Grade 10 · Mathematics
Quarter 2 · Week 1 · TEKS
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This Grade 10 mathematics lesson focuses on describing and performing transformations of figures in a plane using coordinate notation. Students will explore rigid transformations (translations, reflections, rotations) and non-rigid transformations (dilations), including compositions of these transformations. The lesson aligns with TEKS standards and emphasizes understanding how to find images and pre-images of figures under various transformations using coordinate rules.
Students should understand the coordinate plane, plotting points, and basic properties of geometric figures. Familiarity with the concepts of congruence and similarity, as well as basic algebraic operations with coordinates, is helpful.
A transformation moves or changes a figure in some way on the coordinate plane. Transformations can be rigid, which preserve size and shape, or non-rigid, which can change size but preserve similarity or not. Rigid transformations include translations (slides), reflections (flips), and rotations (turns). Non-rigid transformations include dilations (resizing) that can enlarge or reduce figures.
Coordinate notation uses algebraic rules to describe how each point of a figure moves. For example, a translation adds or subtracts values from the x- and y-coordinates. A reflection changes the sign or swaps coordinates depending on the line of reflection. A rotation involves rotating points around the origin or another center by a certain angle, changing coordinates according to rotation rules. A dilation multiplies the distance from the center of dilation by a scale factor, changing coordinates proportionally.
A translation moves every point of a figure the same distance in the same direction. The coordinate rule for translating a point (x, y) by a vector (a, b) is (x + a, y + b). For example, translating (3, 4) by (2, -1) results in (5, 3). This transformation preserves the size and shape of the figure.
A reflection flips a figure over a line called the line of reflection. Common lines of reflection include the x-axis, y-axis, or the line y = x. For example, reflecting a point (x, y) over the x-axis changes it to (x, -y). Reflecting over the y-axis changes (x, y) to (-x, y). Reflecting over y = x swaps coordinates to (y, x). Reflections are rigid transformations that preserve size and shape but reverse orientation.
A rotation turns a figure about a fixed point called the center of rotation. Common rotations are 90°, 180°, and 270° about the origin. For example, rotating (x, y) 90° counterclockwise about the origin results in (-y, x). Rotations preserve size and shape but change orientation. The center of rotation can be any point, not just the origin, which requires translating points relative to that center before applying rotation rules.
A dilation resizes a figure by a scale factor relative to a center of dilation. The coordinate rule for dilation of a point (x, y) with center at the origin and scale factor k is (kx, ky). If the center is at (h, k), the rule is ((x - h)k + h, (y - k)k + k). Dilations with scale factor greater than 1 enlarge the figure; less than 1 reduce it. Dilations preserve similarity but not necessarily congruence.
A composition applies two or more transformations in sequence to a figure. The order matters and can affect the final image. Compositions can combine rigid transformations, non-rigid transformations, or both. For example, a figure can be translated then reflected, or dilated then rotated. Using coordinate notation, each transformation’s rule is applied step-by-step to find the final coordinates of each point.
- Misconception: Dilations always preserve size. Correction: Dilations preserve shape similarity but can change size unless the scale factor is 1. - Misconception: The order of transformations does not matter. Correction: The order can change the final image. - Example: To find the image of point (2, 3) after a translation by (4, -1) followed by a reflection over the y-axis, first translate to (6, 2), then reflect to (-6, 2).
Transformations are used in computer graphics, robotics, and design to move and resize objects. Understanding coordinate notation allows precise control of these movements. For example, resizing a logo (dilation) and then rotating it for placement on a product involves applying these transformations in sequence.
Find the image of point (3, 2) after a translation by the vector (4, -1).
A translation by vector (4, -1) means adding 4 to the x-coordinate and subtracting 1 from the y-coordinate. So, (3 + 4, 2 - 1) = (7, 1).
The image of (3, 2) after the translation is (7, 1).
Reflect the point (5, -3) over the x-axis and find the new coordinates.
Reflection over the x-axis changes the y-coordinate's sign but keeps the x-coordinate the same. So, (5, -3) becomes (5, 3).
The image after reflection over the x-axis is (5, 3).
Rotate the point (2, 4) 90° counterclockwise about the origin.
A 90° counterclockwise rotation about the origin changes (x, y) to (-y, x). So, (2, 4) becomes (-4, 2).
The image after rotation is (-4, 2).
Dilate the point (3, 6) by a scale factor of 2 with the center of dilation at the origin.
Dilation with center at the origin multiplies both coordinates by the scale factor. So, (3*2, 6*2) = (6, 12).
The image after dilation is (6, 12).
Find the image of point (4, 1) after a composition of a translation by (1, 2) followed by a reflection over the y-axis.
First translate: (4 + 1, 1 + 2) = (5, 3). Then reflect over the y-axis: change x to -x, so (5, 3) becomes (-5, 3).
The image after the composition is (-5, 3).
Introduce the concept of transformations by showing a simple figure on a coordinate plane and asking students how they could move or change it while keeping its shape or size.
Students suggest ways to move or flip the figure and describe what happens to its points on the coordinate plane.
Engage students by connecting to their prior knowledge of moving objects and introduce the idea that transformations can be described using coordinates.
Provide students with graph paper and figures to perform translations, reflections, rotations, and dilations using coordinate rules.
Students plot points, apply coordinate rules for each transformation, and record the new coordinates of the image points.
Students explore how each transformation changes coordinates and observe patterns in the coordinate changes.
Explicitly teach the coordinate notation rules for each transformation type and demonstrate compositions of transformations.
Students take notes on the coordinate rules and practice applying them to given points and figures.
Clarify the algebraic rules for transformations and how to apply them step-by-step, including how to handle compositions and different centers of dilation or rotation.
Present problems involving compositions of transformations, including mixed rigid and non-rigid transformations, and ask students to find images or pre-images.
Students solve problems by applying sequences of coordinate rules and explain their reasoning.
Students deepen understanding by working with complex transformations and justifying their steps.
Assign an exit ticket where students describe and perform a transformation or composition using coordinate notation and explain their process.
Students complete the exit ticket individually, showing coordinate calculations and explanations.
Assessment checks students’ ability to describe and perform transformations using coordinate notation and understand compositions.
Students plot points, apply coordinate rules for each transformation, and record the new coordinates of the image points.
Translate the point (1, 5) by the vector (3, -2). What are the coordinates of the image?
(4, 3)
Reflect the point (-2, 7) over the y-axis. What are the coordinates of the image?
(2, 7)
Rotate the point (6, -1) 180° about the origin. What are the coordinates of the image?
(-6, 1)
Dilate the point (4, 3) by a scale factor of 0.5 with center of dilation at the origin. Then translate the image by (-2, 4). What are the final coordinates?
(0, 5.5)
A point (x, y) is reflected over the line y = x, then rotated 90° clockwise about the origin. Write the coordinate rule for the composition and find the image of (3, 2).
The composition rule is (x, y) → (y, x) → (x, -y). The image of (3, 2) is (3, -2).
Provide coordinate grids with labeled points for easier plotting. Use step-by-step guided examples for each transformation type. Allow use of graphing calculators or software for visualization.
Use visuals and gestures to demonstrate transformations. Provide vocabulary lists with translations and diagrams. Use sentence frames to help describe transformations (e.g., 'The point (x, y) moves to ...').
Explore compositions of three or more transformations and their coordinate rules. Investigate transformations with centers not at the origin and derive general coordinate formulas. Analyze the effects of transformations on figures with algebraic expressions for coordinates.
Encourage students to find examples of transformations in their environment, such as reflections in mirrors, rotations of objects, or resizing images on screens, and describe them using coordinate notation if possible.
In this lesson, students learned to describe and perform rigid transformations (translations, reflections, rotations) and non-rigid transformations (dilations) using coordinate notation. They practiced applying coordinate rules to find images and pre-images of figures under single and composed transformations. Understanding how transformations affect coordinates helps students analyze geometric figures precisely and prepares them for more advanced geometry concepts.
Use the exact standards alignment and retrieved-source provenance stored with this enrichment.