“Hidden” Learning Objectives for a Linear Equations Problem or Project

The lesson plans I find most interesting, both to read and to teach from, have both “public” and “hidden” learning objectives. The public objectives focus student attention and help interest students in the problem: they need to be short, to the point, and tightly related to the problem or project at hand.

The “hidden” objectives are the focus of teacher attention. They reflect the skills and concepts that the teacher hopes to see students grappling with, discussing with peers, and mastering over time while working on successive problems. If students are informed about a teacher’s list of objectives in assigning a task, students are likely to use only that list in their work. By not publicizing the teacher’s objective list, students are more likely to try a wider variety of approaches to solving a problem. I think the problem solving process starts with determining which concepts and skills seem relevant to the problem, therefore keeping the teacher’s objective list hidden helps students become better problem solvers.

The list below covers topics typically taught over a large percentage of the school year, so not all objectives are appropriate at any given point in the year. However, by the end of the year hopefully most of the following objectives will have been mastered by most students in a class:

1. Using whole sentences, describe all the ways you can think of determining whether a quantitative relation is “linear” or not.
2. Why would a variable be described as “dependent” or “independent” (or control / response, input / output, domain / range)?  What does each term mean?
3. For a linear relation that is described verbally with enough information to determine two or more points on the line:
   1. Identify the independent and dependent variables, if appropriate
   2. Calculate the rate of change, or “slope”
   3. Determine or calculate the y-intercept, or starting value
4. Recognize linear relations from each of the following starting points by determining if there is a constant slope between all points described:
   1. a description (word problem)
   2. a list of ordered pairs
   3. a graph
   4. an equation
5. Given any of the starting points in (4), be able to confidently and efficiently produce:
   1. an accurate graph
   2. a list of ordered pairs that are consistent with the information provided
   3. an equation in any form that describes the information provided
   4. a point-slope form of an equation that describes the information provided
   5. a slope-intercept form version of the equation
   6. a standard form version of the equation
6. For a linear equation in slope-intercept form, use the context of the situation and whole sentences to describe:
   1. what the variable’s coefficient tells us about the situation
   2. what the constant term tells us about the situation
7. Determine what the value of the dependent variable will be for a given value of the independent variable using:
   1. a graph
   2. an equation
   3. a list of ordered pairs
8. Determine what the value of the independent variable will be for a given value of the dependent variable using:
   1. a graph
   2. an equation
   3. a list of ordered pairs
9. What is meant by a “system of equations”? How do you recognize a “system” when you see one?
   1. In a word problem
   2. On a graph
   3. When working with equations
10. If someone tells you “the solution” to a system of equations, what is it that they have told you?
    1. In terms of a graph
    2. In terms of the equations
    3. In terms of the table of values (ordered pairs) for each equation
11. How can you determine if an ordered pair is the solution to a system of equations:
    1. on a graph?
    2. from the equations?
12. What does someone mean if they describe a system of linear equations as:
    1. Consistent
    2. Inconsistent
    3. Redundant
13. Identify consistent, inconsistent, and redundant systems of linear equations from:
    1. a graph
    2. two equations in slope-intercept form
    3. two equations in point-slope form
    4. two equations in standard form
14. Solve two and three variable linear systems confidently and efficiently using:
    1. a graph
    2. linear combination
    3. substitution
    4. matrices
15. Interpret the solution to a two-variable system of linear equations in the context of the problem.  Use complete sentences to recommend the best course of action when one is:
    1. to the left of the solution
    2. to the right of the solution
    3. at the solution
16. Apply existing knowledge to a new situation, then extend and reflect:
    1. What is familiar about this problem?
    2. What is new or different about this problem?
    3. What existing knowledge applies?
    4. Can I figure out a way to solve the problem?
    5. Can I figure out any alternative approaches to solving the problem?
    6. How can I determine that my solution is correct?
    7. What does my mathematical solution mean in the context of the problem?

If an objective is missing, or if you disagree with some items on the list, I look forward to reading your thoughts in the comment section below.  I will be happy to edit or expand the above list and give credit where credit is due.