How Math Fixed Music
How Math Fixed Music Rational Exponents Sound GREAT Before we dive in, music is primarily defined by what we hear, not by the analysis and insight provided by math. For example, an octave is a note whose frequency is double that of its parent note. The mathematical relationship was discovered after the fact. The following … 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
Cube Roots and Higher Order Roots
other roots Cube Roots and Other Radicals Square roots ask what squared is the radicand. A geometric explanation is that given the area of a square, what’s the side length? A geometric explanation of a cube root is given the volume of a cube, what’s the side length. The way you find the volume of … Read more
Multiplying and Dividing Square Roots, Rationalizing the Denominator
1.7.2 square root operations continued Square Roots Multiplication and Division At some point square roots should no longer be considered an operation but rather the most efficient way to express a number. For example, the best way to write one hundred trillion is 1× 10 14 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgE na0kaaigdacaaIWaWaaWbaaSqabeaacaaIXaGaaGinaaaaaaa@3BE4@ . The … Read more
Addition and Subtraction of Square Roots
add subtract square roots Mathematical Operations and Square Roots Part 1 In this section we will see why we can add things like 5 2 +3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaka aabaGaaGOmaaWcbeaakiabgUcaRiaaiodadaGcaaqaaiaaikdaaSqa baaaaa@3A0D@ but cannot add things like 2 5 +2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaka aabaGaaGynaaWcbeaakiabgUcaRiaaikdadaGcaaqaaiaaiodaaSqa baaaaa@3A0D@ . Later … Read more
The Most Important Component of Quality Teaching is …
What do you think the single most important part of effective teaching, in high school, is? Breaking down classroom management and teaching into a lock and step routine is impossible. People are too variable. And, especially in high school, we are talking about the interactions of 150 – plus people a day! It is because … Read more
Square Roots Part 1
Note: Square roots are pretty tricky to teach and learn because the tendency is to seek answer-getting methods. Patience at the onset, allowing for full development of conceptual understanding is key. Do not revert to tricks and quick “gets” when first learning square roots. Always revert back to the question they ask and how you … Read more
Where the First “Law” of Logarithms Originates – Wednesday’s Why E6
For a PDF of the following, click here. For a PowerPoint of this information, click here. rule 1 Wednesday’s Why – MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbCaqa aaaaaaaaWdbiaa=nbiaaa@3804@ Episode 6 In this week’s episode of Wednesday’s Why we will tackle why the following is “law” of exponents is true in a way that hopefully will … Read more
What Do Grades Really Mean?
What Do Grades Mean The following is highly contentious. Many of the situations discussed here should ultimately be considered on an individual basis. The purpose of this is not to create a rubber-stamp solution to all problems that arise with grade assignment and student ability and or performance, but is to provide a general framework … Read more
How to Teach Exponents – Part 1
Exponents are one of the most difficult topics to teach because once you understand how they work, it seems so obvious. And once you understand something to the point of it being obvious, your memory of how difficult it was to learn and what caused problems is almost entirely erased. To “Get It,” with exponents … Read more
