This lesson really grew from student questions. We ran into trouble with a previous lesson (Perfracimals 2) when students had trouble with basic division. Then, during a remediation lesson (Intervention day-division remediation), students started noticing patterns with division and asking questions about it like "will that always happen?" and "will that happen with any other numbers?" I decided to follow their curiosity into an exploration of terminating and repeating decimals, while at the same time working on our division and fraction sense. In this lesson students will be rotating between about 4 - 5 stations in which they will be looking for division patterns in posters they and their peers have made. The posters provide a concrete reference point for ELL students when they are trying to express their thinking. In one class I used ipads at every table for students to record each other explaining. It's often hard to tell if students are on task when they are engaged in a group discussion activity, because I can't be at each group. Having the recording devices helps maintain the expectation and provides some accountability.
This is where I set up the exploration for students. As soon as they walk in I ask that they put their backpacks in one corner of the room to keep them out of the way. I remind them that last week they organized the results of some division problems into posters and today we are going to take a really close look at a few of them. It's all about what they notice about the division and what they wonder about the division.
I remind them that they started to notice last week some patterns when we divide by 2. You noticed that when we divide an odd number by 2 the decimals all ended in point 5. Some of you wondered if that would always be true. Some of you wondered why it might be true or what might be making that happen. Some of you wondered if that would happen with any other numbers.
I tell them that today they need to focus in the same way on what they are noticing and wondering about. Look for patterns that help you notice what makes a repeating decimal or what makes a decimal end a certain way, what is it about the division (or fraction) that makes these things happen? These three questions will be written on the board. When we are done I am going to ask them to share some of the things they noticed and wondered about.
They have questions at each poster table just to help them start looking for patterns and to get their conversation going. Sentence starters and questions are displayed on the board if they need helps sharing their ideas. These are particularly helpful for ELL students.
I tell them they need to move to the next table when I way "switch".
It's important to give time during the instruction period for student questions when they are being asked to do an activity they have never done before.
Although there will be 8 -9 stations set up students only need to rotate through 4 or 5 of them.
Several of my students had trouble setting up a division problem from the fraction. When I told them to do the division both ways and decide which answer was a reasonable value for the fraction they still had trouble. This station asks them to notice that the decimal will have a value greater than one if the numerator of the fraction is greater than the denominator. Not all of my students need to visit this station, but there are a few that could use the reinforcement even after our remediation. I expect these students to be successful at this station, while they might have a little more difficulty with some of the others. As I circulate to this station if there is a more advanced group I might switch the poster with one that is not organized in such an obvious way, so that they have to really look for it. Or I might switch it with a poster where mistakes are made in the division so they can find them.
Students are asked to decide if there is anything about the division that tells them if the decimal will repeat or not? I am expecting to hear students talking about dividing by 3, 6, and 9, and maybe other multiples of 3. As I circulate to these stations I ask them to show me evidence for and against their claims. "Show me a fraction that supports that claim" or "Does anyone see any evidence against that claim?". I also ask what other numbers they think we could divide by that would make a repeating decimal and suggest they try some out to test it at the "weird, weirder, and weirdest" station when they get there. I am also really listening for any questions students might have that would lead to further investigation for them to test out.
There are three stations where students have a choice of either doing some division to test out something they wondered about at a previous poster station or do the division I have provided. The division I have given them is dividing by 9, 11, and 7. When I circulate to these stations I expect to hear them say they repeat. However, at each station they will find some additional intriguing patterns that they may not notice unless I tell them "that's not the weirdest part of it, look again".
At the nine's station I may ask them what are you dividing 9 into? (1) and what was the result? (point one repeating), or (2) and point 2 repeating, etc. They think this is really weird and I ask what they think might happen if we divided 10, 11, or 12 by 9. I suggest they try it out and see.
At the eleven's station they may notice that it results in a repeating decimal where two digits repeat. I make sure they understand how to place the repeat bar, but I also tell them to look closely at the two digits that repeat and see if they notice a pattern in all three that they did. Some will notice they all add up to 9, others may notice they have an increasing/decreasing pattern.
At the seven's station they start out thinking there is no pattern, but I tell them to persevere it will appear. When it does I ask them how many digits repeat.
By presenting these as weird and whacky division problems I hope to inspire their curiosity about the math. I also want them to experience the joy of discovering a pattern in the math!
At these stations students are examining patterns in terminating decimals. They are asked specifically what happens when dividing by 8, 4, etc. I expect the most detailed discussions at these stations because there are so many different ways numbers can terminate. As I circulate to these stations I am listening for what number they terminate in and what place value they terminate in. I ask students what patterns they notice in the decimals when they divide by 8. I ask if dividing by any other number makes this happen or how this is similar or different from dividing by other numbers. I ask them to show me some evidence for or against their claims and ask them to explain their thinking.
Interesting conversations can come out of this one as they start predicting how other denominators would terminate as decimals. This is especially true if they have already visited the repeating decimals station. They may begin to wonder what will happen if a denominator is a multiple of both 3 and 4.
This is done as a whole class discussion. I ask students to share what their math family group noticed at each station. For example "what did you notice about dividing by 3?" I am not interested in a particular skill at this point. I am more interested in having students practice making a claim, supporting it with evidence, listening to, critiquing, & building on the ideas of others, and increasing their confidence with division. My job is to continue the conversation with probing questions: "what made you say that?", "what do you mean by that?", "can anyone add to that?", "did anyone notice something else about...?", "what do you wonder about that?". Keeping ELL students involved in whole class discussions can be tough. One thing that helps is encouraging students to come up and show us what they mean or what someone else means. Often the ELL student is encouraged by their math brother & sisters to bring up an example by saying "Karla has a good example for that", or "Luis had a good idea".