# #08 Probability (grades 6-10)

Regular price \$20.95

Soft-bound, 64 page book, 28 reproducible task cards, full teaching notes.
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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1/15/15: towels, sweaters or other fabric to muffle sound (optional)
7/105/150: pennies
1/15/15: hand calculators
* 1/8/8: rolls masking tape
* 1/15/15: bottle caps, unbent, all the same brand
1/15/15: empty cereal boxes, 32 ounce Grape-Nut boxes recommended
2/30/30: pieces corrugated cardboard, about 10x12 inches
1/4/8: heavy scissors (optional)
1/15/15: quarts package filler (paper "straw," styrofoam peanuts, or loosely wadded newspaper)
* 1/4/4: rolls clear tape
1/15/15: index cards, 4x6 inch
* 10/150/150: paper clips
1/10/15: size-D batteries, dead or alive
* 1/10/15: thick rubber bands
* 50/750/750: straight pins
1/5/15: metric rulers
* 1/15/15: plastic drinking straws
1/15/15: scissors
1/4/4: wire cutters
* 0.1/1/1: cups modeling clay
1/1/1: package pinto beans or equivalent
1/15/15: plastic sandwich bags or equivalent
4/60/60: jar lids or other shallow containers
2/10/30: dice
1/5/15: styrofoam cups
*10/50/150: flat thumb tacks, all the same brand
1/1/5: recycled phone book with white pages
* 1/1/1: roll black electrical tape
1/2/8: desk calculator with tape
10/150/150: flat wooden toothpicks
1/1/1: scientific calculator (trigonometry text with sine tables)
• 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.
• 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.
• Lesson 3: To apply the techniques of probability analysis studied thus far to a biased "coin."
• Lesson 4: To develop the idea of permutations. To figure the probabilities of various permutations in multiple coin tosses.
• Lesson 5: To develop the idea of combinations. To throw 7 pennies at a time, and record a frequency distribution of combinations.
• Lesson 6: To compare an actual distribution of combinations for a penny toss with its ideal distribution.
• Lesson 7: To begin construction of a "pinball machine" based on Pascal's Triangle.
• Lesson 8: To complete construction of the pinball machine based on Pascal's Triangle. To plot a frequency distribution for 128 outcomes.
• 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.
• Lesson 10: To compare right-left choices in Pascal's Triangle to other binomial probabilities.
• Lesson 11: To generate and evaluate additional bell curves on modified pinball machines.
• Lesson 12: To construct a set of 3 ABC spinners. To determine the probability of the paper clip spinner landing within each wedge.
• Lesson 13: To prepare a permutation tree and sample space for 3 spins of the ABC spinners.
• Lesson 14: To determine probabilities by counting different kinds of permutations in a sample space. To compare most probable outcomes with actual outcomes.
• Lesson 15: To construct a new set of numbered spinners and graph their most probable distribution on an Outcome Sheet.
• Lesson 16: To apply the mathematical ideas of "or" and "not" to a numbered spinner.
• Lesson 17: To apply the mathematical idea of "and" to numbered spinners and other systems of chance.
• Lesson 18: To write a sample space for rolling a pair of dice. To work out probabilities for rolling various permutations.
• 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.
• 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.
• Lesson 21: To construct a random string of binomial outcomes based on phone numbers.
• Lesson 22: To appreciate that runs and patterns are a common occurrence in the fabric of randomness.
• Lesson 23: To examine lead changes between evenly matched binomial outcomes.
• Lesson 24: To graph the frequencies of run lengths on the tooth tape. To compare them with run probabilities on Pascal's triangle.
• 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.
• Lesson 26: To estimate the probability of a randomly tossed penny landing on a grid line. To confirm this probability with simple geometric analysis.
• 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.
• 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.
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.
National Science Education Standards (NRC 1996)

#### 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.