On Tuesday we explored some of the fundamental concepts related to understanding fractions. We explored the idea of the whole in the sense that fractions at the elementary school level have very little value, or meaning, if you don't know what the whole is to which the fraction refers. To test your understanding of this concept decide whether you would rather have half the money I am holding in my right hand or a fourth of the money in my left hand. You would need to use your imagination to do this. The clear choice is that you cannot choose because you don't know how much money I have in each hand. A half is clearly more than a quarter but only if the wholes are the same (or, to be really exact, as long as the whole from which the quarter is taken is not more than twice the size of the whole from which the half is taken). We explored other fraction ideas such as the top number, numerator, is a counting or cardinal number, an adjective, while the bottom number, the denominator, is a nominal or naming number, a noun (teacher knowledge at the elementary school level).
It is critical to help children grasp the fundamental ideas of fractions because they are so counter-intuitive compared with whole numbers. Yesterday we used our understanding of these fundamental concepts to look at some of the procedural knowledge of fractions such as performing the four operations. We looked at finding common denominators using the fraction bars (in the picture) to change 1/3s and 1/4s into 1/12s so we could add them together. We also did this to compare; which is larger, 1/3 or 1/4 (assuming the wholes were the same, of course)? We then looked at multiplication of fractions by exploring simple ones like 1/2 x 2/5 (a half of two-fifths is one fifth) and 3/5 x 5/8 (three fifths of five eighths is three eighths). Try this using the fraction bars. They work so well for this because you can hold up five 1/8 pieces and just take 3 of them to get three eighths.This is where the idea of the one or whole becomes so important because the one or whole of the 3/5 is the 5/8 and the one or whole of the 5/8 is the 1 (red piece in the pic). The question to be answered then is what is the one or whole of the answer, 3/8?
This example uses the equal groups concept of multiplication. We can also conceptualize this problem using squared paper. Draw a rectangle that is 5 x 8. Mark off 3/5 down the 5 side and 5/8 down the 8 side. The rectangle you have now created will be 3/8 of the large rectangle. Once students begin to see the patterns they can generalize to develop the algorithmic rules for multiplying by fractions. Math is the science of pattern after all. It also helps keep my students' attention at 4:25 in the afternoon when I speak with my best American accent.
Try this division one. How much of a 1/2-cookie could you get from 3/8 of a cookie? Isn't that easier than "change the sign and invert the second fraction"? Well done Emily who solved it in less than a second.