by Jeffrey Elkner
It has been my long held belief that a deep understanding of what a number is, and particularly what a fraction is, is required for success in higher level math courses.* It was for that reason that I was so excited about the opportunity to teach a course titled Math Foundations and Pre-algebra this year, since it would permit me to help students get back on the path of success in mathematics by intervening in the most effective spot - with their basic understanding of numbers and number theory.
My approach has been to show students visual representations of fractions in as many contexts as I can, and then require them to decode these visual representations and to convert them into symbolic ones (as ratios of integers). One of my favorite problems of this type consists of an illustration of a line segment with two numeric values at each end, equally spaced marks along the length of the seqment, and a point on one of the marks. I ask students to determine the numeric value of the point.
I needed a way to quickly create the illustrations I needed. The solution I found was to use Turtle Blocks, the Sugar Learning Platform activity that has been ported to the Debian and Ubuntu operating system desktops. When I needed a tool to be able to easily generate the point on line segment diagrams I wanted to help students learn about fractions and how they related to the number line, I wrote this point_on_a_line_segment.ta program in Turtle Blocks:
Using this little program and GIMP to trim the images it produces to size, I can generate point on number line problems in about a minute. I can then use the images in quiz items I give students using our new SchoolTool Quiz application to track student work on these quiz items.
What is the value of the red point?
4/7. The length of the segment is 1 unit (distance between 0 and 1). The 1 unit segment is divided into 7 equal parts, so each part is 1/7 unit. The red point is at the end of the 4th part, so its value is 4/7.
What is the value of the red point?
12. The length of the segment is 3 units (distance between 11 and 14). The 3 unit segment is divided into 9 equal parts, so each part is 3/9, or 1/3 units. The red point is at the end of the 3rd part, so its value is 3/3 + 11, or 12.
What is the value of the red point?
38/3 or 12 2/3. The length of the segment is 3 units (distance between 11 and 14). The 3 unit segment is divided into 9 equal parts, so each part is 3/9, or 1/3 units. The red point is at the end of the 5th part, so its value is 5/3 + 11, which can be written as either 38/3 or 12 2/3.
What is the value of the red point?
7. The length of the segment is 2 units (distance between 6 and 8). The 2 unit segment is divided into 12 equal parts, so each part is 2/12, or 1/6 units. The red point is at the end of the 6th part, so its value is 6 + 6/6 = 6 + 1 = 7.
What is the value of the red point?
3/5. The length of the segment is 1 unit (distance between 0 and 1). The 1 unit segment is divided into 5 equal parts, so each part is 1/5 units. The red point is at the end of the 3rd part, so its value is 3/5.
What is the value of the red point?
47/6 or 7 5/6. The length of the segment is 2 units (distance between 6 and 8). The 2 unit segment is divided into 12 equal parts, so each part is 2/12 or 1/6 units. The red point is at the end of the 11th part, so its value is 6 + 11/6 = 6 + 1 5/6 = 7 5/6.