This is the Mathematics Grade 6 Quarter 4 Module 7 self-learning module, “Making Lists and Diagrams of Outcomes and Telling the Number of Favorable Outcomes and Chances.” It was prepared by DepEd Region VI for home-based learning under the Alternative Delivery Mode (First Edition, 2020). After working through it, learners can find every possible outcome of a situation, such as all the outfits or snack orders you can make, using a list, a tree diagram, or a table.
What learners will learn
After going through this module, learners are expected to:
- make listings and diagrams of outcomes and tell the number of favorable outcomes and chances using these listings and diagrams.
The module builds on earlier lessons, where learners studied probability by doing experiments and recording outcomes.
Lessons in this module
The module has one lesson, Lesson 1: Making Lists and Diagrams of Outcomes and Telling the Number of Favorable Outcomes and Chances. It follows the usual parts of a DepEd self-learning module:
- What I Know: a five-item pretest, with problems about balls and chips in bags, outfits, tossing two coins, and car models and colors.
- What’s In: a review problem about ordering a burger and a drink from a snack house menu, solved by listing and by a tree diagram.
- What’s New: Emilia’s problem of matching three blouses and two skirts for a party.
- What is It: the main discussion of sample spaces and three ways to find them.
- What’s More: a practice problem on James’s snack order.
- What I Have Learned: a short summary of the key ideas.
- What I Can Do: three problems on breakfast choices, rolling a pair of dice, and choosing a car.
- Assessment: five multiple-choice items and one ice cream problem.
- Additional Activities: one problem on Joy’s pants and shirts.
Key ideas: sample spaces and how to find them
In probability, every experiment has a set of possible outcomes called a sample space. A sample space is a list of all the possible outcomes in an activity or experiment. The module teaches three ways to find one.
1. Listing outcomes
Listing is a systematic way of writing the sample space: you simply write down every possible outcome. Counting all the outcomes lets you work out the probability of any particular event.
Example from the module: Roy chooses one drink (tea or coffee) and one snack (brownie, muffin, or pastry). The list is:
- tea, brownie
- tea, muffin
- tea, pastry
- coffee, brownie
- coffee, muffin
- coffee, pastry
That gives 6 possible combinations. In another example, Ben picks two of four fruits (mango, guava, banana, apple), which also gives 6 combinations.
2. Tree diagram
A tree diagram is a drawing with line segments that shows the different possible paths to each outcome. Each first choice branches out into every second choice, and the ends of the branches are the outcomes.
Example from the module: With 2 food choices (burger, siopao) and 4 fruit choices (melon, guava, banana, papaya), each food branches into 4 fruits. This gives 8 possible outcomes. Jemar, with 5 shirts and 2 pairs of pants, gets 10 possible combinations the same way.
3. Table or grid of outcomes
A table or grid of outcomes is a table where the first row and first column show the items to be paired. Each cell holds one outcome.
Example from the module: A boy is chosen from Nory, Kelvin, and Raffy, and a girl from Lucy, Jean, Mary, Nancy, and Millene.
| Boys / Girls | Lucy | Jean | Mary | Nancy | Millene |
|---|---|---|---|---|---|
| Nory | Nory, Lucy | Nory, Jean | Nory, Mary | Nory, Nancy | Nory, Millene |
| Kelvin | Kelvin, Lucy | Kelvin, Jean | Kelvin, Mary | Kelvin, Nancy | Kelvin, Millene |
| Raffy | Raffy, Lucy | Raffy, Jean | Raffy, Mary | Raffy, Nancy | Raffy, Millene |
The grid shows 15 possible outcomes. Using a grid for Emilia’s 3 blouses and 2 skirts gives 6 possible outfits.
A pattern to notice
In every example, the number of outcomes equals the number of first choices times the number of second choices: 2 drinks and 3 snacks give 6, 2 foods and 4 fruits give 8, and 3 boys and 5 girls give 15. The lists, diagrams, and tables let learners check this by seeing every outcome.
How to use this module
- Learners: Start with the pretest (What I Know) to see what you already know. Then go through the parts in order, from What’s In to Additional Activities. Use a separate sheet of paper for all answers and do not write on the module. Answer keys are included for every activity and test; check your work only after you answer, and be honest.
- Parents: Help your child read each problem slowly and draw the list, tree, or table on paper. If your child has trouble, the module encourages asking the teacher or facilitator.
- Teachers: The pretest and Assessment show how well learners can find sample spaces. The What’s More, What I Can Do, and Additional Activities problems let learners choose any of the three methods, so their solutions can be compared.
Frequently asked questions
What grade and quarter is this module for?
It is Mathematics Grade 6, Quarter 4, Module 7. It was published by DepEd Region VI as a First Edition, 2020 self-learning module.
What is a sample space?
A sample space is a list of all the possible outcomes in an activity or experiment. The module shows how to find it by listing, by a tree diagram, or by a table or grid.
Does the module have an answer key?
Yes. The module includes answers for the pretest, What’s In, What’s More, What I Can Do, the Assessment, and the Additional Activities.
How many outcomes are there when you roll a pair of dice?
According to the module’s answer key, there are 36 possible combinations when a pair of dice is rolled.
What is the difference between a tree diagram and a table of outcomes?
A tree diagram uses line segments that branch out to show each possible path to an outcome. A table or grid puts one set of choices in the first row and the other in the first column, with each cell showing one outcome.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 4 – Module 7: Making Lists and Diagrams of Outcomes and Telling the Number of Favorable Outcomes and Chances
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[illustration: students studying outdoors and at home, a teacher helping learners, a child using a lantern, and a globe on a desk]
CO_Q4_Mathematics 6_Module 7
Mathematics – Grade 6 Alternative Delivery Mode Quarter 4 – Module 7: Making Lists and Diagrams of Outcomes and Telling the Number of Favorable Outcomes and Chances 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 book 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.
Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
Development Team of the Module
Author: Estela R. Khey
Editors: Mae Joy M. Tan, Susana L. Lutero, Nora B. Rebadomia, Mary Jane O. Amante, Ma. Pinky C. Arquio, Susan T. Dela Torre
Reviewers: Ivy Joy A. Torres, Ma. Theresa L. Tabotabo, Jem Rymon S. Chien, Mae Joy M. Tan
Illustrator: Eldiardo E. Dela Peña
Layout Artists: Eldiardo E. Dela Peña, Peter A. Alavanza
Management Team: Ramir B. Uytico, Pedro T. Escobarte, Jr., Elena P. Gonzaga, Donald T. Genine, Adonis A. Mosquera, Clarissa G. Zamora, Fevi S. Fanco, Ivy Joy A. Torres, Jason R. Alpay, Mae Joy M. Tan, Jem Rymon S. Chien, Ethel S. Gali
Printed in the Philippines by ________________________
Department of Education – Region VI
Office Address: Duran Street, Iloilo City Telefax: (033)336-2816, (033)509-7653 E-mail Address: region6@deped.gov.ph
6
Mathematics
Quarter 4 – Module 7:
Making Lists and Diagrams of Outcomes and Telling the Number of Favorable Outcomes and Chances
Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.
In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.
Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.
Thank you.
What I Need to Know
This module was designed and written with you in mind. It is here to help you master the skills in making listings and diagrams of outcomes and telling the number of favorable outcomes and chances using these listings and diagrams. The scope of this module allows you to use it in many different learning situations. The language used recognizes your diverse vocabulary level. The lessons are arranged to follow the standard sequence of your course. But the order in which you read them can be changed to match with the textbook you are now using.
After going through this module, you are expected to:
- make listings and diagrams of outcomes and tell the number of favorable outcomes and chances using these listings and diagrams.
What I Know
Directions: Read and answer the following problems. Use appropriate methods like listing, tree diagram and table/grid in solving each item. Write your answers on your answer sheet.
There are four balls (red, blue, orange, green) inside the bag and three chips (yellow, black, white) in another bag. How many possible outcomes will there be if a ball and a chip are drawn from the bags?
Annie has 4 shirts and 2 pairs of pants. How many different outfits can she possibly wear?
Two coins are tossed at the same time. Find all the possible ways the coins can land.
A car manufacturer makes 3 models of cars; Innova, Revo and Corolla. These cars are all available in a choice of 4 colors: red, green, blue, and orange. How many different cars are available?
The table below shows the tops and pants Karen has in her closet. How many possible combinations of outfits Karen can wear?
| Tops | Pants |
|---|---|
| T-Shirt Blouse | Black Blue Red Green |
Lesson 1: Making Lists and Diagrams of Outcomes and Telling the Number of Favorable Outcomes and Chances
In the previous lessons you have learned about probability by performing experiments and recording outcomes. This time we will focus on making listings and diagrams of outcomes and tell the number of favorable outcomes and chances using these listings and diagrams.
What’s In
Directions: Solve the problem using different ways. Write your answers on your answer sheet.
This is a menu posted on the wall of a snack house.
| Burgers | Beverages |
|---|---|
| Hamburger Cheese Burger | Hot Chocolate Iced Tea Orange Juice |
How many different ways can you order for a snack? a. Solve through the listing method b. Solve through a tree diagram
What’s New
Emilia wants to attend a birthday party. She had these outfits in her closet. How many possible combinations of outfits can she wear?
[illustration: three blouses and two skirts]
How will you solve this problem?
What is It
In dealing with probability, every experiment has a set of possible outcomes called a sample space.
A sample space is a list of all the possible outcomes in an activity or experiment. We can identify sample spaces in a number of ways including lists, tree diagrams and tables or grids.
A. Listing Outcomes
Listing Outcomes – is one of the systematic processes of writing the sample spaces. It is simple listing every possible outcome. Listing or counting all the possible outcomes enables us to calculate the probability of any particular even occurring.
Examples:
- Ben has four kinds of fruits namely mango, guava, banana and apple in his fruit basket. He wants to eat two kinds of fruits. What would be the possible combinations of fruits he has?
Possible Combinations:
- mango, guava
- mango, banana
- mango, apple
- guava, banana
- guava, apple
- banana, apple
The list shows that there are 6 possible combinations of fruits.
- Roy goes to a coffee shop. He chooses one drink and one snack. The table below shows what the coffee shop offers.
| Drink | Snack |
|---|---|
| Tea Coffee | Brownie Muffin Pastry |
Write down all the possible combinations of drink and snack that Roy can order.
- tea, brownie
- tea, muffin
- tea, pastry
- coffee, brownie
- coffee, muffin
- coffee, pastry
The list shows that there are 6 possible combinations.
B. Tree Diagram is a tool or drawing with “line segments” to provide visual representation of the different possible “paths” for the outcomes.
Examples:
- How many snacks combinations can you make with one food choice and one fruit choice?
| FOOD CHOICES | FRUIT CHOICES |
|---|---|
| Burger Siopao | Melon Guava Banana Papaya |
Solution:
[chart: tree diagram; “Outcomes” (printed above the right-hand column)]
- Burger
- Melon: Burger, Melon
- Guava: Burger, Guava
- Banana: Burger, Banana
- Papaya: Burger, Papaya
- Siopao
- Melon: Siopao, Melon
- Guava: Siopao, Guava
- Banana: Siopao, Banana
- Papaya: Siopao, Papaya
Based on the tree diagram, there are 8 possible outcomes.
- Jemar is going to a party. He has 5 new shirts and 2 new pairs of pants in his cabinet. How many possible combinations of shirts and pants can he have?
Solution:
[chart: tree diagram; “Outcomes” (printed above the right-hand column)]
- Shirt 1
- Pants A: Shirt 1, Pants A
- Pants B: Shirt 1, Pants B
- Shirt 2
- Pants A: Shirt 2, Pants A
- Pants B: Shirt 2, Pants B
- Shirt 3
- Pants A: Shirt 3, Pants A
- Pants B: Shirt 3, Pants B
- Shirt 4
- Pants A: Shirt 4, Pants A
- Pants B: Shirt 4, Pants B
- Shirt 5
- Pants A: Shirt 5, Pants A
- Pants B: Shirt 5, Pants B
Based on the tree diagram, there are 10 possible combinations of shirts and pants.
C. Table or grid of outcomes is a table where the first row and first column represent the elements that need to be combined.
Examples:
- A boy and a girl are to be randomly chosen to represent their class. The boy is chosen from Nory, Kelvin, and Raffy. The girl is chosen from Lucy, Jean, Mary, Nancy, and Millene. Find the sample spaces for this problem.
Solution:
| Girls / Boys | Lucy | Jean | Mary | Nancy | Millene |
|---|---|---|---|---|---|
| Nory | Nory, Lucy | Nory, Jean | Nory, Mary | Nory, Nancy | Nory, Millene |
| Kelvin | Kelvin, Lucy | Kelvin, Jean | Kelvin, Mary | Kelvin, Nancy | Kelvin, Millene |
| Raffy | Raffy, Lucy | Raffy, Jean | Raffy, Mary | Raffy, Nancy | Raffy, Millene |
[transcriber’s note: the top-left cell is split diagonally; “Girls” is printed at the top and “Boys” at the bottom of that cell]
Based on the table or grid above, there are 15 possible outcomes.
- Now, let us try to answer the problem above.
Emilia wants to attend a birthday party. She has these outfits in her closet. How many possible combinations of outfits can she wear?
[illustration: three blouses and two skirts in Emilia’s closet]
Solution:
[chart: table or grid of outcomes; unnumbered solution grid]
- Column headers, printed as pictures: skirt 1, skirt 2
- Row headers, printed as pictures: blouse 1, blouse 2, blouse 3
- Labels printed in the top-left cell: “Skirt” (upper right) and “Blouse” (lower left)
- Each cell contains a picture of the blouse and skirt pair for that row and column.
[illustration: grid of blouse and skirt pairs]
Based on the table or grid above, there are 6 possible combinations of outfits that Emilia can wear.
What’s More
Directions: Read and answer the problem below. Write your answer on your answer sheet.
James wants to order snacks. Here’s the menu:
Burgers
- Mushroom Burger
- Bean Burger
- Cheese Burger
Beverages
- Green tea
- Buko juice
- Lemon water
- Hot chocolate
- Ginger tea
How many ways can he order snacks? Make a list of the possible outcomes using any of the methods discussed.
What I Have Learned
A sample space is a list of all the possible outcomes in an activity or experiment. It is important to have the knowledge in identifying all of these possible outcomes. We can identify sample spaces in a number of ways including lists, tree diagrams and tables or grids.
Listing Outcomes is one of the systematic processes of writing the sample spaces. It is simply listing every possible outcome. Listing or counting all the possible outcomes enables us to calculate the probability of any particular event occurring.
Tree Diagram is a tool or drawing with “line segments” to provide visual representation of the different possible “paths” for the outcomes.
Table or grid of outcomes is a table where the first row and first column represent the elements that need to be paired to identify all possible outcomes.
What I Can Do
Directions: Read and solve the following problems. Write your answer in your answer sheet.
Gethel wants to eat her breakfast. She chooses between bread and cereal, and four choices of what to drink: coffee, kalamansi juice, tea, or water. List the possible combinations of breakfast Gethel can make.
A pair of dice is rolled. How many possible combinations are there in rolling a pair of dice? Use appropriate method in finding all the possible combinations.
Kezia wants to buy a car. She can buy a Toyota or Ford. The cars available are in gray or black colors. How many options she can choose from? Use appropriate method in finding possible options.
Assessment
A. Read and answer the following problems. Then, write the letter of the correct answer on your answer sheet.
A certain type of bag comes in white or black and in a small or large size. How many combinations are possible?
- A. 2
- B. 3
- C. 4
- D. 6
Gythro picks a number from 1 to 4 and chooses the color red, green, or yellow. What is the total number of outcomes?
- A. 6
- B. 9
- C. 12
- D. 24
A certain brand of shoes comes in five sizes and six colors. Find the number of possible outcomes.
- A. 9
- B. 18
- C. 24
- D. 30
Zina has a choice of 2 colors of pants and 3 colors of shirts. How many different outfits can she wear?
- A. 4
- B. 6
- C. 7
- D. 8
Which shows the sample space for flipping two coins?
- A. H, T
- B. HH, TT
- C. HT, HH
- D. HH, HT, TH, TT
B. Read and answer the problem below. Write your answers on your answer sheet.
A convenient store sells ice cream in cup or in cone in either vanilla or chocolate flavor. Each flavor has three choices of toppings namely: nuts, strawberries and cherries.
How many combinations of ice cream would there be? Solve using listing or diagram.
Additional Activities
Directions: Read and solve the problem below. Write your answer on your answer sheet.
Joy has a black and green pants with white, yellow, blue and red shirts in her cabinet. How many different choices of pants and shirts can she wear?
Write the possible outcomes in either listing, tree diagram and tabular form.
Answer Key
[transcriber’s note: this page is printed upside down; its text is transcribed upright, in the order it reads on the page]
What I Know
- 12
- 8
- 4
- 12
- 8
What’s More
- (mushroom burger, green tea)
- (mushroom burger, buko juice)
- (mushroom burger, lemon water)
- (mushroom burger, hot chocolate)
- (mushroom burger, ginger tea)
- (bean burger, green tea)
- (bean burger, buko juice)
- (bean burger, lemon water)
- (bean burger, hot chocolate)
- (bean burger, ginger tea)
- (cheese burger, green tea)
- (cheese burger, buko juice)
- (cheese burger, lemon water)
- (cheese burger, hot chocolate)
- (cheese burger, ginger tea)
Possible outcomes: 15
What I Can Do
- (bread, coffee)
- (bread, kalamansi juice)
- (bread, tea)
- (bread, water)
- (cereal, coffee)
- (cereal, kalamansi juice)
- (cereal, tea)
- (cereal, water)
- 36
- 4
What’s In
a. Listing Method
- Hamburger, Hot Chocolate
- Hamburger, Iced Tea
- Hamburger, Orange Juice
- Cheeseburger, Hot Chocolate
- Cheeseburger, Iced Tea
- Cheeseburger, Orange Juice
b. Tree Diagram
[chart: tree diagram; “Outcomes” (printed above the right-hand column)]
- Hamburger
- Hot Chocolate: Hamburger, Hot Chocolate
- Iced Tea: Hamburger, Iced Tea
- Orange Juice: Hamburger, Orange Juice
- Cheeseburger
- Hot Chocolate: Cheeseburger, Hot Chocolate
- Iced Tea: Cheeseburger, Iced Tea
- Orange Juice: Cheeseburger, Orange Juice
Possible outcomes: 6
[transcriber’s note: the drink in the “What’s In” problem is read as “Iced Tea”; this word is hard to read in the printed image]
Assessment
A.
- C
- C
- D
- B
- D
B.
By Listing Method
- (Cup, Vanilla, Nuts)
- (Cup, Vanilla, Strawberries)
- (Cup, Vanilla, Cherries)
- (Cup, Chocolate, Nuts)
- (Cup, Chocolate, Strawberries)
- (Cup, Chocolate, Cherries)
- (Cone, Vanilla, Nuts)
- (Cone, Vanilla, Strawberries)
- (Cone, Vanilla, Cherries)
- (Cone, Chocolate, Nuts)
- (Cone, Chocolate, Strawberries)
- (Cone, Chocolate, Cherries)
By Tree Diagram
[chart: tree diagram; “By Tree Diagram”]
- Cup
- Vanilla
- Nuts: Cup, Vanilla, Nuts
- Strawberries: Cup, Vanilla, Strawberries
- Cherries: Cup, Vanilla, Cherries
- Chocolate
- Nuts: Cup, Chocolate, Nuts
- Strawberries: Cup, Chocolate, Strawberries
- Cherries: Cup, Chocolate, Cherries
- Vanilla
- Cone
- Vanilla
- Nuts: Cone, Vanilla, Nuts
- Strawberries: Cone, Vanilla, Strawberries
- Cherries: Cone, Vanilla, Cherries
- Chocolate
- Nuts: Cone, Chocolate, Nuts
- Strawberries: Cone, Chocolate, Strawberries
- Cherries: Cone, Chocolate, Cherries
- Vanilla
[transcriber’s note: this page is printed upside down in the source image; the text below is transcribed in reading order, and the page number and running footer are omitted]
Additional Activities
Answer: 8
A. Listing Outcomes
(black pants, white shirt) (black pants, yellow shirt) (black pants, blue shirt) (black pants, red shirt) (green pants, white shirt) (green pants, yellow shirt) (green pants, blue shirt) (green pants, red shirt)
B. Tree Diagram
[chart: tree diagram; B. Tree Diagram]
- Black Pants
- White shirt — Black Pants, White shirt
- Yellow shirt — Black Pants, Yellow shirt
- Blue shirt — Black Pants, Blue shirt
- Red shirt — Black Pants, Red shirt
- Green Pants
- White shirt — Green Pants, White shirt
- Yellow shirt — Green Pants, Yellow shirt
- Blue shirt — Green Pants, Blue shirt
- Red shirt — Green Pants, Red shirt
C. Table / Grid
| Shirt \ Pants | white shirt | yellow shirt | blue shirt | red shirt |
|---|---|---|---|---|
| Black pants | Black pants, white shirt | Black pants, yellow shirt | Black pants, blue shirt | Black pants, red shirt |
| Green pants | Green pants, white shirt | Green pants, yellow shirt | Green pants, blue shirt | Green pants, red shirt |
References
Most Essential Learning Competencies (MELC) in Mathematics 6 Burgos, J. et al. (2016) 21st Century Mathletes. Philippines: Book Media Press Inc.
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