Fun With Fractions: Day 3

After we wrapped up our summary of how to compare fractions, we moved in to equivalency.

Everyone unfolded all their fraction strips and started finding where they had matching folds. By this time, everyone was pretty clear on how to use their strips and has a better understanding that all of our strips are the same size for a reason. After about thirty seconds, we choose one volunteer to tell us one match they had found. They picked /5 and /10 and the class helped list every fraction that matched on that list. From there we introduced equivalent and had the students spend about 15 minutes in their groups using their strips to list as many matching fractions as they could.

After all of the resistance to this kind of work in my Honors class, this class’ attitude was really refreshing. Neither of us saw any lists of multiplied numbers; they really worked with trying to find matching pairs visually*. We listed a few more on the board at the end of the time, and then go into defining equivalence and multiplying by a whole with any factor.

Then we introduced some other models. I really struggled with when was the appropriate time to do this–I didn’t want to tie students to the strips, but I also didn’t want to have so many things going on that students couldn’t focus on the important conceptual understanding. Ultimately, I decided to put it here, after comparison and wrapping up equivalency. We first did tape diagrams/fraction bars, and then moved on to number lines. We did a little the day before around partitioning number lines, but I saved most of it for today. Its definitely harder to work with eyeballing your own partitions, so we actually first flipped our notebooks sideways so we could make an 8″ + long line that worked with the fraction strips. It was a nice bridge between using something that was already the correct size (the strips) and emphasizing that the placement on the number line needed to be accurate.

I’m out for Day 4 at a training, but hopefully they’re doing their best at ordering fractions on a number line. (Although I already am thinking we could have done better by very intentionally choosing fractions that all use the same partitions for more problems to really get into that idea before making them need several different kinds of partitions (/5 and /8 on the same line). There’s always next year!

How do you help your students order fractions? Any ideas beyond “multiply all of the things”?

*I don’t have any problem with multiplying by a whole, but most of my students don’t have any idea why they’re doing it or what it means.


Number Sense: Whatever That Means…

Next up in Algebra 1 is “Number Sense”–I pushed hard for this one, and we have around three weeks. As I talked with a coworker today, he pointed out that much of number sense can’t exactly be taught–a point I only partially agree with.

Wikipedia says, The term “number sense” involves several concepts of magnitude, ranking, comparison, measurement, rounding, percents, and estimation, including: [10]
  • estimating with large numbers to provide reasonable approximations;
  • judging the degree of precision appropriate to a situation;
  • solving real-life problems involving percentages and decimal portions;
  • rounding (understanding reasons for rounding large numbers and limitations in comparisons);
  • choosing measurement units to make sense for a given situation;
  • comparing physical measurements within and between the U.S. and metric systems; and
  • comparing degrees Fahrenheit and Celsius in real-life situations.[10]

All good points, if technically stated.

I’d call number sense being able to understand relative size and equivalency of numbers, estimate both individual quantities and combinations (like products and sums) and locate numbers on a number line. I know I’m missing a lot here, and I plan on refining that definition a lot (notice something? Let me know in the comments!).

That’s all good, but I need to determine how that translates to the classroom. I talk a lot about fractions below, and have a lot more to say about them, but have been really pressed for time. I worked with my elementary school teacher/math specialist sister to develop a lot of what I did, including rethinking my OWN understanding of fractions, and I’m really excited for where my other class goes with this.

One thing for sure: I want to use Estimation 180 throughout this unit, hopefully daily (discovered through Infinite Sums).

Here’s my very work-in-progress plan (these are topics; some might be several in a day, others might take multiple days):

  • Prime Factorization
  • Place value (naming place values, building numbers, identifying place value of a specific digit and vv, naming numbers correctly)
  • Powers of 10 and equivalent forms (x 10, x 0.001 etc)–we will use the Pyramids again here
  • The Fraction Screener (to give me an idea where my students are)
  • Folding Fraction Strips & Defining Unit Fractions
  • Modeling fractions using fractions strips and number lines
  • Comparing fractions & summarizing comparison rules
  • Locating fractions on a number line and representing fractions several ways
  • Fraction equivalency (Stuck on: how to prove two fractions are equivalent)
  • Ordering fractions, then locating them on a number line, including a gallery walk style group placement activity
  • Adding, Subtracting, Multiplying and Dividing fractions (maybe. I teach 9th grade. My kids have seen this over and over and over–I don’t want to repeat the stuff they’ve already seen/heard/done, so I’m only doing this if I really think I can add value)
  • The ONE (using pattern blocks to determine the whole for different size partitions
  • Converting between fractions and decimals (use a calculator? I’m not sure I see too much value in having them convert to /10, /100, /1000 but I’m open to the idea)
  • Equivalent numbers and expressions (both 3/6 & 2/4 and 4+7 = 2*6-1)
  • Basic probability (relating it to fractions too, since who doesn’t like beating a dead horse)
  • Ordering decimals and large numbers on a # line (including a great world populations number line activity)
  • Distance on a number line (and scale? Will this help prepare for graphing?)
  • Order objects (the online game I found here via Infinite Sums)

This is where the order falls apart a little bit…

  • Classifying Numbers
  • Properties of real numbers
  • Perfect squares
  • Benchmark values (along the lines of estimation 180, the average could reasonably be which value,
  • Other number systems, like Mayan and Egyptian (honestly, because this will be on a test I have to give at the end of the quarter)
  • Perimeter and area; frequency tables (not even number sense. Just on dumb test.)

Overall ideas: Does this make sense? About how much would I expect this to be? That’s really the point of number sense, isn’t it? To understand how much it is, without going through a bunch of rules, especially rules that we understand only as rules and not as concepts. The one thing I don’t want to do is rehash the same old same old they’ve already seen year after year and never quite understood.

What does Number Sense mean to you? What am I missing? Any ideas?