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
Lo how a rose e'er blooming From tender stem hath sprung As I write this blog post, a radio station is playing Chrismas music. One of my favorite Christmas songs is the old German hymn that many of us know as "Lo, How a Rose E're Blooming." I was humming
Throughout my 13 years of teaching, I heard this question time and time again from parents: How can I help? The troubles they specified were varied and numerous (e.g., concern, frustration, bewilderment). They often included the following: I don’t remember algebra. I don’t like math. The way math is taught now differs
Curriculum Pathways has an Algebra 1 Course. That’s right, an entire standards-aligned course! And this resource is much more than a typical math textbook. PLUS, it’s a cross-platform resource! Yes, you read that correctly, cross-platform. That means this resource can be used on the PC, iPad, and Chromebook. And it
Mobile is at the forefront of our edtech development here at SAS Curriculum Pathways—not simply because it's the hottest emerging hardware technology entering classrooms around the world, but because of the exciting possibilities it brings to traditional learning environments. We’ve known for years that simply dropping off hardware in the classroom
Solving equations and inequalities are essential skills in mathematics. And for many students, practicing is the key to success in solving. As students work on the concept of solving equations/inequalities at home, many think, "How will I know if I’m doing it right?" One answer is the Practice Solving Series,
In my article about finding an initial guess for root-finding algorithms, I stated that Newton's root-finding method "might not converge or might converge to a root that is far away from the root that you wanted to find." A reader wanted more information about that statement. I have previously shown
Have you ever seen a problem similar to this on social media sites? 5 - 1 x 0 + 3 ÷ 3 = ? Is the answer 1 or is it 6?* Or did you get something completely different? Here’s another example.
"Daddy, help! Help me! Come quick!" I heard my daughter's screams from the upstairs bathroom and bounded up the stairs two at a time. Was she hurt? Bleeding? Was the toilet overflowing? When I arrived in the doorway, she pointed at the wall and at the floor. The wall was
Equations that involve trigonometric functions can have infinitely many solutions. For example, the solution to the equation tan(θ)=1 is θ = π/4 + kπ, where k is any integer. In order to obtain a unique solution to the equation, we define the "arc" functions: inverse trigonometric functions that return a
Last week I was chatting with some mathematicians and I mentioned the blog post that I wrote last year on the distribution of Pythagorean triples. In my previous article, I showed that there is an algorithm that uses matrix multiplication to generate every primitive Pythagorean triple by starting with the
“Mathematics has beauty and romance. It’s not a boring place to be, the mathematical world. It’s an extraordinary place; it’s worth spending time there.” – Marcus du Sautoy. I agree, there IS beauty in mathematics, and I genuinely love math! Math is vital to everyday life from young kids learning
Pi is the ratio of a circle’s circumference to its diameter. Although it is a non-terminating, non-repeating number, the decimal approximation 3.14 and the fraction 22/7 are typically used to represent the irrational mathematical constant. The Greek letter π, was first used as the symbol for pi in 1706 by
Math can be difficult for students. But learning how to use numbers in flexible ways can be the key building block to understanding. SAS Math Stretch, available for free in the App Store, designed to develop elementary math skills and number sense includes activities focused on counting, number relations and
Mobile devices are arriving in classrooms around the world in soaring numbers. Chromebooks and iPads are everywhere – from 1:1 programs to BYOD to carts classes can share. Why is that important? Because these devices have the power to change the classroom, provided teachers have the right integration strategies and lesson
Mathematics, what a beautiful subject. It requires inquisitiveness, perseverance, and critical thinking. It promotes discourse, challenges the mind, and creates problem-solvers. Unfortunately, it also scares many students and adults. Why is that? How can one subject cause so much joy and at the same time so much discomfort? As a
Math is essential to a successful career in computer science. That’s right! Jobs such as those listed below require algebra, calculus, discrete math, and other courses. Software engineer/developer (every industry) Social network analysis Gaming Anti-terrorism/cybercrime/fraud detection Data scientist (sports/business/finance) Math and computer science share terminology and notation. Computer scientists use math
Pascal's triangle is the name given to the triangular array of binomial coefficients. The nth row is the set of coefficients in the expansion of the binomial expression (1 + x)n. Complicated stuff, right? Well, yes and no. Pascal's triangle is known to many school children who have never heard of polynomials
