This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational. Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!
One thing that I almost never do is post the day’s “learning objectives” on the board. I find it sufficient to say to the class “Today we’re going to learn about…” and then say whatever it is we’re going to learn. That seems to be enough. There are occasions though when I will tell them what type of problem they will be solving at the end of the particular lesson. I did this when teaching the topic of complex fractions to my class of seventh graders.
I announced that at the end of the lesson they would be able to do the following problem (which was a challenge problem that appeared in the JUMP Math teacher’s manual):
I was expecting to hear gasps and exclamations of “No way!” when a boy raised his hand and said “Oh, I know how to solve that.” He then narrated what needed to be done. He had certainly never seen this exact same problem before. He put together basic skills that he learned and saw how they fit together and solved a more complex problem—an example of knowledge transfer. Which is what this lesson is about, though most students will not be able to solve something like this straight off like this student did.
Warm-ups. The warm-ups for this particular lesson should focus on what we have been doing, as well as some word problems:
1. (-2/7 + 5/14) ÷ 3/28 Answer: 1/14÷3/28 = 1/14 x 28/3 = 2/3
2. (2-3)/(2-7) Answer: -1/-5 = 1/5
3. -3/4 x 5 ¼ Answer: -3/4 x 21/4 = -63/16 or -3 15/16
4. How many 2/3 ounce servings are in a 5/6 ounce cup of yogurt?
Answer: 5/6 ÷ 2/3 = 5/6 x 3/2 = 5/4 o 1 ¼ serving
Basic Lesson. Since we have just finished a lesson that covered fractional division, I will ask students to solve something like 2/3 ÷ 4/15 which they can do fairly readily. They have learned in previous lessons that fractions are division. The fraction 6/2 is a division: 6 divided by 2. Similarly 2/3 is a division: 2 divided by 3. Therefore we can represent a fractional division problem as a “complex fraction”. The problem just given can be represented as:
Examples, Worked and Otherwise. We then practice rewriting complex fractions as ordinary fractional division problems, and solving them such as:
After a few of these, the problems can be more complicated:
And the solution to the first problem given (which my student solved without any lesson):
In general, my students enjoy complex fractions and look at them as puzzles. This is now something to add to the repertoire of problems to include on future warm-ups.