Given a rectangular grid with unit spacing, what is the expected distance between two random vertices, where distance is measured in the L1 metric? (Here "random" means "uniformly at random.") I recently needed this answer for some small grids, such as the one to the right, which is a 7 x 6

## Tag: **Math**

Continued fractions show up in surprising places. They are used in the numerical approximations of certain functions, including the evaluation of the normal cumulative distribution function (normal CDF) for large values of x (El-bolkiny, 1995, p. 75-77) and in approximating the Lambert W function, which has applications in the modeling

Did you know Curriculum Pathways resources include an entire online Algebra 1 Course? And it's much more than a typical math textbook. The Algebra 1 Course consists of 46 lessons organized into 10 units. In addition, students can work through two pre-algebra units (7 lessons) in preparation for the course.

Kicking off the new school year, you may be looking for ways to ease your students back into math. Here are a few ways that Curriculum Pathways can help. And remember, all Curriculum Pathways resources, including apps, are free! Check out Crio! Use our brand new drag-and-drop editor to transform your

My colleague Robert Allison recently blogged about using the diameter of Texas as a unit of measurement. The largest distance across Texas is about 801 miles, so Robert wanted to find the set of all points such that the distance from the point to Texas is less than or equal

Back in high school, you probably learned to find the intersection of two lines in the plane. The intersection requires solving a system of two linear equations. There are three cases: (1) the lines intersect in a unique point, (2) the lines are parallel and do not intersect, or (3)

That exciting #edtech day is finally here! The new iPads are unboxed, charged, and configured. The only question left is where to begin? Of course the best answer involves ongoing professional development and a good school-wide education technology plan. And that's no secret. Well-planned and resourced professional development is the

Tap into student interest in athletic competitions with these sports themed lessons. Creating and Annotating Maps Our Interactive Atlas (also available in Spanish) is a great tool for project-based learning. Using our map and annotation tools, students demonstrate mastery of content while engaging their research and writing skills. Consider the following sports related activities:

A new Star Wars movie debuts in just a few weeks! Star Wars: The Last Jedi will be the next chapter in the saga we’ve all come to love. The first film, Star Wars: Return of the Jedi, debuted in 1977. So, math teachers, how old were you when that

Are you ready to code in your ELA, math, science, and social studies classes? We now have lessons to support coding across the curriculum. Our new Show with Code series challenges students to demonstrate mastery of certain concepts through CodeSnaps coding activities. The initial lessons in the series include these titles and disciplines: Show with

What a powerful question! And it's one commonly posed to math teachers all over the world. Need help with an answer? Here are a few to consider. Suppose you're asked the question about a piecewise function. Let's start with the standard definition: "a combination of two or more functions, each represented

More than five years ago, the Curriculum Pathways team released a comprehensive and FREE online Algebra 1 course. It includes 46 engaging lessons that use videos and real-world examples, along with interactive elements to develop students’ problem-solving, reasoning, and critical-thinking skills. Since the course's release, some states and districts have chosen not to

Our Curriculum Pathways team has been hard at work to modernize existing resources and create new ones for you to use in your classroom this year. As always, please let us know if you cannot find a resource on a topic you're teaching. We'd love to add your suggestion to the

The all-new online professional development offerings from Curriculum Pathways enable teachers to learn about new ideas and resources specific to their discipline areas and grade levels. Equally important, teachers focus on resources that model effective technology integration. This model fills a vital need identified by teachers, the National Schools Boards Association, the Center

"The first year of teaching is the toughest.” After making it through my first year as a high school math teacher, I have a much better understanding of why people always say this. There are so many challenges: adapting to a new environment, staying organized, learning curriculum, making professional connections,

Every day in my class, students have about 10 minutes of story time. What grade level do I teach? At the mention of story time, many of you may picture a classroom full of individual carpet spaces with twenty small chairs and desks and even smaller students. But I'm not an elementary school teacher;

Giving students the opportunity to “see” how the parameters of a function affect its graph can be challenging for teachers. So is sitting with a student one-on-one as they solve or simplify multiple problems while other students await the same valuable time. But with the use of graphing, solving, and

Since numbers are its foundation, mathematics is commonly referred to as the one truly universal language. But do you see a difference in these equations? 1.000 + 1.000 = 2.000 1,000 + 1,000 = 2,000 Whether these two equations represent the same value could depend on where you are. That’s

Are you teaching that PEMDAS is flexible? Are your students equipped to handle that flexibility? You remember PEMDAS, right? It’s the acronym we use to help kids understand, or better yet remember, the order of operations. We’ve even added the catchy phrase Please Excuse My Dear Aunt Sally to assist

Year after year, teachers must meet the needs of a wide range of students. Often overlooked, however, are strategies to help English Language Learners (ELL), whose numbers are growing. From 2002-03 to 2012-13, the percentage of ELLs in U.S. public schools increased in all but 11 states. ELLs describe a

This article uses graphical techniques to visualize one of my favorite geometric objects: the surface of a three-dimensional torus. Along the way, this article demonstrates techniques that are useful for visualizing more mundane 3-D point clouds that arise in statistical data analysis. Define points on a torus A torus is

Data--that small four-letter word packs a mighty punch! Data is everywhere. Stores use it to determine how to advertise products, politicians use it to support their campaigns, and Google uses it to improve the search engine that may have brought you to this blog! Data is displayed in newspapers, brochures, billboards, as

Last week I blogged about how to draw the Cantor function in SAS. The Cantor function is used in mathematics as a pathological example of a function that is constant almost everywhere yet somehow manages to "climb upwards," thus earning the nickname "the devil's staircase." The Cantor function has three

Are you teaching a lesson on parallel lines? What about a lesson on the interior and exterior angles of polygons? If so, our Geometry lessons are the perfect resources for you! We released our first one, Parallel Lines and Angles, this past fall. Since then, we've released four more. The lessons

One of my goals a few years back was to become a blogger. I’m not a writer. I deal with numbers not words. So panic often set in when I started to write. But then I realized that—panic or not—I had to step out of my comfort zone if I wanted

Starting a new lesson and need a quick overview of the upcoming concepts? Or are you starting a new lesson and want students to quickly review prerequisite skills? Well, these 4-6 minute tutorials are just what you need. Why are some numbers rational and some irrational? That’s an important question

A moving average (also called a rolling average) is a statistical technique that is used to smooth a time series. Moving averages are used in finance, economics, and quality control. You can overlay a moving average curve on a time series to visualize how each value compares to a rolling

“It is a truth universally acknowledged, that a single man in possession of a good fortune must be in want of a wife.” –Jane Austen, Pride and Prejudice It is almost certainly a truth universally acknowledged that Jane Austen’s opening line from her great novel is one of the most

Recently I blogged about how to compute a weighted mean and showed that you can use a weighted mean to compute the center of mass for a system of N point masses in the plane. That led me to think about a related problem: computing the center of mass (called

As 2015 comes to a close, let's look back on the great year at SAS Curriculum Pathways. Again, we saw record-breaking new users, soaring student usage, and tons of app downloads. While we love all of our 1,500+ resources, here are the ones that teachers and students used the most in