Mathematics Grade 10

42  92  c 2 ; c = 97 km  9.85 km Jose’s house 0,0 to Diego’s house 3,12 32  122  c2 ; c = 153 km  12.37 km Emilio’s house 9,4 to Diego’s house 3,12 62  82  c 2 ; c = 10 km Provide the students opportunity to derive the Distance Formula. Askthem to perform Activity 4. In this activity, the students should be able to come upwith the Distance Formula starting from two given points on the coordinate plane.DEPED COPYActivity 4: Let Me Formulate! y 2. Answer Key y 1. x y x x3. C 8,1. By determining the coordinates of the point of intersection of the two lines AC = 6 units BC = 8 units4. Right Triangle. BC  AC . Hence, the triangle contains a 90-degree angle. Pythagorean Theorem can be applied. AB = 10 units 210 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

5. C x1,y2  y AC = x1  x2 or x2  x1 BC = y1  y2 or y2  y1 AB 2 = x2  x12  y2  y12 AB = x2  x12  y2  y12DEPED COPY x Before proceeding to the next activities, let the students give a briefsummary of the activities done. Provide them an opportunity to relate or connecttheir responses in the activities given to their new lesson. Let the students readand understand some important notes on the distance formula and the midpointformula and in writing coordinate proofs. Tell them to study carefully theexamples given.What to PROCESS In this section, let the students apply the key concepts of the DistanceFormula, Midpoint Formula, and Coordinate Proof. Tell them to use themathematical ideas and the examples presented in the preceding section toanswer the activities provided. Ask the students to perform Activity 5. In this activity, the students willdetermine the distance between two points on the coordinate plane using theDistance Formula. They should be able to explain how to find the distancebetween points that are aligned horizontally, vertically, or neither. 211 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 5: How far are we from each other?Answer Key 1. 8 units 6. 13 units 2. 15 units 7. 10.3 units 3. 11.4 units 8. 11.66 units 4. 13 units 9. 13.6 units 5. 6.4 units 10. 12.81 units a. Regardless of whether points are aligned horizontally or vertically, the distance d between these points can be determined using the Distance Formula, d  x2  x12  y2  y12 . Moreover, the following formulas can also be used. a.1) d = x2  x1 , for the distance d between two points that are aligned horizontally DEPED COPY a.2) d = y2  y1 , for the distance d between two points that are aligned vertically b. The Distance Formula can be used to find the distance between two points on a coordinate plane. Let the students apply the Midpoint Formula in finding the coordinates ofthe midpoint of a segment whose endpoints are given by doing Activity 6. Thisactivity will enhance their skill in proving geometric relationships using coordinateproof and in solving real-life problems involving the midpoint formula.Activity 6: Meet Me Halfway!Answer Key1. 9,9 6.  8,92. 7,8 7.  5,43. 4,4 8.  15 ,15  2 24. 4,1 9.  8,75.  3 , 5   2 2  10.  5,4 212 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Provide the students opportunity to relate the properties of somegeometric figures to the new lesson by performing Activity 7. Ask them to plotsome set of points on the coordinate plane. Then, connect the consecutive pointsby a line segment to form a figure. Tell them to identify the figures formed anduse the distance formula to characterize or describe each. Emphasize to thestudents the different properties of these geometric figures for they need this indetermining the missing coordinates of each figure’s vertices.Activity 7: What figure am I? Answer Key 1. yDEPED COPY 2. y3. x x y x 4. y x 213 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

5. DEPED COPY 6. x x y y x7. x y 8.9. y y x 10. y x 214 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

a. The figures formed in #1, #2, and #3 are triangles. Each figure has three sides. The figures formed in #4, #5, #6, #7, #8, and #9 are quadrilaterals. Each figure has four sides. The figure formed in #10 is a pentagon. It has five sides.b. ΔABC and ΔFUN are isosceles triangles. ΔGOT and ΔFUN are right triangles.c. ΔABC and ΔFUN are isosceles because each has two sides congruent or with equal lengths. ΔGOT and ΔFUN are right triangles because each contains a right angle.d. Quadrilaterals LIKE and LOVE are squares. Quadrilaterals LIKE, DATE, LOVE and SONG are rectangles. Quadrilaterals LIKE, DATE, LOVE, SONG, and BEAT are parallelograms. Quadrilateral WIND is a trapezoid.e. Quadrilaterals LIKE and LOVE are squares because each has four sides congruent and contains four right angles. Quadrilaterals LIKE, DATE, LOVE, and SONG are rectangles because each has two pairs of congruent and parallel sides and contains four right angles. Quadrilaterals LIKE, DATE, LOVE, SONG, and BEAT are parallelograms because each has two pairs of congruent and parallel sides and has opposite angles that are congruent. Quadrilateral WIND is a trapezoid because it has a pair of parallel sides.DEPED COPY An important skill that students need in writing coordinate proof is to namethe missing coordinates of geometric figures drawn on a coordinate plane.Activity 8 provides the students opportunity to develop such skill. In this activity,the students will name the missing coordinates of the vertices of geometricfigures in terms of the given variables.Activity 8: I Missed You But Now I Found You!Answer Key 5. A  a,0 For questions a-d, evaluate D a,d  students’ responses.1. O a  b,c E b,c2. V a,b3. V 3a,0 6. S 0,0 P a,b M 3a,b4. W  b,c What to REFLECT on and UNDERSTAND 215 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Ask the students to take a closer look at some aspects of the DistanceFormula, the Midpoint Formula, and the Coordinate Proof. Provide them withopportunities to think deeply and test further their understanding of the lesson bydoing Activity 9. In this activity, the students will solve problems involving thesemathematics concepts and explain or justify their answers.Activity 9: Think of This Over and Over and Over … Again!Answer Key1. y = 15 or y = -9; The values of x were obtained2. a. x = 21 – if N is in the first quadrant by using the distance formula and the coordinates of the x = -3 – if N is in the second quadrant midpoint were determined by using the midpoint formula. Students may further give explanations to their answers based on the solutions presented.b.DEPED COPY3, 5   23.  7,44. 99 km5. Luisa and Grace are both correct. If the expressions are evaluated,Luisa and Grace will arrive at the same value.6. a. Possible answer: To become more accessible to students comingfrom both buildings.b. 90,70c. The distance between the two buildings is about 357.8 m. Since the study shed is midway between the two school buildings, then it is about 178.9 m away from each. This is obtained by dividing 357.8 by 2.7. a. 100 km b. 5 hours8. No. The triangle is not an equilateral triangle. It is actually an isosceles triangle. The distance between A and C is 2a while the distancebetween A and B or B and C is a 2 .9. a. Yes. FS  c  a2  b  d 2 and AT  a  c2  b  d2 . Since a  c2  c  a2 , then FS = AT.b. Rectangle; The quadrilateral has two pairs of opposite sides that are parallel and congruent and has four right angles. Develop further students’ understanding of Coordinate Proof by askingthem to perform Activity 10. Ask the students to write a coordinate proof to provethe particular geometric relationship. Let them realize the significance of theDistance Formula, the Midpoint Formula, and the different mathematics conceptsalready studied in coming up with the coordinate proof. 216 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 10: Prove that this is True!Answer Key1. Show that PR  QS . If PR  QS , then PR  QS . PR   b  a2  c  02  b2  2ab  a2  c2 PR  a2  2ab  b2  c2DEPED COPY QS  b   a2  c  02  b  a2  c  02  b2  2ab  a2  c2 QS  a2  2ab  b2  c2 Therefore, PR  QS and PR  QS . Hence, the diagonals of an isosceles trapezoid are congruent.2. Show that MC  1 LG . 2 MC   a  0 2   b  02  2   2   a2  b2 4 4 MC  a2  b2 2 LG  0  a2  b  02  a2  b2 1 LG  a2  b2 2 2 Therefore, MC  1 LG . Hence, the median to the hypotenuse of a right 2 triangle is half the hypotenuse. 217 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

3. Show that PQ  QR  RS  PS .PQ   0  b  a 2   c  c 2   2   2 PQ    b a 2   c 2  2   2  b2  2ab  a2  c2 2 DEPED COPYQR   b  a  02   c  02 RS   0   b a 2   0  c 2   2   2   2   2 QR   b  a 2   c 2   b  a 2    c 2  2   2   2   2  b2  2ab  a2  c2 RS  b2  2ab  a2  c2 2 2PS   0   b a 2   c  c 2   2   2 PS   b  a 2   c 2  2   2  b2  2ab  a2  c2 2Therefore, PQ  QR  RS  PS and PQRS is a rhombus. 218 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

4. Show that BT  CS . If BT  CS , then BT  CS .BT  a   a 2   0  b 2  2  2   a  a 2  0  b 2  2  2   3a 2    b 2  2   2 BT DEPED COPY9a2  b2 2CS    a  a 2   0  b 2 Therefore, BT  CS and BT  CS .  2   2  Hence, the medians to the legs of an isosceles triangle are congruent. 2 2   3a     b   2  2CS  9a2  b2 25. Equate the lengths AC and BD to prove that ABCD is a rectangle. AC  BDb  a2  c  02  a  b  02  c  02b2  2ab  a2  c2  a2  2ab  b2  c2b2  2ab  a2  c2  a2  2ab  b2  c2  2ab  2ab 4ab  0 Since a > 0, then b = 0. And that A is along the y – axis. Also, B is along theline parallel to the y-axis. Therefore, ADC is a right angle and ABCD is arectangle. 219 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

6. Show that CG  1 LE 2 LE  b  02  c  02 LE  b2  c2 CG   a  b  a 2   c  02  2 2   2    b 2   c 2 DEPED COPY2   2  CG  b2  c2 2 Therefore, CG  1 LE . 2 Before the students move to the next section of this lesson, give a shorttest (formative test) to find out how well they understood the lesson. Ask themalso to write a journal about their understanding of the distance formula, midpointformula, and the coordinate proof. Refer to the Assessment Map.What to TRANSFER Give the students opportunities to demonstrate their understanding of theDistance Formula, the Midpoint Formula, and the use of Coordinate Proofs bydoing a practical task. Let them perform Activity 11. You can ask the students towork individually or in group. In this activity, the students will make a sketch ofthe map of their municipality, city, or province on a coordinate plane. They willindicate on the map some important landmarks, and then determine thecoordinates of each. Tell them to explain why the landmarks they have indicatedare significant in their community and to write a paragraph explaining how theyselected the coordinates of these landmarks. Using the coordinates assigned tothe different landmarks, the students will formulate then solve problems involvingthe distance formula and the midpoint formula. They will also formulate problemswhich require the use of coordinate proofs.Activity 11: A Map of My Own Answer Key Evaluate students’ answers. You may use the rubric. 220 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Summary/Synthesis/Generalization: This lesson was about the distance formula, the midpoint formula, the useof coordinate proofs, and the applications of these mathematical concepts in reallife. The lesson provided the students with opportunities to derive the distanceformula, find the distance between points, determine the coordinates of themidpoint of a segment, name the missing coordinates of the vertices of somegeometric figures, write a coordinate proof to prove some geometric relationships,and solve problems involving the different concepts learned in this lesson. Moreover,the students were given the opportunities to formulate then solve problemsinvolving the distance formula, the midpoint formula, and the coordinate proof.DEPED COPYLesson 2: The Equation of a CircleWhat to KNOW Find out how much the students have learned about the differentmathematics concepts previously studied and their skills in performingmathematical operations. Checking these will facilitate teaching and students’understanding of the equation of a circle. Tell them that as they go through thislesson, they have to think of this important question: “How does the equation of acircle facilitate finding solutions to real-life problems and making decisions?” Two of the essential mathematics concepts needed by the students inunderstanding the equation of a circle are the perfect square trinomial and thesquare of a binomial. Activity 1 of this lesson will provide them opportunity torecall these concepts. In this activity, the students will determine the number thatmust be added to a given expression to make it a perfect square trinomial andthen express the result as a square of a binomial. They should be able to explainhow they came up with the perfect square trinomial and the square of a binomial.Emphasize to the students that the process they have done in producing aperfect square trinomial is also referred to as completing the square.Activity 1: Make It Perfect!Answer Key a. Add the square of one-half the coefficient of the linear term.1. 4; x  22 b. Factor the perfect square trinomial.2. 25; t  52 c. Use the distributive property of3. 49; r  72 multiplication or FOIL Method.4. 121; r  1125. 324; x 182 221 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Answer Key 9. 1 ; s  1 26. 81; w  9 2 36  6  4  2 10. 9 ; t  3 2 64  8 7. 121;  x  112 4  28. 625 ; v  25 2 4  2 Provide the students opportunity to develop their understanding of theequation of a circle. Ask them to perform Activity 2. In this activity, the studentswill be presented with a situation involving the equation of a circle. Let them findthe distance of the plane from the air traffic controller given the coordinates of thepoint where it is located and the y-coordinate of the position of the plane at aparticular instance if its x-coordinate is given. Furthermore, ask them to describethe path of the plane as it goes around the airport. Challenge them to determinethe equation that would define the path of the plane. Let them realize that thedistance formula is related to the equation defining the plane’s path around theairport.DEPED COPYActivity 2: Is there a traffic in the air?Answer Key1. 50 km2. When x = 5, y = 49.75 or y = -49.75.When x = 10, y = 48.99 or y = -48.99.When x = 15, y = 47.7 or y = -47.4When x = 15, y = 47.7 or y = -47.4When x = -20, y = 45.83 or y = -45.83.Answ3e.rsNWKohe.eyItnisxn=ot-3p0o,ssyib=le4f0or or y = -40. the plane to be at a point whose x  coordinateis 60 because its distance from the air traffic controller would begreater than 50 km.4. The path is circular. x2  y 2  2500 Provide the students opportunity to come up with an equation that can beused in finding the radius of a circle. Ask them to perform Activity 3. In thisactivity, the students should be able to realize that the Distance Formula can beused in finding the radius of a circle. And that the distance of a point from thecenter of a circle is also the radius of the circle. 222 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 3: How far am I from my point of rotation?A.Answer Key 1. 8 units y 2. Yes, the circle will pass through 0,8 ,  8,0, and 0,8 because the distance from these points to x the center of the circle is 8 units. 3. No, because the distance from point M  4,6 to the center of the circleDEPED COPY is less than 8 units. No, because the distance from point N 9,2 to the center of the circle is more than 8 units. 4. 8 units; 8  0 = 8 5. If a point is on the circle, its distance from the center is equal to the radius. 6. Since the distance d of a point from the center of the circle is d  x2  y2 and is equal to the radius r, then r  x2  y2 or x2  y2  r 2. 223 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

B. y Answer Key 1. 61 units or approximately 7.81 units2. Yes, the circle will pass through x  2,7, 8,7 , and  3,4 because the distance from each of these points to the center of the circle is 61 units or approximately 7.81 units.DEPED COPY3. No, because the distance from point M  7,6to the center of the circle is more than 7.81 units.4. 61 units or approximately 7.81 units. Note: Evaluate students’ explanations.5. If the center of the circle is not at the origin, its radius can be determined by using the distance formula, d  x2  x12  y2  y12 . Since the distance of the point from the center of the circle is equal to the radius r, then r  x2  x12  y2  y12 or x2  x12  y2  y12  r 2 . If Px,y  is a point on the circle and Ch,k  is the center, then x2  x12  y2  y12  r 2 becomes x  h2  y  k2  r 2 . Before proceeding to the next activities, let the students give a briefsummary of the activities they have done. Provide them with an opportunity torelate or connect their responses in the activities given to their new lesson,equation of a circle. Let the students read and understand some important noteson equation of a circle. Tell them to study carefully the examples given.What to PROCESS Let the students use the mathematical ideas they have learned about theequation of a circle and the examples presented in the preceding section toperform the succeeding activities. 224 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

In Activity 4, the students will determine the center and the radius of eachcircle, given its equation. Then, the students will be asked to graph the circle. Askthem to explain how they determined the center and the radius of the circle.Furthermore, tell them to explain how to graph a circle given its equation indifferent forms. Strengthen students’ understanding of the graphs of circlesthrough the use of available mathematics freeware like Geogebra.Activity 4: Always Start at This Point!Answer Key 3. Center: 0,0 1. Center: 0,0 Radius: 10 units Radius: 7 unitsDEPED COPYyy xx2. Center: 5,6 4. Center:  7,1 Radius: 9 units Radius: 7 units y y x x 225 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Answer Key y 6. Center: 5,85. Center:  4,3 Radius: 11 units y Radius: 8 units x xDEPED COPY a. Note: Evaluate students’ responses. b. Determine first the center and the radius of the circle defined by the equation, then graph. If the given equation is in the form x 2  y 2  r 2 , the center is at the origin and the radius of the circle is r. If the given equation is in the form x  h2  y  k 2  r 2 , the center is at h,k  and the radius of the circle is r. If the given equation is in the form x2  y 2  Dx  Ey  F  0 , transform it into the form x  h2  y  k 2  r 2 . The center is at h,k  and the radius of the circle is r. Ask the students to perform Activity 5. This time, the students will write theequation of a circle given the center and the radius. Ask them to explain how todetermine the equation of a circle whether or not the center is the origin.Activity 5: What defines me?Answer Key a. Write the equation in the1. x2  y 2  144 form x 2  y 2  r 2 where the origin is the2. x  22  y  62  81 center and r is the radius of the circle.3. x  72  y  22  225 Write the equation in the4. x  42  y  52  505. x 102  y  82  27 form x  h2  y  k2  r 2 where h,k is the center and r is the radius of the circle. b. No, because the two circles have different radii. 226 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activities 6 and 7 provide students opportunities to write equations ofcircles from center-radius form or standard form to general form and vice-versa.At this point, ask them to explain how to transform the equation of a circle fromone form to another form and discuss the mathematics concepts or principlesapplied. Furthermore, challenge them to find a shorter way of transformingequation of a circle from general form to standard form and vice-versa.Activity 6: Turn Me into a General!Answer Key 6. x2  y 2 14x 15  01. x2  y2  4x  8y  16  0 7. x2  y 2  4y  45  0 8. x2  y 2  4x  96  02. x2  y 2  8x 18y  47  0 9. x2  y 2 10x 10y  23  0 10. x2  y 2  8x  8y  03. x2  y 2 12x  2y  44  04. x2  y 2 16x 14y 112  05. x2  y 2 10y 11 0Note: Evaluate students’ explanations.DEPED COPYActivity 7: Don’t Treat this as a Demotion!Answer Key 4. x2  y  42  100 Center: 0,4 1. x 12  y  42  64 Center: 1,4 Radius: 10 units Radius: 8 units 5.  x  2 2   y  1 2  4  3  3 2. x  22  y  22  36 Center:  2 , 1  Center:  2,2  3 3  Radius: 6 units Radius: 2 units 3. x  52  y  22  32 6.  x  5 2   y  3 2  9  2  2 Center:  5,2 Center:  5 , 3  Radius: 4 2 units 2 2 Radius: 3 unitsa. Grouping the terms, then applying completing the square, addition property of equality and factoring. 227 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY b. Completing the square, Addition Property of Equality, Square of a Binomial c. Using the values of D, E, and F in the general equation of a circle, x2  y2  Dx  Ey  F  0 , to find the center (h,k) and radius r. The GeoGebra freeware can also be used for verification.What to REFLECT on and UNDERSTAND: Ask the students to have a closer look at some aspects of the equation ofa circle. Provide them with opportunities to think deeply and test further theirunderstanding of the equation of a circle by doing Activities 8 and 9. Give morefocus on the real-life applications of the equation of a circle.Activity 8: A Circle? Why not? Answer Key 1. No. x2  y2  2x  8y  26  0 can be written as x  12  y  42  9 . Notice that -9 cannot be expressed as a square of another number. 2. Yes. x2  y2  9  4x  10y can be written as x  22  y  52  20. 3. No. x2  y2  6x  8y  32 is not an equation of a circle. Its graph is not also a circle. 4. No. x2  y2  8x  14y  65  0 is merely a point. The radius must be greater than 0 for a circle to exist.Activity 9: Find Out More! Answer Key 1. x  32  y  82  81 2. x 102  y  72  36 or x 102  y  52  36 3.  3x  5y  7 4. x  52  y  52  13 5. a. x  32  y  42  100 b. Yes, because point 11,6 is still within the critical area. c. Follow the advice of PDRRMC. d. (Evaluate students’ responses/explanations.) 228 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Answer Key 6. a. Wise Tower - x  52  y  32  81 Global Tower - x  32  y  62  16 Star Tower - x 122  y  32  36 b. 12,2 - Star Tower  6,7 - Wise Tower 2,8 - Global Tower 1,3 - Wise and Global Tower c. Many possible answers. Evaluate students’ responses. Before the students move to the next section of this lesson, give a short test (formative test) to find out how well they understood the lesson. Ask them also to write a journal about their understanding of the equation of a circle. Refer to the Assessment Map. What to TRANSFER Give the students opportunities to demonstrate their understanding of the equation of a circle by doing a practical task. Let them perform Activity 10. You can ask the students to work individually or in a group. In Activity 10, the students will paste some small pictures of objects on grid paper and position them at different coordinates. Then, the students will draw circles that contain these pictures. Using the pictures and the circles drawn on the grid, they will formulate problems involving the equation of the circle, and then solve them. Activity 10: Let This be a Part of My Scrapbook! Answer Key Evaluate students’ answers. You may use the rubric. 229 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYSummary/Synthesis/Generalization: This lesson was about the equation of circles. The lesson provided thestudents with opportunities to illustrate the center-radius form of the equation of acircle, determine the center and the radius of a circle given its equation and viceversa, write the equation of a circle from standard form to general form and vice-versa, graph circles on the coordinate plane, and solve problems involving theequation of circles. Moreover, they were given the opportunity to formulate andsolve real-life problems involving the equation of a circle through the practical taskperformed. Their understanding of this lesson and other previously learnedmathematics concepts and principles will facilitate their learning of other relatedmathematics concepts. 230 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

SUMMATIVE TESTPart IChoose the letter that you think best answers the question.1. Which of the following is NOT a formula for finding the distance between twopoints on the coordinate plane?A. d  x2  x1 C. d  x2  x12  y2  y12B. d  y2  y1 D. d  x2  x12  y2  y12DEPED COPY2. A map is drawn on a grid where 1 unit is equivalent to 2 km. On the samemap, the coordinates of the point corresponding to San Rafael is (1,4).Suppose San Quintin is 20 km away from San Rafael. Which of the followingcould be the coordinates of the point corresponding to San Rafael?A. (17,16) B. (17,10) C. (9,10) D. (-15,16)3. Let M and N be points on the coordinate plane as shown in the figure below. y xIf the coordinates of M and N are 5,7 and 5,4, which of the followingwould give the distance between the two points?A. 7  4 B. 7  5 C.  4  7 D.  4  54. Point Q is the midpoint of ST . Which of the following is true about ST?A. ST  QS  QT C. ST  2QS  QTB. ST  QS  QT D. ST  2QS  QT5. The distance between points Mx,5 and C5,1 is 10 units. What is the x-coordinate of M if it lies in the second quadrant?A. -7 B. -3 C. -1 D. 13 231 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

6. What is the distance between points D(-10,2) and E(6,10)?A. 16 B. 20 C. 10 2 D. 8 57. Which of the following equation describes a circle on the coordinate plane with a center at 2,3 and a radius of 5 units?A. x  22  y  32  252 C. x  32  y  22  252B. x  22  y  32  52 D. x  22  y  32  528. Which of the following would give the coordinates of the midpoint of P(-6,13) and Q(9,6)?  6 13 9  6  6 13 9  6A.DEPED COPY2 , 2  C.  2 , 2     B.   6 9 , 13  6  D.   6 9 , 13  6   2 2   2 2 9. The endpoints of a segment are (-5,2) and (9,12), respectively. What are thecoordinates of its midpoint?A. (7,5) B. (2,7) C. (-7,5) D. (7,2)10. The coordinates of the vertices of a rectangle are W  2,6, I10,6,N10,3, and D 2,3. What is the length of a diagonal of the rectangle?A. 7.5 B. 9 C. 12 D. 1511. The coordinates of the vertices of a triangle are G 4,2, O5,1 , andT 10,8. What is the length of the segment joining the midpoint of GT andO?A. 2 10 B. 58 C. 3 10 D. 10612. The endpoints of a diameter of a circle are E 6,8 and G4,2. What is thelength of the radius of the circle?A. 10 2 B. 5 2 C. 2 10 D. 1013. What proof uses figures on a coordinate plane to prove geometric properties?A. Indirect Proof C. Coordinate ProofB. Direct Proof D. Two-Column Proof14. What figure is formed when the points K(-2,10), L(8,8), M(6,2), and N(-4,4)are connected consecutively?A. Trapezoid B. Parallelogram C. Square D. Rectangle 232 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

15. Three speed cameras were installed at different points along an expressway.On a map drawn on a coordinate plane, the coordinates of the first speedcamera are (-2,4). Suppose the second camera is exactly between the othertwo and its coordinates are (12,8). What are the coordinates of the third speedcamera?A. (26,12) B. (26,16) C. (22,12) D. (22,16)16. In the equilateral triangle below, what are the coordinates of P? A. 0,2a B. 2a,0  C. 0,a 3  D. 0,a 2DEPED COPY17. Jose, Andres, Emilio, and Juan live in different barangays of Magiting town as shown on the coordinate plane below. Andres Jose Town Hall Juan EmilioWho lives the farthest from the Town Hall if it is located at the origin?A. Jose B. Andres C. Emilio D. Juan 233 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

18. What is the center of the circle x2  y2  4x  6y  36  0 ?A. (9,-3) B. (3,-2) C. (2,-3) D. (2,-10)19. A radius of a circle has endpoints  4,3 and 1,2. What is the equationthat defines the circle if its center is at the second quadrant?A. x  12  y  22  50 C. x  42  y  32  50B. x  12  y  22  50 D. x  42  y  32  5020. A radio signal can transmit messages up to a distance of 5 km. If the radiosignal’s origin is located at a point whose coordinates are (-2,7). What is theequation of the circle that defines the boundary up to which the messagesDEPED COPYcan be transmitted?A. x  22  y  72  25 C. x  22  y  72  25B. x  22  y  72  5 D. x  22  y  72  5Part IIDirections: Solve each of the following problems. Show your complete solutions.1. A tracking device that is installed in a mobile phone indicates that its user is located at a point whose coordinates are (18,14). In the tracking device, each unit on the grid is equivalent to 7 km. If the phone user came from a place whose coordinates are (2,6)? How far has he travelled?2. The equation that represents the transmission boundaries of a cellular phone tower is x2  y2  10x  2y  199  0 . What is the greatest distance, in kilometers, can the signal of the tower be transmitted?Rubric for Problem Solving 4 3 2 1Used an Used an Used an Attempted to solveappropriate appropriate appropriate the problem butstrategy to come strategy to come strategy but came used anup with correct up with a solution, up with an entirely inappropriatesolution and but a part of the wrong solution strategy that led toarrived at a solution led to an that led to an a wrong solutioncorrect answer incorrect answer incorrect answer 234 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Part III A: GRASPS AssessmentPerform the following.Goal: To prepare emergency measures to be undertaken in times of natural calamities and disasters particularly typhoons and floodsRole: Radio Group Chairman of the Municipal Disaster and Risk Management CommitteeAudience: Municipal and Barangay Officials and VolunteersDEPED COPYSituation:Typhoons and floods frequently affect your municipality during rainy seasons. For the past years, losses of lives and damages to properties have occurred. Because of this, your municipal mayor designated you to chair the Radio Group of the Municipal Disaster and Risk Management Committee to warn the residents of your municipality of any imminent natural calamities and disasters like typhoons and floods. The municipal government gave your group a number of two-way radios and antennas to be installed in strategic places in the municipality. These shall be used as the need arises. As chairman of the Radio Group, you were tasked to prepare emergency measures that you will undertake to reduce if not to avoid losses of lives and damages to properties during rainy seasons. These include the positioning of the different two-way radios and antennas for communication and coordination among the members of the Radio Group. You were also asked to prepare a grid map of your municipality showing the positions of the two- way radios and antennas.Products: 1. Emergency Measures to be undertaken in times of natural calamities and disasters 2. Grid map of your municipality showing the locations of the different two-way radios and antennasStandards: The emergency measures must be clear, relevant, and systematic. The grid map of the municipality must be accurate, presentable, and appropriate. 235 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Rubric for the Prepared Emergency Measures 4 3 2 1The emergency The emergency The emergency The emergencymeasures are measures are measures are measures are notclearly presented, clearly presented clearly presented clearly presented,relevant to the and relevant to but not relevant to not relevant to thesituation, and the situation but the situation and situation, and notsystematic. not systematic. not systematic. systematic.Rubric for Grid Map of the MunicipalityDEPED COPY 4 3 2 1The grid map is The grid map is The grid map is The grid map isaccurately made, accurately made not accurately not accuratelyappropriate, and and appropriate made but made and notpresentable. but not appropriate. appropriate. presentable.Part III BUse the prepared grid map of the municipality in Part III A in formulatingproblems involving plane coordinate geometry, then solve.Rubric on Problems Formulated and SolvedScore Descriptors 6 Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, 5 shows in-depth comprehension of the pertinent concepts 4 and/or processes, and provides explanations wherever 3 appropriate. Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in- depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes most significant parts of the solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or 236 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Score Descriptors misinterprets less significant ideas or details. Poses a problem and finishes some significant parts of the 2 solution and communicates ideas unmistakably but shows gaps on theoretical comprehension. 1 Poses a problem but demonstrates minor comprehension, not being able to develop an approach.Source: D.O. #73, s. 2012Answer Key Part I Part II (Use the rubric to rate students’ works/outputs) 1. 56 5 km 1. C 2. 15 km 2. C Part III A (Use the rubric to rate students’ works/outputs) 3. C Part III B (Use the rubric to rate students’ works/outputs) 4. ADEPED COPY 11. A 5. B 6. D 12. B 7. D 13. C 8. B 14. B 9. B 15. A]10. D 16. C 17. C 18. C 19. C 20. CGlossary of TermsCoordinate Proof – a proof that uses figures on a coordinate plane to provegeometric relationships.Distance Formula – an equation that can be used to find the distance betweenany pair of points on the coordinate plane. The distance formula isd  x2  x12  y2  y12 or PQ  x2  x12  y2  y12 , if Px1,y1 andQx2,y2 are points on a coordinate plane.Horizontal Distance (between two points) – the absolute value of the differenceof the x-coordinates of two pointsMidpoint – a point on a line segment that divides the same segment into twoequal parts.Midpoint Formula – a formula that can be used to find the coordinates of themidpoint of a line segment on the coordinate plane. The midpoint of Px1,y1and Qx2 , y2  is  x1  x2 , y1  y2  .  2 2  237 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYThe General Equation of a Circle – the equation of a circle obtained byexpanding x  h2  y  k 2  r 2 . The general equation of a circle isx2  y 2  Dx  Ey  F  0 , where D, E, and F are real numbers.The Standard Equation of a Circle – the equation that defines a circle withcenter at (h, k) and a radius of r units. It is given by  x  h2  y  k 2  r 2.Vertical Distance (between two points) – the absolute value of the difference ofthe y-coordinates of two points.DepEd INSTRUCTIONAL MATERIALS THAT CAN BE USED AS ADDITIONALRESOURCES:1. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third Year Mathematics. Plane Coordinate Geometry. Module 20: Distance and Midpoint Formulae2. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third Year Mathematics. Plane Coordinate Geometry. Module 22: Equation of a Circle3. Distance Learning Module (DLM) 3, Module 3: Plane Coordinate Geometry.4. EASE Modules Year III, Module 2: Plane Coordinate GeometryReferences and Website Links Used in This Module:References:Bass, L. E., Charles, R. I., Hall, B., Johnson, A., & Kennedy, D. (2008) Texas Geometry. Boston, Massachusetts: Pearson Prentice Hall.Bass, L. E., Hall, B.R., Johnson, A., & Wood, D. F. (1998) Prentice Hall Geometry Tools for a Changing World. NJ, USA: Prentice-Hall, Inc.Boyd, C., Malloy, C. & Flores. (2008) Glencoe McGraw-Hill Geometry. USA: The McGraw-Hill Companies, Inc.Callanta, M. M. (2012) Infinity, Worktext in Mathematics III. Makati City: EUREKA Scholastic Publishing, Inc. 238 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYChapin, I., Landau, M. & McCracken. (1997) Prentice Hall Middle Grades Math, Tools for Success. Upper Saddle River, New Jersey: Prentice-Hall, Inc. Cifarelli, V. (2009) cK-12 Geometry, Flexbook Next Generation Textbooks. USA: Creative Commons Attribution-Share Alike. Clemens, S. R., O’Daffer, P. G., Cooney, T. J., & Dossey, J. A. (1990) Addison- Wesley Geometry. USA: Addison-Wesley Publishing Company, Inc. Clements, D. H., Jones, K. W., Moseley, L.G., & Schulman, L. (1999) Math in my World. New York: McGraw-Hill Division. Department of Education. (2012) K to 12 Curriculum Guide Mathematics. Philippines. Gantert, A. X. (2008) AMSCO’s Geometry. NY, USA: AMSCO School Publications, Inc. Renfro, F. L. (1992) Addison-Wesley Geometry Teacher’s Edition. USA: Addison-Wesley Publishing Company, Inc. Rich, B. & Thomas, C. (2009) Schaum’s Outlines Geometry Fourth Edition. USA: The McGraw-Hill Companies, Inc. Smith, S. A., Nelson, C.W., Koss, R. K., Keedy, M. L., & Bittinger, M. L. (1992) Addison-Wesley Informal Geometry. USA: Addison-Wesley Publishing Company, Inc. Wilson, P. S. (1993) Mathematics, Applications and Connections, Course I. Westerville, Ohio: Glencoe Division of Macmillan/McGraw-Hill Publishing Company. 239 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYWebsite Links as References and Sources of Learning Activities:CliffsNotes. Midpoint Formula. (2013). Retrieved fromhttp://www.cliffsnotes.com/math/geometry/coordinate-geometry/midpoint-formulaCliffsNotes. Distance Formula. (2013). Retrieved fromhttp://www.cliffsnotes.com/math/geometry/coordinate-geometry/distance-formulaMath Open Reference. Basic Equation of a Circle (Center at 0,0). (2009).Retrieved from http://www.mathopenref.com/ coordbasiccircle.htmlMath Open Reference. Equation of a Circle, General Form (Center anywhere).(2009). Retrieved from http://www.mathopenref.com/coordgeneralcircle.htmlMath-worksheet.org. Using equations of circles. (2014). Retrieved fromhttp://www.math-worksheet.org/using-equations-of-circlesMath-worksheet.org. Writing equations of circles. (2014). Retrieved fromhttp://www.math-worksheet.org/writing-equations-of-circlesRoberts, Donna. Oswego City School District Regents exam Prep Center.Geometry Lesson Page. Midpoint of a Line Segment. (2012). Retrieved fromhttp://www.regentsprep.org/Regents/ math/geometry/GCG2/ Lmidpoint.htmRoberts, Donna. Oswego City School District Regents exam Prep Center.Geometry Lesson Page. Midpoint of a Line Segment. (2012). Retrieved fromhttp://www.regentsprep.org/Regents/math/geometry/GCG3/ Ldistance.htmStapel, Elizabeth. \"Conics: Circles: Introduction & Drawing.\" Purplemath.Retrieved from http://www.purplemath.com/modules/ circle.htmWebsite Links for Videos:Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrievedfrom https://www.khanacademy.org/math/geometry/ cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theoremKhan Academy. Completing the square to write equation in standard form of acircle. Retrieved from https://www.khanacademy.org/math/ geometry/cc-geometry-circles/equation-of-a-circle/v/completing-the-square-to-write-equation-in-standard-form-of-a-circleKhan Academy. Equation for a circle using the Pythagorean Theorem. Retrievedfrom https://www.khanacademy.org/math/geometry/ cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theorem 240 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYKhan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved from https://www.khanacademy.org/math/geometry/ cc-geometry- circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theorem Ukmathsteacher. Core 1 – Coordinate Geometry (3) – Midpoint and distance formula and Length of Line Segment. Retrieved from http://www.youtube.com/watch?v=qTliFzj4wuc VividMaths.com. Distance Formula. Retrieved from http://www.youtube.com/watch?v=QPIWrQyeuYw Website Links for Images: asiatravel.com. Pangasinan Map. Retrieved from http://www.asiatravel.com/philippines/pangasinan/pangasinanmap.jpg DownTheRoad.org. Pictures of, Chengdu to Kangding, China Photo, Images, Picture from. (2005). Retrieved from http://www.downtheroad.org/Asia/Photo/ 9Sichuan_China_Image/3Chengdu_Kangding_China.htm funcheap.com. globe-map-wallpapers_5921_1600[1]. Retrieved from http://sf.funcheap.com/hostelling-internationals-world-travel-101-santa- clara/globe-map-wallpapers_5921_16001/ Hugh Odom Vertical Consultants. eleven40 theme on Genesis Framework· WordPress. Cell Tower Development – How Are Cell Tower Locations Selected? Retrieved from http://blog.thebrokerlist.com/cell-tower-development-how-are-cell- tower-locations-selected/ LiveViewGPS, Inc. GPS Tracking PT-10 Series. (2014). Retrieved from http://www.liveviewgps.com/gps+tracking+device+pt-10+series.html Sloan, Chris. Current \"1991\" Air Traffic Control Tower at Amsterdam Schiphol Airport – 2012. (2012). Retrieved from http://airchive.com/html/airplanes-and- airports/amsterdam-schipol-airport-the-netherlands-/current-1991-air-traffic- control-tower-at-amsterdam-schiphol-airport-2012-/25510 wordfromthewell.com. Your Mind is Like an Airplane. (2012). Retrieved from http://wordfromthewell.com/2012/11/14/your-mind-is-like-an-airplane/ 241 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYVISIT DEPED TAMBAYANhttp://richardrrr.blogspot.com/1. Center of top breaking headlines and current events related to Department of Education.102. Offers free K-12 Materials you can use and share Mathematics Teacher’s Guide Unit 3 This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at [email protected]. We value your feedback and recommendations. Department of Education Republic of the Philippines All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Mathematics – Grade 10Teacher’s GuideFirst Edition 2015 Republic Act 8293, section 176 states that: No copyright shall subsist in any workof the Government of the Philippines. However, prior approval of the government agency oroffice 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 ofroyalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names,trademarks, etc.) included in this book are owned by their respective copyright holders.DepEd is represented by the Filipinas Copyright Licensing Society (FILCOLS), Inc. in seekingpermission to use these materials from their respective copyright owners. . All means havebeen exhausted in seeking permission to use these materials. The publisher and authors donot represent nor claim ownership over them. Only institutions and companies which have entered an agreement with FILCOLSand only within the agreed framework may copy this Teacher’s Guide. Those who have notentered in an agreement with FILCOLS must, if they wish to copy, contact the publishers andauthors directly. Authors and publishers may email or contact FILCOLS at [email protected] or(02) 439-2204, respectively.Published by the Department of EducationSecretary: Br. Armin A. Luistro FSCUndersecretary: Dina S. Ocampo, PhDDEPED COPY Development Team of the Teacher’s GuideConsultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, andRosemarievic Villena-Diaz, PhDAuthors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D.Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B.Orines, Rowena S. Perez, and Concepcion S. TernidaEditor: Maxima J. Acelajado, PhDReviewers: Carlene P. Arceo, PhD, Rene R. Belecina, PhD, Dolores P.Borja, Maylani L. Galicia, Ma. Corazon P. Loja, Jones A. Tudlong, PhD, andReymond Anthony M. QuanIllustrator: Cyrell T. NavarroLayout Artists: Aro R. Rara, Jose Quirovin Mabuti, and Ronwaldo Victor Ma.A. PagulayanManagement and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr.,Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr.Printed in the Philippines by REX Book StoreDepartment of Education-Instructional Materials Council Secretariat (DepEd-IMCS)Office Address: 5th Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600Telefax: (02) 634-1054, 634-1072E-mail Address: [email protected] All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Introduction This Teacher’s Guide has been prepared to provide teachers of Grade 10 Mathematics with guidelines on how to effectively use the Learner’s Material to ensure that learners will attain the expected content and performance standards. This book consists of four units subdivided into modules which are further subdivided into lessons. Each module contains the content and performance standards and the learning competencies that must be attained and developed by the learners which they could manifest through their products and performances. The special features of this Teacher’s Guide are: A. Learning Outcomes. Each module contains the content and performance standards and the products and/ or performances expected from the learners as a manifestation of their understanding. B. Planning for Assessment. The assessment map indicates the type of assessment and categorized the objectives to be assessed into knowledge, process/skills, understanding, and performance C. Planning for Teaching-Learning. Each lesson has Learning Goals and Targets, a Pre-Assessment, Activities with answers, What to Know, What to Reflect on and Understand, What to Transfer, and Summary / Synthesis / Generalization. D. Summative Test. After each module, answers to the summative test are provided to help the teachers evaluate how much the learners have learned. E. Glossary of Terms. Important terms in the module are defined or clearly described. F. References and Other Materials. This provides the teachers with the list of reference materials used, both print and digital. We hope that this Teacher’s Guide will provide the teachers with the necessary guide and information to be able to teach the lessons in a more creative, engaging, interactive, and effective manner. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Table of Contents Curriculum Guide: Mathematics Grade 10 Unit 3 Module 6: Permutations and Combinations......................................... 242 Learning Outcomes ............................................................................................242 Planning for Assessment....................................................................................243 Planning for Teaching-Learning .........................................................................246 Pre-Assessment .................................................................................................248 Learning Goals and Targets ...............................................................................248 Lesson 1: Permutations ........................................................................................248 Activity 1 ..................................................................................................249 Activity 2 ..................................................................................................250 Activity 3 ..................................................................................................252 Activity 4 ..................................................................................................253 Activity 5 ..................................................................................................255 Activity 6 ..................................................................................................256 Activity 7 ..................................................................................................256 Activity 8 ..................................................................................................257 Activity 9 ..................................................................................................258 Summary/Synthesis/Generalization ...................................................................259 Lesson 2: Combination..........................................................................................259 Activity 1 ..................................................................................................259 Activity 2 ..................................................................................................260 Activity 3 ..................................................................................................261 Activity 4 ..................................................................................................265 Activity 5 ..................................................................................................265 Activity 6 ..................................................................................................265 Activity 7 ..................................................................................................267 Activity 8 ..................................................................................................268 Activity 9 ..................................................................................................269 Summary/Synthesis/Generalization ...................................................................270 Summative Test .......................................................................................................271 Glossary of Terms...................................................................................................277 References and Website Links Used in This Module ....................................278 Module 7: Probability of Compound Events......................................... 279 Learning Outcomes ............................................................................................279 Planning for Assessment....................................................................................280 Planning for Teaching and Learning ..................................................................285 Pre-Assessment .................................................................................................286 Learning Goals and Targets ...............................................................................288 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Lesson 1: Union and Intersection of Events....................................................288 Activity 1 ..................................................................................................288 Activity 2 ..................................................................................................289 Activity 3 ..................................................................................................290 Activity 4 ..................................................................................................291 Activity 5 ..................................................................................................292 Activity 6 ..................................................................................................292 Activity 7 ..................................................................................................293 Activity 8 ..................................................................................................294 Activity 9 ..................................................................................................295 Summary/Synthesis/Generalization ...................................................................295 Lesson 2: Independent and Dependent Events ..............................................296 Activity 1 ..................................................................................................296 Activity 2 ..................................................................................................298 Activity 3 ..................................................................................................298 Activity 4 ..................................................................................................298 Activity 5 ..................................................................................................299 Activity 6 ..................................................................................................299 Summary/Synthesis/Generalization ...................................................................300 Lesson 3: Conditional Probability ......................................................................300 Activity 1 ..................................................................................................300 Activity 2 ..................................................................................................301 Activity 3 ..................................................................................................302 Activity 4 ..................................................................................................303 Activity 5 ..................................................................................................304 Activity 6 ..................................................................................................305 Activity 7 ..................................................................................................305 Summary/Synthesis/Generalization ...................................................................306 Summative Test .......................................................................................................307 Glossary of Terms...................................................................................................312 References and Website Links Used in This Module ....................................313 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.