Student Actions

Teacher Actions

• Students will develop a deep and flexible conceptual understanding by comparing many models of a linear function, including tables, graphs and equations written in various forms to discover that each form (slope-intercept, point-slope, standard, and other forms) are actually equivalent and represent the same data set.

• Students will develop accurate and appropriate procedural fluency by using the same inverse properties of equality learned in pre-algebra to solve for an equation and put into various forms where they maintain their equality.

• Students develop mathematical reasoning as they experience the solutions in more than one representation algebraically and graphically so they can make connections to real situations.  By doing this, they  make connections between the different representations and chose the most appropriate form of the equation for a given situation.

• Students will develop the ability to communicate mathematically by interpreting and translating equations verbally in slope-intercept form and transforming them further into point-slope and standard form in order to choose the form most appropriate to, and most understandable in, a given context.

   

• Use and connect mathematical representations of the different forms of equations by helping students to examine the graphs and tables of  but equivalent, equations in order to visualize that the outcomes are the same.

• Build procedural fluency from conceptual understanding by having the students pay attention to slope and the intercepts that may be involved in an equation. Students can explore all of the relationships for linear equations while transforming equations from one form to another.

• Pose purposeful questions that guide students to  connect students to real-world situations and model the situations using the different forms of linear equations while delving into what linear form is best for a scenario.

• Implement tasks that promote reasoning and problem solving by expecting students to interpret the mathematical relationship in the original context of the problem algebraically and graphically

   

Key Understandings

Misconceptions

• Students can write many forms of an equation to represent proportional linear relationships.
• Students can convert between slope-intercept form, point-slope form and standard form and recognize that each is a form of the same  linear equation.
• Given sufficient information, including a verbal description, students can write a linear equation in many forms. 

Example:

A. What is an equation of the line that passes through the point (3,1) and has a slope of 2?

B. Next, find the equation in slope-intercept form.

(Here students need prior knowledge to use point-slope form first when given a point and the slope.)

Solution:

1. y - 1 = 2(x - 3) is the first answer expressed in point-slope form.

2. Transforming the previous equation,

y - 1 = 2(x - 3)

y -1 = 2x - 6

y = 2x - 5

Example:

Which equation is equivalent to

3x + 4y = 15?

Taken from regents exam prep:

http://www.jmap.org/JMAP/RegentsExamsandQuestions/3-AdobePDFs/WorksheetsByPI-Topic/IntegratedAlgebra/Algebra/A.A.23.TransformingFormulas2.pdf 

• Students only recognize slope-intercept form as linear.
• Students do not realize the coefficients in standard form (Ax + By = C) represented by A and  B,  aredifferent than m (slope) and b (y-intercept) in the slope-intercept form (y = mx + b).
• Students may incorrectly interpret the negative signs in point-slope form [ (y-y1) = m(x - x1) ] for x1 and y1.

   

Example:

(a) dividing the change in x by the change in y

(b) subtracting the x coordinate from the y coordinate