Tag Archives: modeling

Another Blog Post About Fraction Division

Person 1 mentioned on Twitter tonight that there aren’t enough blog posts out there about fraction division.

Person 2 recommended using rectangles to model fraction division.

I decided to help Person 1 using Person 2’s suggestion. Though the meat of this post is in this PDF I made and not in the blog post itself:

Fraction Division (1/29/2015 Stacked all fractions and made a cover page.)

Good enough, I say. I made a lot of examples fairly quickly, so I apologize if there are some errors here and there. Let me know and I can easily fix them and re-post the PDF.

And now there are n + 1 posts on fraction division on the internet. Woot!

Ratios and Rates 3 (#MSMathChat)

One thing students need to understand is that they can take any two quantities and make a ratio out of them.

They can.

It may not mean anything useful, but they can do it. For example, if I have 3 potted plants in my classroom and 28 students, I could say the ratio of potted plants to students is 3:28. Does that mean anything? Do you care?

I can’t say I care a whole lot about this ratio.

However, I can still talk about the relationship between these two quantities. With this ratio, I see that there are nearly 9 times as many students as there are potted plants. Another way to put it would be for every 1 potted plant there are nearly 9 students. Again, is there much meaning to that? Do you care that this relationship exists?

Probably not. I can’t say I really care.

Although, that gets me thinking. I might ponder with my students what our classroom would look like if that relationship were reversed. What if we had nearly 9 times as many potted plants as students? What would that look like? Going back to the original ratio, what if we looked in other classrooms? Would we see a similar ratio? Why or why not?

In the end though, we should realize that just because we can make a ratio, doesn’t mean that we should. Often we analyze ratios because the relationship between the quantities matters to us in some way. This is why teachers wrack their brains coming up with “real-life” examples of ratios. And we should do that!

But let the kids play, especially early on in their learning of ratios.

Let them come up with ratios based on objects in the classroom. If a student sees the ratio of girls to boys in the class is 9:16, ask the students to find another ratio of 9:16 in the class. If they can’t, ask them to find a relationship that’s close to the same. The ratio 9:16 tells me there are nearly ½ as many girls as boys, so look for another pair of objects in the room where there is nearly ½ as many of one as the other.

Give students small objects like race cars and other small toys. Ask them to model a 2:1 ratio using those objects. Challenge them to do it with more than 3 objects. Be sure to ask them to model relationships using different language. Instead of giving a ratio like 3:2, say something like, “For every 3 pencils, put out 2 paperclips.” If a student puts out 3 pencils and 2 paperclips, challenge them to keep that relationship but use more pencils and paperclips.

The more students can understand how this is all about relating quantities, the more they can appreciate why we talk about the real-world examples that we use. Teachers choose these examples because the relationships in them matter. And it’s fun to discuss what may happen if that relationship isn’t maintained. Let’s say some kind of dough requires a 1:5 ratio of wet to dry ingredients. What’s going to happen if we goof up and make a 1:5 ratio of dry to wet ingredients? Even better, bring in some flour and water and model what happens!

Sadako and the Thousand Paper Foldables

Source: Brian Stockus

Gather round everyone. I’m going to paraphrase a story.

“Once upon a time a week or so ago, a teacher made a foldable with her class to help them learn about integer operations. The day before the big unit test, the teacher decided to give the students a quiz first to help them prepare. Even though the students were able to access their interactive notebooks, none of them were using the integer operation foldable that was glued in their notebooks. The class bombed the quiz. The next day on the unit test, the students were again allowed to use their interactive notebooks, and again none of them were using the integer operations foldable. It wasn’t until after the test that the students realized that the foldable contained ALL of the rules they needed for integer operations. They could just open a flap for a particular operation and see what to do.”

I read this story on a blog this week and it struck me because I’ve been in the same situation with my students before. They had a resource in their hands that could help them, and yet they seemed oblivious to using it. Why?

As I said in my previous post, I want to question why everyone is using foldables so much. I’m not necessarily against them, but anecdotes like this make me want to pause and reflect.

What is the inherent advantage of taking the extra time to cut a piece of paper into flaps, write in it, and possibly glue it into a notebook? Some advantages I see:

Motivation. Making the foldable feels more “hands on” than taking notes, so it is possibly more motivating for students to make a foldable.

Source: Brian Stockus

Structure. The design of a foldable gives the content some structure. In the integer operations foldable, I can tell by looking at it that there are four key concepts, and each gets its own window. If students were just taking notes, they may do so haphazardly, losing the structure of the information in the process. I know my own note taking in high school was mostly just writing things down one after the other without any thought about how any of the information went together. I also spent a good chunk of time doodling in the margins, so I can’t say my mind was focused on what I was writing.

Source: Brian Stockus

Focus. Because the foldable usually has some kind of flaps, I have the ability to control the information I am seeing. If I want to learn about adding integers, I can open that flap and focus on that information. Look at how overwhelming it is when all of the flaps are open. That’s probably what it would look like if it was just notes in a notebook.

Source: Brian Stockus

Based on the opening story however, these advantages weren’t obvious to the students. It did not occur to them until AFTER the test that this tool was pretty useful, which tells me the foldable failed. Here are some musings about why that might have happened in this case and why it might happen in other classrooms:

Ownership. From my own experience and from what I’ve been reading, teachers are finding these clever foldables online as a way to summarize key concepts. They’re fun to do and they look attractive glued into interactive notebooks. The problem is that this is a teacher-centric activity. When it comes to summarizing student learning, the teacher has controlled the structure of that summary. She is even controlling the content if all students do is copy her words into their foldables. The students are basically just re-creating the teacher’s work. The activity lacks personal meaning so the students don’t think about the foldable as a tool that can help them later on.

To make foldables more meaningful, I think students need to learn about a variety of foldable templates. Then, after the class has learned about a topic, the teacher can ask students which foldable they want use to summarize their learning. (I might go so far as to say foldables are just one option. Students could also choose to make a graphic organizer or flash cards.) This would be a great discussion to hear students’ thoughts about how to structure the information they learned about. The students might not all make the same foldable, but at least what they make will be personally meaningful for them. It would be great to have students share their foldables afterwards so they can compare with classmates and make changes if they realize they got something wrong or left something out. And if you get to a point where the students feel none of the available options will work, then that’s the time to seek out and introduce a new template(s).

Making, followed by using. So, you spent 20 minutes making a clever foldable with definitions and examples of various mathematical properties. Students glue it in their notebooks, and later on when doing homework and other assignments they’ll turn to it as a helpful resource since it clearly summarizes important knowledge. But they don’t. Why?

My thought is that it has to do with the lack of experience actually using the foldable with any meaningful purpose. Once the foldable is made and glued in the notebook, it must be used. That’s what makes an interactive notebook, you know, interactive. Many students won’t do this naturally either. They need to hear the teacher at various points in class say things like, “Hmm, this problem includes subtracting two negative numbers. I know we just learned that. How can I check to make sure how to do that correctly?” or “This looks like a problem where I’m going to need to use the distributive property. Do I have a resource somewhere that I can use if I get stuck?” It sounds silly to ask these questions, but if students aren’t choosing to use their resources on their own, then obviously they haven’t learned how to ask themselves these questions. They need modeling.

In closing, I’d like to reiterate the point of asking my original question: Be critical of your practice. We’re not Sadako trying to fold 1,000 paper foldables so our education wish will be granted. The reality is that foldables are tools, and we can guide students to choose an appropriate tool for the job and model how to use it effectively.