This is the Mathematics 7 Quarter 1 Module 8 self-learning module (SLM) from DepEd SOCCSKSARGEN Region, First Edition, 2020. It is for Grade 7 learners who already know principal roots and irrational numbers from the previous module. After it, they can estimate the square root of a whole number that is not a perfect square to the nearest hundredth and plot square roots on a number line.
What learners will learn
The module lists two objectives. Learners should be able to:
- Estimate the square root of a whole number to the nearest hundredth.
- Plot irrational numbers (square roots) on a number line.
Lessons in this module
- Lesson 1: Estimating the square root of a whole number to the nearest hundredth. Learners use nearby perfect squares and repeated squaring of decimals to find values such as √7 ≈ 2.65. They then apply this to word problems about square lots, rooms, and murals.
- Lesson 2: Plotting irrational numbers on a number line. Learners find the two consecutive whole numbers a square root lies between, decide which one it is closer to, and mark it on a number line.
Key idea 1: Estimating a square root to the nearest hundredth
The square root of a perfect square is a whole number. The square root of any other whole number has digits after the decimal point. To estimate it, the module uses three steps.
- Find the closest perfect squares below and above the number. For √7, these are 4 and 9, so 2 < √7 < 3. Since 7 is closer to 9, √7 is closer to 3.
- Guess to one decimal place and test by squaring. (2.6)² = 6.76 and (2.7)² = 7.29, so √7 lies between 2.6 and 2.7.
- Narrow down to the hundredths. Square 2.61, 2.62, and so on. (2.64)² = 6.9696 and (2.65)² = 7.0225. Since 7 is closer to 7.0225, √7 ≈ 2.65.
The module works through five examples:
| Square root | Lies between | Estimate to the nearest hundredth |
|---|---|---|
| √7 | 2 and 3 | 2.65 |
| √44 | 6 and 7 | 6.63 |
| √125 | 11 and 12 | 11.18 |
| √263 | 16 and 17 | 16.22 |
| √73 | 8 and 9 | 8.54 |
The lesson opens with a real-life problem. Arlo and JJ help design a baseball infield that should have an area of 4,900 square feet, but the distance between home plate and first base measures 67 feet. Using square roots, the side should be 70 feet, so the field is 3 feet short between each base.
Key idea 2: Plotting square roots on a number line
To plot an irrational number such as a square root, first estimate it and find the two consecutive integers it lies between. Then place the point closer to the integer whose square is nearer to the number. The module’s examples:
- √3 lies between 1 and 2 (the principal roots of 1 and 4). Since 3 is closer to 4, plot √3 closer to 2.
- √21 lies between 4 and 5 (roots of 16 and 25). Since 21 is closer to 25, plot √21 closer to 5.
- √87 lies between 9 and 10 (roots of 81 and 100). Since 87 is closer to 81, plot √87 closer to 9.
The lesson begins with Jose and Daniel, whose houses are √11 km apart. The answer key gives this distance as about 3.32 km, which lies between 3 and 4 on the number line.
Activities in the module
- What I Know: a 15-item pre-assessment.
- Let’s Recall: finding square and cube roots of perfect squares and cubes (Lesson 1), and naming the two consecutive integers a square root lies between (Lesson 2).
- Estimate Me!: ten square roots to estimate, plus a code puzzle that answers the trivia question “What animal has the largest heart?”
- Where Am I? and Locate Me!: matching and plotting square roots on number lines.
- What I Have Learned: short reflection questions, such as how to find the square root of 2.
- What I Can Do: word problems about square lots, rooms, gardens, windows, and carpets.
- Assessment: a 15-item multiple-choice post-test.
- Additional Activities: the maze “Make Your Way!” and “What’s wrong with me?”, where learners cross out the wrong graph of each square root.
How to use this module
- Learners: Start with the pre-assessment and note the items you missed. Then go through Lesson 1 and Lesson 2 in order, doing each activity on a separate sheet of paper, not on the module. Finish with the Assessment to check your learning. The answer key is at the end; use it honestly, only after you answer.
- Parents and guardians: Help your child follow the parts in order and check answers against the answer key after each activity. A calculator can help confirm the squares your child computes.
- Teachers: The Notes to the Teacher encourage letting learners explore their own ideas and guiding them to improve their output. The pre-test and post-test cover the same skills, so you can compare results to see progress.
If learners have trouble with any task, the module asks them to consult their teacher or facilitator.
Frequently asked questions
What grade and quarter is this module for?
It is for Grade 7 Mathematics, Quarter 1, Module 8. It follows the module on principal roots and irrational numbers.
What are the steps to estimate a square root to the nearest hundredth?
Find the perfect squares just below and above the number, then test one-decimal guesses by squaring them. Finally, square numbers to the hundredths and pick the one whose square is closest to the original number.
How do you plot √21 on a number line?
√21 lies between 4 and 5 because 16 < 21 < 25. Since 21 is closer to 25, place the point closer to 5.
Does the module have an answer key?
Yes. The answer key covers the pre-assessment, the activities in both lessons, and the final Assessment.
What is the answer to the trivia question in Lesson 1?
The decoded answer to What animal has the largest heart? is the blue whale.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 1 – Module 8: Estimating the Square Roots of Whole Numbers and Plotting Irrational Numbers
[illustration: teacher and students at a table, a student reading under a tree, a student with a lamp, and a globe]
CO_Q1_MATHEMATICS 7_Module 8
Mathematics – 7 Self-Learning Module (SLM) Quarter 1 – Module 8: Estimating Square Roots of Whole Numbers and Plotting Irrational Numbers First Edition, 2020
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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
Writers: Regilyn M. Magana, Bonifacio D. Suravilla III
Editors: Jay-ar S. Lipura, Nora S. Falle, Shienna Lyn L. Antenor, Ph.D
Reviewers: Evelyn C. Frusa PhD, Noemi E. Parcon, Rolex H. Lotilla and Arvin M. Tejada
Illustrator:
Layout Artist: Precious Joy A. Ubas
Cover Art Designer: Reggie D. Galindez
Management Team: Allan G. Farnazo, CESO IV – Regional DIrector Fiel Y. Almendra, CESO V – Assistant Regional DIrector Crispin A. Soliven Jr., CESE – Schools Division Superintendent Gilbert B. Barrera – Chief, CLMD Arturo D. Tingson Jr. – REPS LRMS Peter Van C. Ang-ug – REPS, ADM Jade T. Pajonar – REPS, Subject Area Supervisor Belen L. Fajemolin PhD – CID Chief Evelyn C. Frusa PhD – EPS –LRMS Bernardita M. Villano – ADM Coordinator Noemi E. Parcon – EPS – Mathematics
Printed in the Philippines by Department of Education – SOCCSKSARGEN Region
Office Address: Regional Center, Brgy. Carpenter Hill, City of Koronadal Telefax: (083) 2288825/ (083) 2281893 E-mail Address: region12@deped.gov.ph
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-test 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 Teachers are also provided to the 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. 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
[illustration: hand touching a circle, icon beside the part title]
This module was designed and written with you in mind. It is here to help you master the nature of Principal Roots and Irrational Numbers. 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.
This module is divided into two lessons, namely:
- Lesson 1 – Estimating the square root of a whole number to the nearest hundredth; and
- Lesson 2 – Plotting irrational numbers on a number line.
OBJECTIVES:
- Estimate the square root of a whole number to the nearest hundredth; and
- Plot irrational numbers (square roots) on a number line.
What I Know
[illustration: hand holding a pencil, icon beside the part title]
PRE-ASSESSMENT
Find out how much you already know about this module. Read each statement carefully. Choose the letter of the best answer and write the chosen letter on a separate sheet of paper. After taking and checking this short test, take note of the items that you were not able to answer correctly. Then, discover the correct answer as you go through this module.
Which whole number is closest to √46?
- A. 5
- B. 6
- C. 7
- D. 8
Which perfect square is closest to √181?
- A. 4
- B. 9
- C. 16
- D. 25
Which of the following is the perfect square number closest but less than 20?
- A. 16
- B. 19
- C. 22
- D. 25
Which of the following is the perfect square number closest but greater than 136?
- A. 100
- B. 138
- C. 144
- D. 169
Between which consecutive numbers does the estimate √10 lie?
- A. 2 and 3
- B. 3 and 4
- C. 4 and 5
- D. 5 and 6
Which of the following is the closest approximation of √51?
- A. 6.64
- B. 6.94
- C. 7.04
- D. 7.14
What is the square root of 15 to the nearest hundredths?
- A. 3.01
- B. 3.27
- C. 3.52
- D. 3.87
Which of the following number is the best estimate of √125?
- A. 10.18
- B. 11.18
- C. 12.18
- D. 13.18
Estimate √78.
- A. 7.38
- B. 7.83
- C. 8.38
- D. 8.83
What is the square root of 60 to the nearest hundredths?
- A. 7.75
- B. 7.76
- C. 7.77
- D. 7.78
For numbers 11 – 12, refer to the given number line below.
[chart: number line; points A, B, C, and D placed on a real number line] Labeled ticks: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 Points, left to right: D, C, B, A Values are not labeled on the chart.
Which letter most accurately represents the location of −√14 on the real number line?
- A. A
- B. B
- C. C
- D. D
Which letter most accurately represents the location of √8 on the real number line?
- A. A
- B. B
- C. C
- D. D
Locate and plot √2 on a number line.
- A. [illustration: number line with ticks 0, 1, 2, 3 and a dot between 1 and 2]
- B. [illustration: number line with ticks 0, 1, 2, 3 and a dot at 2]
- C. [illustration: number line with ticks 0, 1, 2, 3 and a dot between 0 and 1]
- D. [illustration: number line with ticks 0, 1, 2, 3 and a dot between 1 and 2, nearer 2]
Locate and plot √5 on a number line.
- A. [illustration: number line with ticks 0, 1, 2, 3 and a dot between 1 and 2, nearer 2]
- B. [illustration: number line with ticks 0, 1, 2, 3 and a dot near 0]
- C. [illustration: number line with ticks 0, 1, 2, 3 and a dot between 1 and 2]
- D. [illustration: number line with ticks 0, 1, 2, 3 and a dot between 2 and 3]
Locate and plot √23 on a number line.
- A. [illustration: number line with ticks 1, 2, 3, 4, 5 and a dot between 3 and 4]
- B. [illustration: number line with ticks 1, 2, 3, 4, 5 and a dot between 1 and 2]
- C. [illustration: number line with ticks 1, 2, 3, 4, 5 and a dot between 2 and 3]
- D. [illustration: number line with ticks 1, 2, 3, 4, 5 and a dot between 4 and 5]
How was your performance in the pre-assessment? Were you able to answer all the questions? Are all problems familiar to you?
Lesson 1 Estimates the Square Root of Whole Number to the Nearest Hundredths
[illustration: small hand and puzzle piece icon beside the part title]
You have learned Principal roots and Irrational numbers in the previous module and you already know how to evaluate square roots that are perfect squares. But what if the whole number given is not a perfect square? How can you find the square root of this number? What are the different ways/methods to use in looking for the answer?
These are some of the questions which you can answer once you understand the key concepts of estimating square roots of a whole number to the nearest hundredths.
What’s In
Let’s Recall!
Find the square/cube root of each number.
- √25
- √64
- √169
- √100
- √289
- √400
- ∛8
- ∛27
- √196
- ∛1000
From the activity done, were you able to name two consecutive integers between the given square roots? But how about if you are tasked to find the estimate value of the given square roots that are not perfect squares? What are you going to do? You will find this out in activities in the next section. Before performing these activities, read and understand first how they are used in real-life situation.
Notes to the Teacher
Let the student explore his ideas guide him that he can express his skills and talents. Facilitate him to enhance his output
What’s New
Arlo is enjoying his time with the Baseball team in his school. The players and the coaches have all been very friendly to him and Arlo loves helping out. He usually assists the players in the preparation prior to the game. He also hands them out towels and water. Sometimes, he collects and organizes equipment after the game.
The team is also active in their social responsibility to the community by doing a lot of community service. They conduct small league tournaments that encourages the youth to love and play baseball. The team also grants scholarships to those who wish to join. Arlo has been invited to go along with the team in these events.
There is one new ball field being created and the team was tasked to help design the infield so that it has the correct dimensions.
When Arlo and one of the players, JJ, arrived in the field, they measured the distance from home plate to first base. It measures 67 feet.
[illustration: silhouette of a baseball batter swinging a bat]
“That isn’t correct,” JJ tells Arlo. “The area of the infield for league should be 4900 square feet. This measurement is inaccurate. We have to help them fix it.” What is inaccurate about the 67 feet? Arlo is puzzled. Does it have to be a longer distance or a shorter one?
To figure this problem out, Arlo will need to use his knowledge of squares and square roots. This lesson will expand upon what you learned in the previous lesson. Then you can help Arlo figure this problem out at the end of the lesson.
What is It
Estimate the square roots of a whole numbers to the nearest hundredths
When you are finding the square root of a perfect square number using a calculator, you probably notice that the answer is a whole number, on the other hand each time you find the square root of numbers that are not a perfect square, you will see several digits after the decimal point.
This means that the square root of that number is a whole number and some fraction of another.
To determine the approximate value of the square root of a number that is not a perfect square, a series of estimation is made.
Let’s look at an example.
Estimate each square root to the nearest hundredths. A. √7 B. √44 C. √125 D. √263 E. √73
Solution:
A. √7
Step 1: Find the perfect squares closest (less than and greater than) to the given number.
Since 7 is not a perfect square, we need to find the two perfect squares that are closest to 7. They are: 4 and 9 because √4 < √7 < √9 or 2 < √7 <3.
Therefore √7 lies between 2 and 3. And since 7 is much closer to 9 than 4, we know that √7 is closer to 3 than 2.
Step 2: Make an educated guess to 1 decimal place and try it.
Since we know that square root of 7 is closer to 3 than 2, we then square numbers closer to 3 with one decimal place.
- (2.5)² = 6.25
- (2.6)² = 6.76 — The √7 lies between 2.6 and 2.7.
- (2.7)² = 7.29
- (2.8)² = 7.84
Note: Squaring of 2.1, 2.2, 2.3, and 2.4 is not necessary since square root of 7 is closest to 3.
Step 3: Decide which is closer.
Since 7 is closer to 6.76 compared to 7.29 and 6.76 is the square of 2.6 then estimate further by squaring the numbers 2.61, 2.62, 2.63, 2.63, 2.64, and 2.65.
- (2.61)² = 6.8121
- (2.62)² = 6.8644
- (2.63)² = 6.9169
- (2.64)² = 6.9696
- (2.65)² = 7.0225
Note: Squaring 2.66, 2.67, 2.68, and 2.69 is not necessary since √7 is closer to 2.6 than 2.7.
Notice that 7 is closer to 7.0225 compared to 6.9696. Meanwhile, 7.0225 is the square of 2.65.
Hence, √7 is 2.65 when estimated to nearest hundredths.
B. √44
Step 1: Find the perfect squares closest (less than and greater than) to the given number.
Since 44 is not a perfect square, we need to find the two perfect squares that are closest to 44. They are: 36 and 49 because √36 < √44 < √49 or 6 < √44 <7.
Therefore √44 lies between 6 and 7. And since 44 is closer to 49 than 36, we know that √49 is closer to 7 than 6.
Step 2: Make an educated guess to 1 decimal place and try it.
Since we know that the square root of 44 is closer to 7 than 6, we then square numbers closer to 7 with one decimal place.
- (6.5)² = 42.25
- (6.6)² = 43.56 — The √44 lies between 6.6 and 6.7.
- (6.7)² = 44.89
- (6.8)² = 46.24
- (6.9)² = 47.61
Note: Squaring 6.61, 6.62, 6.63, and 6.64 is not necessary since √44 is closest to 7.
Step 3: Decide which is closer.
Since 44 is closer to 43.56 compared to 44.89 and 43.56 is the square of 6.6 then estimate further by squaring the numbers 6.61, 6.62, 6.63, 6.63, and 6.64.
- (6.61)² = 43.6921
- (6.62)² = 43.8244
- (6.63)² = 43.9569
- (6.64)² = 44.0896
Note: Squaring 6.65, 6.66, 6.67, 6.68 and 6.69 is not necessary since √44 is closer to 6.6 than 6.7.
Notice that 44 is closer to 43.9596 compared to 44.0896. Meanwhile, 43.9569 is the square of 6.63.
Hence, √44 is 6.63 when estimated to nearest hundredths.
C. √125
Step 1: Find the perfect squares closest (less than and greater than) to the given number.
Since 125 is not a perfect square, we need to find the two perfect squares that are closest to 125. They are: 121 and 144 because √121 < √125 < √144 or 11 < 1√144 <12.
Therefore √125 lies between 11 and 12. And since 125 is closer to 121 than 144, we know that √125 is closer to 11 than 12.
Step 2: Make an educated guess to 1 decimal place and try it.
Since we know that the square root of 125 is closer to11 than 12, we then square numbers closer to 11 with one decimal place.
- (11.1)² = 123.21
- (11.2)² = 125.44 — The √125 lies between 11.1 and 11.2.
- (11.3)² = 127.69
Note: Squaring of 11.4, 11.5, 11.6, 11.7, 11.8, and 11.9 is not necessary since square root of 125 is closest to 11.
Step 3: Decide which is closer.
Since 125 is closer to 125.44 compared to 123.21 and 125.44 is the square of 11.2 then estimate further by squaring the numbers 11.16, 11.17, 11.18, and 11.19.
- (11.15)² = 124.3225
- (11.16)² = 124.5456
- (11.17)² = 124.7689
- (11.18)² = 124.9924
- (11.19)² = 125.2161
Note: Squaring 11.11, 11.12, 11.13, and 11.14 is not necessary since √125 is closer to 11.2 than 11.1.
Notice that 125 is closer to 124.9924 compared to 125.2161. Meanwhile, 124.9924 is the square of 11.18.
Hence, √125 is 11.18 when estimated to nearest hundredths.
D. √263
Step 1: Find the perfect squares closest (less than and greater than) to the given number.
Since 263 is not a perfect square, we need to find the two perfect squares that are closest to 263. They are: 256 and 289 because √256 < √263 < √289 or 16 < 1√263 <17.
Therefore √263 lies between 16 and 17. And since 263 is closer to 256 than 289, we know that √263 is closer to 16 than 17.
Step 2: Make an educated guess to 1 decimal place and try it.
Since we know that the square root of 263 is closer to16 than 17, we then square numbers closer to 16 with one decimal place.
- (16.1)² = 259.21
- (16.2)² = 262.44 — The √263 lies between 16.2 and 16.3.
- (16.3)² = 265.69
Note: Squaring of 16.4, 16.5, 16.6, 16.7, 16.8, and 16.9 is not necessary since square root of 263 is closest to 16.
Step 3: Decide which is closer.
Since 263 is closer to 261.44 compared to 265.69 and 261.44 is the square of 16.2 then estimate further by squaring the numbers 16.21, 16.22, and 16.23.
- (16.21)² = 262.7641
- (16.22)² = 263.0884
- (16.23)² = 263.4129
Note: Squaring 16.24, 16.25, 16.26, 16.27, 16.28, and 16.29 is not necessary since √263 is closer to 16.2 than 16.3.
Notice that 263is closer to 263.0884 compared to 263.4129. Meanwhile, 263.0884 is the square of 16.22.
Hence, √263 is 16.22 when estimated to nearest hundredths.
E. √73
Step 1: Find the perfect squares closest (less than and greater than) to the given number.
Since 73 is not a perfect square, we need to find the two perfect squares that are closest to 73. They are: 64 and 81 because √64 < √73 < √81 or 8< √73 <9.
Therefore √73 lies between 8 and 9. And since 73 is closer to 81 than 64, we know that √73 is closer to 9 than 8.
Step 2: Make an educated guess to 1 decimal place and try it.
Since we know that the square root of 73 is closer to 9 than 8, we then square numbers closer to 8 with one decimal place.
- (8.5)² = 72.25 — The √73 lies between 8.5 and 8.6.
- (8.6)² = 73.96
- (8.7)² = 75.69
Note: Squaring of 8.1, 8.2, 8.3 and 8.4 is not necessary since square root of 263 is closest to 16.
Step 3: Decide which is closer.
Since 73 is closer to 72.25 compared to 73.96 and 72.25 is the square of 8.5 then estimate further by squaring the numbers 8.51, 8.51, 8.53, 8.54, and 8.55
- (8.51)² = 72.4201
- (8.52)² = 72.5904
- (8.53)² = 72.7609
- (8.54)² = 72.9316
- (8.55)² = 73.1025
Note: Squaring 8.56, 8.57, 8.58, and 8.59 is not necessary since √73 is closer to 8.5 than 8.6.
Notice that 73 is closer to 72.9316 compared to 73.1025. Meanwhile, 72.9316 is the square of 8.54.
Hence, √73 is 8.54 when estimated to nearest hundredths.
What’s More
Estimate Me!
A. Estimate each square root to the nearest hundredths.
- √50
- √89
- √21
- √143
- √250
- √136
- √101
- √3
- √67
- √208
B. Decipher the code below to find the answer to the given trivia. Estimate the square root of a whole number to the nearest hundredths.
“What animal has the largest heart?”
| 8.19 | 14.07 | 6.56 | 6.16 | 11.09 | 14.18 | 9.95 | 14.07 | 6.16 |
- A √99
- B √67
- E √38
- U √43
- W √123
- R √92
- G √87
- O √58
- L √198
- F √39
- H √201
- S √44
[transcriber’s note: the trivia graphic labeled “Trivia Challenge” is decorative and omitted; the number row and the empty answer boxes beneath it are shown as one table, and the letter–expression key is printed in two columns]
8.19 14.07 6.56 6.16 11.09 14.18 9.95 14.07 6.16
From the activity done, were you able to estimate the estimated square root of the given whole number in the nearest hundredths?
What I Have Learned
Answer the following questions. Give your solutions or explanations.
- How do you find the square roots of a whole number which are not perfect squares?
- What are steps to follow in order for you to find the estimation of the given square roots which are not perfect squares?
- How do you find the square root of 2?
- Did you find any difficulty in estimating the square roots of a whole number? Why or Why not?
Were you able to answer all the questions in the activity? Do you have better understanding now on how to estimate the square roots of a whole numbers that are not perfect squares in the nearest hundredths? In the next activity, let’s extend your understanding by applying the concepts in real-life situation.
What I Can Do
Here is another activity that will let you apply what you learned about estimating the square root of a whole number in the nearest hundredths.
Read and solve the following problems:
Ericka owns a square lot with a land area of 237 square feet. What is the side length of the kitchen floor to the nearest hundredths of a foot?
Aevryn wants to use this tile to finish the floor in her room. The room is asquare and has a floor area of 441 square feet. She wants to put8 inches tiles along the edge of the floor.
a. What is the length of one side of the room?
b. How many tiles can she fit along one side?
c. How may tiles does she need in order to cover the floor?
You use one gallon of white paint to apply a base coat of paint on a square wall mural. The paint covers 380 square feet per gallon. Estimate the side length of the mural to the nearest hundredths of a foot.
What’s In
Let’s Recall!
Name the two consecutive integers between which the given square roots lie.
- √2
- √7
- √28
- √150
- √41
- √175
- √84
- √12
- √213
- √423
From the activity done, were you able to name two consecutive integers between the given square roots? But how about if you are tasked to locate the square roots of the given number on a line? What are you going to do? You will find this out in activities in the next section. Before performing these activities, read and understand first how they are used in real-life situation.
Lesson 2 Plotting Irrational Numbers on a Number Line
You have learned in the previous lesson on how to estimate square root of a whole number which are not perfect squares. But what about if you are tasked to plot these square roots on a number line? How are you going to do this?
These are some of the questions which you can answer once you understand the key concepts of estimating square roots of a whole number to the nearest hundredths and how to plot these square roots on a number line.
Notes to the Teacher
Let the student explore his ideas guide him that he can express his skills and talents. Facilitate him to enhance his output
What’s New
How to Plot?
Jose and Daniel planned to do their performance task given by their teacher in Math subject on Saturday. They agreed that Jose will go to the house of Daniel as their meeting place. Daniel said that, his house is √11km away from the house of Jose.
- How far is Daniel’s house from Jose’s house?
- Can you visualize or picture out how far is the distance between their houses?
- How can you plot √11 on a number line?
Can you answer the first question? If yes, how will you answer it? Is it possible for you to use a number line to determine how far their houses from one another? Your goal in the next section is to learn on how to use a number line to help you identify where the square roots are located.
What is It
In plotting irrational numbers on a number line, we estimate first the square root of the given irrational number and to which two consecutive integers it lies in between.
A. Locate and plot √3 on a number line.
This number is between 1 and 2, principal roots of 1 and 4. Since 3 is closer to 4 than to 1, √3 is closer to 2. Plot √3 closer to 2.
[chart: number line; A. Locate and plot √3; tick labels 0 1 2 3; one point marked and labeled √3]
B. Locate and plot √21 on a number line.
This number is between 4 and 5, principal roots of 16 and 25. Since 21 is closer to 25 than to 16, √21 is closer to 5 than to 4. Plot √21 closer to 5.
[chart: number line; B. Locate and plot √21; tick labels 0 1 2 3 4 5; one point marked and labeled √21]
C. Locate and plot √87 on a number line.
This number is between 9 and 10, principal roots of 81 and 100. Since 87 is closer to 81, √87 then is closer to 9 than to 10. Plot √87 closer to 9.
[chart: number line; C. Locate and plot √87; tick labels 5 6 7 8 9 10 11; one point marked and labeled √87]
What’s More
Where Am I?
In this activity, estimate the given square root and find the letter that corresponds to it at the number line.
[chart: number line; Where Am I?; tick labels 0 1 2 3 4 5 6 7 8 9 10; five points labeled A, B, C, D, E above the line]
- √57
- √6
- √99
- √38
- √11
Locate Me!
Plot the points on a number line.
- Point A: √18
[chart: number line; Point A; tick labels 1 2 3 4 5 6]
- Point B: √73
[chart: number line; Point B; tick labels 5 6 7 8 9 10]
- Point C: √30
[chart: number line; Point C; tick labels 3 4 5 6 7 8]
- Point D: √52
[chart: number line; Point D; tick labels 4 5 6 7 8 9]
- Point E: √7
[chart: number line; Point E; tick labels 0 1 2 3 4 9]
Values are not labeled on the chart.
What I Have Learned
Answer the following questions.
- How do you plot √2 on a number line?
- What are your ways/steps in plotting irrational numbers on a number line?
What I Can Do
Here is another activity that will let you apply what you learned about plotting square roots on a number line. Round off your answer to the nearest hundredths.
Ms. Liza is buying a square piece of land which is 548 square meters in area. What is the dimension of the land?
If a square bedroom has an area of 111square feet. What is the length of the side?
Mr. Cruz owns a square garden with an area of 245 square feet. How much fencing material will a gardener need to buy in order to fence the gardenof Mr. Cruz if each foot fencing material cost 55 pesos?
Anna wants to put up a frame for her window. She knows that the area of herwindow is 82 square feet. What is the estimate perimeter of the window?
Aerox bought a square carpet with an area of 95 square feet for his dining room. What is the approximate length of one side of the carpet?
Assessment
Multiple Choice. Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper.
Which whole number is closest to √154?
- A. 10
- B. 11
- C. 12
- D. 13
Identify which perfect squares is closest to 216?
- A. 196
- B. 225
- C. 256
- D. 289
Which of the following is the perfect square number closest but less than 137?
- A. 100
- B. 121
- C. 125
- D. 144
Which of the following is the perfect square number closest but greater than 239?
- A. 196
- B. 225
- C. 256
- D. 289
Between which consecutive numbers is the estimate √57 lie?
- A. 4 and 5
- B. 6 and 7
- C. 7 and 8
- D. 8 and 9
Which of the following is the closest approximation of √109?
- A. 9. 44
- B. 9.45
- C. 10.44
- D. 10.45
What is the square root of 63 to the nearest hundredths?
- A. 7.94
- B. 7.95
- C. 8.05
- D. 8.23
Which of the following number is the best estimate of √146?
- A. 11.08
- B. 11.18
- C. 12.08
- D. 12.18
Estimate √103.
- A. 10. 05
- B. 10.15
- C. 10.21
- D. 10.23
What is the square root of 176 to the nearest hundredths?
- A. 13.19
- B. 13.27
- C. 13.38
- D. 13.64
For numbers 11 – 12. Refer your answer on the given number line below.
[chart: number line; For numbers 11 – 12; tick labels -5 -4 -3 -2 -1 0 1 2 3 4 5; four points labeled D, B, A, C above the line]
Values are not labeled on the chart.
Which letter most accurately represents the location of −√19 on a number line?
- A. A
- B. B
- C. C
- D. D
Which letter most accurately represents the location of √26 on a number line?
- A. A
- B. B
- C. C
- D. D
Locate and plot √32 on a number line.
A.
[chart: number line; item 13 option A; tick labels 3 4 5 6; one point marked]B.
[chart: number line; item 13 option B; tick labels 3 4 5 6; one point marked]C.
[chart: number line; item 13 option C; tick labels 3 4 5 6; one point marked]D.
[chart: number line; item 13 option D; tick labels 3 4 5 6; one point marked]Locate and plot √21 on a number line.
A.
[chart: number line; item 14 option A; tick labels 3 4 5 6; one point marked]B.
[chart: number line; item 14 option B; tick labels 3 4 5 6; one point marked]C.
[chart: number line; item 14 option C; tick labels 3 4 5 6; one point marked]D.
[chart: number line; item 14 option D; tick labels 3 4 5 6; one point marked]Locate and plot √15 on a number line.
A.
[chart: number line; item 15 option A; tick labels 3 4 5 6; one point marked]B.
[chart: number line; item 15 option B; tick labels 3 4 5 6; one point marked]C.
[chart: number line; item 15 option C; tick labels 3 4 5 6; one point marked]D.
[chart: number line; item 15 option D; tick labels 3 4 5 6; one point marked]
Values are not labeled on the chart.
Additional Activities
Make Your Way!
Help Mr. Monkey find his way for his breakfast. By estimating the square root in the nearest hundredths and following the correct answer to end the maze, you can help Mr. Monkey find his banana.
[illustration: Mr. Monkey in the top right corner]
[chart: maze; Make Your Way!; cells and connecting values as printed, listed row by row]
- Row 1 cells: √157, √133, √93, √30
- Row 1 horizontal connectors: 11.63, 9.61, 5.48
- Row 1 to row 2 vertical connectors: 11.12, 11.18, 9.64, 4.46
- Row 1 to row 2 diagonal connectors: 12.82, 9.60, 9.67
- Row 2 cells: √201, √117, √12, √86
- Row 2 horizontal connectors: 14.18, 3.46, 9.25
- Row 2 to row 3 vertical connectors: 14.14, 10.67, 3.64, 9.26
- Row 2 to row 3 diagonal connectors: 10.82, 9.16, 9.37
- Row 3 cells: √53, √112, √82, √103
- Row 3 horizontal connectors: 7.28, 10.58, 9.03
- Row 3 to row 4 vertical connectors: 7.14, 10.56, 9.34, 10.17
- Row 3 to row 4 diagonal connectors: 7.18, 9.06, 14.25
- Row 4 cells: √214, [illustration: bananas], √9.32, √87, √206
- Row 4 horizontal connectors: 14.63, 14.33
transcriber’s note: the maze is printed as a picture; the connector values are listed above in the order they appear, and their exact grid positions are not stated in the text.
[illustration: bananas, the goal of the maze]
What’s wrong with me?
Identify which of the following illustrates the correct graph of a square roots. Cross-out the box of the wrong answer.
| Square root | Graph (left) | Graph (right) |
|---|---|---|
| √105 | [illustration: number line with labels 9.50, 10, 10.50 and a dot] | [illustration: number line with labels 9.50, 10, 10.50 and a dot] |
| √33 | [illustration: number line with labels 5, 5.50, 6 and a dot] | [illustration: number line with labels 5, 5.50, 6 and a dot] |
| √148 | [illustration: number line with labels 12, 12.50, 13 and a dot] | [illustration: number line with labels 12, 12.50, 13 and a dot] |
| √217 | [illustration: number line with labels 14, 14.50, 15 and a dot] | [illustration: number line with labels 14, 14.50, 15 and a dot] |
| √86 | [illustration: number line with labels 9, 9.50, 10 and a dot] | [illustration: number line with labels 9, 9.50, 10 and a dot] |
Answer Key
Lesson 1
What’s In
Let’s Recall
- 5
- 8
- 13
- 10
- 17
- 20
- 2
- 3
- 14
- 10
What’s New
The distance between each of the bases should be 70 feet. With a current distance of 67 feet, the distance is three feet short between each base.
What I Know
Assessment – Pre
- C
- C
- A
- C
- B
- D
- D
- B
- D
- A
- D
- B
- A
- D
- D
What’s More
A. Estimate Me!
- 7.07
- 9.43
- 4.58
- 11.96
- 15.81
- 11.66
- 10.05
- 1.73
- 8.19
- 14.42
B. Trivia
BLUE WHALE
What I Can Do
- 15.39 feet
- a. 21 feet
b. 29 feet - 19.49 feet
Lesson 2
What’s In
Let’s Recall
- 1 and 2
- 2 and 3
- 5 and 6
- 12 and 13
- 6 and 7
- 13 and 14
- 9 and 10
- 3 and 4
- 14 and 15
- 20 and 21
What’s New
- 3.32 km
- 2
- [illustration: number line with labels 2, 3, 4 and a dot]
What’s More
Where I am
- D
- A
- E
- C
- B
What’s More
Locate Me
- [illustration: number line with labels 2 and 3 and a dot]
- [illustration: number line with labels 7 and 8 and a dot]
- [illustration: number line with labels 5 and 6 and a dot]
- [illustration: number line with labels 8 and 9 and a dot]
- [illustration: number line with labels 4 and 5 and a dot]
What I Can Do
- 23.41 m
- 10.54 ft
- P 860.75
- 36.24 ft
- 9.75 ft
Assessment
- C
- B
- B
- C
- C
- C
- A
- C
- B
- B
- D
- C
- B
- A
- A
Additional Activities
Make a Way
[illustration: “Make a Way” puzzle board of square roots and decimal values in rounded boxes and arrows, with several boxes blacked out; the labels are rotated and too small to transcribe reliably]
[transcriber’s note: the puzzle labels are printed sideways and at a size that could not be read reliably, so they are not transcribed]
What’s wrong with me!
[illustration: answer version of “What’s wrong with me?” with a cross drawn over one box in each row]
| Square root | Graph (left) | Graph (right) |
|---|---|---|
| √105 | [illustration: number line with labels 9.50, 10, 10.50 and a dot] | [illustration: number line with labels 9.50, 10, 10.50 and a dot] |
| √33 | [illustration: number line with labels 5, 5.50, 6 and a dot] | [illustration: number line with labels 5, 5.50, 6 and a dot] |
| √148 | [illustration: number line with labels 12, 12.50, 13 and a dot] | [illustration: number line with labels 12, 12.50, 13 and a dot] |
| √217 | [illustration: number line with labels 14, 14.50, 15 and a dot] | [illustration: number line with labels 14, 14.50, 15 and a dot] |
| √86 | [illustration: number line with labels 9, 9.50, 10 and a dot] | [illustration: number line with labels 9, 9.50, 10 and a dot] |
[transcriber’s note: the crossed-out boxes are drawn in the image; the table lists the labels of all ten graphs]
References
Barker, T. 2016. Estimating Irrational Numbers. (online) Available at: https://slideplayer.com/slide/9373261/. (Accessed 1 June 2020).
CK-12’s Math 7. 2016. Approximating Square Roots. (online). Available at: https://www.ck12.org/book/ck-12-middle-school-math-grade-7/section/9.2/ (Accessed 4 June 2020).
Nivera, G. C. (2003). Explorations and Application: Elementary Algebra (87 – 88). Makati City: Salesiana Publisher, Inc.
Orines, F.B, et.al. (2002). New Trends in Math Series First Year: Algebra I Functional Approach. (78 – 81). Quezon City: Vibal Publishing House, Inc.
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