents the solution set to the equation.

9. Gia had 10 songs in a playlist composed of songs from her two favorite artists, Beyonce and Jennifer Lopez. How many songs did she have by each one in the playlist? A. Create an equation using two variables to represent this

situation. Be sure to explain the meaning of each variable.

B. List at least three solutions to the equation you created in Part A.

C. Create a graph that represents the solution set to the equation.

10. Compare your solutions to Exercises 8 and 9. How are they alike? How are they different?


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Practice Graphing Vertical & Horizontal Lines

11. Graph each of the following equations.

A. y = -2 B. x = 3 C. y = 0

D. x = -2 E. y = 3 F. x = 0


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Lesson Summary One way to graph a linear equation is to choose values for one variable and then determine the value for the other variable. Use number sense to help you choose appropriate values.

Graph 1 3 72

y x+ = .

Vertical and Horizontal Lines

Graph Equation Characteristics

Vertical Lines

x = number

• No y-value

• Slope is undefined

Horizontal Lines

Y = number

• No x-value

• Slope is 0

Discrete Versus Continuous Data - Discrete data can be numeric -- like numbers of apples -- but it can also be categorical -- like red or blue, or male or female, or good or bad. Continuous data are not restricted to defined separate values, but can occupy any value over a continuous range. [source: StackExchange Cross Validated]

12

yChoose values for y where

will give an integer value.

x y 2 2 3 -4


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The number of solutions of a linear system of equations depends on the slope and y-intercept of the equations.

One Solution No Solutions Infinite Number of Solutions

Slopes are different Slopes are the same and y-intercepts are different

Slopes are the same and y-intercepts are the same.


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Homework Problem Set 1. Match each equation with its graph. Explain your reasoning.

A. 𝑦𝑦 = 5𝑥𝑥 − 6 B. 𝑥𝑥 + 2𝑦𝑦 = −12

C. 2𝑥𝑥 + 𝑦𝑦 = 4 D. 𝑦𝑦 = 3𝑥𝑥 − 6

E. 𝑥𝑥 = −𝑦𝑦 − 4

-10 -5 5 10

-10

-5

5

10

x

y

-10 -5 5 10

-10

-5

5

10

x

y 2 1

-10 -5 5 10

-10

-5

5

10

x

y 5

-10 -5 5 10

-10

-5

5

10

x

y

-10 -5 5 10

-10

-5

5

10

x

y 4 3


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Graph the solution set in the coordinate plane. Label at least two ordered pairs that are solutions on your graph.

2. 5𝑥𝑥 + 3𝑦𝑦 = 30 3. 𝑦𝑦 = −2𝑥𝑥 + 6


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Hart Interactive – Algebra M2 Lesson 16 ALGEBRA I

4. 2𝑥𝑥 − 3𝑦𝑦 = −6 5. 𝑦𝑦 = −2(𝑥𝑥 + 1)


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6. Mari and Lori are starting a business to make gourmet toffee. They gather the following information from another business about prices for different amounts of toffee. Which equation and which graph are most likely to model the price, 𝑝𝑝, for 𝑥𝑥 pounds of toffee? Justify your reasoning.

Graph 1 Graph 2

Pounds 𝑥𝑥

Price, 𝑝𝑝, for 𝑥𝑥

pounds

0.25 $3.60 0.81 $6.48

1 $7.20 1.44 $8.64

Equation A: 𝑝𝑝 = 5𝑥𝑥 + 2.2

Equation B: 𝑝𝑝 = 7.2√𝑥𝑥

7. Alexis realized that she could use linear equations to graph the first letter of her name. She came up with

the following three equations.

2 1

2 92

y xy xy

= += − +=

A. Graph her three equations and then highlight the “A”.

B. What should she add to her description so that only a part of each line is drawn?

1 2 3 4

5

10

15

20

0 x

y

1 2 3 4

5

10

15

20

0 x

y


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8. Treasure Hunt! Below is a map of Math City. Choose a location on the map and then write two or more equations that would help someone determine your location.

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Math City

Legend

A = Algebra Airport

B = Binomial Bank

C = Coefficient Café

D = Delta the Dog Doctor

E = Ellipse Eye Exams

F = Flipping for Fish Store

G = GCF Groceries

A

B

C

D

E

F

G


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Spiral Review – Rearranging Equations

9. Solve for 𝑚𝑚.

𝑡𝑡 = 𝑚𝑚𝑚𝑚𝑛𝑛

10. Solve for 𝑢𝑢. 1𝑢𝑢 = 1

𝑓𝑓

11. Solve for 𝑚𝑚. 𝐴𝐴 = 𝑚𝑚 + 3𝐴𝐴

12. Solve for ℎ. 𝑉𝑉 = 𝜋𝜋𝑟𝑟2ℎ

13. Solve for 𝑚𝑚. 𝑇𝑇 = 4𝑚𝑚 + 7

14. Solve for 𝑑𝑑.

𝐹𝐹 = 𝐺𝐺𝑚𝑚𝑛𝑛𝑑𝑑

15. Solve for 𝑦𝑦. 𝑎𝑎𝑥𝑥 + 𝑏𝑏𝑦𝑦 = 𝑐𝑐

16. Solve for ℎ.

𝐴𝐴 = 12ℎ(𝑏𝑏1 + 𝑏𝑏2)


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Lesson 16: Polygraph Linear Systems - MR....

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