Getting Started |
| Misconception/Error The student does not understand the slope criterion for perpendicular lines. |
| Examples of Student Work at this Level The student:
- Indicates that he or she does not know how to find the slope of
 .
- Says the slope of is -2,
 , or  .
- Says the slope is some variation on the y-intercept given in the original equation (e.g., 10, -10,
 , or  ).
|
| Questions Eliciting Thinking What is the slope of a line whose equation is y= x + 10?
If is parallel to this line, what is its slope? What if is perpendicular? Could you find its slope? |
| Instructional Implications Provide the student with the graph of several pairs of perpendicular lines and ask him or her to calculate the slopes of the lines in each pair. Have the student write each slope as a fraction and then encourage the student to look for a consistent relationship between the pairs of slopes. Help the student generalize the relationship between the slopes of perpendicular lines by observing that they are always both opposite and reciprocal or their product is always -1.
Consider implementing MFAS task Definition of Perpendicular Lines (G-CO.1.1).
Guide the student through a proof of the criterion for perpendicular lines. Consider implementing MFAS task Proving the Slope Criterion for Perpendicular Lines - 1 (G-GPE.2.5). |
Moving Forward |
| Misconception/Error The student understands the slope criterion for perpendicular lines but cannot find the slope of a line given its equation in standard form. |
| Examples of Student Work at this Level The student:
- Correctly identifies the slope of as 2 but says the slope of
 is or  .
- Indicates that he or she is unable to find the slope of the line given by the equation in standard form.
|
| Questions Eliciting Thinking How did you find the slope of ?
In what form is the equation 2x + 6y = 9 written? Can you read the slope off of this equation?
How were you able to find the slope of the line in the first equation? What could you do to find the slope of the line whose equation is 2x + 6y = 9 ?
Suppose this line had a slope of  ? What would the slope of a line perpendicular to this line be? |
| Instructional Implications Review with the student the different forms of equations of lines. Provide the student with several examples written in each form. Have the student identify the equations written in slope-intercept form. Model rewriting equations given in standard or point-slope form in slope-intercept form.
Provide the student with several examples of equations written in standard form or point-slope. Ask the student to rewrite each equation in slope-intercept form and identify its slope as well as the slope of a line perpendicular to it. |
Making Progress |
| Misconception/Error The student does not know to or is unable to algebraically find the y-intercept of the line whose equation is to be written. |
| Examples of Student Work at this Level The student can find the slope of the line whose equation he or she is writing but is unable to use a given point to complete the equation. Instead, the student:
- Uses the y-intercepts of the original equations as the y-intercepts of the equations of the perpendicular lines.
- Uses the y-coordinate of the given point (-2,7) as the y-intercept of the equation of the perpendicular line.
- Estimates the y-intercept by graphing the line using the given point and the slope.
|
| Questions Eliciting Thinking You said perpendicular lines have slopes that are opposite and reciprocal. What do you know about their y-intercepts? Do they have to be the same?
Why do you suppose you were told the coordinates of B? Is that needed to write the equation of ?
Is (-2, 7) a y-intercept? How can you tell if a point could be a y-intercept?
What if the y-intercept was a rational number such as 11.6? Do you think you could have found it by graphing? Do you know how to find it algebraically? |
| Instructional Implications Have the student graph the line given by y= x + 10 using its slope and y-intercept. Then have the student graph the perpendicular line whose equation is to be written by using its slope, 2, and the given point, (-2, 7). Have the student use the graph to estimate the y-intercept of the perpendicular line. Then guide the student to find its actual value algebraically and to write its equation in slope-intercept form. Ask the student to repeat this exercise with the equation given in the second problem. When the student is finished, ask him or her if there was anything easier about writing the equation of the perpendicular line in the second problem.
Give the student more practice writing the equations of lines given points and equations of both parallel and perpendicular lines written in a variety of forms. |
Almost There |
| Misconception/Error The student makes a minor algebraic error. |
| Examples of Student Work at this Level The student:
- Describes the slope of as 2x instead of 2.
- Leaves the equation in point-slope form instead of writing it in slope-intercept form.
- Substitutes the x-coordinate for the y-value in the equation and the y-coordinate for the x-value in the equation.
- Makes a sign error.
|
| Questions Eliciting Thinking Is slope represented by a number or a term in an equation?
In what form is your equation ? In what form were you asked to write it?
I think you made an error when you wrote this equation. Can you review your work and try to find the error? |
| Instructional Implications Provide specific feedback to the student regarding his or her error and allow the student to revise the work on his or her paper. Give the student a few examples of common errors made when writing equations and have him or her identify and correct those errors. |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student understands the slope criterion for perpendicular lines, correctly finds the slope of each line, and uses the given points to write the equations of the lines in slope-intercept form. The student provides the following answers:
- (a) m = 2
(b) y = 2x + 11
- y = 3x - 4
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| Questions Eliciting Thinking What did you notice about the given point for Question 2?
Can two parallel lines have the same y-intercept? Can two perpendicular lines have the same y-intercept? |
| Instructional Implications Review with the student how to write equations of horizontal and vertical lines. Ask the student to write the equations of the lines containing the sides of a rectangle with opposite vertices at (0, 0) and (5, 4).
Ask the student to prove the slope criterion for perpendicular lines. Consider implementing MFAS task Proving the Slope Criterion for Perpendicular Lines-1 (G-GPE.2.5) or Proving the Slope Criterion for Perpendicular Lines-2 (G-GPE.2.5). |
