This lesson is the first of three in which students use ratio, percent, and graphing to test for proportionality. In this first one students are representing a single population with ratios and on a graph. The focus is on using the tools and emphasizing how they show equivalence. The focus in on multiple methods and representations and the teachers role is to highlight and share student ideas and allow them to lead the learning.
The warm up warm up percent ratio equivalence.docx asks tells students that 60% of the lady bugs in my yard have spots and asks them to figure out how many we would expect to find with spots with different given totals. The first total is 100, which is easy for them because they know that 60% means that 60 out of the 100 total will be spotted. They then need to scale up and simplify to find the number out of a total of 300 and then out of 10. The last one asks how many out of a total of 15 would be spotted.
As always, students work in small math family groups of 3 or 4 and I circulate to highlight multiple methods and good ideas that students come up with. In one of the videos (need for organization) one student recognizes the difficulty in sharing his work when it is disorganized. This is an idea I might share with the class. I may show his "disorganized" work under the document camera and say that "Zavier thinks his work looks a little 'all over the place', so he is going to try to organize it in a table to make it easier to share...what a great idea!"
I expect students to have a little struggle with the last one, because none of the equivalent ratios that we have so far, 60/100, 180/300, & 6/10, can be scaled directly to a denominator of 15. I expect students to realize that they need to simplify to 3/5 before scaling it up.
In solving the last problem in the warm up one of my students began to explain how she found the correct answer: 9/15. Instead of simplifying 6/10 to 3/5 and then scaling up as I expected, she said she added 5 to the 10 and added 3 to the 6. In years past I may have stopped listening as soon as she said "added", but I kept listening and modeled it on the board and saw immediately that in fact her additive method works. We shared it with the class and asked the small groups to discuss why it makes sense that her method works. Sadly, I did not get video, but there were two wonderful connections made. Most students could see that adding 3 and 5 was still maintaining the constant ratio of 3/5 and said that as long as you are adding 3 (spotted lady bugs) for every 5 (total lady bugs) it would stay in proportion. Another group pointed out that, starting with 6/10, she added half of 10, so she has to add half of 6, which is a lot like scaling up or simplifying by the same factor!
After they have finished the warm up I point out how they have actually found 60% of 300, 10, and 15 and then ask them to show what the information looks like in a graph. If they have not used a table for their data and are having trouble figuring out how to graph the information, I suggest organizing their work in the table first.
I provide them with the graph with axes already labeled and numbered, because I didn't want them to be able to fit all the data on the graph. I intend for them to struggle with 180/300 so they have to figure out another way to represent it on the graph. This helps to reinforce the concept of equivalence in proportions. As I circulate I listen for this question. I would expect them to ask if they can simplify, but if they don't I ask what they think they could try. If they still don't suggest it I tell them I will check in with some other groups and see if they have any ideas. If they do suggest simplifying I tell them to give it a try and see where that point ends up on the graph and if it makes sense to them. Some students may ask if it has to be simplified all the way to which I respond "I wonder if that matters? Try it and see where it ends up on the graph". As several students begin asking I draw it to the attention of the class, "several of you are wondering if you can simplify 180/300 to make it fit on the graph, what an interesting idea, I can't wait to see what that looks like!"
I also look to see who might be connecting the points and noticing again that it is a straight line through zero, which is not a new idea to them. When I see students doing this I will show their work to the class and ask if anyone else is noticing this pattern. This helps them to connect to the graph as a tool for showing equivalence and proportionality. Having a variety of tools for communicating their thinking and their solutions is really helpful for my ELL students.
I ask a few follow up questions for whole class discussion that focus in on the tools that can be used to test for proportionality. After they have explained what each point represents or "says" and noticed that all the points they put on their graph lie in a straight line that passes through zero, I ask them to explain what this tells us about the ratios at each point. I expect them to conclude that they are all proportional or equivalent to each other and that they all show the same thing, 60% of the lady bugs are spotted. If students are a little unsure I might ask at each point "does this point say that 60% are spotted?", "how about this point?".
This is the perfect segue into the next questions that I might use to highlight the ratios and percents as tools. I might ask "How did we already know they are proportional before we graphed them?", or "How does this proportionality show in the ratios?" I want to include in the discussion the idea that no matter whether we simplify partially, fully, scale up to the percent or any other total the ratios are equivalent.