This is a fun little activity that you can do in a class period, or just give for homework, and it will get your kids to think carefully about fractions, most likely beyond the junk that is in whatever textbook you are using. It would also be excellent as an assessment of your students’ basic understanding of how fractions work.
The idea is this: do your students have a basic understanding of how fractions work? Do they REALLY? This is how you do it in 5 questions. In questions 1 and 2, we look at whether your student understands a “unit fraction,” that is, a fraction with 1 in the numerator. They should understand that if you know the unit fraction equivalence of a quantity, they should be able to figure out the whole. Thus, if 1/5 of the pizza is a single slice, the pizza has a total of 5 slices. If 1/10 of the book is 12 pages, then 10/10 of the book is 120 pages - this can be “demonstrated” a number of ways, including just adding 1/10th over and over again until you have 10/10, which makes it 120 pages, or by multiplying by 10.
! The next two questions give larger fractions: if you know 2/3 or 5/6 of a number, can you figure out the entire number? This requires two steps: if you know that 2/3 is 8 hours, then 1/3 is 4 hours, so 3/3 is 12 hour. You’ll notice that I’m emphasizing reasoning here, not memorization. That’s because reasoning lasts a lifetime; memorization is ephemeral. A student who comprehends how fractions work will be able to solve these without a “formula.” Thus, if 5/6 of a quantity is 25¢, then 1/6 is 5¢ and 6/6 is 30¢. Capish?
The last problem uses a teen pop star to riff on the other questions: instead of finding the total amount, we’re finding the missing piece. We know that the 14 years is 7/8ths of this kid’s career, so 1/8 must be 2 years, so he only has 1/8th of his career left, or another 2 years. See ya!
If you like these problems, you should definitely have your students write problems like these on their own. One of the things they’ll realize is that they’ll need to have to have the piece divisible by the numerator. If you say that 3/5 of a number is 19, then dividing 19 by 3 to get 1/5 is going to result in the fraction 6 1/3, which means the entire number is 6 1/3 x 5 = 6 x 5 + 1/3 x 5 = 30 + 1 2/3 = 31 2/3. Wow, that wasn’t fun...... but your students will learn how to make problems of their own that “make sense.”