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Students explore slope-intercept, point-slope, and standard forms of linear models. They consider situations in which these forms make fitting a model easier or more challenging, and practice converting between forms.

Lesson Goals

Students will be able to…​

  • Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms.

Student-facing Lesson Goals

  • Let’s learn about how to convert between forms of linear models and when one or the other might make our work more efficient.

Materials

Supplemental Materials

🔗Other Forms of Linear Models

Overview

Students are reminded of the three forms of linear models available to us, discuss when and why we might choose one form over another, and practice translating between them.

Launch

When trying to fit a piece into a puzzle, sometimes we rotate the piece to see it from a different angle. When fitting a model to a dataset, we might prefer to look at the linear relationship from different angles as well!

We will mostly be using Slope-Intercept form of the line in this course, because it’s the simplest form that is defined in terms of the response variable, making it most compatible with the programming environment.

But depending on the information we have available to us - or who we’re writing this model for - we might want to use other forms of linear models, like Point-Slope or Standard Form. Fortunately, we can always translate any model into another!

You may already be familiar with the different forms of linear models available to us:

Slope-Intercept Point-Slope Standard

y = mx + b

y - y1 = m(x - x1)

Ax + By = C

  • m:  slope

  • b:  y-intercept / vertical shift

  • m: slope

  • y1:  y-coordinate of a point

  • x1:  x-coordinate of the same point

  • x-int: C /A

  • y-int: C /B

  • slope: - A /B

Makes it really easy to read the slope and y-intercept.

Makes it easy to find the equation of the line given a single point and slope.

Makes it easy to find the x- and y-intercepts of the line.

Why do we use these letters as stand-ins for the constants?

The letters used in these forms of the line are just conventions people have agreed upon over time!

Standardized conventions have their benefits, but sometimes they also have downsides.

For example, some students might be confused by the fact that:

  • b in the slope-intercept form and B in standard form represent different things

  • b represents the vertical shift in linear models while it represents the base in exponential models

As we’ve thought through what would best support students with learning to model data, we’ve learned that people are in less agreement about how to name constants for non-linear models.

We believe that students are best supported by using letters that highlight the similarities between the various models. Our nonlinear modeling materials will consistently use k for the vertical shift.

Applying this to the point-slope form, we’d get: y = mx + k

You can foreshadow this for your students by discussing the fact that while the letters used are conventions, they are just stand-ins for the patterns we can see with our eyes. Having identified the pattern of the point-slope form, we could just as easily have written it with other letters.

Pose the questions below to assess student understanding of when and why we might choose one form over another.

Why we might choose to use one form over another?

  • Suppose our scatter plot has a state with 0% college enrollment, and another with $0 median income. Which linear form would be easiest to use?

  • Standard Form

  • Suppose we only know the slope of a model, but we know the college graduation rate and median income for Rhode Island. Which form would make it easy to figure out the rest of the model?

  • Point-Slope Form

  • Which form makes it easiest to define our model in Pyret?

  • Slope-Intercept Form

Investigate

Based on the information available to us, it might be easier to use one form over another. But for someone else to read and understand that model, we might choose a different model! Fortunately, we can easily translate back and forth between linear forms!

  • Let’s practice identifying which form of a linear model is easiest to write with the given information and then translating models from each of those forms into Pyret function definitions.

  • Turn to Which Form is Best?

  • Practice translating some of the models we found for predicting median-income from a state’s pct-college-or-higher into other forms on Other Forms of Linear Models.

Synthesize

  • If you needed to draw the graph of a linear model, which form would you like to start from? Why?

  • Answers will vary.

  • If we’re making the graph in Pyret, though, point-slope is definitely easiest because it’s already solved for y!

🔗Additional Exercises

To practice reading linear models and connecting them to graphs:

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). CCbadge Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.

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