The Slope of a Fitted Line

5 min

Narrative

The purpose of this Warm-up is for students to estimate the slope of a line given points that are close to the line, but not on the line. This prepares students for thinking about the model's fit to data in the rest of the lesson.

Launch

Arrange students in groups of 2. Give 1 minute of quiet work time followed by 1 minute to discuss their solution with their partner. Follow with a whole-class discussion. 

Student Task

Estimate the slope of the line. Be prepared to explain your reasoning.

Scatterplot, x, negative 10 to 10 by 5, y, negative 10 to 10 by 5. Points at negative 4 comma negative 6, 5 comma 2, 8 comma 3. Line passes just above first point, just below second point.

Sample Response

Sample response: About 0.8. It is a little less than 89\frac{8}{9}, which is the slope of a segment connecting (-4,-6)(\text{-}4,\text{-}6) and (5,2)(5,2), and a little more than 34\frac{3}{4}, which is the slope of a segment connecting (-4,-6)(\text{-}4,\text{-}6) and (8,3)(8,3).

Synthesis

Poll the class and ask students if their estimated slope was close to their partner's estimate. Select 2–3 groups who had close estimates to share their solutions and explain their reasoning. Display the graph with the single line given in the task and record the students' responses next to the graph for all to see.

If students do not mention that it is better to use points that are far apart rather than close together for estimating the slope, consider displaying this graph for all to see:

<p>Various lines on a coordinate plane.</p>

To remind students of previous work, draw a slope triangle whose horizontal side has a length of 1, demonstrating that the length of the vertical side is equal to the slope of the line.

Standards
Building On
  • 8.EE.6·Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
  • 8.EE.B.6·Use similar triangles to explain why the slope <span class="math">\(m\)</span> is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation <span class="math">\(y = mx\)</span> for a line through the origin and the equation <span class="math">\(y = mx + b\)</span> for a line intercepting the vertical axis at <span class="math">\(b\)</span>.
Building Toward
  • 8.SP.3·Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. <em>For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.</em>
  • 8.SP.A.3·Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. <span>For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.</span>

10 min

10 min

10 min