Linear Patterns

10 min

Narrative

This activity prepares students to write their own linear equations by providing a pattern to examine in the same style they will use in upcoming activities. They are given an image of lines on a coordinate plane and equations for two of the lines. They are then asked to write equations for the other lines. Although the given equations and solutions are in slope-intercept form, it is not important that students use that form. Any equivalent expressions should be accepted.

Student Task

Here is a pattern made from 8 linear equations.

Two of the equations are y=2xy = 2x and y=−12xy = \frac{-1}{2}x. If every line is parallel to one of those two, write an equation for each of the other 6 lines.

Sample Response

y=2x+1,y=2x−1,y=2x+2,y=-12x+1,y=−12x+2,y=-12x−1y = 2x+1, y = 2x-1, y = 2x+2, y = \frac{\text{-}1}{2}x +1, y = \frac{-1}{2}x+2, y = \frac{\text{-}1}{2}x-1 (or equivalent)

Synthesis

The purpose of the discussion is to help students write equations from lines. Consider asking students:
  • “What methods did you use to write additional equations?” (Because the lines are parallel, I knew they would have the same slopes and only needed to use the vertical intercept to write the equation in slope-intercept form.)
  • “Could you write the equations of the lines if the problem did not specify that they are parallel?” (Yes. Each line goes through at least two points on the coordinate grid, so I could find the slope and intercept from that.)
Standards
Addressing
  • A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • HSA-CED.A.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Building Toward
  • F-BF.3·Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  • F-BF.3·Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  • F-BF.3·Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  • F-BF.3·Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  • F-BF.3·Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  • F-BF.3·Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  • HSF-BF.B.3·Identify the effect on the graph of replacing <span class="math">\(f(x)\)</span> by <span class="math">\(f(x) + k\)</span>, <span class="math">\(k f(x)\)</span>, <span class="math">\(f(kx)\)</span>, and <span class="math">\(f(x + k)\)</span> for specific values of <span class="math">\(k\)</span> (both positive and negative); find the value of <span class="math">\(k\)</span> given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. <span>Include recognizing even and odd functions from their graphs and algebraic expressions for them.</span>

20 min

15 min