As students work with new tools for testing for proportionality it is just as important for them to see what it looks like when quantities are NOT in proportion as when they are. In this lesson, again, there are multiple methods students may use and student discussion is encouraged for sharing ideas and answering questions. The teacher's primary role here is to allow student ideas to guide the learning. When students get stuck you will hear me asking them to explain their work so that they can discover the problem for themselves. The key here is asking students to explain what's right as well as what's wrong so that the question itself does not become a clue that something is wrong. When teachers only question students about mistakes they tend to view questioning as an indication that something is wrong. Students are less likely to explain, and ultimately find the problem, if they think their work is wrong.
This warm up Warm up which will find percent.docx surfaces a mistake that I expect students to be aware of and able to explain at this point. We have taken great care to consistently define our terms to prevent such a mistake, but I bring it back here, because I don't wan them to become complacent and sloppy. Students are reminded of the lady bug data from my dad's yard from the previous lesson (day 2 of 3). They are reminded that several different ratios can be written to represent this data, but not all can be used to find the percents. Students need to remember and remind each other why only the part to total ratios can be used to calculate the percents of spotted and non-spotted lady bugs.
Students are given data to show the number of spotted and non-spotted lady bugs in my brother's, my aunt's, and my cousin's yards Family lady bugs.docx and are asked if they are proportional to the one's we have looked at in the previous days' lessons. The difference between this lesson and the previous lesson (day 2 of 3) is that not all the populations of lady bugs are proportional. Students must explain both why one is proportional and why the others are not. As I circulate I encourage students to explain their work and also ask them to show what it looks like graphically. One question I use a lot here is "How can you tell by looking at the ... (ratios/percents/graph)?"
Students get stuck on the simplest form here. They are so used to only comparing ratios with common denominators they forget that they can determine disproportionality if the ratios in simplest form are not equal. I may need to ask them questions like "Is 11/20 in simplest form?" and "can it be simplified further?" (no) and "then can it be equal to 3/5?" In one of the videos you also hear me suggest scaling them up to percents to prove it.
During the exploration phase I look for student examples that are representative of a certain method or are unique. I take note and make sure to call on those students or just pick up their work to show under the document camera. I like to give the class a little time to try to figure out what the student may have done before anyone explains the work. This way students can find connections to the work they did. I may have students explain their own work or someone else's work. When students know that other students may be explaining their work they are more likely to include more evidence and show more work as well as organize it more neatly.
I want students to see the different tools they can use to figure out whether two quantities are proportional. I think that making connections between them helps strengthen their understanding of the tools. Using multiple representations is also really helpful to both special education and ELL students.