The Problem
“Cats and Canaries” is a problem almost similar to Farmer Montague’s goats and chickens with one major difference (see attached handout). The similarities are clear: 4 footed animals and 2 footed animals. And in this case, students have to find out the number of cats Ms. Lang has in her 25 animals. The crucial difference in this problem is that it has to be done in a group. And this makes it more challenging.
This problem should respond to at least 6 of the Common Core Mathematical Practices:
- Make sense of problems and persevere in solving them. Group work should offer support, a place to ask if this makes sense and opportunity to bounce ideas off.
- Reason abstractly and quantitatively. Using different solution methods students will need to come up with a coherent representation of the problem and be able to explain it to group mates and the class as a whole. This problem also is a beginning bridge to abstract reasoning.
- Construct viable arguments and critique the reasoning of others. This practice will be facilitated through group work since they will be forced to continually check in with mates to clarify clues and point out wrong turns.
- Attend to precision. Students will have to communicate clearly with each member of their group and prepare to present to whole class. They will have to label clearly and check their calculations.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
When I was working through this problem I soon realized what a good idea it was to make a table. Suddenly all the data I needed was laid out in an organized format. Noticing the way the numbers changed up and down the sides helped me to quickly fill in my table without actually calculating feet and paws.
Note: Since I did not have enough class time to develop and implement this problem with students, it will be used next semester, probably within first 4 weeks. I will then add student work and reflection on to what I am submitting here.
Students will work together in groups of 3-6. They will draw their work on large sheets of paper and present to whole class. Each group’s work will be taped to the board for display. After each presentation students will be urged to ask a question of the presenters. And after all have presented, students will be asked which method they liked, which seemed to work best and explain. The problem explained: The 1 page handout is set up as six cards to be cut apart. Each group gets the same set of six cards and they are instructed to distribute one or two cards to each member depending on the number of students in the group. Each card ends with the question, “How many cats does she have?” In addition, each card contains one essential piece of information needed for the group to solve the problem. Group members are told not to show their cards. However, they can and should read them aloud to other group members on demand. The rules are distributed with the worksheet and read aloud as a class before beginning. They are repeated below.
How to start
- In a group of 3-6, pass out the clue cards. If there are extras then it is okay for some people to have two cards.
- Read the information aloud to your group mates as many times as you wish. But you cannot show your card to anyone else.
- Work together in your group to figure out the problem. Use any method that you think will help, such as drawing a diagram, making guesses, keeping a list or whatever.
- When you think you have solved it, check to see that it fits all of the clues.
- Compare your methods with other groups. Do you have a favorite?
Meeting the requirements of the assignment: “Cats and Canaries” seems to meet the criteria of the assignment. Students cannot proceed directly to a solution without some struggle and problem solving. Moreover, the value of this problem is enhanced because students are actually forced to work together, checking in with each other to ask, “Please read your clue card again.” They will have to think mathematically using basic math skills of addition, subtraction, multiplication and division as well as draw on prior experience of math work using tables and organizing their data. Some students may solve the problem algebraically.
Planning
To prepare to teach the “Cats and Canaries” problem, I see four steps that I need to complete. First, I should solve the problem myself on my own, taking notes at each twist and turn of my journey. Second, after arriving at what I hope is a solution, I should look back at the places where I felt stuck, stymied or got off on the wrong track. Third, after discovering these places, I should anticipate that the student learners will find this also an obstacle and prepare some questions that will get them back on track without doing the work for them, and still reinforcing the belief that they can do this. Fourth, during the next semester I will teach this problem and complete this assignment.
My Work
To start, I had the disadvantage of not having a group to help me sort it out and focus on the criteria which I feel would have helped me stay on track. On the positive side, I had all six cards in front of me and had no choice but to read all clues. Nevertheless, as you will see from my work which follows, I made several mistakes because I didn’t read carefully – a humbling experience. I began by reading, carefully I thought, the six clue cards. Page 1 of my work notes follows. Listed at the top of my work notes are what I initially felt to be the essentials: she has cats and canaries; 25 heads to pet means 25 animals (not sure why I saw this as important); and the question, “How many cats does she have?” Then a bit overwhelmed by all the criteria to meet, I focused on one which I wrote below a line I drew at the top: Total # of cat paws and the total # of canary feet are each a multiple of 5. Then I decided to use the guess and check process but quickly changed to a Table. Prior knowledge led me to the table. This is a point I am not sure my students will arrive at unless I help them come up with it. As you can see I set up a table, #of cats on 1 side and # of canaries on the other. I began with 0 cats which gives me 25 canaries, a not likely possibility since Ms. Lang appears certain she has some of both. Nevertheless, I began at what I considered the beginning, 0 and 25. Then I followed the table down to the point where I arrived at 25 cats and 0 canaries. Then working with the multiple of 5 I went back up and eliminated the top five lines and the bottom 5 lines thinking that these were not multiples of 5. Then at sea as to what else this table could reveal I went back to the clue cards and searched for one I found most manageable: the bottom left card, “The total of the number of cat paws and canary feet is divisible by 2, 4, 8, 10, 20, 40, and 80.” Yikes, I thought, I have total of 25 – where in the world do the 40 and 80 come from? Then at the bottom of the page you can see that I have written, “Oh wait! This is heads – it is the paws and feet! So I then realized that I had to add to my table on each side, #of cat paws and on the canary side, # of canary feet. So I started a new page and a new table. Page 2 of my work follows page 1. On page 2, I wrote the 2nd criteria that I was going to use -- “The total of the number of cat paws and canary feet is divisible by 2, 4, 8, 10, 20, 40, and 80.” Then I completed a second table, adding # of cat paws to the left side and # of canary feet to the right side. So, with 0 cats I had 0 cat paws and 25 canaries I had 50 canary feet. And so on down ending with 100 cat paws, 25 cats and 0 canaries and of course 0 canary feet. After completing my table I looked at my clues, saw that I needed a number that would be divisible by 80 at most and then zeroed in on 80 cat paws – a natural mistake I suppose. But I discovered it only when I added 80 and 20 together to get the total and realized that 100 is no longer evenly divisible by 80 (or 40). You will see the “Oh wait! comment right below where I added 80 and 20 to get 100, on the right side of the sheet. Ah, now what? Then I reasoned that I had to subtract 20 so that my total of paws and feet would be 80 so that led me to what I thought was my final answer, 60 cat paws and 20 canary feet. And yes, they meet another clue of 3 times as many cat paws as canary feet. It was only today, as I prepared to type this up that I realized the final answer to the question asked, “How many cats does she have” is 15.
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