What is a "typical" grape?
Does it matter that grapes are round and that strawberries are different on the top and bottom?
ARE grapes round?
That's right--that is a GRAPE on the left! The kids were in AWE. We talked about the idea of a "benchmark" object to help us know the relative size of things and how "big" grape to one person might not be a big grape to someone else. This all, however, was fine and good but didn't get us to our main discussion--that a fractional amount is totally and completely dependent on the size of the "whole". I then asked them who ate more pizza last night--me, eating 2/3 of a pizza or my husband eating 1/2 of a pizza. They immediately declared me the winner . . . until I sat quietly looking pensive. A few seconds later someone chimed out, "Wait! Were the pizzas the same size?" and we were off to the races.
This leads me to today's investigation . . . today I reminded the students about grapes and pizzas and we reviewed some of the "truths" about fractions that we had learned last week--that fractions refer to equal pieces, that the top number (numerator) refers to the number of pieces we are working with (NOT shading!), and the bottom number (denominator) refers to the number of groups total. I then passed out 6, 3" x 4.5" rectangles of paper to each student and asked them to remember what they know about folding fractions to create models of "one whole", halves, thirds, fourths, sixths, and eighths. Believe it or not, this was still not easy for some of them! I could really tell which students were beginning to develop some better fraction number sense. When I saw them struggling to get six and eight equal parts, I finally broke down and asked the question:
"What did you do to make fourths?"
The students talked at their tables and we shared our thinking. Overall, most of them were NOT randomly trying to make four equal parts--they made halves and then used the halves to make fourths.
One of my learning targets for today was to make sure I stressed the concept of "unit fractions"--that a "whole" can be divided into smaller, equal parts that we can "count" with...just like we can count with whole numbers. We added a definition of unit fractions to our interactive notebooks and then I asked the students to glue their "wholes" down and record number sentences reflecting how they can be decomposed (this is a BIG concept in the Core, people--they need to be able to see how fractions are made of other fractions . . . today was the first step in developing that understanding). Here's what we did:
First I threw some fraction bars under the document camera and modeled how the pieces worked together to make "wholes"
Then I asked what would happen if I separated each "whole" into its respective "parts" . . .
I then modeled how you could write a "number sentence" to match what had just been modeled. I needed them to understand that those "unit fractions" can be counted to make a whole--and that a "whole" can be made up of any number of equal parts.
You see where I was going? I wanted them to understand that 8 "eighths" makes up a "whole" (This is multiplication of fractions, by the way!) and that 4 "fourths" makes up a "whole". Over time, we will be able to get more sophisticated with this, but this was where I wanted to start. Overall, the students seemed to get this concept--though I am sure that I will certainly get some 1/2 + 1/2 = 2/4 along the way. I'm a realist.
The one thing that REALLY shocked me happened way back in the dividing of the rectangles. . . some of my students clearly had some major misconceptions about what "equal parts" means. For example, one of my math rock stars (really--one of my BEST thinkers) drew this to represent thirds:
Yup. I was shocked too. I've never had someone do that before! I've had circles divided incorrectly, but not rectangles! So, being the pushy, "what ELSE don't they know?" kind of teacher, I threw a few things up on the board. . .
|Trying to show 3/5!|
|Also trying to show 3/5!|
This blog post is now a part of my comprehensive fraction unit available by clicking the image below. Hundreds of teachers have now used it to change the way they teach fractions!