Once a set of learning objectives have been settled on for an activity, problem, or project, what should the problem’s context be? Since linear equations model situations where there is a constant rate of change, common contexts for linear equation projects often include the following:
- Steepness, height, angle
Examples: road grade, hillside, roof, skateboard park element, tide height over the two weeks before (or after) a full moon, sun angle at noon over a six month period
- Estimating time to complete a task (setup plus completion)
Examples: mowing a lawn, painting a wall, writing a research paper
- Purchase and delivery costs of bulk materials
Examples: mulch, gravel, lumber
- Purchasing a service that charges by consumption
Examples: cell phone, electricity, water, movie rental, etc.
- Total earnings over time from differing wage and bonus plan structures
Examples: hiring bonuses, longevity bonuses
- Energy use over time
Examples: calories burned, electricity, heating oil, gasoline
- Game points accumulated over time
Examples: by a professional athlete, a team, a video game player
- Pollutant levels over time Continue reading Linear Equation Activity Ideas
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:
Continue reading “Hidden” Learning Objectives for a Linear Equations Problem or Project
The greatest value of GeoGebra, or Geometer’s Sketchpad, or other such packages, in my eyes is their ability to help students dynamically visualize the effect each constant has on the graph of an equation. I find them an invaluable “thinking aid” as I ponder a new equation form, and they help me formulate my own answers to questions such as “why does it do that?”
Check out applets that help students explore the relationship between function parameters and their graphs:
GeoGebraBook: Exploring Linear Functions, which contains
GeoGebraBook of Quadratic Applets, which contains
Exponential and Logarithmic functions:
GeoGebraBook of Exponential and Logarithmic Applets, which contains
GeoGebraBook for Exploring Trig Functions, which contains
Unit Circle Symmetries:
GeoGebraBook of Unit Circle Symmetry applets, which contains:
Towards the end of the unit(s) on Linear Equations and their graphs, students can feel a bit overwhelmed. The following is an attempt to summarize and link the key concepts you need to be comfortable with.
What is the least amount of information you need to know in order to be able to identify a line exactly? Two pieces of information: a point that the line passes through, as well as either a second point on the line or the line’s slope.
Continue reading Analyzing Linear Equations: a summary