Ohio Resource Center
Lessons
The Smithville Families
Discipline
Mathematics
Grades
6, 7, 8
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Professional Commentary

First, students review Pascal's Triangle by completing and discussing the entries of the first eight rows. They then determine the total number of possible girl/boy combinations in a five-child family. This is accomplished by having students investigate the make-up of different five-child families that could be born in a town called Smithville. A coin is used to simulate the births of different children. If the coin shows a head, the child is a girl and if it shows a tail, the child is a boy. The different combinations are presented in an organized manner so that students can discover patterns that will enable them to identify all possibilities. Students are encouraged to look for patterns that will assist them in generating the numbers in subsequent rows of Pascal's Triangle. Finally, students work collaboratively to address and analyze questions regarding the theoretical probabilities of other multiple-child families using Pascal's Triangle. In addition to the lesson plan, the site includes activity sheets, answer sheets, ideas for teacher discussion, extensions of the lesson, and additional resources. (author/sw)

Common Core State Standards for Mathematics
Grade 7
Statistics and Probability
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.7
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
1. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
2. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7.SP.C.8
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
1. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
2. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
3. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
Ohio Mathematics Academic Content Standards (2001)
Data Analysis and Probability Standard
Benchmarks (5–7)
H.
Find all possible outcomes of simple experiments or problem situations, using methods such as lists, arrays and tree diagrams.
I.
Describe the probability of an event using ratios, including fractional notation.
J.
Compare experimental and theoretical results for a variety of simple experiments.
Benchmarks (8–10)
H.
Use counting techniques, such as permutations and combinations, to determine the total number of options and possible outcomes.
I.
Design an experiment to test a theoretical probability, and record and explain results.
J.
Compute probabilities of compound events, independent events, and simple dependent events.
Grade Level Indicators (Grade 5)
7.
List and explain all possible outcomes in a given situation.
9.
Use 0,1 and ratios between 0 and 1 to represent the probability of outcomes for an event, and associate the ratio with the likelihood of the outcome.
10.
Compare what should happen (theoretical/expected results) with what did happen (experimental/actual results) in a simple experiment.
Grade Level Indicators (Grade 6)
7.
Design an experiment to test a theoretical probability and explain how the results may vary.
Grade Level Indicators (Grade 7)
7.
Compute probabilities of compound events; e.g., multiple coin tosses or multiple rolls of number cubes, using such methods as organized lists, tree diagrams and area models.
Grade Level Indicators (Grade 8)
10.
Calculate the number of possible outcomes for a situation, recognizing and accounting for when items may occur more than once or when order is important.
11.
Demonstrate an understanding that the probability of either of two disjoint events occurring can be found by adding the probabilities for each and that the probability of one independent event following another can be found by multiplying the probabilities.
Grade Level Indicators (Grade 9)
8.
Describe, create and analyze a sample space and use it to calculate probability.
9.
Identify situations involving independent and dependent events, and explain differences between and common misconceptions about probabilities associated with those events.
10.
Use theoretical and experimental probability, including simulations or random numbers, to estimate probabilities and to solve problems dealing with uncertainty; e.g., compound events, independent events, simple dependent events.
Principles and Standards for School Mathematics
Data Analysis and Probability Standard
Understand and apply basic concepts of probability
Expectations (6–8)
use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations;
compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.
Expectations (9–12)
understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases;
use simulations to construct empirical probability distributions;
understand how to compute the probability of a compound event.