Peach Rationing

When looking for problems for my students I came across this problem:

Whenever the monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. I wonder how long he could make his peaches last for? Here are his rules: 
 •Each fraction must be in its simplest form and must be less than 1 . 
•The denominator is never the same as the number of peaches left (for example, if there were 45 peaches left, he would not be allowed to keep 44 45 of them). 
Can you start with fewer than 100 peaches and choose fractions so that there is at least one peach left after a week? What is the longest that you can make them last, starting with fewer than 100 peaches? Send us your solutions showing how many peaches you started with and the fractions you used each day.

Link to the problem here: http://nrich.maths.org/2312

So, I decide I would try this problem on my own. So first thing I picked the largest number that was less than 100, but wasn't prime. I determine that prime numbers would not work based on the rules given. If you tried 11, the only denominator you could use is 11 and that is not possible. So I took 99 and kept 32/33 of the peaches. I picked 33 for the denominator because it is the largest factor of 99. To find the numerator, I took 33 and subtracted 1 from it. This gave me 96 and I subtracted 1 from it to get 95. So I did the same thing using 18/19. This result was 90 and after subtracting 1, I got 89. 89 is a prime number and I had to stop here. So I got stuck here.

Instead of trying to start over, I went back to 95 and tried 17/19(one number less on the numerator than 18 I used previously). This gave me 85 minus which is 84 and did not give me a prime number. So from this point on I tried used the greatest factor of the number given and double checked each one before going through with the multiplication. So here were the days I got:
Day 1 99*32/33->96-1->95
Day 2 95*17/19->85-1->84
Day 3 84*41/42->82-1->81
Day 4 81*26/27->78-1->77
Day 5 77*10/11->70-1->69
Day 6 69*22/23->66-1->65
Day 7 65*11/13->55-1->54 (12/13 gave a final answer of 59 which is prime)
Day 8 54*26/27->52-1->51
Day 9 51*15/17->45-1->44 (16/17 gave a final answer of 47 which is prime)
Day 10 44*10/11->40-1->39 (21/22 gave a final answer of 41 which is prime)
Day 11 39*12/13->36-1->35
Day 12 35*4/5->28-1->27 (6/7 gave a final answer of 29 which is prime)
Day 13 27*7/9->21-1->19 (8/9 gave a final answer of 23 which is prime)
Day 14 20*4/5->16-1->15 (9/10 gave a final answer of 17 which is prime)

I continued this pattern at this for Day 15 and 16, but I ran into a problem.

Day 15 15*2/3->10-1->9
Day 16 9*2/3->6-1->5

I wanted to get my final answer down to one. I could not do this with 5 being prime. I knew that I could get my number of peaches down to 4, then I could get a 1 for my final answer. Since I could not do anything with 9, I went back a day to Day 15. I used the fraction 1/3.

Therefore,
Day 15 15*1/3->5-1->4
Day 16 4*1/2->2-1->1

So I was able to get sixteen days with the peaches. This did leave me a couple of questions at the end. The first question was there other numbers that would work? A simple answer would be yes. With being able to pick the fraction, you could come up with many numbers that would still give up more than one peach after a week. The second question I had was could you find a number that lasted more than 16 days. On the website there are no answers, but I think there might be a number that lasts longer. It would take many calculations to do this though.

The struggle in mathematics of women

At the college level, a study was done to find out what percent of students would have to take a developmental mathematics class. Nearly 55% of all students who attend a community college will have to enroll in a developmental mathematics course. 58% of women are will have to take one of this courses while only 50% of men will enroll in a developmental mathematics course.

This made me think about my students. In the two quarters that I have taught in this academic year I have had 145 students between 6 classes. Of 145 students, 112 of those students were female. When I started thinking about it, I remember most of my students were female. So I decide to go back as far as I could with the records I had. Since the fall quarter of 2012, I have taught 18 classes. In those 18 classes, I taught 423 students with 324 of them being female (77% of my students). All, but one of my classes had more than 60% of students who were female. The one class that didn't was at 59% (17 out of 29).

The first thing I asked myself after seeing those numbers are why are my percents much higher than 58%. The main reason is the gender difference. Women are more likely to ask for help if they need it. When students are registering at Baker, students find out that they will need to their mathematics courses. Most of the people who deal with registration will recommend students to take my mathematics classes if they ask or mention that they struggle with mathematics. Since females are more likely to point this out or ask who should they take, I am more likely to have a higher percentage of females in my classes.

Now that I know that I have higher percentages than normal. I need to make sure that I gear my teaching to provide opportunities for my students. First, I can show students that you can still enhance your knowledge and improve their capabilities. Students are not born into they capabilities and cannot expanded beyond them. They need the encouragement to know they can learn how to do mathematics. Second, I should give students feedback that focuses on strategies, effort and the process of learning. I need to identify where the student has gained improvement in their use of strategies or errors in their problem solving. This will enhance their own beliefs about their abilities. This leads to improvements in persistence and performance on tasks given. Finally, I need to create interest in math by using activities that connect math with careers that do not reinforce existing stereotypes of women. If you reinforce the stereotype, women will have less interest in mathematics. You need get students interested in mathematics. Using stereotypes, will cause female students to lose interest.

Critical thinking with my students

I came across this video ( http://www.youtube.com/watch?v=vKA4w2O61Xo#t=237 ). After viewing this, I made the connections between what I saw and what had been discussed in class. So I decide to try this with my classes that I teach. I gave the students the same introduction as they did in the video using the same rule he gave to the people. So I told I have a rule and the numbers 2,4,8 followed that rule.

I also tried this with my sister who is also an English teacher to get an idea how this would work with my students. My sister was able to do figure it out in three tries. She started tried 3,6,12 and then 1,2,3. After she found that those worked, she tried 3,2,1 and she concluded the answer from there.

Then I told students to give me three numbers and I would tell them if they followed the rule or not. To help students to see this, I also wrote the numbers on the board so they could refer back to them. When working with my students, there were several things that I noticed with the students.

The first thing I noticed is they gave more right answers than wrong. When we discussed this after the activity, they wanted their sequence to be correct and not wrong. Students want the right answers, but as mathematics, the null hypothesis is generally more important to us than the hypothesis. One example we worry about is dividing by zero. We stress to students that we can not divide by zero so we look for instances where we will end up dividing by zero.

The second thing I noticed was students mimicked what answers the other students gave as answers. Once they saw that 11,12,13 worked, they would use three consecutive numbers. Now, they were not sure why they worked, but they didn't want to get the answer wrong. They also could not get the idea out of their minds that of the order. Most of the answers at the beginning were just ascending order. The students could not see that they could change the order of the numbers.

The biggest discussion came with one of the classes. Earlier in the discussion, 3,6,9 came up a correct solution. Now, as the activity, went on a couple of students had figured out the solution, but I told them not say anything to the class. They gave 9,6,3 as a solution and this caused a big stir in the class. Many students wanted to say that these were the same sequence. They focused on the numbers and not the order. As the students started talking back and forth, they realized what the rule was.

Based the results of this activity and the lack of reasoning in these sequences, I decided that I would do this for the rest of the quarter. I would spend time with every class doing an activity like this at the beginning of class. So the next class, I used the rule of even odd even and gave the students the sequence of 4,5,6. This time around the students do much better thinking abut the types of sequences they used. In fact, one of the students started with 6,5,4 just to see if ascending had anything to do with it. I'm looking forward to doing this with my students for the rest of the quarter.

UPDATE:
So I was able to continue doing this with my class three more times (only 3 weeks left in the quarter).  I tried a few more rules: even, odd, even; one number is prime; the sum of the numbers is even.  With each class, students were able to come up with the rules faster and faster.  When a student mentioned a rule they thought worked, I would ask that student to share with the class their thought process and what made them believe that this was their rule.  Some students still did not understand why they had to do this.  They just wanted to give me a sequence that worked.  Overall, I could see much that students were thinking through the process with their sequences they proposed. 

Proportional thinking in my classroom

With my students, they are introduced to ratios, rates, and proportions. My favorite problem that I pose to them is about a car I bought.

Here is the prompt they would get for this problem. I was in the market to get a new car. After test driving several different cars, I decided on getting 2001 Buick Century. While driving around after I bought it, I noticed that the gas gauge stays at full until I have about a gallon of gas left and then it goes to empty. So, I went to my mechanic and asked him about it. He told me it would be around $500 to fix it. I did not want to spend the money to fix it so I came up with a solution. What did I do so I would not have to spend the money to fix it?

At this point, students are asked to brainstorm ideas on what to do. Some students say that they would just fix it, but I remind them that I don't want to spend the money. We talk about what information do you know and what type of information could you find out. We eventually get to the gas tank itself and we could find out the mileage that the car gets. So, I give them that I went 300 miles on 15 gallons of gas. I pose this question, how does this help us?

Students point out that we now know how many miles per gallon get. The students will point out that we can find how many gallons the tank is from the owner's manual which is 17 gallons. Then the students set up a proportion to find 340 miles. At this point, I ask the students what does that 340 miles mean. Some students see that this is how far you can go on a tank of gas, but many students still do not see what this information means.

After discussing this problem, we look at how the unit rate could change. I tell the students that I check it once a month and ask why? After discussing that it could change, we talk about what types of things could change the rate.

Many of these students really enjoy this problem. They like that you can come up with a solution to this without spending the money. Some of the students actually have come across this problem with a car they have already encountered. I like this problem for my students because they get to look at different concepts like unit rates and proportions. It's also an excellent real-life example that students may encounter or already have encountered.

Gradual Release

When we discussed Gradual Release in class on Wednesday, I had really only heard of it. It was one of those buzz words in education that I had heard and knew what it generally meant. Though I knew that it was about have students get more control of learning in the classroom, I did not know how the actually process worked. When looking at the handout of Gradual Release, I was actually quite suprised. It was basically the same process that I use in my classroom. When talking about most concepts in the classroom, I use a three step approach. Most of the concepts I cover in class are either concepts my students forgot or really did not learn or retention from school. Since they struggle with the concepts, I start talking about the concept either making connections or possibly looking at a problem in general. I will go though the concepts possibly with a sample problem or two. Then I do blur the second part between sharing and guiding the students. Depending on the concepts, I may model the concept. I may guide them through the problem using leading questions. If it is an important concept or a hard concept, I will using modeling first, then guiding them through the problem. Once I have done this, I give the students one or two problems or even an activity for students to do on their own or in groups. I will then walk around and see how students were doing and check their work. By walking around I get to check to see if there is any information that I forgot to explain or general confusion. It also allows me to work with individual students if they are struggling. I can teach them other methods that may work best for them. Overall, my students like this process because they get to see the concepts on the board, but get to try it on their own to see if they understand the problems. I like it because I get to know the students and where the strengths and weaknesses are in their learning.

My classroom situation

For the most part in my classes, I have been given plenty of resources to be successful. For my classes, the students have numerous resources that are available to them. The students have access to an online textbook, online lecture videos, study plans which tracks each individual student's progress, and different ways to get help on problems from their homework. For individual problems, the students have examples, a help me solve this feature, short videos, access to the section of the textbook, and direct e-mail with the instructor for help. As for instructor resources, there are not as many as for the students. All instructors are given access to a faculty guide. The faculty guide gives the instructors an idea of how they should structure the class. For some topics, instructors have submitted ideas or activities that can be used for the class. Mostly, the instructors are left to find things that work best for them. It gives you the freedom to teach things in ways that you want, but you have little ideas to be given. Also the instructors are not given much guidance. As a new instructor, you are not given a mentor instructor. You may be given a name of instructor who can help you, but it is left up to you to determine most things on your own. Most instructors do not get to interact with other instructors on a day to day basis. I occasionally see other instructors throughout the day, but do not get to have much interaction. We generally get to meet twice a quarter, once at the beginning of the quarter and once in the middle of the quarter. These meetings are recommended, but not required therefore not all of the faculty show up to these meetings. The biggest challenge I have in my classroom is with the students. Most of my students are motivated which is a big difference from when I worked with secondary students. My biggest problem is the level of my students. There is a big difference in the levels of my students. Some of my students need just a refresher in mathematics since they have not taken it in years, but some of my students struggle with the basic concepts of adding and subtracting whole numbers. With having these great differences, sometimes it is hard to keep the higher level students engaged while not losing the lower level students to confusion.