I came up with this lesson after overhearing a conversation between two students in a previous lesson (Poster Patterns day 3). Angelina and Austin had noticed that dividing by multiples of 3 seemed to repeat and dividing by 4 terminated with a 5 in the hundredths place. They wondered what would happen if we divided by a multiple of both 3 & 4, like 12. This was a great opportunity to follow student inquiry and delve a little deeper into an exploration of repeating and terminating decimals. It is important for the teacher to listen for these opportunities and take the time to explore student inquiry. Students take more ownership when the questions & ideas come from them. I am hoping that my division phobic students gain greater insight into the patterns of division. If they come to expect the answer to repeat or terminate in the thousandths place then maybe they won't give up before they finish the division when it starts to look weird. (MP1) I expect them to develop better number sense when they see the relationships between the numbers. It is always important when you have student discussions that ELL students have two things: something visual and concrete to refer to like the place value chart, and at least one other student who is bilingual.
I start the lesson by relating it to the previous lesson activities (Poster Patterns days 3 & 4) and reviewing some of the patterns we found when dividing by 2, 3, 4, etc. I tell them that while I was watching the videos of their group conversations in the day 3 lesson I came across an interesting discussion between two of their peers which asked: If we expect dividing by 3 to make a repeating decimal and dividing by 4 to result in a decimal that terminates with a 5 in the hundredths place, what will dividing by 12 do? 12 is a multiple of both 3 and 4, so which pattern might it follow? Then I played the recorded video for the class. I do this to give credit to the two students, but also to highlight the kind of thoughtful curiosity that I want to inspire. Because it is novel to see video of themselves and their peers in class I tell them I'm going to play it again and ask them to look at what Angelina starts to do right at the end. When someone points out that she starts to do some division to test it out and see what happens I say "what a great idea, let's try that! I can't wait to see what happens." I expect students to try out different problems dividing by 12 and get different results. This is a great way to engage them in argumentation when they notice they are getting different types of decimals. "It repeats!", "No, it doesn't mine terminated.", etc.
Once students have found that dividing by 12 sometimes produces repeating decimals and sometimes terminating decimals I want to give them a way to organize their results so they can begin looking at patterns. I want them to start exploring the patterns that cause one or the other type of decimal. I tell students we are going to try to figure out when dividing by 12 will repeat and when and how it will terminate. They are going to look for patterns that will give us a clue. In the exploration worksheet Dominant Division Traits.docx in addition to showing the division and stating whether it terminates or repeats I ask students to do two more things. I give them the division as a fraction first (1/12, 2/12, ... 11/12) and ask them to simplify it if possible. I also ask them to describe how the terminating decimals terminate. This question needs some scaffolding because it is a little vague. I tell them I want to know what number it terminates with and where it terminates, for example: "when I divide by 2 the decimal terminates with a 5 in the tenths place".
Many of the division problems on the sheet students have already come up with and completed on their own in the warm up, so it doesn't take that long for them to complete the sheet. I circulate just to make sure they are simplifying correctly and that they are describing the terminating decimal sufficiently (with what number & where). Otherwise it will be harder for them to find patterns.
I remind them they are looking for clues that will allow us to predict when dividing by 12 will cause a repeating or a terminating decimal. As I circulate I ask them what they are thinking or noticing and I ask probing questions to get them to look deeper. "Are you saying...?", "Is that always true?", "did they terminate in the same way, or is there something different about it?", "What is the same about all the division that repeated?", etc. It is really important for the teacher to really listen for what students are trying to articulate and help them articulate it. For ELL students it will be really helpful to keep the place value chart available for reference. When they can point to the chart they can convey their meaning while you or a student can provide the vocabulary they need.
I do this exit ticket as a whole class discussion because I want them to practice articulating their ideas, using evidence. I also want them to really listen to and build upon or counter the arguments provided by their peers. This is not a lesson to prepare them for a test, I am not looking for skills. I am more interested today in students making observations, articulating ideas, and using mathematical evidence.
I simply restate the question from the beginning: "If dividing by 3 produces repeating decimals and dividing by 4 produces decimals that terminate with a 5 in the hundredths place, what will happen when we divide by 12, which is 3x4?"
As students share their conjectures I ask for a show of hands to see who noticed the same thing. I ask if anyone has anything to add to that? "Did anyone notice something similar?" or "Does anyone have another way of saying that", or "Does anyone have another way of looking at that".
What I would like students to see is that when the denominator simplifies to a 3 the division will result in repeating, when it simplifies to a 4 it will follow the pattern of dividing by 4, and more simply that there are patterns that cause repeating and terminating. More importantly I want to inspire them to explore their own curiosity and test out their own conjectures.