Sunday, August 24, 2014

Dimensional Analysis: Start with what the students know

Want your students, especially those that struggle in math, to be successful in dimensional analysis all year long? In that case, I highly recommend that chemistry teachers start off using units that are relevant to students, as well as incorporating dimensional analysis on a continual basis (more on the latter later). When I first started teaching chemistry I began the dimensional analysis (DA) unit with metric problems. The challenge here in the USA is that we don't use the metric system like other countries do and so not only was I facing a battle teaching DA, I was also battling student knowledge concerning the metric system. I was frustrated. The students were frustrated. It was a lose-lose situation. 

I decided to change my approach my second year of teaching based on what I believed most, if not all, students would be familiar with - money. My students were grouped in fours and each group received a bag of coins...yes, actual coins and I always received the exact amount back. 
Of course this could be done with fake money or without it as well; however, I wanted my students interacting with objects and discussing - so decide fake or real $$$ if you wish to do this. I would start off by asking the students to make me a dollar out of each set of coins. They all knew how to do this (100 pennies, 20 nickels, 10 dimes, 4 quarters).
I would then choose a set of coins (for example, quarters and dimes or nickels and pennies) and ask the students to discuss in their groups for approximately 30 seconds to 1 minute if these measured the same quantity. Group answers were always yes.  Then I would ask students to try to write a mathematical relationship to show this. 
Most groups would write 4 quarters = 10 dimes or 20 nickels = 100 pennies. I would then write the equality as a fraction on the board (both ways as shown in the two graphics above) and ask the students to discuss in their groups for approximately 30 seconds to 1 minute if both fractions were the same. As I walked around I would hear some groups agreeing and others disagreeing. I would then pull the class back and use Popsicle sticks to call random students to share (either they could share what they believed or if they were not confident in their response they could share what their group discussed). After a few student responses, I would talk about conversion factors and equalities (still using the illustrations shown above). I would ask students to write a conversion factor for $1 using dimes and nickels just to ensure they were understanding and following along. 

Next, I would move to another topic students were familiar with - time. I don't know about you, but students are always asking for the time or when class is over or when something is beginning; it's probably the reason I didn't have a classroom clock. Anyway, to see if students understood the money concept, I would ask them to write a conversion factor for 1 hour and 60 minutes (shown below). 
Other years I've written the conversion factors on the board and asked if the fractions were the same. I would challenge groups to come up with as many time conversion factors as possible within a 1-2 minute period (examples:  24 hours = 1 day;  365 days = 1 year;  10 years = 1 decade). After this, I would move into the heart of dimensional analysis (DA). My main focus was ensuring that students understood and could show the steps of the process. Even if they calculated the answer incorrectly but set up the problem correctly that was okay (at least for now as that was a technical issue). 
I had a magnetic white board and printed out colored pieces similar to the illustration above and used magnetic tape (you can use different colored paper versus color ink). I would then tell the students: Criss-cross apple sauce. At this point I would get laughed at and also felt a little silly no matter how many times I had said this in the past; however, it was an instructional tool I used throughout the lesson. I would mention criss-cross apple sauce and ask students to look at the magnetic pieces on the board (shown above).  Of course, I purposely mixed some of the units around to show students the idea of canceling units that were similar in the numerator and denominator as it does not matter mathematically where they are at in the problem. However, if you think that might confuse them, then put the units in order from given to unit wanted
At this stage of the game I would incorporate the Gradual Release model (I do, We do, You do) and work a few problems making sure I said Criss-cross apple sauce as often as possible while working problems. I should note that I had large student white boards and each student practiced on them during this part of instruction. Additionally, for each problem, I would give students the conversion factors (example shown above) and stress that they could only use a conversion factor once and to cross it out when they did. I would stress this because sometimes students would use the same conversion factor twice in the problem (written in both forms - see examples at top for reference). Some teachers might be against this, however, my goal is to build comprehension and capacity for solving these problems. In the beginning I want students to show me they understand the concept/skill of dimensional analysis, not whether they can memorize a bunch of conversion factors - at least that is not the standard here in Texas and probably not the standard in many places.

Last but not least, is how I grade student work. As we practice (during instruction or a warm-up), I go over the points system and show students how they earn points per problem. This problem is worth six points: 1pt. per conversion factor step, 1pt. for the correct answer, 1pt. for the answer written with the correct number of significant figures, and 1pt. for the unit. By the time students take a quiz or test they realize the importance of showing their work and it usually is not a hassle. 

I continually incorporate DA warm-ups and tricks throughout the fall semester so students have had plenty of practice by the time stoichiometry comes around.  I like to be evil during labs and tell students to weigh out a unit of a substance not in grams (for example 50mg of sucrose) and quietly laugh as students go back and start weighing. Some groups will realize right away that the balances only measure in grams, while others will not; but it's a great way to incorporate DA throughout the school year.  :p

Whew! That was a lot. I hope this helps. I would be remiss if I didn't highlight a few of my products geared toward this topic.  The first one is a Dimensional Analysis: Time Conversions Cut & Paste activity for the Interactive Notebook.  The other is a bundle of items for a measurement unit - Numbers in Science for the Chemistry Interactive Notebook and has a metric cut & paste activity, as well as dimensional analysis notes and problems. Check out the reviews and if you have any questions, feel free to comment here or leave a message at Bond with James: Q&A (click on the tab)

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