Pesky Fractions

Using patterns to conquer those pesky fractions!

Quick note: Before you begin reading this scintillating discussion, if you really just want some practical ideas to get going on today, the math fairy would suggest that you look at “Fraction War & Fraction Blackjack” in the kids section or “Making 1’s” in the parent section. Back to math fairy ramblings….

Really, the Math Fairy is not confident that the word “conquer” should be used here. Fractions are very, very tricky for children. Why? Well, I’m sure there are many reasons. I think fractions are so difficult because they are parts of things and, let’s be realistic, who wants a partof a cookie?! I don’t! I want the whole cookie! Don’t you? So, that’s our first problem when we teach kids about fractions. It’s all this annoying sharing nonsense. And, maybe it’s because I’m an oldest child, but truly, deep down, I hate sharing! Then there are those algorithms. Really! Can’t anyone think of a better way to add and subtract fractions? How about multiplying mixed numbers? Even with cross-cancelling it’s a pain-in-the-you-know-what! And don’t even get me started on division of fractions. It’s cool and all. There are some fun rap songs out there about keep-change-flip. But, seriously, turn the fraction upside down??? What’s up with that? That’s a rhetorical question because, of course, if pressed, I can explain to you why but I always have to give it a great deal of thought in order to correctly explain it. It’s just plain counterintuitive.

The difficulty comes, especially in the United States where we use a non-metric measurement system, that we actually do use fractions all the time. We use them in cooking, sewing, manufacturing, auto mechanics, time, distance, the list goes on. So, we do need to know how to use them and manipulate them. And, more importantly, we really need to understand what we are doing and if our answer makes sense.

Patterns can help! As I said, I think it depends upon the child and their connection with fractions whether it will help them conquer fractions, but here’s one example of how fractions work with patterns.

Fraction BarsIf you take a look at the fractions strips and start to play with them, there are lots of things to notice.

For example, notice where the lines match up in the middle: ½, 2/4, 3/6, 4/8 and so on are all the same size. This is a great tool for understanding equivalent fractions.

If you rewrite the fractions so that they are 1/8, 2/8, 3/8 and so on, children start to notice that the largest pieces, such as 7/8 or 9/10 have a numerator and denominator that are close in size. This is a huge understanding for comparing fractions. Then it becomes quite obvious that ¼ is less than 7/8 because I have 1 piece in the first fraction, but 7 pieces in the second. Even though the piece size is smaller in 7/8, playing around with the patterns of growth in fractions can help me see their relative size.

As mentioned in the “having fun” piece, the Math Fairy recommends that children play with fraction bars and explore with them rather than lots of instructing. Get to know the fraction bars. Become their friend. Sleep with them. (Not really, but you get the idea!) It’s really exciting when kids find patterns we’ve never thought of! Communicating about what they’ve found and sharing their ideas with others really expands their ability to understand fractions. There are no “wrong” answers when we’re looking for patterns in mathematics. Rather, there are stronger, easier to communicate patterns vs. less obvious, more obscure patterns that will require greater explanations in order to help others see them.

For example, at this moment the Math Fairy is struggling to figure out how to explain the connection she sees between the fraction bars and Pascal’s Triangle.

Pascal's TriangleDo you see it? I think if you took the fraction strips and just used one of each size bar, i.e. the whole number 1, then 1/2 , then 1/3, then ¼ and so on. Line them up so that they make an inverted fraction with the largest bar at the top. Do you see it now? Both Pascal’s triangle and the fraction bars remind me of the ball drop machine at the science center, aka The Galton Machine.

Galton machine From Wikipedia: “The bean machine, also known as the quincunx or Galton box, is a device invented by Sir Francis Galton[1] to demonstrate the central limit theorem, in particular that the normal distribution is approximate to the binomial distribution.

The machine consists of a vertical board with interleaved rows of pins. Balls are dropped from the top, and bounce left and right as they hit the pins. Eventually, they are collected into one-ball-wide bins at the bottom. The height of ball columns in the bins approximates a bell curve.Overlaying Pascal’s triangle onto the pins shows the number of different paths that can be taken to get to each bin.A large-scale working model of this device can be seen at the Museum of Science, Boston in the Mathematica exhibit.”

So cool! Notice the connection to Pascal’s Triangle. You could really have some fun learning about the mathematics that connect the Galton Machine and Pascal’s triangle. Connecting back to the pictures you can make with fraction bars helps us to understand the deeper meaning of fractions. They’re not some weird, isolated, cookie sharing nonsense, but rather a piece of the greater world of mathematical patterns.

Do you see where patterns can take you? On a mathematical adventure! Start exploring and see if fractions can become something fun and interesting instead of something to dread (or, worse yet, to share!).

Have fun with math every day!