Grade 8 · Mathematics
Quarter 2 · Week 2 · TEKS
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This lesson guides Grade 8 students to understand and explain how applying a positive rational scale factor to two-dimensional figures on a coordinate plane, with the origin as the center of dilation, changes the figures. Students will use algebraic expressions to represent these dilations and interpret their effects on coordinates, connecting proportionality concepts to geometric transformations.
Students should understand coordinate planes, plotting points, and basic multiplication of rational numbers. Familiarity with proportional relationships and the concept of scale factors will support learning.
Dilation is a transformation that changes the size of a figure but not its shape. The figure is enlarged or reduced by a scale factor, which is a positive rational number. When the center of dilation is the origin (0,0) on the coordinate plane, each point of the figure is moved along a line from the origin through the point, and its distance from the origin is multiplied by the scale factor.
If a point on the coordinate plane has coordinates (x, y), then after a dilation with scale factor k (where k > 0) and the origin as the center, the new coordinates become (kx, ky). This means both the x- and y-coordinates are multiplied by the scale factor. This algebraic rule applies to every point in the figure.
- If k > 1, the figure enlarges, and all points move farther from the origin. - If 0 < k < 1, the figure reduces in size, and all points move closer to the origin. - The shape remains similar because all distances from the origin are scaled proportionally. - The coordinates of the image are proportional to the coordinates of the pre-image by the scale factor k.
Students can write expressions such as (kx, ky) to represent the dilation of a point (x, y). For example, if k = 2 and the original point is (3, 4), the dilated point is (2*3, 2*4) = (6, 8). This algebraic approach helps explain how the figure changes size and position relative to the origin.
- Confusing dilation with translation or rotation; dilation changes size but keeps shape and orientation. - Believing the center of dilation can be any point without changing the algebraic rule; the algebraic rule (kx, ky) applies only when the origin is the center. - Thinking scale factors can be negative; scale factors in this lesson are positive rational numbers, so the figure does not reflect or flip.
Given a point A at (4, 3) on the coordinate plane, find the coordinates of point A' after a dilation with scale factor 3 and the origin as the center.
To find the dilated point A', multiply each coordinate of A by the scale factor 3. The x-coordinate becomes 3 * 4 = 12, and the y-coordinate becomes 3 * 3 = 9. So, A' is at (12, 9).
The coordinates of A' are (12, 9).
A triangle has vertices at (2, 1), (4, 1), and (3, 3). Find the coordinates of the triangle after a dilation with scale factor 1/2 centered at the origin.
Multiply each coordinate by 1/2. The new vertices are: (2 * 1/2, 1 * 1/2) = (1, 0.5) (4 * 1/2, 1 * 1/2) = (2, 0.5) (3 * 1/2, 3 * 1/2) = (1.5, 1.5)
The dilated triangle has vertices at (1, 0.5), (2, 0.5), and (1.5, 1.5).
Explain why the figure remains similar after dilation with the origin as the center and a positive scale factor.
Because all points are multiplied by the same positive scale factor, the distances between points change proportionally, preserving angles and shape. The figure enlarges or reduces but keeps the same shape, making it similar to the original.
The figure remains similar because dilation scales all distances proportionally, preserving shape and angles.
Introduce the concept of dilation by showing a simple figure on the coordinate plane and asking students what happens if we multiply the coordinates by 2.
Observe the figure and predict how its size and position will change if coordinates are doubled.
Multiplying coordinates by a scale factor changes the size of the figure but keeps its shape and orientation, centered at the origin.
Provide students with graph paper and a figure plotted on the coordinate plane. Ask them to apply a scale factor of 1/2 to each point and plot the new figure.
Calculate new coordinates by multiplying original coordinates by 1/2 and plot the dilated figure.
Students see that multiplying coordinates by 1/2 reduces the figure's size, moving points closer to the origin.
Demonstrate the algebraic rule for dilation: (x, y) becomes (kx, ky). Show examples with different scale factors.
Write algebraic expressions for dilations of given points and explain the effect of the scale factor.
The algebraic rule shows that dilation multiplies each coordinate by the scale factor, scaling the figure proportionally from the origin.
Present real-world problems where dilation applies, such as resizing a blueprint or map. Ask students to use algebraic representations to solve.
Apply the dilation rule to solve problems involving resizing figures on the coordinate plane.
Using algebraic expressions to represent dilation helps solve practical problems involving proportional resizing.
Ask students to explain in writing or orally how a given scale factor changes the coordinates of a figure on the coordinate plane.
Demonstrate understanding by explaining the dilation process using algebraic expressions and coordinate changes.
Students consolidate their understanding by articulating the algebraic effect of dilation on coordinates.
Calculate new coordinates by multiplying original coordinates by 1/2 and plot the dilated figure.
If a point is at (5, 2), what are its coordinates after a dilation with scale factor 2 and the origin as the center?
The coordinates after dilation are (10, 4).
A rectangle has vertices at (1, 3), (1, 6), (4, 6), and (4, 3). Find the coordinates after a dilation with scale factor 3/2 centered at the origin.
The new vertices are (1.5, 4.5), (1.5, 9), (6, 9), and (6, 4.5).
Explain how the coordinates of a point change after a dilation with scale factor 0.5 and the origin as center.
Each coordinate is multiplied by 0.5, so the point moves halfway closer to the origin, reducing the figure's size by half while keeping the same shape.
A triangle has vertices at (2, 2), (4, 2), and (3, 5). After a dilation with scale factor k, the new vertices are (6, 6), (12, 6), and (9, 15). What is the scale factor k?
The scale factor k is 3 because each coordinate is multiplied by 3.
If a figure is dilated with scale factor k > 0 centered at the origin, and a point originally at (x, y) moves to (kx, ky), prove that the slope between any two points remains unchanged after dilation.
The slope between two points after dilation is (ky2 - ky1)/(kx2 - kx1) = k(y2 - y1)/k(x2 - x1) = (y2 - y1)/(x2 - x1), so the slope remains the same, preserving the figure's shape.
Use graph paper with grid lines to help students plot points accurately. Provide step-by-step guided examples multiplying coordinates by scale factors. Use physical manipulatives or dynamic geometry software to visualize dilation.
Use visual aids and gestures to demonstrate dilation. Provide vocabulary cards with definitions and examples of key terms. Encourage peer discussion to explain concepts in simpler language.
Challenge students to prove properties of dilation such as slope preservation. Explore dilations with scale factors greater than 1 and between 0 and 1 and compare effects. Investigate dilations with centers other than the origin and discuss algebraic differences.
Ask students to find examples of dilation in real life, such as resizing images or maps, and describe how the size changes while the shape remains the same. Encourage them to explain the concept to a family member using algebraic expressions.
In this lesson, students learned that dilation with the origin as the center and a positive rational scale factor multiplies the coordinates of points in a figure by that scale factor. This transformation changes the size of the figure while preserving its shape, resulting in a similar figure. Using algebraic expressions like (kx, ky) helps explain and predict the effect of dilation on any point in the coordinate plane.
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