Math 8 Q1 Module 9 - Mathematics Learning Material PDF

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MathematicsQuarter 1 – Module 9:Slope of a Line8Mathematics – Grade 8 Self-Learning Module (SLM) Quarter 1 – Module 9: Slope of a Line First Edition, 2020Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval o...

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8 Mathematics Quarter 1 – Module 9: Slope of a Line

Mathematics – Grade 8 Self-Learning Module (SLM) Quarter 1 – Module 9: Slope of a Line First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

Development Team of the Module Writer: Leslie A. Aban Editor: Margie T. Javier, Floramae A. Dullano Reviewers: Zaida N. Abiera, Mark R. Bubungan Illustrator: Leslie A. Aban Layout Artists: Jaydan March C. Panis Cover Art Designer: Reggie D. Galindez Management Team: Allan G. Farnazo, CESO IV – Regional Director Fiel Y. Almendra, CESO V – Assistant Regional Director Romelito G. Flores, CESO V – Schools Division Superintendent Mario M. Bermudez, CESO VI – Assist. Schools Division Superintendent Gilbert B. Barrera – Chief, CLMD Arturo D. Tingson Jr. – REPS, LRMS Peter Van C. Ang-ug – REPS, ADM Jade T. Palomar – REPS, Mathematics Juliet F. Lastimosa – CID Chief Sally A. Palomo – Division EPS In- Charge of LRMS Gregorio O. Ruales – Division ADM Coordinator Zaida N. Abiera – Division EPS, Mathematics

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Regional Center, Brgy. Carpenter Hill, City of Koronadal (083) 2288825/ (083) 2281893 [email protected]

8 Mathematics Quarter 1 – Module 9: Slope of a Line

Introductory Message For the facilitator: Welcome to the Mathematics 8 Self-Learning Module (SLM) on Slope of a Line! This module was collaboratively designed, developed and reviewed by educators both from public and private institutions to assist you, the teacher or facilitator in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling. This learning resource hopes to engage the learners into guided and independent learning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills while taking into consideration their needs and circumstances. In addition to the material in the main text, you will also see this box in the body of the module:

Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners.

As a facilitator you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module.

For the learner: Welcome to the Mathematics 8 Self-Learning Module (SLM) on Slope of a Line! The hand is one of the most symbolized part of the human body. It is often used to depict skill, action and purpose. Through our hands we may learn, create and accomplish. Hence, the hand in this learning resource signifies that you as a learner is capable and empowered to successfully achieve the relevant competencies and skills at your own pace and time. Your academic success lies in your own hands! This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner. This module has the following parts and corresponding icons: What I Need to Know

This will give you an idea of the skills or competencies you are expected to learn in the module.

What I Know

This part includes an activity that aims to check what you already know about the lesson to take. If you get all the answers correct (100% ), you may decide to skip this module.

What’s In

This is a brief drill or review to help you link the current lesson with the previous one.

What’s New

In this portion, the new lesson will be introduced to you in various ways such as a story, a song, a poem, a problem opener, an activity or a situation.

What is It

This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills.

What’s More

This comprises activities for independent practice to solidify your understanding and skills of the topic. You may check the answers to the exercises using the Answer Key at the end of the module.

What I Have Learned

This includes questions or blank sentence/paragraph to be filled in to process what you learned from the lesson.

What I Can Do

This section provides an activity which will help you transfer your new knowledge or skill into real life situations or concerns.

Assessment

This is a task which aims to evaluate your level of mastery in achieving the learning competency.

Additional Activities

In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. This also tends retention of learned concepts.

Answer Key

This contains answers to all activities in the module.

At the end of this module you will also find:

References

This is a list of all sources used in developing this module.

The following are some reminders in using this module: 1. Use the module with care. Do not put unnecessary mark/s on any part of the module. Use a separate sheet of paper in answering the exercises. 2. Don’t forget to answer What I Know before moving on to the other activities included in the module. 3. Read the instruction carefully before doing each task. 4. Observe honesty and integrity in doing the tasks and checking your answers. 5. Finish the task at hand before proceeding to the next. 6. Return this module to your teacher/facilitator once you are through with it. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it!

What I Need to Know This module was designed and written with you in mind. It is here to help you master the Slope of a Line. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. In this module, you will be able to:  illustrate and find slope of a line given two points, equation and graph. Specifically, you are expected to: 1. illustrate a slope of a line; and 2. find the slope of a line given two points, equation and graph.

What I Know Let us check your prior knowledge about linear equation in two variables and slope of a line by answering the questions below. Direction: Encircle the letter of the correct answer. 1. Which of a. b. c. d.

the following words is essential in finding the missing coordinate? Point Line Slope Graph

2. What do you call the equation of a line in the form of y = mx + b? a. Standard Form b. Slope – intercept Form c. Point – slope Form d. General Form For item numbers 3 – 6, refer to the figure at the right. 3. Which line has a positive slope? a. s b. m z c. z d. q

m

s 4. Which line has a negative slope? a. s b. m c. z d. q 5. Which line has an undefined slope? a. s b. m c. z d. q 6. Which line has a zero slope? a. s b. m c. z d. q

q

7. Find the slope of the line that passes through the points (0, 8) and (8, 0). a. 1 b. 0 c. -1 d. undefined 8. What is the slope of the roof indicated at the right? 2 a. b. c. d.

3 2

5 1 5 5 2

9. Determine the slope of the line 2y – 3x = 7. 2 A. B.

3 3 2

C.

−

D.

−

2 3 3 2

10. Find the slope given the graph at the right. A. 1 B. 2 C. -1 D. -2 11. Find the value of x so that the slope of the line containing the points P(2, – 3) 5 and Q(5, x) is 3. A. B. C. D.

5 3 2 1

12. Find the slope of the given points (-1, 1) and (0, 2). A. 1 B. 0 C. -1 D. Undefined 13. Find the slope of the line 2y – 3 = 8x + 7. A. 4 B. -4 C. 5 D. -5

14. Find the slope given the graph below. 1 A. 6 1

B.

−

C. D.

6 -6

6

15. Find the value of 𝑥 such that the slope of the line containing the points, A (9, x) and B (6, -2) is -1. A. 1 B. 0 C. -1 D. -2

Lesson

9

Slope of a Line

Do you know that there are a lot of real-life experiences that you can apply using slope of a line? Have you ever asked yourself how the steepness of the mountain affects the speed of a mountaineer? How can the value of a quantity given the rate of change be predicted? Roads, hills, stairways, or a tilted chair-back are all application of slope.

What’s In I know you already have the knowledge of the basic concepts on Rectangular Coordinate Plane. To review our previous lesson, try to answer the Activity 1 below. Activity 1: What is your point? Direction: Answer the following questions correctly based on the given figures at the right. Write your answers on the space provided. 1. Name the point that coordinates of: a. (5, 0) __________ b. (-5, 4) __________ c. (-2, -4) __________

has

the

2. Write the coordinates of each point. a. H __________ b. E __________ c. G __________ d. F __________ 3. In what quadrant/axis is each point located? a. C __________ b. B __________ c. L __________

L

I

N

What’s New

Have you tried riding a bicycle? Is there any difficulty you have encountered in riding one? Here is an activity in which you can relate. Activity 2: Pedal Up, Speed On Due to COVID-19 pandemic, families are mandated to stay at home. To buy his family’s essential needs, Laurenz is using his bike. He goes up two different hills in order for him to reach the nearest convenience store.

Hill #1

Hill #2

Hill #3

Hill #4

Probing questions: 1. Which hill will be harder for Laurenz to pedal up? ___________________________________________________________________________ ___________________________________________________________________________

2. Which hill will Laurenz gain more speed if he is going down two hills? ___________________________________________________________________________ ___________________________________________________________________________

Roads, hills, stairways, or a tilted chair-back are all application of our next lesson.

What is It Below are important terminologies, notations and symbols that you must learn and remember about slope of a line in this module. Definition of Slope The steepness of a line can be measured by the ratio of the change in vertical distance, 𝑦2 − 𝑦1 to the change in the horizontal 𝑥2 − 𝑥1 , between any two points on the line. This numerical value called the slope of a line represented by m.

𝒔𝒍𝒐𝒑𝒆 (𝒎) =

𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒉𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆

𝒎=

𝒚𝟐 − 𝒚𝟏

𝒓𝒊𝒔𝒆

𝒓𝒖𝒏 𝒚𝟐 − 𝒚𝟏 𝒎= 𝒙𝟐 − 𝒙𝟏

𝒙𝟐 − 𝒙𝟏

Finding the Slope Given Two Points We can get the slope of a line without looking at the graph. In order to get the rise over run, subtract the y – coordinates to get the “rise”, and subtract the x coordinates of two given points to get the “run”. Examples: 1. Find the slope of the line passing through the points (2, 3) and (4, -5).

𝑚= = =

𝑦2 − 𝑦1 𝑥2 − 𝑥1 −5 − 3 4−2 −8 2

𝒎 = −𝟒

Step 1: Use the formula. Please note that regardless which value is your y 1y2 and x1x2, you will still arrive with the same answer.

𝑚=

Step 2: Substitute the values of the coordinates to the formula.

=

Step 3: Simplify.

=

Step 4: Reduce to lowest terms, if possible.

𝑦2 − 𝑦1 𝑥2 − 𝑥1

3 − (−5) 2−4 8 −2

𝒎 = −𝟒

2. Find the slope of the line passing through the points (0, 1) and (1, 0).

𝑚=

𝑦2 − 𝑦1 𝑥2 − 𝑥1

=

𝑚=

1−0 0−1

=

1 an Equation Finding the Slope Given =

𝑦2 − 𝑦1 𝑥2 − 𝑥1 0−1 1−0

=

−1

−1 1

If given an equation, there are two ways to find the slope. First, we identify the 𝒎 = −𝟏 𝒎 = −𝟏 slope given an equation by rewriting the given equation to the slope – intercept form, y = mx + b, where m is the slope. Example: 1. Identify the slope of the equation, 3𝑥 − 2𝑦 − 10 = 0.

3𝑥 − 2𝑦 = 10 −2𝑦 = −3𝑥 + 10 −2𝑦 −3𝑥 10 = + −2 −2 −2 𝟑 𝒚= 𝒙−𝟓 𝟐

𝟑

Therefore, 𝒎 = . 𝟐

Another method to find the slope if given an equation of a line is to write the 𝑨

equation in the form Ax + By = C, where the slope (m) = − . 𝑩

Examples: 2. Identify the slope of the equation 8𝑥 + 11𝑦 = 7. A=8

B = 11

Identify A and B. 𝐴

𝟖

Substitute to 𝑚 = − 𝐵, then simplify.

𝒎 = − 𝟏𝟏.

3. Identify the slope of the equation -6x = 2y + 14. Method 1: −2𝑦 = 6𝑥 + 14

−2𝑦 −2

=

6𝑥 + 14 −2

𝑦 = −3𝑥 − 7 Therefore, 𝒎 = −𝟑.

Rewrite in the form y = mx + b. Add – 2y and 6x to both sides of the equation. Addition Property of Equality (APE) Divide both sides of the equation by -2. (DPE) Simplify.

Method 2: − 6𝑥 − 2𝑦 = 14

𝐴 = −6, 𝐵 = −2 −6

𝑚 = − −2

Rewrite in the Ax + By = C. Add – 2y to both sides of the equation. Addition Property of Equality (APE) Identify A and B. 𝐴

Substitute to 𝑚 = − , then simplify. 𝐵

Therefore, 𝒎 = −𝟑.

Finding the Slope Given Graphs To find the slope of a line from its graph, select any two arbitrary points on the line. Remem Remember! ber! The line never changes, so any pair of points you will get in the line will result to the same slope. Remem Remember! ber! After choosing the two points of the line, calculate the rise and run. Remem Remember! ber! You can do “rise” (upward or downward), and you can only do “run” to the right. If the point rise upward, the sign will always be positive, otherwise, it will be negative.

Examples: Find the slope of a line. Solution:

1.

Step 1: Select any two points on the line. A (-2, 0)

B (0, -3)

Step 2: Calculate the rise and run.

A

3 “rise” downward = -3 2 “run” to the right = 2

B

Step 3: Use the formula in getting the slope. 𝒎=

𝒓𝒊𝒔𝒆 𝒓𝒖𝒏

=

−𝟑 𝟐

=−

𝟑 𝟐

Solution: 2.

Step 1: Select any two points on the line. A (0, 0)

B

B (-4, 1)

Step 2: Calculate the rise and run.

A

1 “rise” downward = -1 4 “run” to the right = 4 Step 3: Use the formula in getting the slope. 𝒎=

𝒓𝒊𝒔𝒆 𝒓𝒖𝒏

=−

𝟏 𝟒

Given these examples, what have you observed between the direction of the line and its slope? ___________________________________________________________________________ ___________________________________________________________________________ Let us have more examples.

Solution: 3.

A

B

Step 1: Select any two points on the line. A (-3, 3)

B (3, 3)

Step 2: Calculate the rise and run. No upward/downward = 0

“rise”

6 “run” to the right = 6 Step 3: Use the formula in getting the slope. 𝒎=

𝒓𝒊𝒔𝒆 𝒓𝒖𝒏

=

𝟎 𝟔

=𝟎

Solution:

A

4.

Step 1: Select any two points on the line. A (2, 3)

B (2, -2)

Step 2: Calculate the rise and run. 5 upward/downward = 5

B

“rise”

No “run” to the right = 0 Step 3: Use the formula in getting the slope. 𝒎=

𝒓𝒊𝒔𝒆 𝒓𝒖𝒏

=

𝟓 𝟎

= 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅

What have you observed with the slope of the line if it is parallel to the x – axis? __________________________________________________________________________________ __________________________________________________________________________________

What about if it is parallel to y – axis? __________________________________________________________________________________ __________________________________________________________________________________

Please take note of the different slope trends below for you to master the concept more. Slope Trends

An increasing line from left to right defines a positive slope.

A decreasing line from left to right defines a negative slope.

A horizontal line defines a zero slope.

A vertical line defines an undefined slope.

What’s More

It is now the perfect time to practice what you have learned. The next activity will lead you to the correct answers if you follow the provided step by step process. Activity 3: Lead Me The Steps! Direction: Use the guided steps provided for you to solve the following problems. Write your answers on the space provided. A. Find the slope of the line passing through (2, 3) and (4, -5).

𝑚=

𝑦2 − 𝑦1 𝑥2 − 𝑥1

Step 1: Use the formula.

=

Step 2: Substitute the values of the coordinates to the formula.

=

Step 3: Simplify.

𝒎 = ________

Step 4: Reduce to lowest terms, if possible.

B. Find the slope of this equation 3x + 5y = 10. Method 1: _____________________

Rewrite in the form y = mx + b. Add –3x to both sides of the equation. Addition Property of Equality (APE)

_____________________

Divide both sides of the equation by 5. (DPE)

_____________________

Simplify.

Therefore, 𝒎 = _________.

Method 2: _____________________

Is it written in the form Ax + By = C? Yes or No?

_____________________

Identify A and B.

_____________________

Substitute and simplify.

Therefore, 𝒎 = _________.

C. Find the slope of the following graphs given at the right. 1. Blue Line Step 1: Select two arbitrary points. ______________________ Step 2: Calculate the rise and run. ______________________ ______________________ 𝑟𝑖𝑠𝑒

Step 3: Use the formula 𝑚 = 𝑟𝑢𝑛 , Then, substitute the values of rise and run. ______________________

2. Red Line Step 1: Select two arbitrary points. ______________________ Step 2: Calculate the rise and run. ______________________ __________...