The question above should be asked by math teachers after almost every problem. During my bell ringer today we had unexpected, crazy, rich discussion over two problems that I never thought of as rich in mathematical ideas. The problems were :
In a certain school, 45 percent of the students purchased a yearbook. If 540 students purchased yearbooks, how many students did not buy a yearbook?
If a machine produces 240 thingamabobs per hour, how many minutes are needed for the machine to produce 30 thingamabobs?
Solve these and see what you come up with. Keep track of what you do. In my classroom, I randomly choose students to come up and show their methods. Sometimes they are wrong and we use incorrect answers as an opportunity to show common mistakes and I do everything in my power to show that these mistakes are helping everyone in class grow their brain (Thanks youcubed.org!!!) In this first problem, many students were blurting 243. In this case students were multiplying 0.45 and 540 or switching up their proportional reasoning. To handle this I told them to slow down and read the question again. Does that answer make sense? Also, please stop blurting answers! You are stealing someones opportunity to think.
Problem 1 method 1 The first student came up and told me that you first take 540 students divided by 45 percent to find the number of students per percent. Then take that answer (12) times 55 percent for 660. I asked them how they came up with 55 percent and they were able to tell me that 100 – 45 = 55 which is the percent of students who did not buy the yearbook. Then I asked, “Who thought of this differently?” Surprisingly, many hands flew up.
Problem 1 Method 2 The next method was the straight up algebra. If x is the total number of students, then 0.45x = 540. Solve this and 540/.45 = 1200. Since there are 1200 students we know that 1200-540 = 660 student that did not buy the yearbook. Each time a new method popped up I would give it the name of the student who produced it.
Problem 1 Method 3 In the third method a student divided 45 and 540 by 9. This student then proceeded to tell me that because of this every 5 percent is 60 students, so 11 times 60 is 660. Here the 11 felt like witchcraft for some of the kids, so I had the student explain where the 11 came from and he was able to tell everyone that 55% has 11 groups of 5. This method is similar to method 1 and this student may just have wanted to come to the board, but that is ok. Kudos for thinking deeply.
Problem 1 Method 4 A student kept blurting out FISH. At this students previous school they had been taught that proportional reasoning was a method called fish. When you set up the proportion 45/540 = 55/x , where x is the total number of students. We then proceed to cross multiply and divide. When you trace out the pattern of this the Christian fish symbol is produced. .
In all cases I was able to show students the connections between their solutions and use the language of mathematics. In the word percent, per is divide and cent is 100. Using labels helps you keep everything organized. I had one student use 45/100 = x/540 to come up with 243. Again, when I added the labels for each numbers 45( purchasers)/100 (entire class) = x (entire class)/540 (purchasers) several students in the class had an aha moment.
Problem 2 method 1 By the time we made it to the second problem, I had students practically running up to the front of the class to show what they did. So I didn’t have to randomly pick them. The first student told me it takes 8 boxes of 30 thingamabobs to package 60 minutes worth of product, so 60/8 means that it takes 7.5 minutes to produce one box.
Problem 2 method 2 On the second method a student told us that if 240 thingamabobs are produced in 60 minutes, then 120 are produced in 30 minutes. Following this reasoning then 60 are produced in 15 minutes and 30 are produced 7.5 minutes.
Problem 2 method 3 It surprised me that the algorithm we teach is always the last method shown. My personal opinion is that when we follow these “efficient” algorithms it slows down thinking . In this method the student lets x = minutes for 30 thingamabobs. Then 240(thingamabobs)/60(minutes) = 30(thingamabobs)/x(minutes). Then students cross multiply and divide to get 7.5 minutes. 240x=1800 then 1800/240 = 7.5 I think “cross multiply and divide” is a “trick” that students don’t really understand. I try and always show that it is really just common denominators and the multiplication of the entire equation by that denominator in order to simplify the fraction.
All of this illustrates a point. If I had not asked the question, “Who thought of this differently?”, then none of this rich discussion would have followed. Maybe some of the students would have left the problems feeling dumb for not answering them correctly, or thinking that their method for the correct answer was not a “good” method. I try to also ask, “Who had an incorrect answer?” and “What did you do wrong?” If they don’t know then we work together in front of the class to figure it out or I have them explain their thinking and inevitably there are several others in the class following the same line of thinking. I had expected to spend about 10 minutes on these problems. We spent 20 minutes. When we were finished the kids seemed energized and excited. The students who presented had a feeling of ownership and pride in what they had accomplished. I don’t always produced class periods like this, but when I do… It feels magical.