Probability #08

(grades 6-10)
Soft-bound, 64 page book, 28 reproducible task cards, full teaching notes.

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Involve your students in the mathematics of chance and the science of statistical analysis. Flip coins, spin an alphabet wheel, build a pinball machine based on Pascal’s Triangle, count permutations, tally combinations, plot frequency distributions, calculate odds on a paper-clip spinner, toss a pi without creating a mess!

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Table of Contents for #08 Probability:

Preparation and Support

A TOPS Model for Effective Science Teaching • Getting Ready • Gathering Materials • Sequencing Task Cards • Long Range Objectives • Review / Test Questions

Activities and Lesson Notes

    CORE CURRICULUM
  1. 1. Heads Or Tails?
  2. 2. Even Odds
  3. 3. Uneven Odds
  4. 4. Permutations
  5. 5. Combinations
  6. 6. Bell-Shaped Curve
  7. 7. Pascal's Triangle
  8. 8. Penny Pinball
  9. 9. Biased or Fair?
  10. 10. Pinball Puzzles
  11. 11. More Pinball
  12. 12. Simple as ABC
  13. 13. Sample Space
  14. 14. Single, Double, Triple
  15. 15. Four Probabilities
  16. 16. Or / Not
  17. 17. And
  18. 18. The Lucky Dice
  19. 19. A Tacky Experiment
  20. 20. Spinner Speak

  21. ENRICHMENT CURRICULUM
  22. 21. Tooth Tape
  23. 22. Pattern Search
  24. 23. Lead Changes
  25. 24. Do Run Run
  26. 25. Do Do Run Run
  27. 26. Hit or Miss?
  28. 27. Random Toothpicks
  29. 28. Average Angles

Supplementary Pages

probability grid • permutation tree • bar graphs • Pascal's triangle • angle gauge • letter circle • toothpick protractor • outcome sheet • ABC circles • number circles • line grid

 

Complete Master List for #08 Probability:

Key: (1st/2nd/3rd) denote needed quantities: (1st) enough for 1 student doing all activities; (2nd) enough for 30 students working in 10 lab groups all self-paced; (3rd) enough for 30 students working in 10 lab groups, all doing the same lesson. Starred* items may be purchased below.

  1. 1/15/15: towels, sweaters or other fabric to muffle sound (optional)
  2. 7/105/150: pennies
  3. 1/15/15: hand calculators
  4. * 1/8/8: rolls masking tape
  5. * 1/15/15: bottle caps, unbent, all the same brand
  6. 1/15/15: empty cereal boxes, 32 ounce Grape-Nut boxes recommended
  7. 2/30/30: pieces corrugated cardboard, about 10x12 inches
  8. 1/4/8: heavy scissors (optional)
  9. 1/15/15: quarts package filler (paper "straw," styrofoam peanuts, or loosely wadded newspaper)
  10. * 1/4/4: rolls clear tape

  11. 1/15/15: index cards, 4x6 inch
  12. * 10/150/150: paper clips
  13. 1/10/15: size-D batteries, dead or alive
  14. * 1/10/15: thick rubber bands
  15. * 50/750/750: straight pins
  16. 1/5/15: metric rulers
  17. * 1/15/15: plastic drinking straws
  18. 1/15/15: scissors
  19. 1/4/4: wire cutters
  20. * 0.1/1/1: cups modeling clay

  21. 1/1/1: package pinto beans or equivalent
  22. 1/15/15: plastic sandwich bags or equivalent
  23. 4/60/60: jar lids or other shallow containers
  24. 2/10/30: dice
  25. 1/5/15: styrofoam cups
  26. 10/50/150: flat thumb tacks, all the same brand
  27. 1/1/5: recycled phone book with white pages
  28. * 1/1/1: roll black electrical tape
  29. 1/2/8: desk calculator with tape
  30. 10/150/150: flat wooden toothpicks
  31. 1/1/1: scientific calculator (trigonometry text with sine tables)

Convenient Shopping:

Bottle Caps

crimped-edge, new

Ideal for counting, uniform weights, and mini-containers to hold small portions of chemicals. These unused caps flare more widely than recycled caps. Used bottle caps, if unbent/undamaged, are an excellent free alternative.

Product #1159 not available.

Paper Clips

size #1, steel, box of 100

Paper clips have 1001 uses in TOPS experiments, and science in general. Feel free to use paper clips you already have, but be aware that different brands come in different sizes and weights. In experiments where uniformity is important, don't mix brands.

Stopper - rubber

black rubber, size #6, no hole

Used in #09 Floating and Sinking, #14 Kinetic Model, and #22 Machines.

Straight Pins

steel, one and 1/16 inch long

Used in many TOPS experiments. Sometimes required for their magnetic properties. Don't purchase aluminum straight pins by mistake.

Straws - straight

plastic, thin

Any length straw, between 0.20 and 0.25 inches in diameter is suitable. Grocery stores generally carry straws with flexible "elbows." You can use those if you cut off the bendable section before using.

Tape - clear

3/4 inch x 1000 inch roll

Your standard desk tape with matte write-on surface.

Product #1012 not available.

Tape - masking

3/4 inch x 55 yd roll

A handy science supply used in most TOPS modules.

Teaching Tips for #08 Probability:

We encourage improvisation - it's one of the main goals of our hands-on approach! You and your students might invent a simpler, sturdier or more accurate system; might ask a better question; might design a better extension. Hooray for ingenuity! When this occurs, we'd love to hear about it and share it with other educators. Please send ideas and photos to tops@canby.com.

Lesson by Lesson Objectives for #08 Probability:

  1. Lesson 1: To toss a coin and record the outcomes as an up-and-down graph line. To appreciate that win-loss records that appear to favor one team or the other may really be a matter of chance.
  2. Lesson 2: To convert a running total of heads and tails into a cumulative ratio of outcomes. To appreciate that this ratio is predictable over the long run, even though individual outcomes are not.
  3. Lesson 3: To apply the techniques of probability analysis studied thus far to a biased "coin."
  4. Lesson 4: To develop the idea of permutations. To figure the probabilities of various permutations in multiple coin tosses.
  5. Lesson 5: To develop the idea of combinations. To throw 7 pennies at a time, and record a frequency distribution of combinations.
  6. Lesson 6: To compare an actual distribution of combinations for a penny toss with its ideal distribution.
  7. Lesson 7: To begin construction of a "pinball machine" based on Pascal's Triangle.
  8. Lesson 8: To complete construction of the pinball machine based on Pascal's Triangle. To plot a frequency distribution for 128 outcomes.
  9. Lesson 9: To calculate probabilities for a penny hitting pins on Pascal's triangle. To graph the most probable outcome distribution and compare it with previous results.
  10. Lesson 10: To compare right-left choices in Pascal's Triangle to other binomial probabilities.
  11. Lesson 11: To generate and evaluate additional bell curves on modified pinball machines.
  12. Lesson 12: To construct a set of 3 ABC spinners. To determine the probability of the paper clip spinner landing within each wedge.
  13. Lesson 13: To prepare a permutation tree and sample space for 3 spins of the ABC spinners.
  14. Lesson 14: To determine probabilities by counting different kinds of permutations in a sample space. To compare most probable outcomes with actual outcomes.
  15. Lesson 15: To construct a new set of numbered spinners and graph their most probable distribution on an Outcome Sheet.
  16. Lesson 16: To apply the mathematical ideas of "or" and "not" to a numbered spinner.
  17. Lesson 17: To apply the mathematical idea of "and" to numbered spinners and other systems of chance.
  18. Lesson 18: To write a sample space for rolling a pair of dice. To work out probabilities for rolling various permutations.
  19. Lesson 19: To develop a probability distribution for 10 randomly tossed tacks landing upright. To express the central tendency as a mode, median and mean.
  20. Lesson 20: To construct an alphabet spinner. To distinguish between the high probability of a class of events happening and the low probability of a specific event happening within that class.
  21. Lesson 21: To construct a random string of binomial outcomes based on phone numbers.
  22. Lesson 22: To appreciate that runs and patterns are a common occurrence in the fabric of randomness.
  23. Lesson 23: To examine lead changes between evenly matched binomial outcomes.
  24. Lesson 24: To graph the frequencies of run lengths on the tooth tape. To compare them with run probabilities on Pascal's triangle.
  25. Lesson 25: To compare an actual frequency distribution of runs to its ideal distribution as predicted by the "and" rule. To estimate run frequencies in a phone book.
  26. Lesson 26: To estimate the probability of a randomly tossed penny landing on a grid line. To confirm this probability with simple geometric analysis.
  27. Lesson 27: To estimate the probability of a randomly tossed toothpick landing on a grid line. To confirm that this probability approaches 2 divided by pi.
  28. Lesson 28: To estimate the probability of a toothpick hitting a grid line by averaging the sines of the angles it makes with the grid. To approach a limiting value of 2 divided by pi.

National Science Education Standards (NRC 1996) for #08 Probability:

TEACHING Standards

These 28 task cards promote excellence in science teaching by these NSES criteria:
Teachers of science...
A: ...plan an inquiry-based science program. (p. 30)
B: ...guide and facilitate learning. (p. 32)
C: ...engage in ongoing assessment of their teaching and of student learning. (p. 37)
D: ...design and manage learning environments that provide students with the time, space, and resources needed for learning science. (p. 43)

CONTENT Standards

These 28 task cards contain fundamental content as defined by these NSES guidelines (p. 109).
• Represent a central event or phenomenon in the natural world.
• Represent a central scientific idea and organizing principle.
• Have rich explanatory power.
• Guide fruitful investigations.
• Apply to situations and contexts common to everyday experiences.
• Can be linked to meaningful learning experiences.
• Are developmentally appropriate for students at the grade level specified.

Unifying Concepts and Processes

NSES Framework: Systems, order, and organization • Evidence, models and explanation • Constancy, change, and measurement
Core Concepts/Processes: Sample spaces predict ideal frequency distributions, but only after many rolls of the dice.

Science as Inquiry (content standard A)

NSES Framework: Identify questions that can be answered through scientific investigations. • Design and conduct a scientific investigation. • Use appropriate tools and techniques to gather, analyze, and interpret data. • Develop descriptions, explanations, predictions, and models using evidence. • Think critically and logically to connect evidence and explanations. • Recognize and analyze alternative explanations and predictions. • Communicate scientific procedures and explanations. • Use mathematics in all aspects of scientific inquiry.
Core Inquiries: Count permutations • tally combinations • plot frequency distributions • calculate probabilities. • Decide whether a tack landing "heads" or "tails" is fair or biased.

Science in Personal and Social Perspectives (content standard F)

NSES Framework: Risks and benefits • Personal and community health
Core Content: A gambler claims that a pair of dice are "hot" or "cold." If the dice are fair, can this be true?

History and Nature of Science (content standard G)

NSES Framework: Science as a human endeavor • History of science
Core Content: Develop a frequency distribution based on Pascal's triangle.