Which one doesn’t belong? Can you find more than one?
Which one doesn’t belong? Can you find more than one?
One great thing about blogs, you can reflect on what you have done in class, and remind you when you haven’t done a follow up you intended to do.
So, I needed to go back to What is 3/5. My students had made some good connections during that week and I wanted to see how much of it was retained. I went with a basic warmup today.
I decided on this because I really want students to expand their thinking past one basic shape for fraction representation. As I expected however, the first two diagrams students drew were circles and rectangles. I was pleasantly surprised when one student drew a pentagon.
At first I was worried when I was walking around the room, typical comments on why their figure was 3/5 included “3 out of 5 are shaded.” I am thinking, WHAT? We covered that fully just a month ago, how could they backtrack again? My worries were unfounded when for the last explanation students showed me they were indeed thinking of EQUAL parts, not just 3 parts out of 5. In the first responses it is easier to “explain” that way without fully relating everything. When I asked students to revise it so that I would know exactly what they were thinking, they came up with what a few students already had (as you can see in the student work photo):
3 out of 5 equal pieces are shaded
I am proud of them for remembering the concept and how to explain it so I could understand their thinking, and told them so.
Nicora has a great post on multiplying fractions, check it out here. This is something similar that I end up reviewing with my 8th graders. She was wondering about what I have found so here it is….
I start with AREA, Chris(@trianglemancsd) may call it an array, whatever works. I started approaching multiplication this way since they are pounded with Area problems in 7th grade. The concept and procedure is ingrained and ready to be built upon.
I start with a multiplication problem I am pretty confident all of my students will know, such as 3×5. Students will quickly tell me it is 15. I ask how they can represent that math question and answer, and I get a variety of methods. I will get at least one student who shows me an area problem. I pick that example and we talk about area and what it means, how to represent it and I ask them to draw a diagram on graph paper showing it. This is what I get:
This is exactly what I want to see, even though I kinda manipulated the students into showing me a representation I want to build upon. I then ask them to show me representations of 3 and 5 using the graph paper grid, and then I create the following construction to connect to their previous representation of 3×5.
This is not a big stretch from what they already do and know, so I have them practice it a bit and come up with new problems in order to try and stump their friends or even me. Students try and get tricky, and at least one will introduce the idea of fractions into the mix. Once again, they are leading their learning where I would like them to go.
Students make the grid naturally like they did for 3×5. When asked why, I have had some who say “because we did it for the other problem.” A lot of mathematical thinking there, although some will pick up on it and say it is a representation of the area of 3×5. I will ask for other ideas, and students will tell me that the reason we have a grid of 15 is because it is the LCM of the denominators. Surprisingly, the shading of the grid has never been an issue with any of the students I have worked with. Typical student talk is “you can only shade on the first row, but only three of those.” When I ask why, students have said that it is the area model of 1×3. A few students will reply “because it’s a double grid Mr. Anderson, you have a grid of unshaded squares and a grid of shaded squares.” Some students have even suggested that I separate the drawings of shaded and unshaded squares- creating area models of the numerator and denominator. This really relates to the old rule of “multiply numerators and multiply denominators”, and students made that connection on their own.
The thing that I have to be careful of with this is that they use two fractions that are less than one. As long as they do, this works out very well for them. Students do try to do things like 3x(1/5) and get confused on the outcome, which would also look like 3/15 or 1/5. When I have student discussion about this, I always have one student who brings up that they can change the representation of 3 to 3/1. They then draw their separate area models for the numerator and denominator to find their answer of 3/5.
Show them this clip and this picture:
What questions do you have after seeing this Video and Picture?
Ask students to write down their questions, I normally ask students to find at least 3. When I observe that most students have questions written, I ask them to share those questions with their neighbor. I then throw up a Microsoft Word document and start typing down questions students supply. Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t. I am looking for a key question or questions to start this lesson. If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.
Any of these type of questions will lead students down the inquiry I hope to explore with them.
This part can take a few different paths, depending on your focus for the day (you could use this idea with Proportions, Scatter Plots and Statistics). I started this part by asking what type of mathematical operation or ideas they would need to find an answer. The most common ideas students came up with are:
These are all excellent ideas and I place them on the board as reminders to where we want to go throughout the investigation. Depending on what focus I use this for, students normally fall into 3 strategies for finding our answer.
Some other considerations: The length of the trailer which will determine how big the shoe length is, can be varied. I normally go with what the students discover- many will do searches on trailer sizes. The most common lengths I get in class are 12′, 14′,16′, 18′ and 20′. This variation of the trailer size could be a extension of the problem.
Here are the statistics for this year: 8th grade shoe 13-14
These are the results my students got this year:
Some great discussions resulted in the difference in these estimates for height.
This past weekend, Tracy Zager tweeted a picture that 70% of 7th graders answered incorrectly:
This led to a great conversation which also included Christopher Danielson, Michael Pershan and Curmedgeon. It also sparked perplexity in me. I found I could devise many different variations to this problem to explore different student thinking, but instead I took this week to determine if students understood fractions as equal parts or if they fell back on a quick rule to find an answer. This was the results from day 1:
The results: 46.4% of my 8th grade students got it wrong, a much higher number than I anticipated. Looking at the reasoning given by my students, it appeared that they were following an old rule they partially remembers of counting shaded parts. I was not totally convinced this was the case or if it was because of the even partitioning that was done on the triangle. I wanted to check their understanding of a fraction so I gave them this for day 2:
When I looked at this, I expected small unit fractions and circle diagrams. I was not sure what they would write explaining what the fraction was or an example of it outside of the classroom. I was hoping they would talk about equal parts. This is how they answered:
One-half was the overwhelming fraction used, and they expressed that using a circle. All diagrams had pictures where students created equal parts, many times painstakingly so (evidenced with many eraser marks). Most students simply wrote it is one-half of the circle, none referenced equal parts. For the real life example, a majority of my students expressed the fraction as pertaining to a circular food; such as pizza, pie, cake or even a cheeseburger!
This last part really told me about how students have learned fractions. To gain student interest, teachers have fallen back on the standby of “the quickest way to student interest is food!” That students chose pizza or pie was no surprise, but that students see fractional parts in terms of circle graphs was. I need to make a point of expressing fractions with a different diagram- having students with such a narrow view of how fractions are used outside of the classroom limits them. It is no surprise that when fractions are introduced as slope in 8th grade, students have a total disconnect.
I still had a suspicion that students were only looking at the partitioning of the base instead of the area represented by them, I decided to use a diagram I thought of and that Curmudgeon reproduced and posted on his site:
I was hopeful that students would not only compare the widths of the fractional parts, but also the length. I was right and 77% of students answered the question correctly, here are the results.
I was very happy and proud of the results, and then another thing happened that blew that away. My students started questioning me as to why we were doing these warm-ups, what the correct answers were and were trying to anticipate what I was going to give them tomorrow! I had not gone over anything in class and had not told them what the correct answers were. They tried to pry information out of me, but I told them I had one more warm-up for them tomorrow and then we would discuss the week. They started a student-lead classroom discussion about the problems, what was correct and were sharing ideas about why people answered a certain way. I sat back and allowed them this time, it was great to see.
After class that day, I still had a nagging doubt that students only did so well because they were able to compare two diagrams, there was still an underlying issue of following this rule of counting three out of five pieces shaded. So the next day when they entered the room, this is what they found:
As I suspected, only 20% of them were able to answer it correctly. Those student noticed that one piece was 1/4th the size of the circle, the rest came back with the old standby of three out of five pieces shaded. I was crushed. I asked in each class whether this represented 3/5ths and the first and immediate response was not from the people who said yes, but those that said no. I said, “If this is true that the figure does not represent 3/5ths, why would that be? Take a minute to think about your answer.” I would call on someone who I knew answered incorrectly and when they contemplated the possibility of it being wrong, told me that one piece was bigger than the rest. We then went back and revisited the problems over the week, and students realized that the triangle did not have equal parts either. My students once again really dug into this discussion and other than a guiding question or two- took control and explored the problems and what the correct answers were. I was very happy with the outcome of that day, and I hope the students made some connections that will follow them throughout their mathematical careers. I will recheck their knowledge in two weeks and see if this experience will make a difference in their understanding of the concept.
Today I find that Bridging the Gap stole my thunder (sort of) by posting on a method I have been using in class recently. We have been looking at compound fractions when writing equivalent forms of equations. Changing a standard form equation such as 2x + 1/2y = 4 to slope-intercept has really thrown a curve to my students, and it’s all about one concept- fractions. My students admit they struggle with fractions and then to create a compound fraction? Most just put the pencil down, worry about other things and try to nap in class. I have tackled the problem by using pictures of fractions to help students visualize what is going on.
I first start with an improper fraction example:
We work on dialogue on what 4/2 means, and students came up with the wording you now see on the board. They then draw 4 circles and split them into groups of 2.
We then look at an example when there is a fraction in the denominator:
Students once again write their wording to the fraction and attempt to complete it as instructed. I have had students struggle on how to make groups of 1/2, but when you ask them to draw 1/2 they easily give you a semi-circle. They they have that aha! moment and quickly draw lines bisecting the four wholes. They created eight 1/2 pictures.
Students really grasp the first two examples quickly, and we then throw them the curve, a fraction in the numerator:
Many students automatically say they can’t split a half of a circle into a group of four, so I will go back and change the fraction to a whole number, 4. they quickly draw four lines between the wholes and say that this problem is nothing like the one I asked them. I have them explain their process to find 4/4, and then write it as 1/4. Some students will see that you can cut the circle into four pieces, but others will still struggle- and this is where as an instructor I have to realize that inadvertently I gave them a specific context in which to look at creating groups. I gave them a circle, so I ask the students if we need to start with a circle, and many say no. They draw a rectangle on the board and are able to make it into a group of four:
We do a few other problems where students practice with both cases, and then I ask students to find a pattern. Although they do not tell me to multiply by the reciprocal, students understand what numbers to combine and why they are doing so.
The next step, of course is to extend this concept beyond the unit fraction…