This lesson has a focus on attention to precision. Many students will still be working on writing ratios and calculating percents in this lesson. I rely on peer instruction within the math family groups to help each other with their calculations. The main thing for this lesson will be surfacing a common error and emphasizing that a percent is a part to whole ratio. Students will write part to part and part to whole ratios for this lesson and I expect some may mistakenly use the part to part ratios to calculate the percent. While there will be some argument at the end of class about which way is correct I don't expect to completely resolve it during this lesson. Students will need time in upcoming lessons to practicing refining their definition of percent and choosing the correct ratios to scale up to calculate the percent.
In this warm up Warm up lady bug ratios.docx I tell students that I counted 50 lady bugs in my yard. 40 of them had spots and 10 had no spots. I ask student to write several ratios and percents to accurately report this data. I provide them with sample statements accurate reporting.docx for a different set of data that we looked at in previous lessons (Accurate reporting & Dueling data) just to give them an idea of the different types of statements they could make with ratio and percent. Some students will be able to come up with several statements on their own, but I expect many of them to need more scaffolding, especially my ELL population. Sentence frames are really helpful for them and I provide frames ratio statement frames for lady bug data.docx specifically suited to this set of data to help them create accurate data statements.
One mistake that I expect to arise is that students will use the wrong ratio (part:part) to calculate the percentages. Some students will find that 20% of the lady bugs have no spots and some will calculate 25%. Those that said 20% correctly used the part:total ratio (10 out of a total of 50) and those who said 25% will have mistakenly scaled up the part:part ratio (10 with no spots:40 with spots).
The key here is realizing that a percent is a part to whole ratio ("out of a total of 100"). One step that I encourage my students to take is to clearly define their terms. "what do each of the numbers represent or count?" It is really helpful for my ELL students and also my students with auditory processing disorders if I give them visual cueing. I may do this here by circling a number and drawing a line to what it represents and circle that too. For example I would circle the number 40 and the word "spots" and draw a line between them. As I see good organizational ideas from students like putting the information in a table I like to highlight that. I may say "Look at Makenna's idea" and show it to the class under the document camera.
In the exploration I tell them that I counted 20 lady bugs with spots and 5 without spots in my neighbor's yard and ask if the ratio is proportional to my yard. I expect some of the students to take the time to define their terms and identify what each number represents and some students will not. The students who do take the time are more likely to notice that the total number of lady bugs is not given. The students who have not defined their terms are more likely to mistakenly use 20 as the total number. In this case they are more likely to calculate the percent with spots as either 15/20, which equals 3/4 & 75% or as 5/20, which is 1/4 & 25%. At this point one of two things can happen. There will either be disagreement within a group or all the group members will have done the same thing, wrong or right. If there is disagreement in a group I encourage each member to share what they did and why to decide who is right. If everyone in the group did it wrong I spend some time defining the terms with them and asking them to see where they may have gone wrong. If everyone in the group did it right I ask them to decide why using a denominator of 20 would be wrong.
I do not expect to fully resolve this with all students during this lesson. They need to understand that a percent is a part to whole ratio and that in order to end up with the ratio comparing the number of spotted lady bugs out of a total of 100, you must start with the same ratio out of the total we are given. This is especially difficult for my ELL students because they must use the context clues in the problem in order to define what the terms represent.
The last few minutes of class are spent discussing where the argument stands? I ask students whether they think the neighbor's lady bugs are proportional to mine. Some will say yes, because they have the same percentage of lady bugs with spots and some will say no, because the percentages are different. I want to surface the disagreement and record their mathematical evidence and end with the question of which one is right. The next couple of lessons (Writing Percents & Writing percents and defining our terms) will focus on some skills practice and then we will resolve this question in the following lesson (My neighbor's lady bugs).