This lesson follows an exploration of division (intervention lesson division remediation) in which students began to notice that decimals can terminate in a number of different place value positions and that others repeat. This lesson provides more division practice and basic fraction sense while at the same time helps them notice patterns in the division that gives rise to the decimals. Students are given a variety of fractions and are asked to find a way to organize them both before and after converting them to decimals. This activity forces students to really look more closely at the numbers than they otherwise would have and start to notice patterns. Some students will want to order them from least to greatest. While this does not really allow patterns to arise I did let them do it, because of it helps them develop their fraction and decimal sense. They will be looking at patterns on other posters in later lessons (day 3 & 4).
ELL students really benefit from group work as long as their are other students in their group that they can interact with in both English and their primary language. Grouping the fraction cards gives them another way to communicate their math understanding as well.
Students work with their math family groups of 3-4 and are given a set of fraction cards Fraction to decimal sorting cards.docx. Students are asked to organize the fractions in groups in any way that make sense to them. I really want them to notice similarities and differences in the fractions so they look at the decimals in the same way after they convert the fractions. It is important to give them just enough time to really look at the fractions, discuss options in their group, make a decision, and arrange the fractions. If not enough time is given they will not do as thoughtful a job on the decimals, if too much time is given they will not have as much time to be thoughtful with the decimals. As I circulate I remind them that they are looking for ways to group them and ask them what they are thinking. Some students may want to put them in order from least to greatest, but the patterns will not emerge if they do this with the fractions and I tell them to look for a way to put them in separate groups.
I ask each group to share their organizational method with the class. Some methods are by denominator, odd or even denominator, bigger or smaller than one, dividing odd by even, etc. After each group shares I ask if anyone organized theirs in the same way and then ask for a new method. Then I tell them that the next step is to find a way to organize them by their decimal value. If I don't allow them to first focus on the fraction sort I think they might not look as closely at the decimals. I really want them to examine both and not just one or the other.
When I introduce this section I tell them we are going to look at the decimal equivalents for each fraction and do the same thing. There are 15 fractions for which they need to do the division. I remind them that they can divvy up the fractions between their group members, but to be sure to double check each other's work because they will be making posters to present their work. I also remind them that they need to do their work neatly so their math family can check it more easily and help them find mistakes. They also need to keep the decimals recorded with their fractions because they need to be together on their final posters. While groups are doing the division I expect students to ask each other for help for questions like "did I do this right?", "how do I write the repeating decimal?", etc. If they ask me I always first turn the question back to the group to encourage them to share their ideas and also to develop comfort with "not knowing".
While I circulate I am doing two things. I am helping students with their division difficulties. Some are more mechanical and I have them work on the grid paper to keep the place values lined up. The primary problem is still doing the division backwards and not checking the answer for reasonableness. This is something that we worked on in the previous lesson (Intervention lesson division remediation) so I turn the question to the group. "Is the fraction more or less than one?", "so we expect the decimal to be..?", "Is the division turning out the way we expected so far?", "should we keep going or try another way?" Students who have more difficulty with fraction sense may need a visual reminder with fraction circles or with a diagram. When students do make this mistake they might notice the yuckiness that is dividing by 7 if the do 7/8ths backwards. When this happens I will point out that "this group here just accidently discovered that something yucky happens when you divide by 7". Not only does this help normalize the mistakes it gets the class focused on patterns when dividing by certain numbers like "I am noticing that when we divide by 3 we get a repeating decimal", etc.
As students are finishing up the division I get their attention and tell them that they can begin looking for a new way to organize the decimal numbers and start planning their posters. I tell them they need to have their names on everything and it will be collected together and returned to them tomorrow.