Tag: Lines

Linear Equation Activity Ideas

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

“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:

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

Function Dilations: How to recognize and analyze them

This post was written in 2010. For my current approach to this topic, which uses transformation equations, please follow this link: Function Transformations: Dilation

This post explores one type of function transformation: “dilation”. If you are not familiar with “translation”, which is a simpler type of transformation, you may wish to read Function Translations: How to recognize and analyze them first.

A function has been “dilated” (note the spelling… it is not spelled or pronounced “dialated”) when it has been stretched away from an axis or compressed toward an axis.

Imagine a graph that has been drawn on elastic graph paper, and fastened to a solid surface along one of the axes. Now grasp the elastic paper with both hands, one hand on each side of the axis that is fixed to the surface, and pull both sides of the paper away from the axis. Doing so “dilates” the graph, causing all points to move away from the axis to a multiple of their original distance from the axis. As an example of this, consider the following graph:

Vertical Dilation (no Translation)

The graph above shows a function before and after a vertical dilation. The coordinates of two Continue reading Function Dilations: How to recognize and analyze them

Function Translations: How to recognize and analyze them

This post was written 8 years ago. For my current approach to this topic, which uses transformation equations, please follow this link: Function Transformations: Translation

A function has been “translated” when it has been moved in a way that does not change its shape or rotate it in any way. A function can be translated either vertically, horizontally, or both. Other possible “transformations” of a function include dilation, reflection, and rotation.

Imagine a graph drawn on tracing paper or a transparency, then placed over a separate set of axes. If you move the graph left or right in the direction of the horizontal axis, without rotating it, you are “translating” the graph horizontally. Move the graph straight up or down in the direction of the vertical axis, and you are translating the graph vertically.

In the text that follows, we will explore how we know that the graph of a function like

g(x)=x^2-6x+10\\*~\\*~~~~~~~=(x-3)^2+1

which is the blue curve on the graph above, can be described as a translation of the graph of the green curve above:

f(x)=x^2 ~~~~\text{translated right by 3, and up by 1}

Describing g(x) as a translation of a simpler-looking (and more familiar) function like f(x) makes it easier to understand and predict its behavior, and  Continue reading Function Translations: How to recognize and analyze them

GeoGebra Applets That Help Understand Equation Behavior

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:

Linear functions:

GeoGebraBook: Exploring Linear Functions, which contains

Quadratic functions:

GeoGebraBook of Quadratic Applets, which contains

Exponential and Logarithmic functions:

GeoGebraBook of Exponential and Logarithmic Applets, which contains

Rational functions:

Trigonometric functions:

GeoGebraBook for Exploring Trig Functions, which contains

Unit Circle Symmetries:

GeoGebraBook of Unit Circle Symmetry applets, which contains:

Summary: Algebra

When faced with an algebraic expression or equation, there are only two types of things you can do to it without changing the quantitative relationship that it describes.

Re-write one or more terms in an equivalent form

This can be done to any expression (no equal sign) or equation (with an equal sign) at any time.

There are three common ways in which this is done: by Continue reading Summary: Algebra

Analyzing Linear Equations: a summary

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.

Lines

What is the least amount of information you need to Continue reading Analyzing Linear Equations: a summary