This is the Grade 9 Mathematics Quarter 2 Module 10 self-learning module, Solving Problems Involving Radicals, First Edition 2020, developed by DepEd National Capital Region. It is for Grade 9 learners who already know how to simplify radicals and solve radical equations from the previous module. In it, they apply those skills to word problems about right triangles, areas, electricity, falling objects, and consecutive integers.
On this page 6 sections
What learners will learn
The module has one learning competency:
- Solve problems involving radicals (M9AL-IIj-1)
What is inside the module
The module has one lesson, “Solving Problems Involving Radicals”, organized in the usual self-learning module parts:
| Part | What it contains |
|---|---|
| What I Know | A 10-item multiple-choice pretest |
| What’s In | A short history of the radical sign and the word “radical” (from the Latin radix, meaning root) |
| What’s New | A 10-item review: square roots, simplifying radicals, rationalizing denominators, adding and multiplying radicals, and solving radical equations |
| Problem-solving maps | Two worked examples using the Math-Breaker Map and the Multi-Rule Map |
| What is It | Four fully worked word problems |
| What’s More | Three practice problems and a summary of the skills needed |
| What I Can Do | Two problems to fill in using problem-solving maps |
| Assessment | A 10-item multiple-choice post-test |
| Additional Activities | A reflection on experiences that brought big changes in your life |
| PISA-based worksheet | A real-life shortcut problem with five analysis questions |
| Answer Key | Answers for the tests and activities |
Key ideas and worked examples
Skills you need for radical word problems
The module lists the skills learners must use:
- Simplify radicals: take the square root of all perfect squares and leave the other factors under the radical sign.
- Rationalize the denominator: multiply the numerator and denominator by an expression that makes a perfect nth root. For binomial denominators, multiply by the conjugate.
- Add and subtract radicals: simplify first, then combine similar radicals (same radicand and same index).
- Multiply radicals: use ⁿ√a · ⁿ√b = ⁿ√(ab), and use the FOIL method for two binomial radical expressions.
- Divide radicals: use ⁿ√a / ⁿ√b = ⁿ√(a/b), or rationalize the denominator.
- Use the Pythagorean Theorem: c² = a² + b², where c is the hypotenuse of a right triangle.
- Solve radical equations: isolate the radical, square both sides, solve, and check for extraneous solutions.
Problem-solving maps
The Math-Breaker Map (MBM) breaks a problem into four steps: represent the problem, form an equation, solve for the unknown, and write the final answer. For example, a right triangle with legs 3 cm and 3√3 cm gives x² = 9 + 27 = 36, so the hypotenuse is 6.
The Multi-Rule Map (MRM) writes the rule used beside each step. In the example, the square root of the sum of two consecutive integers is 7: √(2x + 1) = 7, so 2x + 1 = 49 and x = 24. The integers are 24 and 25.
The four worked examples
- Consecutive odd integers: the square root of their sum is 8. Squaring √(2x + 2) = 8 gives 2x + 2 = 64, so x = 31 and the larger integer is 33.
- Area of a square: a side of (√5 + 3) units gives A = (√5 + 3)² = (14 + 6√5) square units, using the FOIL method.
- Electric circuit: with I = √(W/R), a current of 20 amperes and a resistance of 12 ohms give a power of 4800 watts.
- Fire ladder: a 10-meter ladder whose foot is 2 meters from the wall reaches a window b = √(10² − 2²) = √96 = 4√6 meters high.
The practice problems use the same ideas: the frame of a square painting with an area of 2700 cm², the resistance of a resistor, the fall time of a dropped ball, and the height of a kite on a 110-foot rope. The PISA-based worksheet asks how far Juan walks if he cuts diagonally across a plaza instead of following two perpendicular streets.
How to use this module
- Answer the pretest (What I Know) first. Note the items you missed and look for the right answers as you go.
- Read What’s In and answer What’s New to review the radical skills from earlier modules.
- Study the problem-solving maps and the four examples in What is It step by step.
- Solve the What’s More and What I Can Do problems, then take the Assessment.
- Finish the Additional Activities and the PISA-based worksheet, then check your work with the Answer Key.
Learners: write your answers on a separate sheet of paper, do not mark the module, and use the answer key honestly. Ask your teacher or facilitator when a task is hard.
Parents and guardians: the module invites you to check your child’s pretest, and it encourages learners to ask an adult for help.
Teachers: the post-test and the What I Can Do problems show whether learners can set up and solve radical equations on their own, and the PISA-based worksheet works well for discussion.
Frequently asked questions
Which grade and quarter is this module for?
It is Grade 9 Mathematics, Quarter 2, Module 10. It covers the competency solve problems involving radicals (M9AL-IIj-1).
Does the module have an answer key?
Yes. The answer key covers the pretest, the review exercises, What’s More, What I Can Do, the assessment, and the PISA-based worksheet.
What kinds of word problems are in the module?
The problems involve right triangles and the Pythagorean Theorem, the area of a square, electric circuits, falling objects, volumes of cylinders and spheres, and consecutive integers.
What are the Math-Breaker Map and the Multi-Rule Map?
They are problem-solving maps. The Math-Breaker Map guides you from representing the problem to writing the final answer, and the Multi-Rule Map shows the rule used in each step of the solution.
What should I learn before using this module?
You should know how to simplify radicals, operate on them, and solve radical equations, which the previous module covers. The What’s New part reviews these skills.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 2 – Module 10: Solving Problems Involving Radicals
CO_Q2_Mathematics 9 _ Module 10
Mathematics – Grade 9 Alternative Delivery Mode Quarter 2 – Module 10 :Solving Problems Involving Radicals First Edition, 2020
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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
Development Team of the Module
Writer: ARjohn A. Atienza, Cecile D. Respall, Lhea Pendon, Myla V. Generalo
Editors: Juvy G. Delos Santos, Maita G. Camilon, Sally C. Caleja
Reviewer: Remylinda T. Soriano, George B. Borromeo, Angelita Z. Modesto
Illustrator:
Layout Artist: Reymark L. Miraples, Jhunness Bhaby A. Villalobos, Rosel P. Patangan
Management Team: Malcolm S. Garma, Genia V. Santos, Dennis M. Mendoza, Maria Magdalena M. Lim, Aida H. Rondilla, Lucky S. Carpio
Printed in the Philippines by ________________________
Department of Education – National Capital Region Office Address: Misamis St., Brgy. Bago Bantay, Quezon City Telefax: (632) 8926-2213 /8929-4330 /8920-1490 and 8929-4348 E-mail Address: ncr@deped.gov.ph
Mathematics Quarter 2 – Module 10: Solving Problems Involving Radicals
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
LEARNING COMPETENCY In this module, you will be able to:
- solve problems involving radicals. (M9AL-IIj-1)
What I Know
Find out how much you already know about this lesson. Choose the letter of the correct answer. Please answer all items. Take note of the items that you were not able to answer correctly and find out the right answer as you go through this module.
Luis walks 5 km due east and 8 km due north. How far is he from the starting point?
a. √89 km
b. √64 km
c. √39 km
d. √25 kmFind the length of an edge of a cube given its surface area of 72 m².
a. 6√3 m
b. 6√12 m
c. 2√3 m
d. √2 mA taut rope starting from the top of a flagpole and tied to the ground is 15 m long. If the pole is 7 m high, how far is the rope from the base of the flagpole?
a. 4 √11 m
b. 16 √11 m
c. 11 √4 m
d. 16√4 mThe volume of a cylinder is represented by V = π r²h, where r is the radius of the base and h is the height of the cylinder. If the volume of a cylinder is 120 cubic meters and the height is 5 meters, what is the radius of the base? (Use π = 3.14)
a. 2.76 m
b. 3.76 m
c. 4.82 m
d. 7.64 mFive times the square root of 1 less than a number is equal to 3 more than the same number. Find the numbers.
a. 15 and 4
b. -15 and -4
c. 17 and 2
d. -17 and -2The square root of the sum of two consecutive odd integers is equal to 6. Find the larger integer.
a. 16
b. 18
c. 17
d. 19For an airplane flying above the earth, the approximate distance d (in miles) to the horizon is given by the equation d = √(8000a), where a is the altitude of the plane in miles. Determine the altitude of the plane if its distance to the horizon is 150 miles.
a. 1 13/16 mi
b. 2 13/16 mi
c. 3 13/16 mi
d. 4 13/16 miThe formula t = 1.11√L gives the time t (in seconds) required for a pendulum of length L feet to swing through one back-and-forth cycle. A grandfather’s pendulum completes one cycle in 1.28 seconds. To the nearest hundredths in feet, how long is the pendulum?
a. 1.63 ft
b. 1.23 ft
c. 1.22 ft
d. 1.33 ftFind the length of the side of a square if its diagonal is √2 cm.
a. 10 cm
b. 2 cm
c. 1 cm
d. 0.1 cmThe radius of a cone with volume V and height h is r = √(3V/(πh)). Suppose a cone has a radius of 10 cm and a height of 25 cm. Find its volume. (Use π = 3.14)
a. 260 cm³
b. 2616.67 cm³
c. 8263.03 cm³
d. 2616 cm³
Impressive! You were able to finish answering the pre-test. You may ask your facilitator/guardian to check your work. Congratulations and keep on learning!
Lesson 1
SOLVING PROBLEMS INVOLVING RADICALS
In the previous module, you learned about radical equations and how to find its solution. In this module, you will learn how to solve word problems involving radicals.
What’s In
The radical sign is used to denote the positive or the principal root of a number. The word “radical” comes from the Latin word “radix”, meaning “root” was used in the 13th century or so, and was abbreviated as “R” or R with a slash through the right leg of the R, like the Rx symbol at pharmacies.
The symbol that looks like a check (radical sign without the “roof”) originated in Germany, in the 1500’s. It started out looking quite like a musical note. If you had a long expression under the radical sign, the expression was put in parentheses, and later, placed with a line over it. This is where the current symbol came from. Descartes in his La Geometrie (1637) seems to be the first to place the line on top for grouping.
The other use of “radical” means “by the roots” or “from the roots”, such as a “radical” change in a person means changing your inner self to have a dramatic change in appearance, opinion, values, goal or personality, like changing your total outlook in life. Most often, it is a result of an experience that is tragic or a success.
Now, why is a square root called a root in the first place?
Answer: a root is the source of something; if you square a number, the number it came from is the root, as if the square grew from it.
Nivera, G.C. & Lapinid, M. C. (2013)
Previously, you learned that a radical equation is an equation in which a variable is radicand or inside a radical symbol. You also learned how to solve radical equations by performing operations on radicals, applying the laws of radicals and simplifying them.
Now, try to answer the given questions by applying the concepts you have learned in the previous lessons on radicals.
What’s New
What is the square root of:
_____ a. √64
_____ b. √(9/25)
_____ c. √0.36Simplify:
_____ a. √((−4x²y)²)
_____ b. √((x² + 3)²)
_____ c. −√(121x²)Simplify:
_____ a. √400
_____ b. √32
_____ c. −5√18Simplify:
_____ a. √(x⁸)
_____ b. √(4x²y⁴z⁶)
_____ c. ∛(27x⁹y³z¹²)Simplify:
_____ a. √(b⁵)
_____ b. √(16x³y⁵)
_____ c. ∛(−54x⁴y⁸)Simplify:
_____ a. √(75/36)
_____ b. √(18x³/(49xy²))
_____ c. √(x⁵y²/(125xy))Rationalize the denominator:
_____ a. 4/√x
_____ b. 2/∛2
_____ c. ⁶√(64/y)Simplify:
_____ a. √8 + √18 − √32
_____ b. √(8x³) + √(2x⁵) − 3√(50x)Simplify:
_____ a. √(5x) √15
_____ b. (2√(12a)) (3√(2a))Solve for x.
_____ a. 2√(x − 3) = 12
_____ b. √(3x − 2) = √(x + 4)
Another way of solving word problems is using the Problem-Solving-Maps. In the following examples, the MBM (Math-Breaker-Map) and the MRM (Multi-Rule-Map) are used. Study each one of them.
Example 1: MATH-BREAKER MAP (MBM)
[chart: problem-solving map; Example 1: MATH-BREAKER MAP (MBM)]
The length of the legs of a right triangle are 3 cm and 3√3 cm. Find the length of the hypotenuse.
REPRESENTATION OF THE PROBLEM: Let x = the length of the hypotenuse
FORM AN EQUATION: Using Pythagorean Theorem because the given is a right triangle. c² = a² + b²
SOLVE FOR THE UNKNOWN: x² = 9 + 27; x² = 36; √x² = √36; x = 6
WRITE THE FINAL ANSWER: The length of the hypotenuse is 6 units.
Example 2: MULTI-RULE MAP (MRM)
[chart: problem-solving map; Example 2: MULTI-RULE MAP (MRM)]
The square root of the sum of two consecutive integers is 7. Find the two integers.
Let x = be the 1st integer; x + 1 = the second integer
√(x + x + 1) = 7
√(2x + 1) = 7
(√(2x + 1))² = 7²
2x + 1 = 49
2x = 48
x = 24
The 2 consecutive integers are 24 and 25
Rules listed beside the map: Representation of the Problem; Form an Equation; Combine similar terms; Square both sides of an equation; Apply laws of radicals; Addition Property of Equality; Multiplication Property of Equality
What is It
This time, you will apply what you have learned from the previous topics on how to solve problems involving radicals. Let us study the following examples.
Example 1: The square root of the sum of two consecutive odd integers is equal to 8. Find the larger integer.
Solution:
a. Understand the problem: Let x = be the smaller odd integer x+2 = be the larger odd integer We will equate the square root of the sum of these integers to 8.
b. Write the equation: √(x + (x + 2)) = 8
c. Solve:
| √(x + (x + 2)) = 8 | Equation from the given data |
|---|---|
| √(2x + 2) = 8 | Combine similar terms |
| (√(2x + 2))² = 8² | Square both sides of the equation |
| 2x + 2 = 64 | |
| 2x = 62 | Addition Property of Equality |
| x = 31 | Multiplication Property of Equality |
| x = 31 | The smaller odd integer |
| x+2 = 31+2 = 33 | The larger odd integer |
| Since the question is, “find the larger integer”, thus the answer is 33. |
Example 2: The side of a square measures (√5 + 3) units, solve for its area.
Solution:
| A = s² | Formula for the area of a square |
|---|---|
| A = [(√5 + 3) units]² | Substitute the given in the formula |
| A = [(√5 + 3) units] [(√5 + 3) units] | Expand |
| A = [√25 + (3√5) + (3√5) + 9] sq. units | Multiplication of radicals (foil method) |
| A = (5 + 6√5 + 9) sq. units | Simplify |
| A = (14 + 6√5) sq. units | Combine similar terms |
| Therefore, the Area of the square is (14 + 6√5) sq. units |
Example 3: In an electrical circuit, the amount of current measured in amperes (I), the amount of power in watts (W), and the resistance (R) in ohms are related by the formula; I = √(W/R). Find the wattage in a system with 20 amperes and 12 ohms.
Solution:
a. Understand the problem.
b. Identify the given: I = 20 ampere, R = 12 ohms, W =?
c. Solve:
| I = √(W/R) | Given Formula |
|---|---|
| 20 amp = √(W/12 ohm) | Substitute the given values of I and R |
| 20 amp = (√w/√12 ohm)(√12 ohm/√12 ohm) | Rationalize the denominator |
| 20 amp = √((W)12 ohm) / √(144 ohm²) | Multiplication of Radicals |
| 20 amp = √((W)(12 ohm)) / 12 ohm | Get the square root of the denominator |
| (20 amp)(12 ohm) = √((W)(12 ohm)) | Multiplication Property of Equality |
| (240 amp. ohm)² = √((W)(12 ohm))² 57600 (amp)²(ohm)² = (W)(12 ohm) | Square both sides of the equation |
| 57600 (amp)²(ohm)² / 12 ohm = (W)(12 ohm) / 12 ohm | Divide both sides by 12 ohm or MPE |
| 4800 (amp)² (ohm) = W | Simplify the equations |
| W = 4800 (amp)²(ohm) ⟹ W = 4800 watts (Note that IIR = Power (watts)) |
Example 4: To save a woman from a burning building, the firemen placed a 10-meter ladder against the window. The foot of the ladder was 2 meters from the wall. Approximately, how high is the window?
[illustration: a ladder leaning against a multi-storey building with windows; labeled 10 m (ladder) and 2 m (base distance from the wall)]
Solution:
a. Draw a diagram that represents the problem.
[chart: diagram; right triangle with labels c = 10 m (hypotenuse), b = ? (vertical leg), a = 2m (horizontal leg)]
b. Determine the theory involved in the figure. The figure formed is a right triangle; we will use the Pythagorean Theorem in solving the problem. c² = a² + b², in the figure one of the legs or sides is denoted by b, which is the height of the window. We will derive the formula as: b = √(c² − a²) when b is unknown. When a is unknown: a = √(c² − b²). When c is unknown: c = √(a² + b²)
c. Solve: a = 2m; c = 10m; b = ?
| b = √(c² − a²) | Use the derived formula for b |
|---|---|
| b = √((10m)² − (2m)²) | Substitute the given values to the formula |
| b = √(100m² − 4m²) | Square the numbers |
| b = ±√96m² | Perform the operation |
| b = ±(√16)(√6)(√m²) | Factor the radicand to simplify a perfect square factor |
| b = ±4(√6) m | Simplify by getting the square root |
| b = 4√6 meters | Get the principal root for the final answer |
| The height of the window is 4√6 meters. |
What’s More
Solve the following problems.
Lori recently purchased a square painting with an area of 2700 cm². How much wood does Lori need to make the frame?
The maximum number of volts (E), that can be placed across the resistor is given by E = √PR, where P is the power in watts and R is the resistance in ohms. If a 2-watt resistor can have at most 40 volts of electricity across it. Find the number of ohms of resistance of this resistor.
The time (t) in seconds, it takes a free-falling object to fall (h) in feet is given by the formula t = √(h/16). If a ball is dropped from a height of 100 m, how long would it take the ball to hit the ground in seconds?
In solving problems involving radicals, we must know how to:
| Simplify Radicals | -Take the square root of all perfect squares – Leave the other factors under the radical sign |
|---|---|
| Rationalize the Denominator | -Multiply both numerator and denominator by an appropriate expression such that the product will be a perfect nth root. -For binomial denominators, multiply by their conjugates |
| Add and Subtract Radicals | -Simplify all radicals -Add or subtract similar radicals. (Similar radicals have the same radicands and indices.) |
| Multiply Radicals | -Apply the property ⁿ√a · ⁿ√b = ⁿ√(ab), then simplify – In multiplying two radical expressions, apply the FOIL method. |
| Divide Radicals | -Simplify the radical expressions or rationalize the denominator -Apply the property ⁿ√a / ⁿ√b = ⁿ√(a/b), then simplify |
| Use Pythagorean Theorem | -The square of the length of the hypotenuse (c) is equal to the sum of the squares of the length of the legs (a and b) of any right triangle. That is, c² = a² + b² |
| Solve Radical Equation | -Isolate the term with radicals on one side of the equation then, square both sides of the equation. -Solve the resulting equation. -Check for extraneous solution. |
What I Can Do
Try to solve these problems to check your readiness in answering your post-test.
- Find the length of the hypotenuse of a right triangle whose legs are 6 cm and 12 cm.
[chart: problem-solving map; problem 1 boxes, in flow order: “Find the length of the hypotenuse of a right triangle whose legs are 6 cm and 12 cm.” → “Representation of the Problem” (blank box) → “Form an Equation” (blank box) → “Solve for the Unknown” (blank box) → “Write the Final Answer” (blank box)]
- A kite is secured to a rope that is tied to a ground. A breeze blows the kite so that the rope is taut while the kite is directly above the flagpole that is 30 ft from where the rope is staked down. Find the altitude of the kite if the rope is 110 ft. long.
[chart: problem-solving map; problem 2 flow, in order: “Let x be the altitude of the kite.” → “From Pythagorean Theorem: c² = a² + b²; 110² = 30² + x²” → three blank boxes → clouds beside the flow: blank, blank, “Square the numbers”, “Addition Property of Equality”, “Take the square roots of both sides”]
[transcriber’s note: the first two cloud labels and the three blank flow boxes are empty in the printed module]
Assessment
Read the questions carefully. Write the letter of the correct answer in your answer sheet.
A man walks 4 meters to the east going to school and then walks 10 meters northward going to the church. How far is he from the starting point?
a. 5√29 m
b. 4√29 m
c. 3√29 m
d. 2√29 mIf each side of a square garden is increased by 4 m, its area becomes 144 m². What is the length of the side of the original square garden?
a. 6 m
b. 8 m
c. 16 m
d. 19 mA 26-foot ladder rests against a building and reaches a windowsill 24 feet above the ground. How far from the wall is the base of the ladder?
a. 10 ft
b. 11 ft
c. 12 ft.
d. 13 ftThe volume of a spherical balloon is 113.04 cm³. Find the radius of the balloon. (π = 3.14 and volume of a sphere = 4/3 πr³).
a. 2 cm
b. 3 cm
c. 4 cm
d. 5 cmFind the number such that twice its square root is 14.
a. 7
b. 8
c. 49
d. 144The square root of the sum of two consecutive odd integers is 6√2. Find the larger integer.
a. 34
b. 35
c. 36
d. 37The distance (d) in miles that a person can see to the horizon is represented by the equation d = √(3h/2), where h is the height where the person is. (1 mile = 1609.3 m). How far can you see to the horizon through an airplane window at a height of 8000 meters? (Note: Convert first the given height of 8000 meters to miles.)
a. 2.73 miles
b. 2.74 miles
c. 7.46 miles
d. 7.64 milesThe time (t) in seconds it takes a free-falling object to fall is given by the formula, t = √(h/16). How high in feet, will an airplane can reach in 15 seconds?
a. 36 ft
b. 60 ft
c. 3600 ft
d. 6000 ftThe side of a square is 8√7 cm. Find its diagonal.
a. 8√14 cm
b. 4√81 cm
c. 4√18 cm
d. 8√41 cmThe volume of a can is 602.88 cm³. The height of the can is three times the radius of the base. Find the radius of the base of the cylinder given its formula as V = π r²h. (Use π = 3.14)
a. 4 cm
b. 4.5 cm
c. 5 cm
d. 5.5 cm
Additional Activities
- List down three experiences that you had brought out profound changes in you.
- Do some self-check and think if those changes made a positive or a negative impact in your life. Write the things that you have learned from that experienced.
- Write a thought process wherein you consider getting a rewind of an event in your life and what should you change in that particular event.
E-Search
To further explore the concept learned today and if it’s possible to connect the internet, you may visit the following links:
PISA BASED WORKSHEET
Juan is going to Nene’s house to do a school project. Instead of walking two perpendicular streets to his classmates’ house, Juan will take a short-cut diagonal path through the city plaza. Juan is 15 meters away from Nene’s street. The distance from the intersection of the two streets to Nene’s house is 8 meters.
Let’s Analyze
- How will you illustrate the problem?
- How far will Juan travel along the shortcut? Show your solution.
- How many meters will he save by taking the short cut rather than walking along the sidewalks.
- If one of the distances increases or decreases, what might happen to the distance of the short cut? Justify your answer.
- What Mathematical concepts did you use?
Answer Key
[illustration: answer key emblem (key in a circle)]
[transcriber’s note: this page is printed upside down; the text below is transcribed in reading order after rotating it. Some expressions were hard to read and are marked [?].]
WHAT I KNOW
- A
- C
- A
- A
- C
- D
- B
- D
- C
- B
WHAT’S IN
1a. 8 1b. 3/5 1c. 0.6 2a. −4x²y 2b. x²+3 2c. −11x 3a. 20 3b. 4√2 3c. −15√2 4a. x⁴ [?] 4b. 2xy²z³ [?] 4c. 3x³yz⁴ [?] 5a. b²√b [?] 5b. 4xy²√(xy) [?] 5c. −3xy²∛(2xy²) [?] 6a. 5√3 [?] 6b. 3x√2 [?] 6c. [?] 7a. 4√x/x [?] 7b. 8√4 [?] 7c. [?] 8a. [?] 8b. √2x(x²+2x−15) [?] 8c. [?] 9a. 5√3x [?] 9b. 12√6 [?] 9c. [?] 10a. x = 3 [?] 10b. x = 39 [?] 10c. [?]
WHAT’S MORE
| Statement | Reason |
|---|---|
| A = 2700 cm² | Given value |
| s² = 2700 | Substitution |
| s = 30√3 cm | Take the square roots of both sides |
| P = 4(30√3 cm) | Substitution to the formula of Perimeter |
| P = 120√3 cm | Multiply |
[transcriber’s note: the printed table has no header row; the header above is supplied for the table’s two columns]
Lori needs 120√3 cm of wood to make a frame.
| Statement | Reason |
|---|---|
| B = √(P/R) [?] | Given formula |
| 40 = √(2R) | Substitution |
| 1600 = 2R | Square both sides of the equation |
| 800 = R | Multiplication Property of Equality |
[transcriber’s note: the printed table has no header row; the header above is supplied for the table’s two columns]
The resistor needs 800 ohms for the resistance.
| Statement | Reason |
|---|---|
| t = √(h/16) [?] | Given formula |
| t = √(100/16) | Substitution |
| t = 5/2 [?] | Take the square roots of the fraction |
[transcriber’s note: the printed table has no header row; the header above is supplied for the table’s two columns]
It will take 2.5 seconds for the ball to hit the ground.
WHAT I CAN DO
[chart: flowchart; WHAT I CAN DO]
Problem 1: Find the length of the hypotenuse of a right triangle whose legs are 6 cm and 12 cm.
Steps, in order:
- Represent the Problem: Let x = the length of the hypotenuse.
- Form an Equation: Using the Pythagorean Theorem, c² = a² + b², x² = (6 cm)² + (12 cm)².
- Solve for the Unknown: x² = 36 cm² + 144 cm²; x² = 180 cm²; √x² = √180 cm²; x = 6√5 cm.
- Write the final answer: The length of the hypotenuse is 6√5 cm.
Problem 2: A kite is secured to a rope that is taut while the kite is directly above the flagpole that is 30 ft from where the rope is staked down. A breeze blows the kite so the rope is taut. Find the altitude of the kite if the rope is 110 ft long.
Steps, in order:
- Let x be the altitude of the kite.
- From Pythagorean Theorem: c² = a² + b²; 110² = 30² + x²
- 12100 = 90 + x²
- Square the numbers
- Addition Property of Equality
- 11200 = x²
- Take the square roots of both sides
- x = 40√7
[transcriber’s note: the expression “12100 = 90 + x²” is transcribed as printed]
ASSESSMENT
- D
- B
- A
- B
- C
- D
- A
- C
- A
- A
ADDITIONAL ACTIVITIES
Answers may vary.
PISA Based Worksheet
- The problem can be illustrated using a right triangle.
- Juan will travel 17 meters.
- Juan will save 6 meters by taking the short cut.
- If the distance of the shortcut increases or decreases, the length of the sides also increases or decreases in direct variation.
- Pythagorean Theorem (Solving Problems involving Radicals).
References
Nivera, G.C. & Lapinid, M. C. (2013). Grade 9 Mathematics Patterns and Practicalities. Salesiana Books by Don Bosco Press
Bryant, M.L. (2014) Mathematics Learner’s Material 9. Department of Education
http://www.pballew.net/etyindex.html
http://jeff560.tripod.com/mathword.html
For inquiries or feedback, please write or call:
Department of Education – Bureau of Learning Resources (DepEd-BLR)
Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600
Telefax: (632) 8634-1072; 8634-1054; 8631-4985
Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph
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