In this lesson students are finding the percent of a number using a box diagram. They are also reviewing prior knowledge of place value to help them convert percents to decimals and visa versa. Because of the wide range of knowledge gaps it is important for the teacher to be ready with scaffolding materials like a place value chart or explanations that help connect to or develop prerequisite knowledge. It is important to plan for potential scaffolding needs for a couple of reasons. One, so that it takes less time and flows more smoothly and the teacher is not caught off guard and, two, so that the student doesn't feel as though their lack of knowledge is abnormal.
The first two problems on the warm up warm up box diagram to find a percent ask students to find 25% of 12 and of 0.8. Some students will use a box diagram and some will not. I encourage multiple methods and ask that they explain their methods to each other. As I circulate I share out ideas that the students may have. "Fatima is saying that we really need to know what fraction we are looking for when it asks for 25%", "Samantha remembers its the denominator that tells us how many equal parts". Some students may simplify the percent fraction then scale it up to the new total. It is important for students to understand the idea of parts and totals. The part we are looking for is equivalent to 25% which is the part out of the total of 100. Listen for student explanations and models that bring this to the surface and spend time having students make sense of it. For a really nice student model see the reflection section.
The third question asks students to read 0.25 outloud without saying the word "point". I have a place value chart on the board behind a screen. I want to know how many of my students can do this without seeing it, but when I get a quick idea of how many can't I apologize that I forgot to raise the screen and I know its hard to remember things from a long time ago. If the majority of them can't remember, I am pretty sure they don't really understand the patterns in the chart. Some scaffolding might be needed for students to be able to convert decimals to percents. warm up box diagram to find a percent notes.
Once they have read 0.25 as twenty five hundredths I say "hey, I think we can show that in a couple of other ways too!" and I tell them to take out their white boards. I write 0.25 on the board and write "decimal number" above it. I ask them to show another way to write twenty five hundredths on their board. When they raise up their boards I expect to see some with 25/100, some with 25% and some with both. I might see 1/4 but I don't wait for it. I write these up and label above each "fraction" and "percent". I ask if it would be correct to say they are all equal and ask them to explain why. (hundredths means out of 100, which means percent)
I tell students I will give them one decimal number at a time and I want them to write both the fraction and percent equivalent on their boards.
I start out with double digit decimals like 0.11 and 0.75. Each time I ask students how they know to reinforce the prior knowledge. I am also looking for a couple of mistakes (0.11/100 and 0.11%). When I see this I remind them that 0.11 is 11 hundredths and ask how we know (the hundredths place tells us). Then I ask what part of the fraction and the percent tells us it's out of 100 (the denominator & the percent sign), "so do we need the decimal placement to tell us it's out of 100 if the denominator/percent sign already tells us this?" I don't expect many students to persist in this mistake, but if they do, I would give more examples like these first two.
Then I move on to decimals with zero in them like 0.03 and 0.30 and 0.3. In all my years of teaching, this has been a major source of confusion for students. I give the numbers in this order, because students are less likely to make a mistake if they start with 0.03. If they start with 0.3 or any other single digit decimal they are more likely to make a mistake without even thinking about it. If I start with 0.03 and then move on to 0.30 or 0.3 they are more likely to hesitate and question their response. "Hmm, this can't be the same as the last one, there is something different about it, it's not as easy as it looks, I'm going to have to think about this one" Without the first one for reference there is nothing to indicate a need for thought. They are more likely to check with each other and check with the chart if they think there is a possibility they are wrong. Surfacing a common error by highlighting what makes it confusing is a good way to engage students in argumentation.