In the previous lesson (Perfracimals 1) students used the place value chart to figure out why they can slide the decimal two spaces to the right to change a decimal number into a percent. Today they are asked how they might write a decimal in the thousandths place and a repeating decimal as a percent. I do not tell them anything, because I think allowing them to struggle, makes sense of it together, and "discover" it on their own makes for a more powerful learning experience. They take more ownership if the knowledge and make an emotional connection to it, which helps them remember it.
Having the place value chart helps students make connections to prior knowledge and helps ELL better express their thinking and understand the thinking of others. This allows them to refer to a visual model when they aren't sure what words to use. When this happens another student will usually offer them a word.
This warm up Warm up perfracimals 2.docx helps students see the connections between place value, fraction, and percent notations. The first problem is a visual box diagram for 200/5. The visual representation often helps students figure out "which way" to do the division, because they get a sense of the expected size of the solution beforehand. One problem still is reading the problem backwards, "5 divided by 200", which may contribute to the confusion.
The second problem asks students to show 3 mathematical ways to write "15 hundredths". I record these on the board (.15, 15/100, 15%). Establishing this up front helps to scaffold their learning in the exploration section when they are faced with triple digit decimals.
The third question asks them to do the division 3/8. This helps them practice their long division and sets up the exploration when they are asked how they might write 0.375 as a percent.
Students are asked to figure out how to write 0.375 as a percent. I expect them to come up with and share multiple method. This is a good way to get them to use mathematical evidence to convince each other of their method. Also, seeing the connections between the different models helps them develop better rational number sense.
Several students might slide the decimal two spaces to the right, a pattern they discovered in the previous lesson (Perfracimals 1) and then try to make sense of the .5%. One way to help scaffold is to refer to the 3 ways we found in the warm up to write 15 hundredths. if the number in the hundredths place shows the whole percent, then the number in the thousandths place is less than one percent. I may cover the 5 in .375 and ask what percent that is (37%), so this (.375) is a tiny bit more than 37%.
Other students may write .375 as a fraction out of 1000 and try to divide. Students will get confused by a couple of things here. Some will try to write it over 100 and not be able to figure out what's wrong. This is an easy fix - just ask them to read the decimal. Some will say that 1000 can be divided by 10 to get 100, but 375 can't be. This is when we do the division together to get 37.5/100. They think it is weird, but it makes sense to them.
By the end of this they realize that the same pattern they found in the previous lesson still applies here.
Students work on individual white boards, but can work together with their math family groups. Everyone raises their boards at the same time so I can give corrective feedback if necessary and no one can opt out. They love this because I find all kinds of ways to say "fabulous" and the feedback is either telling them what they did that was wrong or what they still need to do to make it right.
First I give a couple of decimal numbers in the thousandths place one at a time (0.125, 0.875) and ask them to write each as a percent. If they get these pretty quickly I move on to a repeating decimal (0.999...). I write it out to the thousandths place before putting the repeat bar because I want them to get this first one right. It will help several of them avoid a common mistake in the next one (.3...), which I only write in the tenths place with the repeat bar. Some students may still write 30%, but several will have realized that it is just like the last problem and correctly write 33.3...% with a repeat bar. This will generate some argumentation and I ask students to explain why they disagree with 30%. I like to play dumb here and say "Look, I moved the decimal two spaces. I know it's not 3%, because I have to take it to the hundredths place." This has the added benefit of pointing out what the "wrong" people did right. I try to get a student to explain that the repeat bar tells us that the next number will be a 3, not a zero. If they don't I keep asking questions, "where did this 3 come from?", "what told you to put a 3 here instead of a zero?", "what do you see that tells you that?". Having students come up and point to the repeat bar on the board is very helpful for ELL students. After students explain I do one more repeating decimal and then I give them some percents to write as decimals. (13.5%, 4.5%, 125%, 33.3...%). I ask them to check with their math families at each problem to engage them in peer instruction, but I don't expect any difficulties. If they have any trouble I ask them to read their decimal and circle the number in the hundredths to make sure it matches the original percent. Additional scaffolding for percent to decimal.docx
One note about the homework: I tell them they will get an extra night to complete it if they have trouble with the division. The next lesson is a division intervention lesson