Some may have had trouble using my GeoGebra applets in their browsers. I have moved all of them to GeoGebraTube, which will hopefully fix the problem. You may search for them by typing “MathMaine” into the GeoGebraTube search box.
Links to all updated interactive graph applets are below. Comments and suggestions are always welcome!
GeoGebraBook: Exploring Linear Functions,
Interactive Linear Function Graph: Slope-Intercept Form
Interactive Linear Function Graph: Point-Slope Form
Continue reading Interactive Graphs for Linear, Quadratic, Rational, and Trig Functions Moved to GeoGebraTube
What is the difference between a Problem and a Project? While it is difficult to draw a definitive line that separates one from the other, the attributes of each and their differences as I see them are:
- Require less student time to complete (usually less than an hour)
- Focus on a single task, with fewer than 10 questions relating to it
- Can involve open-ended questions, but more often does not
- Are often one of a series of problems relating to a topic
- Look similar to many exam questions
- Can be used to introduce new concepts (Exeter Math)
- Can be used as practice on previously introduced concepts (most math texts)
- Require more student time to complete (hours to weeks)
- Focus on a theme, but with many tasks and questions to complete
- Provide an opportunity to acquire and demonstrate mastery
- Ask students to demonstrate a greater depth of understanding
- Ask students to reach and defend a conclusion, to connect ideas or procedures
- Can introduce new ideas or situations in a more scaffolded manner
Why Use Problems?
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
What was “the best” course you ever took? Probably one for which you had to work quite hard, one that you perceived as challenging from the outset, one for which you rose to the challenge. The course probably had a reputation as a tough course, so you probably added it to your schedule with care and made sure you did not take another really challenging course at the same time.
Major time commitments are regularly called for in schools: for musical or dramatic performances, athletic seasons, and some classes too. Could we improve the way such opportunities are scheduled so that students can experience as many as possible each year without creating a killer workload for themselves at critical times during the year?
What if schools offered “challenges” that lasted for either half or a full semester? Each student could be required to be enrolled in two challenges at all times. A research project, art or engineering project, dramatic or musical performance could each count as a challenge, as could a varsity sport, as could any number of academic and extra-curricular offerings. To qualify as a “challenge”, an offering would have to:
- Culminate in a public performance, presentation, or display of student work.
- Involve extensive Continue reading Scheduling for Curricular Depth and Challenge
What if most activities in school asked students to “reach and defend a conclusion”?
- in Math, about quantitative or geometric relationships, about measurements of worldly phenomena, etc.
- in Music, about the effect of a melody line, about a particular mix of instruments, etc.
- in English, about effective use of language or metaphor, about storytelling techniques, etc.
- in Visual Arts, about the effective use of color or negative space, about how a work can be interpreted, etc.
- in History, about a set of events, about relationships between societies, etc.
- in Physical Education, about the effects of various activities on the human body, about the effectiveness of various strategies in a sport, etc.
- in Science about whether two measurements are related in some way, why they might be related, the consistency with which they seem related, about cause and effect, etc.
What might our schools look like under such an approach?
A continuum of activity types are used in classrooms around the world. They range in duration from long (weeks or months) to short (seconds or minutes), from “projects” to “problems”. There are differing styles of activities, ranging from context-rich to almost context-free. There are also differing roles for activities in a curriculum: they can serve as warm-ups, practice, assessment, and/or a primary means of instruction (as at Phillips Exeter Academy and elsewhere).
The activities that seem most effective to me tend to require at least 15 minutes, if not more, to complete unless the context for the activity has been previously introduced. Longer (or recycled from the recent past) activities require fewer mental transitions for students, and hopefully lead to greater focus on the core concepts and skills. Obviously short duration activities cannot include as many of the attributes below as longer ones might. The “laundry list” of attributes I consider when creating or modifying an activity now includes:
1) A warm-up section which:
– engages the Continue reading Eight Attributes of Effective Activities, Problems, or Projects