Fractions and My Brother

Man changing stock quotes on a chalk board. NYPL Digital Gallery

My brother, Zack, was here visiting the other night. My 12 year old son had a question about his fraction homework, and this sparked a conversation between Zack and me about how fractions are taught in school. 

Zack said, “Fractions are a useless implementation of decimals.” 

Hmm. Zack worked in finance for many years as a derivatives trader. I never really understood what he did besides the fact that it entailed making split second decisions involving lots of money and using insane spreadsheets. Zack is still my go-to person when I need help writing a really complicated spreadsheet formula or macro. Not only does he understand what I’m saying, but he has the answer in .9 of a second or less.

Here is a video of Zack sharing his thoughts on fractions. I hope it sparks some interesting conversation for readers of this blog. I brought this up with my colleague, Penny Stearns, who expressed a completely different perspective from Zack. Perhaps she will star in a video rebuttal at some point. Professor Tim Whiteford weighed in with still more to think about. What an excellent topic for students to question and debate in order to deepen their understanding of the application of fractions and decimals in different contexts.

Math and the iPad

 

I have a new iPad and I’m looking for good math apps.

Lee Orlando has a new iPad, too. She loves trying new things and is already way ahead of me on using the iPad in math class. This is from a recent email:

This weekend I bought an adapter for my iPad so I can hook it up to the LCD projector, and I also got a wireless keyboard.  Today, I went into school to try it out.  It was so cool to see the iPad screen projected and to sit at the back of the room (or any place in the room for that matter) and see the text appear.

This was soon followed by another email:

I just found some awesome free apps for the iPad.  All are from "Mathtappers" and the three that I downloaded involve placing numbers (including rational numbers) on a number line and finding equivalent fractions.  All games are designed for three levels of play.  I've been trying them out at each of the levels, and they get pretty challenging at the highest level.  However, the easiest level is well within the ability of fifth graders.

...these apps look like a great way to engage students in a whole-class warm-up activity/discussion.   I had made up my own number line activity using a sketching app that I have... the kids' attention level goes WAY high when they get to come up and draw on the iPad!  Getting that adapter for the LCD projector may have been the best investment I have made in a long time.

Later, I ran into Lee at school and she showed me how she’d photographed a piece of graph paper to use as the background of her sketching app, so that students could draw arrays and fractions with the aid of the grid. What a great idea!

I hunted around a bit and found some other useful apps. My favorite so far is Sketchpad Explorer. If it’s been awhile since you contemplated the Pythagorean theorem, you’ll enjoy the Getting Started screen, which allows you to drag right and non-right triangles around to see the theorem in action.

The real fun, though, comes when you touch the little book icon in the lower left corner of the screen. Choose “Elementary Mathematics” and Sketchpad Explorer presents you with a suite of eight activities involving symmetry, triangles, fractions, decimals, multiples, and volume. There is even a logic game which gives less than and greater than clues to find an unknown number. Sketchpad Explorer’s creativity and nice, clean graphics are appealing. At first glance, there seems to be a wealth of resources and lesson ideas for teachers on the website. I can’t wait to try these with students.

The Powers of Ten

Lee Orlando, fifth grade teacher, kindly contributed this article.

Our Bridges November calendar focuses on decimals and "base-ten fractions," and we've been having some good discussions around the nature of decimal numbers.

Yesterday, we were ordering some numbers that included whole numbers and mixed numbers (expressed as decimals)... one student confidently announced:  "3 is greater than 33.45 because any decimal is smaller than a whole number."   The salient feature for this student was the decimal point, and she was working under the misconception that whenever you see a decimal point, the number is automatically smaller than any whole number.  Just that comment alone kept the students talking and debating for quite a while!

I also did a quick formative assessment: Instead of giving students decimal numbers already arranged appropriately (in a column) in order to add/subtract, I simply gave them the numbers and had them arrange them in order to add (there were four numbers). Perhaps you can guess what many students did. They applied their understanding of place value of whole numbers (ones, tens, hundreds, etc, lining up the numbers from right to left) and totally disregarded the decimal point.  Here were all these decimal numbers, neatly lined up as if they were whole numbers, with the decimal points totally misaligned. These were students who have been adding and subtracting decimal numbers in our weekly math computation practice, but when given the numbers separately - not pre-arranged in a column - their lack of conceptual understanding about decimals and their values was completely transparent.

All this has led to lots of discussions about the power of zero which we had already explored in our Great Wall of Base Ten, but which was now coming back in light of decimal numbers.  I have been digging up some cool resources around this, including this awesome video which you may already know: "The Powers of Ten."

http://www.powersof10.com/film