In this lesson students are asked to verify and critique the accuracy of claims. It is important for the teacher to continue to ask students to define their terms and identify what the numbers are representing. This helps students make sense of the problem and also helps them explain their mathematical evidence. Students will be looking at several different claims that are based on survey data some of which are inaccurate. Students need to decide which claims accurately report the survey data and which don't. This can be a real challenge for English Language Learners and having them work in small groups is really helpful for their language acquisition as well as for their content understanding. It allows them to practice their English and listen to the explanations of others. I include at least one other member of the group who speaks the language because if they learned the math in another language they can express their understanding in that primary language and then hear it in English as well. This kind of activity can also be challenging for students who have been successful with the more algorithmic teaching styles and they may try to solve the problems without the context, which can lead to mistakes.
This warm up Warm up teen crime 1.docx reports the results of a survey in which adults think that 43% of violent crimes are committed by teens. Students are asked to figure out how many crimes would have to be committed by teens if there are 4,000 total violent crimes. Students may need a little scaffolding to enter this problem, because they may not recognize that 43% means a total of 100 violent crimes. They may have the background knowledge required, but they may not be able to access it and realize that it is needed here. I ask them first to decide in their math family groups what 43% represents. As soon as I hear "43 out of 100" I will bring this to the attention of the class, "Donovan has just pointed out that 43% is actually two numbers, decide what each of them represents" "what is each one counting or comparing?" I can model their responses in a table form to help then scale up to 4,000. When they figure out that 1,720 of the crimes would have to have been committed by teens I put up the second part of the problem Warm up teen crime 2.docx which shows them that 520 were actually committed by teens and I ask them if the adults that were surveyed were right? Were they close?
This warm up sets the stage for the process of testing and verifying claims as well as highlights the importance of using the context to help make sense of the problem.
This is a small group activity in which groups of 3 or 4 students are given several statements and are asked to decide if they accurately report the results of a given survey. Students are required to give mathematical evidence. The teacher's role here is to help students navigate confusion and disagreement in their conversations and to continually remind them to use the context. You will hear me asking them what the numbers represent as a way to help them explain their thinking. I will try to paraphrase for them what they are saying and ask other members of the group if they are convinced. If they are not convinced I try to help them ask for clarification. Drawing lines and circles around the numbers and the phrases that define them on the actual sentences is really helpful for ELL students. Color coding can also help.
There are a few common stumbling blocks I expect in this lesson for which defining their terms (what the numbers represent) can be really helpful. I know students will get confused when they are given part to part and part to whole ratios particularly when they are deciding between ratios of 3:7 and 3/10. They will also get mixed up when the language is changed and they are asked about ratios of 3:7 and 7:3. Asking them to explain what each number represents helps them think their way through it. I also know that students will confuse 3% with 3/10ths because this often happens with single digit percents. Sometimes just repeating their statements back to the group will be enough for someone to catch the mistake.
One problem with small group work is that all the wonderful mistakes and explanations are not visible to the whole class. The next lesson (Dueling Data) will highlight a few of these for the whole class.
At the very end I bring back the warm up Warm up teen crime 2.docx question that gives the FBI data on violent crimes. Students have already determined that the adult's perception that 43% of violent crimes are committed by teens was inaccurate and I want to ask the students to figure out the actual percentage. I ask students to figure out what they would say to and show the adults in the survey to convince them they are wrong.