This article discusses how to restrict a multivariate function to a linear subspace. This is a useful technique in many situations, including visualizing an objective function that is constrained by linear equalities. For example, the graph to the right is from a previous article about how to evaluate quadratic polynomials.
There are several different kinds of means. They all try to find an average value from among a set of numbers. Although the most popular mean is the arithmetic mean, the geometric mean can be useful for problems in statistics, finance, and biology. A common application of the geometric mean
What is this math good for, anyway? –Every student, everywhere I am a professional applied mathematician, yet many of the mathematical and statistical techniques that I use every day are not from advanced university courses but are based on simple ideas taught in high school or even in grade school.
Early math skills—such as number sense, number relations, pattern identification, estimation, and measurement—are the foundation for future math achievement. By mastering these basic concepts, young students can engage in more complex cognitive processing and develop higher-level math skills. Also, early math achievement has been linked to an increased likelihood to take
The eigenvalues of a matrix are not easy to compute. It is remarkable, therefore, that with relatively simple mental arithmetic, you can obtain bounds for the eigenvalues of a matrix of any size. The bounds are provided by using a marvelous mathematical result known as Gershgorin's Disc Theorem. For certain
(Editor's note: This guest blog post was written by Buhl High School, Idaho math and science teacher Pat Van Patten. When asked why she stays in the classroom after 30 years, Pat replied, “I still love teaching. I love watching students learn; when the light turns on for them and they
In honor of Black History Month, let’s showcase a few African American mathematicians who have made their mark in the teaching profession. Elbert Frank Cox (1895-1969) was the first black person in the world to receive a Ph.D. in mathematics. In 1917, Cox earned his undergraduate degree from the University of
In June, we launched our free, online lesson-building tool, Crio. Since then many teachers have created and shared their lessons. But for those yet to jump in, I’ll bet there are at least three questions you've asked yourself. So, let me answer them! Why should I create Crio lessons? Because with
Last week I compared the overhand shuffle to the riffle shuffle. I used random operations to simulate both kinds of shuffles and then compared how well they mix cards. The article caused one my colleague and fellow blogger, Rob Pratt, to ask if I was familiar with a bit of
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
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 SAS can help. And remember, all of these resources, including apps, are free! CodeSnaps Have some fun coding a Sphero robot using CodeSnaps! With the
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
