10-Minute Math

I wanted a nice region bounded by two curves (who wouldn’t?), so I graphed a parabola and a reciprocal function passing through (0,1):

y = -x² + 1
y = 1/(x + 1)

The second intersection looked vaguely familiar… 0.61803398875… where had I seen that before?

Oh yeah, the golden ratio!  Phi = 1.61803398875…

An interesting fact is that Phi – 1 = 1/Phi … are there any other numbers that have this quality?

As an enrichment activity I had a student fill a large sheet of graph paper with Pascal’s triangle – just the ones digits, anyway.  Then he colored the digits in different ways to discover patterns.  After doing this by hand, the student was interested in saving some time with Excel, so I devised this method:

1. Make a grid of 0’s.  Really you just need a border of 0’s, but filling an area is faster.  It should be twice as wide as tall.
2. Select cell B2, and give it the formula =MOD(A1+c1,10).
   
3. Copy B2 and paste it into all the cells in the grid except for the top row and the left and right columns.
4. Now they are all adding the values of the two cells diagonally above them.  Just put a “1″ in the middle of the top row and see what happens.
   
5. To color the different digits, click Format, Conditional Formatting, and make some stuff up.
   
6. I changed the font color to white and clicked File > Web Page Preview:
   

You could do some fun experiments with this, leading kids toward the idea of cellular automata!

An extra credit assignment for my calculus class:

A pack of zombie dogs (ZDs) is reproducing quickly and killing humans. Scientists and calculus students have determined that the the total number of human deaths due to ZDs t days after the start of 2010 is given by

1. At the end of January 1st (t = 1), how many deaths have occurred?

2. At the end of January, how many deaths will have occurred?

3. Determine the number of deaths/day after t days: h′(t).
Hint: you may want to go “outside-in” with the chain rule.

4. Graph both h(t) and h’(t) on the same axes:

5. On what day will 90% of the human population (6.8 billion) be annihilated by ZDs?

6. When does the death rate reach half a billion per day?

7. When does the death rate hit its peak?

I collected a set of various nonlinear graphs and hid the labels and scales.  They’re good for discussions (I’m teaching exponential growth now).  Here’s the Word file.

Can you match them all? Height of a projectile, iTunes sales, World Population of gorillas, Transfer speed versus message size on several networks, Heights of a group of men, Population of fruit flies in a container, Decay of radioactive atoms, Heights of babies over time.

I created a review game with this online tool today:

If you want to play it, here it is.

Please let me know if you use any of these in your classroom!

1. Teacher Soundboard
   Add  random sound effects to your instruction.  Negligible educational value.
   
2. Brown Blobs
   Knock the brown spots off the graph with an appropriate linear equation.
   Let me know what happens when you get to level 100 – no one has gotten there yet.
   
3. Read the rest of this entry »