Helping students learn how to add, subtract, multiply and divide fractions with true understanding is a challenge. When I was a student, just like many of you, I was taught procedures but I didn’t really understand what I was doing. I memorized the steps and applied them but when ‘the math’ got too hard I couldn’t extend the algorithm or apply it to new situations. Over my career I am getting better finding different ways to help students understand fraction operations but I know that I have much room to improve.
Last year I met Cathy Fosnot (She publishes all of her books as Catherine Twomey Fosnot) at a professional development session in Edmonton and I was intrigued by her ideas. She has a series of books called Young Mathematicians at Work. Volume 1 addresses addition and subtraction; Volume 2 covers multiplication and division and Volume 3 in this series deals with fractions, decimals, and percents. Cathy and her co-author, Maarten Dolk, focus on how children can investigate fractions, decimals and percents and how they construct their own meaning of these rational numbers.
I have also read the book, Minilessons for Operations with Fractions, Decimals and Percents which is a companion book to her kits. As I worked through this particular book I got quite excited because of so many aha moments and the possibilities with students. Then I felt sad to think that it took me 20 years in the profession to learn about some of her mathematical models. How sad is that!
For fraction addition and subtraction she profiles this sequence of models: Money; Clock; Choose your model: money or clock; Double number line; Choose your price and Choose your model: money, clock, double number line or price. For each model she explains how it can be used along with math strings or mini-lessons that teachers can use with students. The learning is scaffolded and offers children opportunities to create their own meaning in a community of learners. To demonstrate why I think these visual representations are so fantastic I’ll introduce you to one of the models, the clock, an analog clock.
Let’s add 1/2 and 1/3. How many minutes are in 1/2 of an hour? 30 minutes. How many minutes are in 1/3 of an hour? 20 minutes. Add these two amounts together and we can write that we have 50/60 of an hour which reduces to 5/6. Easy isn’t it?
Let’s try another one – 1/6 + 1/4. 1/6 of an hour is 10 minutes. 1/4 of an hour is 15 minutes. So 10 minutes plus 15 minutes results in 25/60 of an hour. This reduces to 5/12 of an hour. Tada! This approach does assume that students can do basic calculations, but given that, I believe all students can be successful with this method.
Even when the questions get more complicated I believe that this model is extremely helpful. Try this: 7/12 + 3/5. 1/12 of an hour is 5 minutes so 7/12 would be 35 minutes. 1/5 of an hour is 12 minutes so 3/5 is 36 minutes. Add these times together and we get 71/60 of an hour which really is 1 11/60 of an hour. Voila!
Fractions that are typically difficult to rename/convert as decimals or percents work really well with the clock model such as thirds, sixths, twelfths, etc. However, not all fractions work easily with the clock, and for this reason Cathy Fosnot provides several models (see the list above). Ultimately the students choose the models that are meaningful to them based on the fractions’ denominators. This approach empowers students to choose the right model for the right situation. I see that it is similar to students renaming a fraction to a decimal or a percent to make ‘the math easier’. However, I think that the visual representation of the clock adds another dimension to our understanding.
As you can see, I love the clock model! Without using the traditional algorithm of finding common denominators we can add or subtract fractions quickly and easily with mental math. It builds on students’ intuitive understanding of number and the visual representation of the clock. Students will be able to carry this mental model with them for the rest of their lives 🙂