May 4th, 2012
This morning Gail is talking about thinking about fractions as unit fractions. The information here is from the work done by H. Wu out of Berkley. Fraction instruction starts on a number line – not with a pie diagram or other physical model. She is showing us an Nspire document that is counting by 1 fourths. The idea is multiples of unit fractions. So in the app when you are at 2 on the number line, the app shows “10/5″ as well as “ten 1/5″. We have ten copies of 1/5.
Equality is defined as having the same place on the number line. When we start with pie diagrams we are creating misconceptions. For one – we wind up not being able to be greater than one. Also we need a different circle for each fraction type.
Gail takes a minute to tell us that words in the common core matter much more than they ever did before. When the core talks about “unit” it really means it.
Next we talk about unit squares. How could we prove 15/6 = 5/2. A number line model or a unit square model will work. Now think about a unit square with 3/4 of it filled in. That is your unit. Then a half filled in unit square would be 2/3. A unit square split into 8 rectangles with 3 filled in has a value of 1/2. One and one half filled in unit squares has a value of 2. You can see these examples here.
On to addition: – we think about additional as copies of unit fractions. Multiplication is sort of a traditional area model.
Here are Gail’s slides from her presentation:
May 6th, 2011
I had to get to an elementary session. I love elementary mathematics and especially fractions. The fraction strand for grades 3-5 in the Common Core has changed significantly.
We started by folding fraction strips into halves, fourths, eighths, thirds, sixths, and we left one whole. We discussed different ways that students might make these folds and how we can encourage accurate folding. We also discussed why we might want kids to fold thirds before you fold sixths. What connections can be made by students as they to fold these quantities. Then we might want the students to start comparing thirds and fourths, or other type of unit fractions. The Core places great emphasis on unit fractions.
Next we moved on to comparing 5/6 to 5/8 and talked about comparing like numerators. Getting students to line up their fraction strips is a nice strategy to help visualize the comparison. If we compare something like 4/6 and 2/3 we can start to get at equivalency. Comparing something like 7/8 to 5/6 helps us think about how far from 1 whole that we are. How do we get kids to see that eighths are smaller than sixths (unit fractions!) and help then transfer that to a problem like 7/8 compared to 5/6?
Consider the standard: Construct viable arguments and critique the reasoning of others. What does this really look like in the classroom? Students would be explaining their reasoning – not just telling you the steps that they went through. We need to listen to each other and very importantly teachers need to listen to what their students are saying. Not listening for what we want to hear, but listening to what the students are actually saying.
A key idea in these documents is the idea that although something may be taught in a certain place, those skills are expected to be used in later grades in different contexts. Not retaught, but spiraled through and built upon. This is going to require a great amount of collaboration across all grade levels to find those places and plan those units of instruction. Dr. Kepner mentioned that, for example, in third grade we should not even be touching computation but rather be building fraction sense in a hundred different ways. Then we start computation in fourth grade the students will have an idea of the concept of a fraction as a single number.
