Grade 9 · Mathematics
Quarter 2 · Week 6 · TEKS
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This lesson guides Grade 9 students through writing linear functions that fit given data sets, both with and without technology, to estimate solutions and make predictions for real-world problems. Students will learn to analyze data, determine reasonable linear models, and apply these models to solve practical problems, aligned with TEKS.111.39.04C.
Students should understand the concept of linear functions, including slope and y-intercept, and be able to calculate slope from two points or a table of values. Familiarity with graphing linear equations and basic use of graphing technology is helpful.
A linear function is a function whose graph is a straight line. It can be written in the form y = mx + b, where m is the slope and b is the y-intercept. When given a set of data points, the goal is to find a linear function that fits the data reasonably well, meaning it models the relationship between the variables so we can estimate or predict values not in the data set.
To write a linear function without technology, students can use two data points from the table to calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Then, using one point and the slope, they can solve for the y-intercept (b) by substituting into y = mx + b. This process creates a linear function that fits the data points exactly, which can be used to estimate other values.
Technology such as graphing calculators or software can analyze a whole data set and find the line of best fit, which minimizes the overall distance between the data points and the line. This line may not pass through any data points exactly but provides a reasonable model for prediction. Students should learn how to input data into technology and interpret the resulting linear function.
The slope represents the rate of change between the variables, such as speed or cost per item. The y-intercept represents the starting value when the independent variable is zero. Understanding these helps students explain what the linear function means in the real-world problem and make informed predictions.
Once a linear function is established, students can substitute values into the function to estimate unknown quantities. For example, if the function models cost over time, students can predict the cost at a future time. Emphasize checking if predictions are reasonable based on the context and data range.
Given the data points (1, 3) and (4, 15), write the linear function that fits these points without technology.
Calculate the slope: m = (15 - 3) / (4 - 1) = 12 / 3 = 4. Use point (1,3) to find b: 3 = 4(1) + b, so b = 3 - 4 = -1. The linear function is y = 4x - 1.
y = 4x - 1
Use a graphing calculator to find the line of best fit for the data: (1, 2), (2, 5), (3, 7), (4, 10).
Input the data into the calculator's statistics mode and select linear regression. The calculator outputs the equation y = 2.6x - 0.4, which models the data with minimal error.
y = 2.6x - 0.4
Interpret the slope and y-intercept of the function y = 3x + 2 in the context of a problem where x is hours worked and y is dollars earned.
The slope 3 means the person earns $3 per hour worked. The y-intercept 2 means the person earns $2 even if no hours are worked, possibly a base pay or fee.
Slope: $3 per hour; Intercept: $2 base pay.
Present a real-world scenario, such as tracking the distance a car travels over time, and show a small data table of time and distance.
Discuss what the data might represent and predict what the distance might be at a time not listed in the table.
This engages students by connecting data to a familiar context and introduces the idea of using data to make predictions.
Provide students with a data table showing two variables with a linear trend. Guide them to calculate the slope using two points and write the linear function without technology.
Calculate slope, find the y-intercept, and write the linear function that fits the data points.
This phase allows students to practice writing linear functions manually, reinforcing understanding of slope and intercept.
Demonstrate how to use graphing technology to input data and find a line of best fit. Explain differences between exact fit and best fit.
Use graphing calculators or software to enter data and obtain a linear function that fits the data.
Students learn to use technology to find linear functions that model data, understanding that the line of best fit may not pass through all points but represents the overall trend.
Present a real-world problem requiring prediction, such as estimating sales or temperature on a future date, using the linear function derived.
Use the linear function to estimate unknown values and explain the meaning of slope and intercept in the problem context.
This phase connects the mathematical model to practical use, reinforcing interpretation and application skills.
Ask students to write a linear function from a new data set without technology and then check their work using technology. Have them explain their reasoning and predictions.
Write the linear function manually, use technology to verify, and explain how the function fits the data and predictions.
This evaluation confirms student mastery of writing linear functions and applying them to real-world problems, both manually and with technology.
Calculate slope, find the y-intercept, and write the linear function that fits the data points.
Calculate the slope of the line passing through points (2, 7) and (5, 16).
Slope m = (16 - 7) / (5 - 2) = 9 / 3 = 3.
Write the linear function in slope-intercept form for the points (0, 4) and (3, 13).
y = 3x + 4
Using technology, find the line of best fit for the data points: (1, 2), (2, 4), (3, 5), (4, 7).
y = 1.5x + 0.5
Explain why the line of best fit might not pass through any of the data points but still be a good model.
It minimizes the overall distance from all points, balancing errors to provide the best overall prediction.
Given a data set with some outliers, describe how technology can help you decide if a linear function is a reasonable fit and how to handle the outliers.
Technology calculates correlation and residuals to assess fit; outliers with large residuals can be excluded or handled with robust fitting methods.
Provide step-by-step guides for calculating slope and intercept. Use graph paper and rulers to help visualize linear relationships. Offer simpler data sets with fewer points for practice.
Use visuals and gestures to explain slope and intercept. Provide vocabulary cards with definitions and examples. Pair students for peer discussion and explanation.
Challenge students to compare different methods of fitting linear functions. Explore correlation coefficients and their interpretation. Investigate non-linear data and discuss when linear models are not appropriate.
Ask students to collect data from a real-world situation at home, such as temperature over several days or distance walked over time, and write a linear function to model the data. They can use technology at home if available or write the function manually.
Students learn to write linear functions that fit data sets both manually and using technology. They understand how to interpret slope and intercept in context and use these functions to estimate and predict values in real-world problems. This skill supports analyzing relationships and making informed decisions based on data.
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