What do you do when 4-5 students in a precalculus course are asking what a cubic or even worse a quadratic equation is? Me? My jaw dropped. I was nearly speechless. My response was on the verge of disrespect/disbelief I simply couldn't wrap my head around the idea.

How in the world have they made it this far in math and they can't identify a cubic, quadratic or exponential equation?

For a good while I felt like the most awful teacher. Here I had them looking at limits and how fast functions approach infinity and they don't know the difference between a quadratic and a cubic function. Then I thought, wait I didn't teach them those functions... I can blame the other guy. With the finger pointing done, I thought about how cool it was they were understanding complex ideas like limits!

Adding to that list I think this is, to some extent, an indication of the state of math education. I care about vocabulary only so far as it allows my students and I to have a conversation using words not just symbols. My students know how to deal with quadratics on paper they just don't know what they're called.

I have never done "matching" on a test. I think my next quiz will have a vocab matching section...

## April 26, 2011

## April 19, 2011

### Follow Up: Sequence Intro

Oh how a trip to Target can lead to good things in the classroom!

I came up with a few ideas for many students to play with to get them started playing with sequences. A little tactile learning if you will... They even went a bit into the idea of series with out them really knowing (or caring).

I had them try to make a stack of ping pong balls 4 layers high with the least number of ping pong balls. (I was very careful in the write up to always say "ping pong balls" not just "balls." Just a bit of classroom management.) This of course led them all to the same solution of creating a triangular pyramid. They looked at the number of ping pong balls in each layer and how the number in each layer increased. I really like the idea of setting a challenge that would lead them to a solution, sure wish I could have figured out more of those!

I also had them look at "stacking circles" (i.e. poker chips). They placed one chip and then surrounded it with rings of different colored chips (just for visual separation between layers).

Both of the ping pong balls and poker chips were fun and tactile, but the patterns were a little difficult for them to describe in a more formal way (read: with an equation). The poker chips were difficult if you include the original chip... So ala Jo Boaler (a book well worth reading) I had them move on to geometric patterns using foam squares, made from foam sheets I picked up at Target in the craft and scrap booking section.

With the foam squares they could get a little more analytical, as the patterns followed function forms they were familiar with (linear, quadratic and exponential). They made tables of numbers, graphed the results and final were able to create equations to describe the patterns.

The handout I gave the students is a google doc.

I came up with a few ideas for many students to play with to get them started playing with sequences. A little tactile learning if you will... They even went a bit into the idea of series with out them really knowing (or caring).

I had them try to make a stack of ping pong balls 4 layers high with the least number of ping pong balls. (I was very careful in the write up to always say "ping pong balls" not just "balls." Just a bit of classroom management.) This of course led them all to the same solution of creating a triangular pyramid. They looked at the number of ping pong balls in each layer and how the number in each layer increased. I really like the idea of setting a challenge that would lead them to a solution, sure wish I could have figured out more of those!

Stacked ping-pong balls |

Poker Chips in rings |

**Reflection on the process**: A few students really enjoyed the change to a bit of tactile learning, something I really need to remember. The process of making connections from squares to table to graph to equation (i.e. from concrete to abstract) was, I think, highly valuable. That process is at least closer to "real math" or "real science." It also served to slow down some of the folk who like to jump straight to the equation, thinking equations are what math is all about.The handout I gave the students is a google doc.

## April 04, 2011

### Exploring patterns

Back from spring break, time to get chugging...

I am getting close to starting sequences and series with my kids, probably do that by the end of the week. I have my usual inquiry style group work put together from last year, but its pretty boring. Not as boring as me standing up in front of the kids for 2 periods, but not awesome. I want to find something more interesting to start with, then I'll go back to what I did (successfully) last year.

I've been searching the inter-webs for ideas. Patterns the kids can build or observe on the desk in front of them. I see the usual suspects of the sea shells and Fibonacci...

Textbooks suggest the always exciting interest rate based problems. I can already see the eye lids drooping and the drool starting to flow. The books also had some brick stacking problems (which might work) or the oh-so inspiring stadium seating problems...

I'm looking for something that can be posed as a challenge to create something that's not 100% obvious. I want the kids to create something, have I said that yet? So far my candidates are:

I am getting close to starting sequences and series with my kids, probably do that by the end of the week. I have my usual inquiry style group work put together from last year, but its pretty boring. Not as boring as me standing up in front of the kids for 2 periods, but not awesome. I want to find something more interesting to start with, then I'll go back to what I did (successfully) last year.

I've been searching the inter-webs for ideas. Patterns the kids can build or observe on the desk in front of them. I see the usual suspects of the sea shells and Fibonacci...

The curve doesn't fit... How am I suppose to convince students it does? And why would I? |

Textbooks suggest the always exciting interest rate based problems. I can already see the eye lids drooping and the drool starting to flow. The books also had some brick stacking problems (which might work) or the oh-so inspiring stadium seating problems...

I'm looking for something that can be posed as a challenge to create something that's not 100% obvious. I want the kids to create something, have I said that yet? So far my candidates are:

- Sphere stacking - Stack
*x*number of spheres to create the tallest structure. - Packing circles (creative I know) - start with one penny, completely surround it, surround those...
- Dilution of liquid - dy/dan style I love the potential to explore the idea of limits
- Mowing lawn or harvesting hay in increasing or decreasing circles
- Stacking blocks - Stack
*x*number of blocks on top of a single block to create the shortest possible structure - Creating a spiral out of square blocks

I may also explore going in different directions. Such as giving an equation and asking them to create a visual representation of the pattern. This could be done with linear, quadratic equations even exponential...

Another option would be for me to create a pattern on paper, have the students expand the pattern and create a mathematical model...

Any thoughts and particularly any ideas would be greatly appreciated.

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