There is so much to talk about with respect to fractions! I hope you enjoy today’s fraction lesson where we look for misconceptions and more!
It was a FASCINATING discussion–try it with your students! Which is larger–1/2 of a grape or 1/2 of a strawberry? You should have HEARD the ideas they had! You should have heard the LOGIC they tried to flip out as “truth”! We had all sorts of conversations about things such as:
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.
“What did we just discuss that could help you in your quest to make sixths and eighths?” A flurry of activity started and I waited for them to finish.
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”
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” make up a “whole” (This is multiplication of fractions, by the way!) and that 4 “fourths”make 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. . .
and I asked, “Which of these are thirds?” I gave them time to talk and then we voted. The little numbers in red show how many of my 22 students thought each shape WAS in thirds. Interesting, eh? So . . . I have a few little petunias trying to find ways to prove/disprove these tomorrow–but the biggest reminder for ME was to never assume that students really understand. When we always present them with pre-divided circles and fraction bars, we don’t always encourage the kind of reasoning needed to decide whether or not equal parts really exist. So I pulled out an exit slip I had planned on doing later in the week so I could check my theory . . . check out some of these misconceptions…sorry they are so blurry!
Trying to show 3/5!
So, know I know where I need to head tomorrow. I need to dig a little deeper on the misconceptions they have about dividing certain shapes. (And a little review on working “precisely” will accompany it, as you can see!) I’m not quite sure yet how to tackle this one–but I think I may ask the students to try to prove it on their own without giving them resources yet! We’ll see what they come up with! Next steps? If I give them a fraction like 1/4, can they draw the “whole”–and how many versions of that “whole” can they find? It should be interesting! Hope I haven’t scared too many of you away yet . . . but be careful what questions you ask–because you might find out what you DIDN’T want to know–and you will need to act upon it! Thanks for stopping back for day 5!