Quick Reference: Adding Fractions with Different Denominators

To add fractions with different denominators you just need to multiply each fraction to get the same denominator:

Take:
\frac{3}{5} + \frac{5}{9}

The easiest common denominator will be the product of both denominators ( 5 × 9 = 45 ). So multiply each fraction.

\frac{3}{5}\times\frac{9}{9} + \frac{5}{9} \times\frac{5}{5}

Notice that you’re really multiplying each fraction by 1 (since 9/9 = 1 and 5/5 = 1) and anything multiplied by one remains the same number. So you’re not changing the value of the fraction, just how it looks.

Now doing the multiplication gives:
\frac{3}{5}\times\frac{9}{9} + \frac{5}{9} \times\frac{5}{5} = \frac{27}{45} + \frac{25}{45}

Which we can add because we now have a common denominator:
\frac{27}{45} + \frac{25}{45} = \frac{52}{45}

And simplify to give a mixed number:
\frac{52}{45} = 1\frac{7}{45}

Resonance Frequencies: MythBusters investigate Tesla’s Earthquake Machine

The whole episode is worth watching, but this little section (at 10:52) of MythBusters’ attempt to build an earthquake machine there demonstrate the resonance frequency in a water tank provides a nice visualization.

Rethinking Classrooms with Pedagogy in Mind

OnlineUniversities.com has compiled an interesting list of ways what we’ve learned from neuroscience is being used to rethink school and the classroom.

From 9 Signs That Neuroscience Has Entered the Classroom via OnlineUniversities.com.

Much of the list consists of simpler, practical things that are straightforward (if not easy) to implement: like starting school later (for the adolescents), emphasizing more group work, and stressing the importance of the emotion (positive affect).

But the article also points out some of the newer technology based approaches, such as interactive “cognitive tutoring”.

It’s a useful introductory reference.

↬ K.Cole.

Landing the Mars Rover: 7 Minutes of Terror

NASA gets dramatic. But the drama is oh so appropriate when you see what they have to do to land a rover on Mars. There are so many steps to the landing — heat shields, atmospheric friction, parachute, rockets — that it’ll be amazing if it works, and the video is a wonderful “strike the imagination” introduction to the physics of forces.

Positive to Negative Feedback: Three to One (at least)

Three positives for every one negative is the minimum ratio required for people to flourish, according to the work of Marcial Losada (and others).

To flourish means to live within an optimal range of human functioning, one that connotes goodness, generativity, growth, and resilience.

— Fredrickson and Losada: Positive Affect and the Complex Dynamics of Human Flourishing (ᔥ PubMed), in American Psychologist (2005).

Why do we need more positives than negatives? Because we’re impacted more by negatives than positives, so we need more positives to offset. Note that people tend toward happiness on average.

Why are positive feelings good? Positivity increases:

  • the scope of your attention. Making it possible to see the bigger picture (see Hirsh and Anderson, 2007 (pdf));
  • intuition;
  • creativity;
  • physical healing;
  • the immune system (at least in conjunction with mindfulness meditation: see Davidson et al, 2003);
  • resilience to adversity;
  • happiness;
  • psychological growth;
  • cortisol (positivity reduces cortisol levels — cortisol is a stress hormone; see Steptoe et al., 2004);
  • resistance to physical pain;
  • how long you’ll live.
  • how much you learn.

In terms of education:

… initially positive attitudes—like interest and curiosity—produce more accurate subsequent knowledge than do initially negative attitudes—like boredom and cynicism. Positivity, by prompting approach and exploration, creates experiential learning opportunities that confirm or correct initial expectations. By contrast, because negativity promotes avoidance, opportunities to correct false impressions are passed by …. positive affect—by broadening exploratory behavior in the moment—over time builds more accurate cognitive maps of what is good and bad in the environment. This greater knowledge becomes a lasting personal resource.

— — Fredrickson and Losada: Positive Affect and the Complex Dynamics of Human Flourishing (ᔥ PubMed), in American Psychologist (2005).

So, once again, research shows how important it is to have a positive (happy) learning environment, and to be able to “spark the imagination” at the beginning of a lesson.

Cumulatively, there is the Losada Zone, a range from 3:1 to 11:1 of positive to negative feelings that are indicative of complex (good) interaction in groups, which separates people who flourish from those who languish.