Subject: Capacity 3

Posted by Rob Quaden on 12/4/2010
In Reply To:Capacity 3 Posted by Tim Joy on 12/4/2010

Message:

Hi,

Great list George... I would like to add the following process-oriented aspects:
- Deductive reasoning and proofs: defending statements based on logic, rules, and internal consistency, rather than by opinion, assertion or popularity.
- Looking for patterns, inductive reasoning, connections that are not expected, wonder.
- Communication: clarity of thinking that can be followed by others and shown to be incorrect without personal attack. Economy of thought,elegance.


To move from theory to practice:
For the past 5 years I have been using the materials from Phillips Exeter Academy in my 8th grade math classes. The material is available for free to anyone and in my opinion provide a quantum leap in the direction of student centered learning and maybe a model for spreading system dynamics.

The process is as follows:
1. Students try the problems on their own without teacher explanations.
2. Students come to class and work in small teams, explain their thinking and try to come to a consensus.
3. Students present to the class and discuss ways of solving the problem.
4. The teacher works on the structure of the class, facilitates, encourages students to present, asks questions, and does some 'mini' lessons. The teacher rarely lectures. The teacher does not provide answers to the problems, but observes in order to direct the conversation. (Note: while the teacher is not doing many of the traditional tasks, this is hard work, but also incredibly rewarding.)


Below is the first page of problems that my 8th grade students tackle.


Enjoy,

Rob


Mathematics 1 1.

1. Light travels at 186282 miles per second, and the Sun is about 93 million miles from the Earth. How many minutes does it take light to reach the Earth from the Sun?
2. How long would it take you to count nonstop to one billion, if you counted by ones? First, write a guess into your notebook. Now try to solve the problem. One approach is to actually do it and have someone time you. There is another way to approach the problem, however. What do you need to know? What assumptions are you making?
3. It takes 1.25 seconds for light to travel from the Moon to the Earth. How many miles away is the Moon?
4. Many major-league baseball pitchers can throw the ball at 90 miles per hour. At that speed, how long does it take a pitch to travel from the pitcher’s mound to home plate, a distance of 60 feet 6 inches? Give your answer to the nearest hundredth of a second. There are 5280 feet in a mile.
5. You have perhaps heard the saying, “A journey of 1000 miles begins with a single step.” How many steps would you take to finish a journey of 1000 miles? What information do you need in order to answer this question? Find a reasonable answer. What would your answer be if the journey were 1000 kilometers?
6. In an offshore pipeline, a cylindrical mechanism called a “pig” is run through the pipes periodically to clean them. These pigs travel at 2 feet per second. What is this speed, expressed in miles per hour?
7. Your class sponsors a benefit concert and prices the tickets at $8 each. Dale sells 12 tickets, Andy 16, Morgan 17, and Pat 13. Compute the total revenue brought in by these four persons. Notice that there are two ways to do the calculation.
8. Kelly telephoned Brook about a homework problem. Kelly said, “Four plus three times two is 14, isn’t it?” Brook replied, “No, it’s 10.” Did someone make a mistake? Can you explain where these two answers came from?
9. It is customary in algebra to omit multiplication symbols whenever possible. For example, 11x means the same thing as 11 • x. Which of the following can be condensed by leaving out a multiplication symbol?
(a) 4• 1 (b) 1.08•p (c) 24•52 (d) 5•(2+x) 3
10. Wes bought some school supplies at an outlet store in Maine, a state that has a 6.5% sales tax. Including the sales tax, how much did Wes pay for two blazers priced at $49.95 each and 3 pairs of pants priced at $17.50 each?
11. (Continuation) A familiar feature of arithmetic is that multiplication distributes over addition. Written in algebraic code, this property looks like a(b+c) = ab+ac. Because of this property, there are two equivalent methods that can be used to compute the answer in the previous problem. Explain, using words and complete sentences.

July 2010 1 Phillips Exeter Academy

Follow Ups:

Capacity 3 - Pedro D. Almaguer Prado 12/4/2010