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.
Saturday, March 29, 2014
Wednesday, March 19, 2014
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.
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.
Wednesday, March 5, 2014
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.
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.
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