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APPLYING LOGIC IN
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Math classes at Urban run from pre-algebra to pre-calculus, from computer programming to probability and statistics. The menu of science classes bears less conventional titles: Physics of Sports, Horticultural Research, Waves, Cycles, and Vibrations, and Microbiology I/II. The math and science proficiencies required for graduation focus on logic and applications. One of the tasks asked of students in math, for example, involves solving a logical puzzle with limited information. In science, students must design and conduct an experiment and then prepare an analytic lab report that they defend before a committee of science and math teachers.

Below we take a closer look at teaching and learning in two classes: trigonometry and a science course called “Chemical Puzzles.”


TRIGONOMETRY    See also
   Chemical Puzzles

He tied two students together with a string, and that was the ten-minute puzzle of the day, to think about with logic. All we had to do was reverse hands. But he wouldn’t let me get out, not even after ten minutes. I was so mad, but when I finally got it, I would have felt like such an idiot if I had let him tell me the answer. — Mika

When Urban Academy math teacher Terry Weber says in his course description that students will solve problems outside the classroom, he really means it. “In trig, they go outside eight times,” he says. “We ask them to figure out how far is it across the East River, and how tall the smokestacks are on the other side. Or they go to the Flatiron Building, which is a triangular solid, and estimate its volume. They figure out the distance between Manhattan and Staten Island, knowing the height of the Statue of Liberty—if they don’t know its height they have to figure it out.”

But using familiar reference points to gain practical skills isn’t all that Weber is after. “My philosophy is that people have to understand why they do stuff, not just how to do it,” he says.

To help them gain that understanding, Weber’s students fill out a “practical procedure” protocol (see below). It begins: “Anyone who reads your practical procedure should be able to figure out exactly what you did and why you did it.” Each page is graded, but the last section of page 2 is graded the most: “Does your answer make sense? Why or why not? Explain, using logic and examples of how you could estimate the answer.”

Weber wants his students to be able to estimate whether their answer is in the ballpark, within the realm of possibility. So, when they estimate the height of the street lamp using trigonometric functions, for example, if their estimate doesn’t make sense after they do a scale drawing, they have to figure out why.

Urban Academy Course Catalog
Spring 2002

Trigonometry (Terry W.)

This class will cover identities, proofs, triangles, equations, patterns, and graphing. You will plan solutions to practical problems and will use a hands-on approach to problems outside the classroom. You will work mainly in small groups and will have regular homework and periodic quizzes and tests.

Prerequisite: Teacher recommendation

In the same vein, Weber has been known to give his students problems that don’t work, without telling them in advance. “Maybe it’s a triangle where one of the sides is longer than the hypotenuse, so the answer they come up with is a negative square root,” he explains. “At that level of math you can’t do that! Whatever the problem, they always have to be asking why something isn’t possible.”

What Weber aims to show, he says, “is that in math there is room for students to put their own opinions into it.” For whatever problem he throws their way, he wants his students thinking, “Could I do this another way?”

CLASSROOM NOTES| 10.15.02

On an autumn morning, Lashawn and Andy are computing the width of the East River as it skirts mid-Manhattan, using the distance between a couple of garbage cans and a lamppost they can see on the other side. Creating a simple sketch to scale on graph paper, they explain their procedure in writing:

Lashawn measured 55 feet by using ruler walking from one garbage can to another. Andy measured 80 degrees from garbage can to top of lamppost across the river by using a protractor horizontally.

After working out the equations and re-checking their figures, Lashawn and Andy find that their solution for x, the distance across the river, equals 312 feet. “Uncommon answer,” they note skeptically and add:

    The answer seems a little off. Only 312 feet for the width of the river? I think the angle measurements are a bit off, maybe by 5 degrees at most. I did a check, and found answers that didn’t exactly match. They are not proportionate to each other (angle 80 to 10, and 55 to 312).

    The only difficulty we had was maintaining a precise measurement of the angles. In addition, the distance between the two garbage cans was small, compared to the large distance across the East River. This added to our inaccuracy.

“You need to tell us more about why you think the answer doesn’t make sense,” their teacher tells them as he reviews their written work. “And you also should critique the instruments used to measure the angle, because looking through a straw is not precise.”

By the end of class, Lashawn and Andy are onto an alternative way to solve the problem, using a tangent method.

PROCEDURES AND PROFICIENCIES

1.0 “Practical procedure”

Note: Anyone who reads your practical procedure should be able to understand exactly what you did and why you did it.

Page 1:
Explain in detail exactly what your procedure was:
Who did what? What tools did you use and how did you use them?

Page 2:
Show all of your formulas and calculations NEATLY.
Does your answer make sense? Why or why not? Explain using logic and examples of how could you estimate the answer.

Page 3: (Graph paper)
Sketch to scale, as best you can, the picture of the problem. Be sure to label everything and state the scale.

Page 4:
What difficulties did you have at any point in your procedure? How could you have avoided them?
What other way, could you have “done” this problem? Be specific in your explanation.

2.0 Sample pre-requisites and proficiencies

Students in Urban math classes routinely complete a step-by-step series of “pre-requisite” assignments and short proficiency tests that ask them to demonstrate and apply what they are learning. Below are two examples. Click here for sample student answers to these examples.

    Logic puzzle

    1) 8 students live in the houses A B C D E F G H. Some are connected by phone lines (direct lines between the oval houses).
    2) The 4-letter name people are not connected to each other.
    3) Vera is not connected to Gabriel.
    4) The 7-letter name people are not connected to each other.
    5) Venus has EXACTLY two connections.
    6) Paul is not connected to Tanisha.
    7) The 5-letter name people are not connected to each other.
    8) Nija is not in F.
    9) The V-names are not connected.
    10) Sean is not connected to Maria.

Part 1: Solve the puzzle. (Place the people in the correct houses)
Part 2: Prove that your solution is the only solution—that no other solution works. Use the reference sentences 1-10 in the body of your proof or as footnotes. The proof steps should be in good order.

    Proficiency problem
You wish to find the volume of a building. The base is a parallelogram and the sides are vertical. The roof is also a parallelogram parallel to the base.

You may not climb the building nor throw anything on it or at it.
You have with you:
a yardstick
tent pegs
tuna sandwiches
a scientific calculator
a backpack
a roll of electric tape
scissors
a sextant
a machete
a level
graph paper and colored pencils
a camera (Polaroid) and film
trig tables
an axe
a thermos of tea
a hammer
a box of straws
a flashlight with batteries
2 compasses (1 of each kind)
a ball of twine
Find the volume efficiently (minimize time and energy).

  • Draw appropriate pictures and label everything.
  • State the formulas to be used and show how to solve them.
  • Show what you will measure and how.

    CHEMICAL PUZZLES

    It’s good when a teacher allows you to fail and accepts something even though it’s not right, but you’re on the road to something right. There’s a lot of pressure on high schoolers to get it right or else not hand it in, because you think it’s stupid. Lots of teachers will give you every step leading up to the answer. But you’re not really learning anything, you’re just reciting it.—Vance

    Terri Gross’s enthusiasm for discovery and that of her students find common ground in her semester-long chemistry course. “Using puzzles works because they really want to know the answers,” explains Gross. “It’s compelling, so by the end of the semester students get in the habit of asking questions. You can put something down in front of them—a liquid solution, for example—and you don’t have to say anything. They automatically start asking questions.”

    She starts the semester with “Mystery Powders” (which range from baking soda to Plaster of Paris) and three liquids (water, dilute vinegar, and dilute iodine) which students group, test, heat as they work to identify them. Answers in hand, they then create a logic puzzle.

    Next Gross presents students with eight clear, colorless liquid solutions, along with a list of names and chemical formulas. “First they group the solutions as to how they react—do they change colors? Turn cloudy?” she says. “Then they do Internet research about the chemicals, learning the chemical formulae, elements, atomic structure.”

    Urban Academy Course Catalog
    Spring 2002

    Chemical Puzzles (Terri G.)

    Labels have fallen off several bottles each containing different solutions of chemicals. How do you find out which solution is in which bottle?

    You’re given a test tube containing a solution. How do you find out what it is? How do you find out if it’s a solution of a single chemical or a mixture of several chemicals?

    These are some of the puzzles students will try to solve by designing and carrying out appropriate experiments. Each student will be required to record observations and data collected in the laboratory notebooks. Class time will also be spent on analyzing experimental results. Students will be expected to become familiar with the chemists’ shorthand of writing formulas and reactions.

    Students will be evaluated according to the following criteria:
    Ability to work independently or in small groups
    Quality of laboratory notebooks
    Quality of written assignments

    One major project will be to identify a series of “unknowns.”

    Her second puzzle involves only four clear, colorless liquid solutions. “Which is which?” Terri asks. The third puzzle asks which of three liquid solutions is the strongest and introduces the topics of pH and neutralization. By the fourth puzzle, students try to resolve why a penny is floating in a beaker of clear hydrochloric acid. “By then they have all the tools to answer it,” Terri says.

    In this unit’s final puzzle, students make up their own experimental question about a rusting nail in an agar solution with potassium ferrous cyanide. “A good question has to be something they can test within the constraints of the classroom,” Terri explains. Students have learned from the succession of puzzles that it could be quantifiable (like the third puzzle) or qualitative (like the first and the fourth). “It has to have a narrow focus, but be broad enough to be interesting, and I prefer it to have some importance,” she says.

    “Sometimes you can’t tell until the very end if it was a good question,” Gross adds. One foolproof indicator: “If it raises a lot of other questions, it was a good question.”

    HOMEWORK AND FINAL EXAM

    Students in “Chemical Puzzles” wrestle with weekly assignments and experiments that build on each other and that ask students to make new connections in what they’ve been learning and to think like a scientist. The final exam has two parts: a written section and a practical application. Gross gives students, in advance of the exam, an outline of what to expect. Below is a sample homework assignment and final exam explanation.

    Homework Assignment #12
    Assigned: Thursday, May 30
    Due: Monday, June 3

    PART ONE: Solving a “chemical” puzzle
    In class today, you watched the movie, The Andromeda Strain. In it, the scientists were trying to solve a very important puzzle. Throughout this semester, you have been using different kinds of evidence to solve puzzles that you’ve been given. What do you think of the procedure that the scientists used to solve their puzzle? In a short essay (approximately 2 written pages), please respond to the following:

  • Define the questions that the researchers had to answer in order to solve the puzzle.
  • Explain what clues the researchers had and what these clues told them about the puzzle.
  • In logical order (think of a logic puzzle), explain how the scientists solved the puzzle.

    PART TWO: Prussian (or Turnbull’s) Blue
    Beginning next week, pairs of students will be working on an experiment. The experiment must be based on the chemical Prussian Blue and the blueprinting process. This experiment may be focused around the generation of the pigment (i.e. how can we make more of this?) or the use of the pigment (i.e. how can we use this best?). Several experimental questions have been proposed. You may choose one of the following, or generate one on your own:

  • How does _________________affect the amount of Prussian Blue that we get from a chemical reaction? (Some ideas might be concentration, light, pH, ratios of potassium ferricyanide to ferric ammonium citrate, and others)
  • How do different materials react to blueprinting?
  • How can the color from blueprinting be removed?
  • How can the color of Prussian Blue be changed?

    We will be generally using the process described in the attached reading.

    By Monday, you and your partner need to have chosen a question, developed an experimental design and written that design out carefully. One design per group is fine.

    Chemical Puzzles:
    Explanation of Final Exam

    This is the explanation of your final examination — Identification of an Unknown.

    You will be given two unknown solutions. You will get 30 ml of each and no more! If you spill it or use it all up, you will not get any more! Each will be a compound from this list:

      BaCl2
      Pb(NO3)2
      (NH4)2CO3
      Na2HPO4
      Na2SO4
      AgNO3
      NaI
      CaCl2
      (Barium Chloride)
      (Lead Nitrate)
      (Ammonium Carbonate)
      (Sodium Phosphate)
      (Sodium Sulfate)
      (Silver Nitrate)
      (Sodium Iodide)
      (Calcium Chloride)
    You will be working in pairs. Each member of the pair needs to have his or her own scheme of analysis. Any student who does not have a complete scheme of analysis will be penalized. These schemes should be different. You need to follow both schemes to find out what the unknowns are. In other words, for each unknown, you need to use two different methods to find out what it is. You will have 0.1 M solutions of these chemicals available to you for testing:

    BaCl2
    Pb(NO3)2
    (NH4)2CO3
    Na2HPO4
    Na2SO4
    AgNO3
    NaI
    CaCl2
    (Barium Chloride)
    (Lead Nitrate)
    (Ammonium Carbonate)
    (Sodium Phosphate)
    (Sodium Sulfate)
    (Silver Nitrate)
    (Sodium Iodide)
    (Calcium Chloride)
    NaOH
    ZnSO4
    Ch3COOH
    CuCl2
    K2CrO4
    NaCl
    Ba(NO3)2
    MgSO4
    (Sodium Hydroxide)
    (Zinc Sulfate)
    (Acetic Acid)
    (Copper Chloride)
    (Potassium Chromate)
    (Sodium Chloride)
    (Barium Nitrate)
    (Magnesium Sulfate)

    (NOTE: If your scheme requires a chemical that is not on this list, you must adjust your scheme. We will not have access to any other chemicals during the class period! I suggest that you adjust the scheme before class!)

    You will then choose one of the unknowns and determine the molarity (moles per liter) of the solution that you have been given.

    You will need to work efficiently in order to complete this assignment on time. There will be no extensions. The write-up will be due on the last day of class. At that time, you will take the written part of the final examination.

    The Write-Up
    Please follow this outline to write up your results. You can simply number each section.

    1. Title: Identification of an unknown chemical.
    2. In a very neat, orderly manner, include both complete schemes of analysis.
    3. For each unknown, show the results of the tests and your identification of the compound.
    4. If you got different results with the two different analyses, explain what may have happened to give you those results. Based on your schemes and the actual procedure that you followed, state which of your results you feel is more accurate. Of course, explain your choice.
    5. Describe your procedure for finding the concentration of the one unknown. This should be detailed enough for someone else to follow it.
    6. Clearly show the results that got after following the procedure. Show your calculations and your determination of the molarity of the compound.
    7. List the sources of error that may have affected your calculations. Based on your errors, do you think that your results are greater or less than the actual number? Explain your reasons.

    The Written Exam
    Your final examination is open book. That means that you can use any notes or handouts that you have in your notebook. I will not supply Periodic Tables of Elements or solubility charts. You will not be able to use textbooks or look at anyone else’s notebooks. These are the topics that will be covered by the written part of the final examination:

    1. The structure of the atom. Given an element symbol, you should be able to tell me how many protons, neutrons and electrons are in a single atom.
    2. Given the chemical formula of a compound, you should be able to determine the charge on each of the atoms in the compound.
    3. Given information about what a compound is soluble and insoluble with, you should be able to determine what the compound is.
    4. Given the mass of a compound that is added to a volume of water, you should be able to determine the molarity of the resulting solution.
    5. Given information about the volume and molarity of a base that is used to neutralize an acid, you should be able to find out the molarity of the acid. You should be able to do the same given information about an acid used to neutralize a base.
    6. Given several compounds to choose from, you should be able to develop a scheme of analysis to differentiate between them.


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