Tuesday, December 29, 2009

How to Solve Absolute Value Inequalities

  • Find the solution of a single variable inequality equation.
  • The at you know how find the value of an inequality equation with a single variable.
  • Write the problem as two variations to remove the absolute value.
  • Solve as normal.
Since absolute values can contain either a positive or negative number that is always evaluated to a positive number. We need to write the equation twice since we don't know if it is positive or negative. Let's use the example:
| 9 + x | < 7

| 9 + x | < 7
(9 + x ) < 7
9 - 9 + x > 7 - 9
x > -2
-(9 + x ) < 7
-(9 + x )/-1 < 7/-1
(9 + x ) > -7
9 + x > -7
9 - 9 + x > -7 - 9
x > -16
x < -2 and x > -16
-16 < x > -2

Many text books have students memorize the fact that you change the inequality sign and make the right side negative and positive but I have found that the method shown here helps students to make fewer mistakes.

Things to remember when solving absolute value inequalities.

  • Isolate the absolute value to one side of the equation first.
  • Remember to change the inequality sign when multiplying or dividing by a negative number.
  • Less than (<) are usually and statements.
  • Greater than (>) are usually or statements.
  • Watch out for the exceptions such as |x| < 0 or any other value less than 0, since absolute values always evaluate to be positive it will never be less than 0.
  • Watch out for the exception |x| > -1 which is all values of x. Again since the absolute value always evaluates to be positive any number that you insert will always work.
  • And statements may be written two different ways whereas or statements may only be written one way.

Monday, December 28, 2009

How to Create Box-and-whisker Graph

Are your students struggling with creating box and whisker graphs? Here are some directions and presentations that can help students quickly learn how to put the right numbers in the right place. The first version is only in PDF and the second contains the first and is fully customizable, or you can just read through the steps listed below.
The OpenOffice/PowerPoint versions contain simple animations and timings to help the presentation flow smoothly but the PDF version can work if you don't have access to OpenOffice or MS PowerPoint.

The middle value when all of the data values are placed in order.

One forth or one quarter of the data.

Value that is far from the rest of the data.
The largest and smallest value.

Example Problem
The owner of a gas station wants you to make a box-and-whisker graph to represent the number of people who came purchased gas each day.

17 21 16 22 24 26 18 28 25 29 21 18 14 23 25 18 26 24 22 23

Follow these steps:
  1. Order the data
  2. Find extremes
  3. Find the median
  4. Find the quartiles
  5. Make your graph
Step 1: Order the data
A line plot or stem-and-leaf plot are the quickest ways to get your data in ascending order.
14 16 17 18 18 18 21 21 22 22 23 23 24 24 25 25 26 26 28 29

Step 2: Find the extremes
The lower extreme is 14, the smallest number.
The upper extreme is 29, the largest number.

Step 3: Find the median
Count the number of data items, which in this set happens to be 20.
Then divide the data into two equal parts, 10 on each side.
14 16 17 18 18 18 21 21 22 22
23 23 24 24 25 25 26 26 28 29

Since there is no middle value we need to take the average of one value from each side, 22 and 23. To find the average evaluate the following expression:


The median is 22.5.

Step 4: Find the quartiles
Now divide each of the halves in half. Since each half contains 10 (even) items the middle is between the 5th and 6th value.

14 16 17 18 18
18 21 21 22 22
23 23 24 24 25
25 26 26 28 29

Take the average of the two values.

The lower quartile is 18.
The upper quartile is 25.

Step 5: Use the information to make your graph.

Tuesday, December 22, 2009

Math Facts Effectiveness Study

Students have been using Math Facts for almost a year now and enough information has been collected to determine its effectiveness. Here I will share the data from an couple of anonymous student that has been faithfully using the Math Facts program.

Determine if :
  • answers are entered faster
  • accuracy is improved
These goals are based off of a previous Math Facts study that indicated that faster times and better Algebra I scores go together not just better accuracy.

Case Study of Student A:
Student A information:
  • Used the Math Facts program for the fall of 2009.
  • This student is a good candidate for extra assistance.
  • 204 attempts made.
  • 184 attempts with a score greater than 70% (note that for the smaller tables 100% is required to get above 70%)
  • 2131 questions answered of attempts with final score greater than 70%.
  • 3.73 second average question response time.
  • 3-5 grade level.
The graph below shows the students average question response time. Each mark on the graph indicates the average response time that it took to respond to each question in a given attempt. Two outliers have been removed from the graph but are still included in the trendline and trendline equation in the graph below. Their values were 14.9 seconds on 9/2/09 and21.1 seconds on 10/21/09. These values probably have more to do with the students ability to focus on the activity at hand than the ability to do the problems.

The problems that student A struggles with can be seen in the table below.

3+4 4 42 46 08.70%
4x2 4 26 30 13.33%
5x2 2 12 14 14.29%
4x3 3 17 20 15.00%
3x3 6 26 32 18.75%

The graph below shows the students response time for the question 3x3. Again one outlier has been included in the calculations but removed from the graph to make the graph more readable. The red marks indicate incorrect responses.

Similar to the previous graph the student shows an overall progress towards more faster and accurate responses.

Case Study of Student B:
Student B information:
  • Used the Math Facts program during the spring and fall of 2009.
  • Average student
  • 459 attempts made.
  • 450 attempts with a score greater than 70% (note that for the smaller tables 100% is required to get above 70%)
  • 20233 questions answered of attempts with final score greater than 70%.
  • 1.47 second average question response time.
  • 6-8 grade level.
The table below shows the problems that this student finds most difficult.

4+12 2 9 11 18.18%
9+7 9 40 49 18.37%
8+3 16 60 76 21.05%
3+8 17 53 70 24.29%
7+9 12 36 48 25.00%
8+9 11 30 41 26.83%
9+8 14 30 44 31.82%
4X12 4 8 12 33.33%
3X12 7 6 13 53.85%
5X12 8 3 11 72.73%
A couple of interesting points here are that the accuracy of 3+8 and 8+3 are similar but 9+7 and 7+8 has a difference of 6.63%.

Below is the graph of this students average question response time. Each marker indicates the average response time for that one attempt. Two outliers at about 3.6 and 4 were removed from the graph but not the trendline.

From the table above it appears that this student stuggles with 8+9. The graph below indicates this students response times to the question 8+9. The gap between the two sets of data is summer break.

As you can also see from the graph the number off incorrect responses has also decreased.

As you can see from analyzing these students overall and specific question responses they were able to improve Math Fact abilities with the use of my Math Facts program.

Advantages of this Math Facts program:
  • Students get immediate visual feedback of whether they got the problem correct or incorrect without it being distracting.
  • Automated grading
  • Record of all information allowing detail analysis.
  • Visual stimulation of student to do better and complete as many of their multiplication tables as possible.

Monday, December 14, 2009

Blended Learning Presentation

As I was browsing the web in search of lesson plans for Scribus I discovered the following presentation at the edubuntu.org Lessons wiki, by JimHutchinson. I thought it was very well done. Thanks for sharing Jim.

Sunday, December 6, 2009

New Random Student Selector

I created a new Random Student Selector that is based on Flash instead of server scripting. This should allow the random selections to work faster and reduce the load on my server, providing you with better service. The program works using the same logic as the server scripted version with the added benefit that if a student comes in late you can add their name to the list. They will be entered into the list of selectable names as though they had just answer the last question before they entered the classroom.

It appears that the server scripted version has been very popular. Since I don't particularly care to have advertising when I am using a web page in my classroom I'm going to try an experiment and see if people will donate to keep the ad's off this page.

Thursday, December 3, 2009

Classroom in a Blog: A Quick Introduction to Inkscape

In the image below I've outlined the sections of tools that you need to know to help you get up and running with Inkscape.

The numbered list below high lights each of the numbered tools in the image above.
  1. New Document, Opening, Printing & Importing

    Tools from left to right: New, Open, Save/Save As, Print, Import, and Export
    Inkscape has the ability to open and save a wide variety of file types: svg, PDF, Adobe Illustrator (ai), and more.
    Inkscape can also send files directly to the printer.
    Inkscape can import even more file types that it can open.
    Inkscape can only export one file type - portable network graphic (png).

  2. Zoom tools

    Tools from left to right: Zoom to selection, zoom to all objects, and zoom to page.
    • The first tool allows you to zoom so that what you currently have selected fills the screen.
    • The second tool zooms so that you can see everything that has been drawn. This can be handy if you are trying to find a missing object.
    • The last tool zooms so that your page fills the page.

  3. Useful dialogs

    Dialogs from left to right: Stroke/fill, text, XML, and alignment.

  4. Quick transformations and in-layer ordering

    Tools from left to right: Rotate 90 counter clockwise, 90 clockwise, mirror along the vertical axis, mirror horizontal, move to back, move back, move up, and move to front.

  5. Drawing tools
    • Select & Transform
      Use to move objects around and change their size and rotation. You can also do a few distortions such as skewing.
    • Edit Path
      This tool allows you to readjust key points within an object called nodes.
    • Tweak
      Manipulate objects by pushing them around.
    • Magnifying glass
      Use this tool if the keyboard shortcuts for zooming in and out aren't enough.
    • Square/Rectangle
    • 3D Box
    • Circle/Ellipse
    • Star/Polygon
    • Spiral
    • Freehand
    • Bezier curve & straight line tool
      Bezier curves are based on points and how they affect a line. The line drawn does not necessarily pass though the point that controls what a bezier curve line looks like.
    • Calligraphy
    • Fill
    • Text
    • Diagram Connect
    • Gradient edit
    • Eye Dropper

  6. Drawing area
    The drawing area is all of the white space in the center of the screen referred to as the canvas. The box in the center is a page on the canvas. Because Inkscape is a drawing program not a desktop publishing program such as Scribus, Quark, Adobe InDesign, or MS Publisher, you only get one page to work on.

  7. Color palette

    There are two main things to think about with the color palette the fill and stroke. The fill as the name implies is the color inside of an object and the stroke is the outline color. In the image above you can see that with the current tool that is selected I don't have a fill or stroke color selected because N/A appears instead of a color.
Coming Next
A lesson that will briefly introduce you to the majority of the tools shown here.

Wednesday, December 2, 2009

Classroom in a Blog: Inkscape

This blog series will help introduce you to the open source vector base drawing program Inkscape. The title obviously is a spin-off from the well known series by Adobe Systems.

Note: This series is based on Inkscape version 0.46 and newer. Since Inkscape has not yet matured to version 1.0 we will focus only the features that are considered stable. My focus is to have lessons that can be used in the classroom without having to install extras and work around solutions or cause the program to crash.

Here is a rough table of contents for what this series will contain. I will come back and update this with links as I progress.
  1. Quick Start
  2. Creating and Editing Shapes
  3. Selecting & Aligning
  4. Transforming Objects
  5. Drawing Tools
  6. Color and Painting
  7. Working with Type
  8. Working with Layers
  9. XML
  10. Combining Inkscape with other Applications
  11. SVG
Other Reading:
Book/Website: Inkscape: Guide to a Vector Drawing Program