Teaching Square Roots Conceptually

Teaching Square Roots Conceptually   teaching square roots How to Teach Square Roots Conceptually If you have taught for any length of time, you’ll surely have seen one of these two things below. 24−−√=62–√   or     4–√=2–√24=62   or     4=2 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIYaGaaGinaaWcbeaakiabg2da9iaaiAdadaGcaaqaaiaaikdaaSqabaGccaqGGaGaaeiiaiaabccacaqGVbGaaeOCaiaabccacaaMc8UaaGPaVlaaykW7caaMc8+aaOaaaeaacaaI0aaaleqaaOGaeyypa0ZaaOaaaeaacaaIYaaaleqaaaaa@479C@  Sure, this can be corrected procedurally.  But, over time, they’ll forget the procedure and revert back to following whatever misconception … Read more

Best Practices

Philosophy and Best Practices Cart Before the Horse The easy part of teaching any curriculum is the curriculum.  The hard part of learning to teach a new curriculum is figuring out how to teach it most effectively.  For the sake of clarity, effective teaching will develop student conceptual understanding and problem solving.  Earmarks of quality … Read more

Student Skills and Tools

The biggest hurdle in transitioning from Middle School to High School is the lacking set of student skills possessed by incoming Freshmen.  Students come in failing to appreciate the importance of homework, struggle to think independently, cannot communicate mathematical thinking, and are easily frustrated to the point of quitting. This observation is not a knock … Read more

Thoughts on Teaching

Foundation 1. The goal: If the question, “When am I going to use this in my real life,” derails your class, there’s a problem with your purpose and goal. The truth is, almost nothing after taught 5th grade is knowledge used daily. The purpose of education is not to teach MLA formatting or how to … Read more

Teaching, Learning and Habits

How Habits and Education Collide   The best definition I have come across for a habit is, “action without thought.”  A quick search on the internet says that a habit is, a settled or regular tendency or practice, especially on that is hard to give up … We certainly need habits, especially in education.  Students, … Read more

Vestiges of the Past Making Math Confusing

Something in Math HAS to Change Convention is a beautiful thing.  It allows us to use symbols to convey little things like direction or a sound.  We can piece those things together to make larger things, and eventually use it to create something like what you’re reading now.  There are no inherent meanings to these … Read more

Rational Exponents and Logarithmic Counting …

For a PDF copy of the following, please click here. rational exponents Rational Exponents In the last section we looked at some expressions like, “What is the third root of twenty-seven, squared?” The math is kind of ugly looking. 27 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIYaGaaG4namaaCaaaleqabaGaaGOmaaaaaeaacaaIZaaaaaaa@392A@ The procedures are clunky and it … Read more

The Smallest Things Can Cause Huge Problems for Students

preemptive Pre-Emptive Explanation It is often the case, for the mathematically-insecure, that the slightest point of confusion can completely undermine their determination. Consider a beginning Algebra student that is learning how to evaluate functions like: f( x )=3x− x 2 +1 f( 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaG4maiaadIha cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa qaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaaaaaa@43D9@ A confident … Read more

Why Teaching Properties of Real Numbers is Important

If you are going to do a fraction review, the lesson here might be of some help.  I believe things are best reviewed in context, but this is a decent set of information that also introduces the real numbers and some other basics of math. The PDF icon to the left has a lesson outline … Read more

Sets of Numbers and the Problem with Zero Chapter 1 – Section 1

1.1 3 Sets of Numbers and the Problem with Zero and Division We will begin with the various types of numbers called Real Numbers. Together, these numbers can be ordered and create a solid line, without gaps. Ø  Natural Numbers: These are counting numbers, the smallest of which is 1. There is not a largest … Read more