What Is A Ray In Math?
A ray in math is a part of a line with a fixed starting point but that has no endpoint.
In other words, it is a part of a line that starts at a given point (rather counterintuitively called its endpoint) but continues forever, or in mathematical terms to infinity.
When drawn properly, a ray consists of three parts:
- A point, representing its origin or endpoint
- A line
- And an arrow, representing how the ray goes on forever
An example of a ray can be seen in the diagram below.
Interestingly, because a ray starts but is essentially endless, it can’t be measured.
It can, however, have an infinite number of points across it, which can be individually named and the distance between them measured.
So, what is the difference between a ray and a line?
Although they seem similar and are often confused by students, lines and rays are two very different things in math.
A line’s length is finite – it has two endpoints, a point where it starts and a point where it stops.
Both endpoints can be named and, because it ends, it can be measured.
In contrast, as we’ve mentioned, a ray has no end and it goes on forever in one direction.
It has one endpoint, which is named, and cannot be measured.
How are rays named?
Naming a ray is fairly simple.
A ray is named using its origin point. The ray in the example below would be named A.
If a ray passes through some other points along its way to infinity, its name will begin with its origin point and then follow this with the name of each subsequent point in sequence.
This can be seen in the example below, which represents ray AB.
When writing the name down, it’s important that students draw a small arrow over their letters, which lets everyone know that they are discussing a ray, such as below.
What Topic Or Concept Are Rays A Part Of?
Rays are a fundamental concept in elementary school geometry.
The reason for this is that when two rays share a single endpoint (called a vertex) they form an angle (the rays themselves become known as the angle’s sides), and so the concept of a ray is usually a jumping off point for the study of angles.
Once students are comfortable with the concept, they’ll generally progress to learning and practicing the different parts of an angle, common angle types, angle estimation and measurement and angle naming conventions.
What Grades Are Rays Usually Taught In?
Rays are usually taught as part of an overall lesson or unit in lines, rays and angles.
Under Common Core standards they are taught as part of 4th grade geometry (code standard 4.G.A.1 to be precise).
When it comes to homeschooling math curriculum that aren’t Common Core aligned, however, their position in an overall scope and sequence can be a little different.
Some more advanced programs, such as Beast Academy or certain programs based on the Singapore method for example, may introduce lines, rays and angles a little earlier, often in grade 3.
Other, more slower paced programs, may introduce them in grade 5.
Where Do Students Usually Struggle With The Concept?
Although rays aren’t the most difficult concept to grasp in geometry, they are fundamental to understanding angles and they sometimes are a little challenging for students to get their heads around.
The most common areas that students get tripped up with, and that homeschooling parents should pay particular attention to during lessons, is in:
- Differentiating rays from line segments
- Ray terminology, particularly with the term endpoint
- Naming rays and being able to write and read their notation in a problem
- The concept of indefinite or infinite length
Homeschool Math Curricula That Explain Lines, Rays And Angles Very Clearly
The often finicky topic of angles, lines and rays isn’t always the easiest subject for homeschooling parents to teach, particularly if they haven’t taught math before or are a bit rusty themselves in their math skills.
Luckily, there are a number of really excellent math programs out there that make learning and practicing geometry a lot easier and more effective.
Over the course of our many reviews, two that stood out to us the most when explaining this topic and that we feel are well-suited to uncertain homeschooling parents, are Math Mammoth and Teaching Textbooks.
Math Mammoth is a fairly rigorous, self-study homeschool curriculum that emphasizes the development of a strong conceptual understanding of math.
The program spends a good deal of time working on getting students to understand the why of math, rather than just how to solve problems, and so spends a good deal of time working on the logic of a concept, using visual models, offering different strategies or perspectives and practicing using progressively complex and challenging problem sets.
Lines, rays and angles are covered in the program’s Light Blue series (a full math curriculum) at around the grade 4 level (Book 4B), as well as in the more targeted Geometry 1 from the company’s skill-development focused Blue series.
A self-study program, lessons are aimed at the student and are taught using a combination of simple language and plenty of visual diagrams and drawings, which can be very helpful when it comes to shapes and angles.
Lessons start off with a general introduction to the concept of rays, which are taught more or less alongside lines and angles, using the concept of the sun’s rays going on forever to link the principle to something students will immediately recognize.
Students then learn how to name rays and angles, identify them, estimate them and so on, before being encouraged to try their hand at exploring the concept deeper with protractor work and exercises.
Interestingly, Math Mammoth also offers free video instruction to go alongside their books, where the author of the series dives deeply into math concepts and explains them pretty clearly and in a very straightforward manner.
An example of Math Mammoth specifically teaching the concept of rays, lines and angles can be found below.
For more information about Math Mammoth and its approach to other math topics, check out our in-depth review of the program, or check out the Math Mammoth website and its impressive range of free sample lessons.
Teaching Textbooks is another homeschool math program that we feel explains rays, lines, angles and other concepts in geometry very well.
A self-study, digital homeschool curriculum, Teaching Textbooks teaches math concepts to students using a combination of short, straightforward, amusing and often interactive animated videos and a good number of automatically scored on-screen questions and exercises to immediately reinforce the learning.
At the same time, the program’s detailed progress reporting allows parents to evaluate skill and knowledge development down to the individual concept (such as rays, angles and lines, in this instance).
While the program tends to take a straightforward and procedural approach to math, it also tends to include a good deal of logic and reasoning skills development in most lessons, which is something that many other homeschool math programs often lack.
When it comes to lines, angles and rays, much like many other topics in the program, Teaching Textbooks has its own specific lesson on the concept.
Students are first introduced to the definition of a ray in easy to understand math terms (like Math Mammoth leaning heavily on the sun’s rays as a simile) before being introduced to a more expansive formal definition.
They then learn proper ray naming conventions with simple, step by step animations that carry them through the endpoint, additional ray points and all the proper notation required.
Following this, students have the opportunity to practice their newfound knowledge with 24 or so practice problems to help make sure they remember what lines, rays and angles are all about.
Key Takeaways on Rays
- A ray is a line segment with a starting point but no end point
- Because it is infinite, a ray in math cannot be measured
- The name of a ray starts with its origin point
- Students must place a short arrow over the name of a ray to denote its status as a ray
- Two rays sharing a common origin point form an angle
A ray in math is a fairly straightforward but important concept that forms the basis for a student’s future study of angles and, later, polygons.
Having a proper understanding of a ray, and mastering the ability to name and/or label them, can have a direct effect on a student’s later comfort in geometry.
There are a number of homeschool curricula out there (such as Math Mammoth and Teaching Textbooks) that can make teaching rays and other critical math topics easy, even for parents who have no teaching experience or are themselves uncomfortable with math.
About the Author
David Belenky is a freelance writer, former science and math tutor and a tech enthusiast. When he’s not writing about educational tech, he likes to chill out with his family and dog at home.