This is a really remedial lesson because my students have serious difficulties with division as well as fraction sense. This lesson addresses both. It is sometimes really shocking how low some of their gaps go, but I feel an obligation to try my best to help them fill as many gaps as I can along the way. They really seem to appreciate it and they treat it like any other learning experience. Many of them are relieved to finally start understanding what to do when they get stuck.
It is really important here to pay attention to their thinking so you can figure out how to help bridge their gaps. So long as the teacher can maintain an atmosphere of normalcy the students may never realize they are working so far below grade level, which is important for keeping a positive attitude.
It's a good idea to have some grid paper handy to help them keep their division neat which is the cause of some common errors like inserting zeros where they don't belong or putting the numbers in the wrong decimal place. This is something one of my elementary teacher friends taught me. Another common error you can spot while circulating is students not noticing when they have a remainder that is larger than the divisor. The most common mistake, however, is doing the division backwards. This gives us the opportunity to teach students how to check for reasonableness and support fraction sense as well.
The Warm up presents two division problems for students to complete, 5/8 and 8/5. While students are working I circulate to help troubleshoot division problems. Several issues that crop up are not knowing how to mark the end of the whole number with a decimal point and continuing to divide. Others might make mistakes when they don't realize what it means when they get a remainder larger than the divisor and they keep dividing. These students I work with one on one or make sure one of their peers is working effectively with them. Some students make mistakes by inserting zeros into the solution rather than bringing them down. Still others are so sloppy in their work that they lose track of where the decimal point is. Teaching them to use grid paper can help students who are making these later two mistakes by helping them organize better and reminding them that if 8 goes into 5 zero times then a zero goes above the 5 in front of the decimal point and the 6 goes over the 50 behind the decimal point, etc.
When we go over these first two problems I model them on grid paper. In order to show the benefits of the grid paper I may display someone's division that is really messy to show the benefit. I never publically call out a particular student, they must volunteer otherwise I will not use anyone's work. If my students had not been trained for a long period of time on the benefits of learning from and sharing our mistakes I would never do this. I may ask someone privately to volunteer ahead of time while I am circulating earlier if I notice some really messy work, especially if the person was able to correct it by being neater. If I don't pick someone out earlier I may ask if anyone has an example that is truly messy so that the class can see the benefit of being neat. It also has to be a student with a lot of confidence and a good sense of humor about themselves. I keep a positive and grateful attitude while sharing it. "Angelina is so kind and brave to share her masterpiece of sloppiness, please be sure to show your appreciation of her excellent mess" and then I treat the paper as a priceless masterpiece.
The last question sets up the exploration by asking which of the division problems is equivalent to 5/8, which gets to students biggest difficulty with division.
The most common mistake is doing the division backwards and not noticing. Most of my students have such weak fraction and division sense that they don't know which answer is reasonable. When I ask them which answer makes sense for the division 5/8 most of them will choose 1.6 and their explanations usually involve trying to remember an algorithm. They talk about transformers, and cars, and "if I tilt my head this way....or is it that way", and "we always divide the bigger number by the smaller number". Unfortunately, when we try to teach students to simply remember things they can't.
It was only after I asked them draw a diagram of the fraction and then asked if it showed one whole that they figured out which answer was reasonable.
So, when my students ask me which number goes in the "house" I say I won't tell them, because they will just forget again. But I tell them I will give them a way to figure it out for themselves. I tell them to look at the fraction and decide if it has a value greater or less than one to predict the value of the division. At first they complain that they have to do the math twice, but I tell them they should be able to tell right away if the division is turning out the way they expect.
Students work individually on white boards, but have access to their math family groups. Students are encouraged to check with each other before they raise their boards. They all raise their boards at the same time so I can give corrective feedback when necessary and no one can opt out.
For these problems I tell them it is really important to take a moment before they start the division to decide what type of answer they expect, more or less than one and then to ask themselves as they do the division if the problem is working out the way they expect. This is how they can tell if they are doing the division right. As I circulate I ask if the group took the time to predict whether they should get an answer greater or less than one. If not I have them copy the fraction and decide. If they have trouble I suggest they draw a diagram of the fraction to help. I make sure to give examples that will turn out both ways: 1/8, 9/2, 1/3, 6/5.
I check in with students as they are doing the division and ask them if the division is turning out the way they expected or if they should try the division another way. I also ask what they expect and why. I am sure to ask these questions whether they are doing the division correctly or not. If I only asked them questions when they were doing something wrong they would pick up on this and never learn to explain their answers. I have noticed at the beginning of the year when I ask students questions about their work they will often just pick up their erasers without responding to my questions or they would apologize or act as though they just noticed their mistake (when there isn't a mistake to notice). They have learned that teachers only question mistakes. When this happens I will ask why they erased their work, or tell them they are forgiven and ask what they are apologizing for.
I ask them how they would prove to someone which answer 0.83... (with a repeat bar over 3) or 1.2 is a reasonable answer for 5/6 .
I am looking to see who is and is not able to draw a diagram or explain how they can tell 5/6 is less than one. Some students may still be focused on the algorithm and others may need more scaffolding with fractions. The students who talk more about a trick they learned (transformers & cars) or about how they know which number is supposed to go in the "house" are more likely to not check the reasonableness of their answers. While others may really need more scaffolding with fraction sense. These are the ones who try to draw a diagram but are unable.