Sunday, November 22, 2009

Fractions on a Clock Face.

Please click on the thumbnails to enlarge the photographs.
One great way to get students to see that they can list several names for one fraction is by exploring what we could name each section of a clock face that has had the hands placed on different numbers. We always leave one hand at 12, and our clocks have equal length arms so we never know know whether we are looking at an hour hand or a minute hand.
Starting with the bottom clock, we see that one hand is on the 12 and one hand is on the 2. Most students first see this as 2 out of 12 hours, and they write the fraction 2/12. Many also see that this could be 10 out of 60 minutes, or 10/60. From that point, many notice that neither of these fractions is stated in its lowest terms. Most then "chop-chop" 2/12 (divide by 2/2) to get the fraction 1/6. Many then can say that 16 2/3% of the clock face has been "covered up" or "rotated through".
The next picture up shows one hand on the 12 and one on the 8. This shows 8/12 or 40/60. Again, neither fraction is in lowest terms. Many students will just see this as 2/3, as 4/12 looks like 1/3, but to prove it, one could divide 40/60 by 10/10 to get 4/6. Then one could divide 4/6 by 2/2 to get 2/3. We have mathematical PROOF! Most students now see that 8/12 is the same as 2/3, and that is the same as 66 2/3%. SWEET!
The goal in all of this is to be able to add fractions with unlike denominators. The third picture up illustrates the addition of 1/4 + 2/3. 1/4 is seen to be equivalent to 3/12 and 2/3 was proven to be equivalent to 8/12. So, we get 3/12 + 8/12 = 11/12.
We might also get 25% + 66 2/3% = 91 2/3% but 91 2/3 / 100 is not in its lowest terms, and most middle school teachers would simply freak-out if the answer is presented in a percent. So, even though the % is correct, I would emphasize finding the answer as a fraction in its lowest terms.
One last note, please remember that any fraction can be stated as another equivalent fraction simply by multiplying or dividing the fraction by the number 1, and any fraction where the numerator and denominator are the same (N/N) equals 1.
So, 32/48 divided by 16/16 , still has the value of 32/48, but it is more universally recognized as 2/3. 400/500 = 4/5 not because "you can drop the zeros", but because 400/500 divided by 100/100 = 4/5!
Similarly, 3/4 = 9/12, because 3/4 X 3/3 = 9/12. 5/6 = 10/12, because 5/6 X 2/2 =10/12!
I hope you are not confused, but if you are, please send me your comments :0}

Monday, November 9, 2009

Fractions Are Instructions to Divide! Sir!

Please click on the thumbnail to make the photo larger.
Just how do you go about solving a question like: What is 3/12 of 36 penguins? There seems to be an almost endless list of strategies that don't work well. However, there is one strategy that works VERY well.
In my class, I have the kids repeat, military style, "Fractions are instructions to divide by the denominator and then multiply by the numerator! Sir!"
Almost all kids can remember this"direct order". Many do not, however, know what needs dividing up, or into how many groups. In the problem above, the 36 penguins need to be divided into 12 even groups, and that puts 3 penguins in each group. 3 would be a fine answer to the question, "What is 1/12 of 36 penguins?", but is not a good answer for 3/12. This is where the "and multiply by the numerator" comes into play. We need to take into account 3 of the 12 equal groups of penguins, or 3 X 3 = 9 penguins.
The work also shows an array of 36 "penguins", and it would be quite correct to say that 3/12 could be understood as 3 out of every 12. This is a great model for small numbers, but I would not want to split up 3600 penguins into an array!
So, "Fractions are an instruction to divide by the denominator and then multiply the answer by the numerator! Sir!"

Monday, November 2, 2009

Fraction Equivalent Strips

Please click on the thumbnails to enlarge the photos.
Please take a look at the fraction strips above. The goal of the above was to match about 20 landmark fractions to their equivalent percents. The goal was not rely on memory, but to strategically think about the placement of these fractions. For example, most students immediately know that 1/2 = 50%. They can use that knowledge to "discover" that 1/4 = 25%, and then to find that 1/8 = 12.5%. Even the more difficult fractions like 1/6 become easy to place when there are landmark percents already present. 16 2/3 % is easy to place if 10% and 20% are already listed.