Grade 9 · Mathematics
Quarter 2 · Week 2 · TEKS
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This lesson guides Grade 9 students through determining the slope of a line using various representations: tables of values, graphs, two points on the line, and equations in forms such as y = mx + b, Ax + By = C, and point-slope form y - y1 = m(x - x1). Students will also calculate the rate of change of linear functions and graph linear functions identifying key features like intercepts and zeros, connecting these concepts to real-world contexts.
Students should understand coordinate plane basics, ordered pairs, and have experience plotting points. Familiarity with linear relationships and basic algebraic manipulation is helpful.
Slope measures the steepness or rate of change of a line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). A positive slope means the line rises from left to right; a negative slope means it falls. Zero slope means a horizontal line; undefined slope means a vertical line.
Given a table with x and y values, slope is found by selecting two points and calculating the change in y divided by the change in x. For example, if the table shows (1, 3) and (4, 9), slope \(m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2\). Consistent slope values between pairs confirm linearity.
On a graph, identify two clear points on the line. Count the vertical units between points (rise) and horizontal units (run). Divide rise by run to find slope. For example, moving from (2, 5) to (5, 11) shows rise 6 and run 3, so slope is 2. Use grid lines to count units accurately.
Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) with coordinates of two points. For example, points (3, 7) and (6, 13) give slope \(m = \frac{13 - 7}{6 - 3} = \frac{6}{3} = 2\). Emphasize subtracting y-values first, then x-values, and simplifying the fraction.
- Slope-Intercept Form (y = mx + b): The coefficient m is the slope directly. - Standard Form (Ax + By = C): Rearrange to slope-intercept form by solving for y: \(y = -\frac{A}{B}x + \frac{C}{B}\). The slope is \(-\frac{A}{B}\). - Point-Slope Form (y - y1 = m(x - x1)): The coefficient m is the slope. Explain how to isolate y to identify slope when needed.
Rate of change in a linear function is the slope. It represents how much y changes for each unit increase in x. In real-world problems, this could represent speed, cost per item, or other rates. For example, if a car travels 60 miles in 2 hours, the rate of change (speed) is 30 miles per hour.
Plot the y-intercept (where x=0) on the graph. Use the slope to find another point by moving rise over run from the y-intercept. Draw the line through these points. Identify x-intercept (where y=0) by solving the equation for x when y=0. Zeros of the function are x-values where y=0. Label slope, intercepts, and zeros clearly.
Given the table of values: x: 1, 3; y: 4, 10. Calculate the slope of the line.
Calculate the change in y: 10 - 4 = 6. Calculate the change in x: 3 - 1 = 2. Divide change in y by change in x: 6 / 2 = 3. The slope is 3.
The slope of the line is 3.
Find the slope of the line passing through points (2, 5) and (6, 13).
Use the slope formula: m = (13 - 5) / (6 - 2) = 8 / 4 = 2. The slope is 2.
The slope of the line is 2.
Identify the slope from the equation y = -4x + 7.
In slope-intercept form y = mx + b, the coefficient of x is the slope. Here, m = -4.
The slope is -4.
Given the equation 3x + 2y = 6, find the slope.
Rewrite in slope-intercept form: 2y = -3x + 6, then y = (-3/2)x + 3. The slope is -3/2.
The slope is -3/2.
From the point-slope form equation y - 2 = 5(x - 1), determine the slope.
The coefficient of (x - x1) is the slope, which is 5 here.
The slope is 5.
Present a real-world scenario such as a taxi fare that charges a fixed base fee plus a cost per mile. Show a table of fares for different miles.
Discuss with a partner how the fare changes as miles increase. Predict if the fare increases steadily or not.
This scenario introduces the idea of rate of change and slope as the cost per mile, helping students connect abstract slope to a real-world context.
Provide students with a table of values representing a linear function. Guide them to calculate the slope between pairs of points.
Calculate the slope from the table by finding differences in y and x values and dividing.
Students practice finding slope from numerical data, reinforcing the concept of rate of change.
Demonstrate how to find slope from a graph by selecting two points and counting rise over run. Show how to find slope from two points using the formula. Explain identifying slope from different equation forms.
Practice finding slope from a graph, two points, and equations in slope-intercept, standard, and point-slope forms.
This phase clarifies multiple methods for determining slope, connecting visual, numerical, and algebraic representations.
Present a real-world problem involving a linear function, such as calculating total cost based on quantity purchased. Have students graph the function and identify slope and intercepts.
Graph the function, label key features, and interpret the meaning of slope and intercepts in context.
Students apply their understanding to real-world scenarios, reinforcing the meaning of slope and intercepts beyond abstract numbers.
Give students a mixed set of problems requiring slope determination from tables, graphs, points, and equations. Observe and provide feedback.
Complete problems independently to demonstrate mastery of slope concepts.
Assessment of student understanding through varied problem types ensures comprehensive skill development.
Calculate the slope from the table by finding differences in y and x values and dividing.
Calculate the slope from the table: x = 1, 4; y = 2, 8.
Slope = (8 - 2) / (4 - 1) = 6 / 3 = 2.
Find the slope of the line through points (3, 5) and (7, 13).
Slope = (13 - 5) / (7 - 3) = 8 / 4 = 2.
Identify the slope in y = -2x + 4.
Slope is -2.
Find the slope from 5x + 3y = 15.
Rewrite: 3y = -5x + 15, y = (-5/3)x + 5, slope = -5/3.
From y - 1 = 4(x - 2), what is the slope?
Slope is 4.
Provide step-by-step guided notes with examples for calculating slope. Use graph paper with grids to help count rise and run visually. Offer sentence frames for explaining slope in words.
Use visuals and gestures to explain rise and run. Provide bilingual glossaries for key vocabulary. Use real-world examples familiar to students' experiences.
Challenge students to derive slope from equations in less common forms. Explore how slope relates to parallel and perpendicular lines. Investigate rate of change in non-linear contexts for comparison.
Ask students to find examples of linear relationships at home or in their community, such as speed vs. time or cost vs. quantity, and describe the slope and intercepts in those examples.
Students learned to determine the slope of a line from tables, graphs, two points, and equations in various forms. They connected slope to rate of change and real-world contexts, practiced graphing linear functions, and identified key features such as intercepts and zeros. Mastery of these skills supports understanding of linear functions and their applications.
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