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Comprehensive Grade 10 Mathematics Lesson on Transformations in the Cartesian Plane

Explore Cartesian plane points, and transformations including translations, reflections, and rotations with practical activities and art integration for Grade 10 learners.

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Lesson Exemplar for Mathematics Lesson 4 10 Quarter 1
Lesson Exemplar for Mathematics Grade 10 Quarter 1: Lesson 4 (Week 4) SY 2026-2027 This material is intended exclusively for the use of teachers participating in the pilot implementation of the MATATAG K to 10 Curriculum during the School Year 2024-2025. It aims to assist in delivering the curriculum content, standards, and lesson competencies. Any unauthorized reproduction, distribution, modification, or utilization of this material beyond the designated scope is strictly prohibited and may result in appropriate legal actions and disciplinary measures. Borrowed contents included in this material are owned by their respective copyright holders. Every effort has been made to locate and obtain permission to use these materials from their respective copyright owners. The publisher and development team do not represent nor claim ownership over them. Every care has been taken to ensure the accuracy of the information provided in this material. For inquiries or feedback, please write or call the Office of the Director of the Bureau of Learning Resources via telephone numbers (02) 8634-1072 and 8631-6922 or by email at blr.od@deped.gov.ph. Development Team Writer: • PNU - RITQ Development Team Validator: • PNU - RITQ Development Team Management Team Philippine Normal University Research Institute for Teacher Quality SiMERR National Research Centre
1 MATHEMATICS / QUARTER 1 / GRADE 10 I. CURRICULUM CONTENT, STANDARDS, AND LESSON COMPETENCIES A. Content Standards The learners demonstrate knowledge and understanding of the laws of sines and the laws of cosines. B. Performance Standards By the end of the quarter, the learners are able to find sides and angles in oblique triangles using the laws of sines and the laws of cosines. (MG) C. Learning Competencies and Objectives Learning Competency 1. Describe the position of points in the Cartesian plane. 2. Describe translations, reflections, and rotations in the Cartesian plane using coordinates. Learning Objectives At the end of the lesson, the students are expected to: 1. Describe the position of points in the Cartesian plane relative to its coordinates. 2. Describe how to translate, reflect, or rotate a given figure on a Cartesian plane. 3. Evaluate whether a certain transformation is a translation, reflection, or rotation. 4. Design an artwork using translation, reflection, rotation, or any combination of the three. D. Content Topic: Transformation Sub-Topic 1: Points on the Cartesian Plane Sub-Topic 2: Translations, Reflections and Rotation E. Integration Art Designs II. LEARNING RESOURCES Kumar, A., & Kumar, A. (2024, February 18). What is Transformation Geometry? Definition, Types, Rule, Example. SplashLearn - Math Vocabulary. https://www.splashlearn.com/math-vocabulary/geometry/transformation-geometry Kuta Software. Geometry. Translation. https://cdn.kutasoftware.com/Worksheets/Geo/12-Translations.pdf
2 Kuta Software. Infinite Geometry. Reflection https://cdn.kutasoftware.com/Worksheets/Geo/12-Reflections.pdf Kuta Software. Infinite Geometry. Rotations https://cdn.kutasoftware.com/Worksheets/Geo/12-Rotations.pdf Transformations in Math and Art – NCMALEarn. (n.d.). https://learn.ncartmuseum.org/lesson-plans/transformations-in-math-and-art/ Math-Aids.Com. WORKSHEET: The Cartesian Plane https://ellengillett.weebly.com/uploads/9/9/0/8/99088664/8_maths_coordinate_geometry_worksheets_combined.pdf TRANSFORMATIONS RULE SHEET. https://www.rcsdk12.org/site/handlers/filedownload.ashx?moduleinstanceid=53114&dataid=41679&FileName=Transformations%20Cheat%20Sheet.pd f Transformation. (2024). In Merriam-Webster Dictionary. https://www.merriam-webster.com/dictionary/transformation III. TEACHING AND LEARNING PROCEDURE NOTES TO TEACHERS A. Activating Prior Knowledge Day 1 1. Short Review Activity 1. Remember the Plot Plot each of the given points on the same Cartesian plane. Name the quadrant that the point lies in. A(1, 5) Quadrant: I B(7, 0) Quadrant: x-axis C(-1, 3) Quadrant: _II____ D(-5, -9) Quadrant: __III___ E(2, -4) Quadrant: __IV___ F(0, 1) Quadrant: __y-axis___ G(-8, 6) Quadrant: ___II__ H(6, 10) Quadrant: _I____ I(-7, -7) Quadrant: __III___ J(0, 0) Quadrant: _origin____ The teacher might need to discuss in detail the concepts shown in each figure for students who have already forgotten them. Note: This concept was taken up in Grade 8. Answer Key:
3 2. Feedback The answers will be discussed in the Worked Example part. B. Establishing Lesson Purpose 1. Lesson Purpose The concept of transformation of a figure, whether in its size, position, or orientation, can be expressed mathematically. The concepts of translation, reflection, and rotation are abundantly manifested all around us, especially in nature, the arts, and designs. 2. Unlocking Content Vocabulary Transformation - a change in any form or appearance. - the operation of changing (as by rotation or mapping) one configuration or expression into another in accordance with a mathematical rule especially: a change of variables or coordinates in which a function of new variables or coordinates is substituted for each original variable or coordinate - maps an initial image, called a pre-image, onto a final image, called an image. a. Translation -is a change that moves all points of a figure (image) the same distance in the same direction. b. Reflection - is a change represented by a flip of a figure (image) over a “mirror,” which usually is a line but may also be a point, or a plane (which are called lines of symmetry, points of symmetry, or planes of symmetry (literally mirrors). c. Rotation - is a change that turns every point of a preimage through a specified angle and direction about a fixed point (called the center of rotation). d. Dilation - is a change in the size of a figure that requires a center point and a scale factor. (This will not be discussed in detail in this lesson.) The teacher may guide students in defining the terms using the layman’s definition of the term (i.e. mentioning Transformers, the movie series). https://www.merriam- webster.com/dictionary/transf ormation C. Developing and Deepening Understanding SUB-TOPIC 1: Points on the Cartesian Plane 1. Explicitation Key concepts: 1. So-called because of Rene Descartes. 2. The main parts are the x- and y-axes, and the origin. (Figure 1) 3. An ordered pair (x, y) means a location (a point) on the Cartesian plane. (Figure 2)
4 4. The Cartesian plane is divided into 4 quadrants. Quadrants are labeled in a counterclockwise manner. Note the signs of the coordinates of a point in each quadrant. (Figure 3) Figure 1 Figure 2 Figure 3 2. Worked Example Plot each of the given points on the same Cartesian plane. Name the quadrant that the point lies in. a. A(1, 5) Quadrant: I b. B(7, 0) Quadrant: none, x-axis c. C(-1, 3) Quadrant: II d. D(-5, -9) Quadrant: III e. E(2, -4) Quadrant: IV f. F(0, 1) Quadrant: none, y-axis g. G(-8, 6) Quadrant: II h. H(6, 10) Quadrant: I i. I(-7, -7) Quadrant: III j. J(0, 0) Quadrant: none, origin Note that in b, f, and j, the points fall on the x- or y-axis, so, they do not belong to any quadrant, name the axis itself or its specific name (i.e., origin), if there is any.
5 3. Lesson Activity Activity 2: Label Me! I. Label the parts of the Cartesian plane below using the boxes provided. Then, identify the ordered pair for each point. II. In each of the items below, the blue figure is the preimage (initial image) and the green figure is the image (transformed image). Describe how the change was made. Choose from the words translation, rotation, reflection, or dilation for the answer (using the layman’s definition of each of the words*). 1. 2. 3. 4. Answer to Activity 2: Label Me! *if the content vocabulary has not been discussed yet Allow the student to guess. Answers: 1. Rotation 2. Reflection 3. Dilation 4. Dilation 5. Translation 6. Translation 7. Reflection 8. Reflection or Rotation 9. Reflection** 10. Reflection or Rotation **Rotation is not an answer.
6 5. 6. 7. 8. 9. 10. DAY 2 SUB-TOPIC 2: Translations 1. Explicitation From Part II of yesterday’s Activity 2, justify the answer for each item. 1. Rotate 900 counterclockwise 2. Flip or reflect over the x-axis 3. Reduce in size (dilation) 4. Enlarge the figure
7 5. Move downwards to the left 6. Move upwards to the right 8. Flip or reflect over the y-axis*** 7. Flip or reflect over the y-axis*** 9. Flip or reflect over y = x 10. Flip or reflect over the y-axis Translation - is a change that moves all points of a figure (image) the same distance in the same direction. Examples are numbers 5 and 6. Notice that the blue figures are moved in a certain direction, where the green figure is. 2. Worked Example Translate the following figure accordingly. Then write the coordinates of the new figure. a. ***line of symmetry cuts across the middle sun horizontally (reflection), or rotate 1800.
8 b. c. d. Write a rule to describe the transformation. translation: one unit to the left, one unit down or (x, y) → (x - 1, y - 1) or (-1, -1)
9 3. Lesson Activity Activity 3. Translation A. Translate the following figure accordingly. Then write the coordinates of the new figure. 1. 3. 2. 4. translation: (x, y) → (x - 2, y - 4) X(-1, 1), K (3, 3), T(3, -1) B. Write a rule to describe the transformation. 5.
10 Day 3 SUB-TOPIC 3: Reflections 1. Explicitation Reflection - is a change represented by a flip of a figure (image) over a “mirror,” which usually is a line but may also be a point, or a plane (which are called lines of symmetry, points of symmetry, or planes of symmetry (literally mirrors). Examples are 2, 7, 8, 9, and 10 of Activity 2. Notice that the figures are mirror images of each other. The “mirror” is called the line of symmetry. It is where the fold can be made to create mirror images. Below are some more examples. 2. Worked Example a. Graph the image of the figure using the given transformation. 1. 2. In the additional examples (car brand logos), explain that the broken lines are the lines of symmetry. Think of the lines of symmetry as a mirror or the fold line so it will result in two same halves of the figure. Note also that there might be more than one mirror.
11 3. b. Write a rule to describe the transformation. Reflection across y = 2 3. Lesson Activity a. Graph the image of the figure using the given transformation. 1. 2.
12 b. Write a rule to describe each transformation. 1. 3. 2. 4. DAY 4 SUB-TOPIC 4: Rotation 1. Explicitation Rotation - is a change that turns every point of a preimage through a specified angle and direction about a fixed point (called the center of rotation). Examples are 1, 7, 8, and 10 of Activity 2. If the fixed point is the origin (-0, 0) and the specified angle is: a. 1800 - (x, y) → (-x, -y), the numbers stay but take the opposite sign of the number b. 900 or 2700 - (x, y) → (y, x), the numbers are interchange and the sign will be dependent on which quadrant the point lies. Note that 2700 can also mean 900 c. 900 counterclockwise.
13 2. Worked Example a. Graph the image of the figure using the given transformation. 1. 2. 3. 3. Lesson Activity Students will work for items 3 and 4 of Part II to identify the solutions to the given oblique triangles.
14 k. Making Generalizations Learners’ Takeaways What must be given in an oblique triangle to solve for the unknown using Sine Law? How will you determine if in a given oblique triangle there can be one, two or no solution? Reflection on Learning Teachers may use this self-assessment activity by asking students to rate their level of understanding. IV. EVALUATING LEARNING: FORMATIVE ASSESSMENT AND TEACHER’S REFLECTION NOTES TO TEACHERS A. Evaluating Learning Formative Assessment Choose items from the links provided in the section Notes Teachers. 1. Homework (Optional) Choose items from the links provided in the section Notes Teachers. For formative assessment and homework, the teacher may ask the students to answer the activities found in these links: https://cdn.kutasoftware.c om/Worksheets/Geo/12- Translations.pdf https://cdn.kutasoftware.c om/Worksheets/Geo/12- Reflections.pdf https://cdn.kutasoftware.c om/Worksheets/Geo/12- Rotations.pdf Fully Understand Confused No Mastery
15 B. Teacher’s Remarks Note observations on any of the following areas: Effective Practices Problems Encountered The teacher may take note of some observations related to the effective practices and problems encountered after utilizing the different strategies, materials used, learner engagement, and other related stuff. Teachers may also suggest ways to improve the different activities explored/lesson exemplar. strategies explored materials used learner engagement/ interaction Others C. Teacher’s Reflection Reflection guide or prompt can be on: ● principles behind the teaching What principles and beliefs informed my lesson? Why did I teach the lesson the way I did? ● students What roles did my students play in my lesson? What did my students learn? How did they learn? ● ways forward What could I have done differently? What can I explore in the next lesson? Teacher’s reflection in every lesson conducted/facilitated is essential and necessary to improve practice. You may also consider this as an input for the LAC/Collab sessions.