This is the second in a series of three lesson in which students use ratio, percent, and graphing to test for proportionality. In this lesson students are asked to compare separate populations, all of which turn out to be proportional. The focus is on students figuring it out together in small groups, asking each other questions and exploring their own ideas. They practice explaining how each tool shows proportionality. Having several different mathematical tools at their disposal helps ELL students both express their reasoning as well as see connections between the different tools. As they refer to their work or to the graphs with their small groups they can get help with the necessary vocabulary. It's much easier for other students to provide needed terminology when the student is pointing to some visual aid.
The warm up Warm up my parents lady bugs.docx tells students about lady bugs found in my mom's and dad's yards and asks students to organize the information in a table. If they do not take the time to do this they are more likely to start with the wrong ratio (part to part) and calculate the percents incorrectly. While students are working on the warm up I return the tables and graphs they started in the previous lesson (My family's lady bugs day 1). They know this is a continuation and many of them will probably begin calculating the percents for the spotted lady bugs in the warm up. We do not spend anytime going over the warm up. It is just meant to give them the information they need in the exploration section.
Students work in their math family groups to decide whether the proportions of spotted lady bugs are the same for my parents and for my yard. Students have already explored several different tools for figuring out and expressing proportionality and this lesson gives them the chance to use them. Some students may find the percents, some may scale ratios up to common denominators, some may write them in simplest form, some may graph them, etc.
As questions and ideas arise in the groups the teacher's role is to highlight them for the class and let their ideas guide the learning. "Ruben is going to use a table", "Austin did something I didn't think of, I can't wait for you to see it", "A question has just come up at this family...", "Look how Mariella decided to organize her work". As students begin coming to conclusions I ask them to try another method to double check. "If you used ratios to figure it out please show what that would look like on the graph", etc. As they work together and ask each other questions they get new ideas from each other and are exposed to multiple methods. Periodically I ask who is using ratios, percent, or graphs and encourage them to make sure to share their ideas with their math brothers and sisters. To encourage greater participation by my ELL students I like to offer up their work for discussion in the group. "Let's take a look at what Misael has shown", "Salvador, can you show us what you did with ratios?". I put their work in the center of the group and have everyone lean in.
Making decisions and figuring things out together without so much direct instruction from the teacher is a really effective way to increase engagement, motivation, and intellectual pride even amongst the most disengaged student populations. The class can be a little noisy, but its worth it when I see them getting excited about using the math.
The last few minutes of class I invite students to come up and share their conclusions and explain how they figured it out. During the exploration section I keep track of who has a unique approach or who has an example that is representative of one common approach and I make sure to call on them. If they don't want to share in front of the class I just show their work under the document camera and ask if they will explain from their seat. I may give the class a few moments to try to figure out what they did before having the student explain. My purpose here is for students to see all the different tools for determining proportionality and also to see the connections between them. I will ask students to point out similarities between the methods. For example if one student shows the work she did to calculate the percent another student may also notice several ratios that are the same as another student's work.