This lesson follows a previous one (day 1) in which students noticed a pattern when finding 10% of different numbers by sliding the decimal one space to the left. It was important for them to connect the movement of the decimal point to the place value chart so they understand why the shortcut works. If they understand that it is actually the digits that are moving since the numbers' value is becoming ten times less they are less likely to forget. This lesson goes much more quickly because it capitalizes on the connection to the previous lesson. Students predict that they will find a shortcut and focus their attention immediately on the decimal point. The teacher's role is to make sure they can make sense of why it is working. If we are finding 1% we are finding 1/100th, therefore dividing by 100. If the number is getting 100 times smaller it makes sense that the decimal point will move two spaces because each space is 10 times smaller than the last. It is really important to make this connection. Otherwise this becomes just one more trick they have to remember.
This warm up Warm up one percent.docx tells students that the bank will pay me $1 for every $100 I keep in my savings account. The first question asks what percent that is. I ask this question so that students can make sense of what the numbers in 1% represent. When presented with a percent most of my students would not be able to tell what the numbers represent without this prompting and, I noticed in the previous lesson (day 1), this can cause problems in setting up the next problem. The next question asks students how much the bank will pay me if I have $200 in my account after 1 year, 2 years.
As I circulate and notice some students have finished quickly I ask them to figure out how much I would have in the account after 5 years. This is a nice way of differentiating so I can work with others who are still having trouble setting up the problem. As I check back in with those groups I look to see if any of them said $10. I ask these students what happened to my original $200 ("is the bank stealing from me?").
After we go over this problem I ask students to predict what pattern they might expect when we find 1% of every number and why.
After students make their prediction about the pattern we will find I ask them how we can test it and see. I expect them to suggest that we actually find 1% of some numbers by dividing them by 100. It is important that we test it on different types of numbers so, as they suggest some numbers to try I label them as small, big, odd, even, whole, terminating in the tenths, etc. to encourage them to suggest different types. We come up with a list of between 7-10 numbers so that each group member can do a couple and still have time to give and get help with the division. Once students verify that their prediction works we spend some time making sense of why the decimal point would move two spaces when we find 1/100th of a number.
Students do their work on individual white boards, but are allowed to work together to take advantage of peer instruction. Students aren't allowed to tell each other answers, but are able to help each other get started or explain how and why they take certain steps. All students raise their white boards to show their solutions at the same time.
I start right in with the "cup half full" diagram from the previous lesson (day 1). I write 100% on the left at the top and 4000 ml. across from it on the right side and tell them that when my cup is 100% full it has 4000 ml. of water in it. I draw a water line down at 10% and ask how many ml. of water are in my cup. I ask them, "if I can find 10% what other percent can I find?" I expect them to suggest any multiple of 10, so I ask them to find one of those and ask how they figured it out. Some will say they added 10% repeatedly and some will say they multiplied.
Then I draw the water line at 1% and ask them how much water is in the cup. I take suggestions for what other percent we can find if we know 1% and we explore some of those. If they don't suggest any combinations of 10% and 1% like 21%, I will. They may not think of subtraction or a percent over 100%, so I might ask them for 99% and 101% as well.
At some point I will tell them we need more of a challenge and suggest that we make the 100% capacity 50 ml. At first they think it will be easier because the number is smaller, but they will be working on their decimal operations at the same time. This is one simple way of assessing how much or little they can do with simple decimals while at the same time giving them access to peer instruction. I can see if they have trouble adding or multiplying decimals. Or even if they don't make sense of 0.5 as half.