Representation in: Fractions
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1.Let's do some math! Which number is bigger: 2/3 or 4/9? Click here after you figure out your answer.
Are you able to find the answer in more than one way? How many different strategies can you think of?
How do you think your students would answer this questions? How do you think their approach would be different than yours?
What do you need to understand in order to use the strategies you thought of?
Where could one go wrong in each of the different approaches?
2. Now do the math with a representation.
How could you use those representations to support each of the following arguments?
I think 2/3 is bigger since it is bigger than 1/2 and 4/9 is smaller than 1/2.
If you draw 2/3 and cut each piece into 3 smaller pieces, then you see it's the same as 6/9, which is 2 pieces more than 4/9.
I know that 2/3 is the same as 4/6 and that has to be bigger than 4/9 since they have the same number of pieces, but 6ths are bigger than 9ths since the whole is shared by fewer people.
3. Dig a little deeper.
4. Enduring Understandings: What conceptual understanding are we seeking to help our students develop?
4. Enduring Understandings: What conceptual understanding are we seeking to help our students develop?
a. Fractions arise when a unit or a set is partitioned (cut into equal pieces). The denominator of a fraction tells us how many pieces the whole is cut into. Students should work with this idea in multiple contexts, with the area model, fraction strips and the number line as well sets of different numbers of objects. They should be able to draw pictures of, and compare the sizes of, different unit fractions (1/b), and they should be able to create a whole, or a whole set, given a representation of a unit fraction.
a. Fractions arise when a unit or a set is partitioned (cut into equal pieces). The denominator of a fraction tells us how many pieces the whole is cut into. Students should work with this idea in multiple contexts, with the area model, fraction strips and the number line as well sets of different numbers of objects. They should be able to draw pictures of, and compare the sizes of, different unit fractions (1/b), and they should be able to create a whole, or a whole set, given a representation of a unit fraction.
b. Fractions are composed of unit fractions. For example, the fraction 3/5 is simply 3 copies of the fraction 1/5. This is similar to whole numbers, which we view as being composed of powers of ten (i.e. the number 300 is 3 copies of the number 100). Students can use this idea to create a representation of any fraction a/b given a representation of 1/b. Viewing a fraction in this way ensures that students can move beyond the idea of fractions as a part of a whole, and enables them to see that fractions can be greater than 1.
c. Fractions are numbers. A common misconception is to apply whole-number reasoning to fractions based on their numerators and denominators separately. Plotting fractions on the number line helps students recognize that each fraction has a specific value, and equivalent fractions are simple two different ways of representing the same value.
5. Let's build some collective understanding through discussion. We know that reflection and connection helps to deepen knowing. Please take a few minutes to open up the Google Doc below and share your thoughts.
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