Habits of Mind Problem #8

Calculators and Classrooms

Due October 27, 2011

A Math teacher has two classes. One has 16 students, the other class has 11 students in it. Both groups have their Math classes in the same class room. The school bought 20 calculators for the two groups, so that they all could use them. The teacher gave some of the calculators to individual students, who can take them home, but have to bring them to every Math class. The rest of the calculators stay in the class room, and those who did not receive their own calculators will get one for each Math class. (Only one student may use the same calculator in class.)

  1. How many students may get a calculator to take home in each class if their teacher wants to give a calculator to the same number of students in each class, so that there would be enough left over in the class room so that everybody could use one in class?
  2. How many students can get a calculator in each class if the teacher does not mind that different number of students got one in each class?
  3. Is there more than one solution to parts 1 and 2 above? What is the best solution to each of the problems 1 and 2 above? Identify why it is the best.
  4. Identify the variables that used in this problem, along with the set of values that the variables may take.
  5. Identify expressions, formulas, or equations that you used in solving this problem.
  6. How would you make this problem more general, and how would the answers to parts 4 and 5 make the solution in the general case easier?
  1. Let C be the number of calculators distributed to each class. Then 2C is an expression for the total number of calculators distributed, and 20 – 2C is the number of calculators left in the classroom. The large class will have (20-2C)+C = 20 - Ccalculators available to use, and this must exceed the number of students in the class room, so 20 – C >= 16, or C <= 4. Likewise, we require for the small class that (20-2C)+ C >= 11, or 20-C >= 11, or C <= 9. The distribution that satisfies both requirements is C <= 4.
  2. Let C be the number of calculators distributed to the large class of 16. Let D be the number of calculators distributed to the small class of 11. Then C+D is an expression for the total number of calculators distributed, and 20 – C - D is the number of calculators left in the classroom. The large class will have (20-C-D)+C = 20 - Dcalculators available to use, and this must exceed the number of students in the class room, so 20 – D >= 16, or D <= 4. Likewise, we require for the small class that (20-C-D)+ D >= 11, or 20-C >= 11, or C <= 9. The distribution that satisfies both requirements is C <= 9 and D <= 4.
  3. Yes, additional solutions are anything from C = 0 (no calculators to take home) to C = 1,2,3,4. For the part 2, there are 40 solutions, any paired combination of D = 0,1,2,3,4 and C = 0,1,2,3,4,5,6,7,8,9. We might declare that a solution that lalows the maximum number of students to take home a calculator is the best, so C = 4 for the first and C = 9, D = 4 for the second.
  4. C, D with 0 <= C <= 20, 0<=D<=20.
  5. Expressions 20-2C, 20-C, 20-C-D. This problem used inequalities not equations, and inequalities are a more general relation among expressions than equations.
  6. Let X be the size of the first classs, let Y be the size of the second class. Let the number of calculators be Z. Even more general, allow for 3 or more classes, each of unspecified size.

Statement of Authorship

My (our) signature(s) below indicates that:

1) I (we) did not use any resources such as the web or books other than our textbook.

2) I (we) did not get any help from any individual other than my peers in the class and my instructor.

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3) Below, I (we) have identified any other classmates with whom we discussed this problem whether they gave us help or we gave them help.

______

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Individual Responsibility When Group Work is Permitted

Opportunities for group work include both situations when several people sign their name to the same assignment and when one individual receives help from a classmate even when they do not submit assignments together. Group work is permitted for Habits of Mind problems, projects, group presentations, etc. This is done in the belief that, done right, it supports learning, minimizes stress when a mistake is made, and results in a culture of achievement. Many of us recognize that it is often easier to work on something that is challenging if we get to share the challenge (and the success) with peers. At the same time, the learning process can be undermined when one person tells another the solution to a problem (instead of offering a helpful suggestion) or when someone signs their name to an assignment when they did not contribute their fair share of effort.

When students work together on a HoM problem, each member of the group should sign their name to this cover page. Your signature certifies that

i) you have participated to a fair degree in the effort to solve the problem, and thus you are entitled to your share of the credit for the project;

ii) you understand your group’s solution and will present it if requested; and

iii) everyone you permit to sign the report has also earned the right to sign the solution.