Essential Questions:
Lesson Plan C.5: Determining Probability of Event
Lesson Seed C.5: Take a Chance
Lesson Seed C.7: Fair vs. Unfair
Unit Overview
Content Emphasis By Clusters in Grade 7
Progressions from Common Core State Standards in Mathematics
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Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.
This unit introduces students to foundational probability concepts, such as understanding that the likelihood of a chance event occurring is represented by a number between 0 and 1; approximating the probability of a chance event by collecting data on the process that causes the event and using that data to predict the relative frequency of the event based on its probability of occurring; developing models to determine the probabilities of various events; and using models such as simulations, lists, tables, and tree diagrams to find the probability of compound events.
At the completion of this unit on using random sampling to draw inferences about a population, the student will understand that::
Students will be able to:
The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.
Students may
Interdisciplinary Connections:
Interdisciplinary connections fall into a number of related categories:
Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include:
Interventions/Enrichments/PD: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)
Vocabulary/Terminology/Concepts: This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.
Part I – Focus Cluster Investigate Chance Processes and Develop, Use, and Evaluate Probability Models
probability: A population is whole set of individuals, items, or data from which information, or a statistical sample, is drawn.
chance event: A chance event in probability is one of a collection of possible outcomes. When a coin is tossed, “heads” and “tails” are chance events that can possibly occur. When a baby is born, “male” and “female” are the two chance events. When tossing a number cube, the chance events are represented by the digit on each face of the cube (“1, 2, 3, 4, 5,” and “6”).
relative frequency: When a collection of data is separated into several categories, the number of items in a given category is the absolute frequency. The absolute frequency divided by the total number of items is the relative frequency. Out of 50 middle school students, 18 are sixth graders; 18 is the absolute frequency. The relative frequency is or 0.36
uniform probability model: A uniform probability model describes an event in which all outcomes theoretically have the same probability of occurring. For example, tossing a coin or a number cube model uniform probability; the heads and tails sides of a coin each have a 1:2 probability of landing face up, while each face of a number cube has a 1:6 probability of landing face up.
outcome: A possible end result of a probability experiment is referred to as an outcome.
event: An event is a collection of possible outcomes.
simple event: A simple event is a single outcome of an experiment. For example, tossing two number cubes and getting a 7 is an event. Getting a 7, specifically with a 2 and a 5, is considered a simple event.
compound event: An event is a collection of possible outcomes. An event that consists of two or more events is a compound event. The probability of a compound event can be determined by multiplying the probability of one event by the probability of a second event. Some compound events (independent events) do not affect each other's outcomes, such as rolling a number cube and tossing a coin. Other compound events do affect each other's outputs (dependent events). For example, if you take two cards from a deck of playing cards, the likelihood of second card having a certain quality is altered by the fact that the first card has already been removed from the deck.
sample space: Sample space is the collection of all possible outcomes in a probability experiment.
simulation: Simulation is a technique used for answering real-world questions or making decision in complex situations where an element of chance is involved.
Part II – Instructional Connections outside the Focus Cluster
proportional relationship: A proportional relationship is a collection of pairs of numbers that are in equivalent ratios (Example: . A proportional relationship also can be described by an equation of the form y = kx, where k is a positive constant (often called a “constant of proportionality”).
within ratios: A ratio of two measures in the same setting is within ratios. Example: If a person walks 4 miles in 2 hours, how many miles will she walk in 3 hours? Let m = miles.
Within Ratios:
between ratios: A ratio of two corresponding measures in different situations is considered between ratios. Example: If a person walks 4 miles in 2 hours, how many miles will she walk in 3 hours?Let m = miles.
Between Ratios:
additive reasoning versus multiplicative reasoning: Proportional situations are based on multiplicative relationships. Equal ratios result from multiplication or division, not addition or subtraction. Consider any proportion, for example What operation was used to convert? Was additive reasoning used:? Or was multiplicative reasoning used:
The example below shows an application of multiplicative reasoning: Last month, two bean plants were measured at 9 inches tall and 15 inches tall. Today they are 12 inches tall and 18 inches tall, respectively. Which bean plant grew more during the month, the 9-inch bean plant or the 15-inch bean plant?
Using additive reasoning, we know that each plant added 3 inches in a month. To determine plant grew more, however, we use multiplicative reasoning. In other words, we want to know what proportion of the original plant height is represented by 3 inches. The 3-inch increase represents of the first plant’s growth, whereas the 3-inch increase represents of the second plant’s growth. Since the 9-inch plant grew more.
identify that a proportional relationship intersects (0, 0): As shown on the graphs below, when the value of x is 0, the value of y is also 0, or (0, 0). For example, (a) when zero electricity is being generated, zero pounds of carbon are emitted; (b) when zero ounces of oregano are weighed, the cost is zero dollars; (c) when a cone has a height of zero, the cone flattens into a 2-dimensional circle, and thus, has volume of zero.
a. b. c.
determine other points using (1, r), where r is the unit rate: In the ordered pair (1, r), x-value “1” represents one standard of measurement. The y-value “r” represents a given specific quantity.
percent error: We often assume that each measurement we make in mathematics or science is true and accurate. However, sources of error often prevent us from being as accurate as we would like. Percent error calculations are used to determine how close to the true values our experimental values actually are. The value that we derive from measuring is called the experimental, or observed, value. A true, ortheoretical, value can be found in reference tables.
The percent error can be determined when the theoretical value is compared to the experimental value according to the equation below:
percent error =
Example: A student measured the volume of a 2.50 liter container to be 2.38 liters. What is the percent error in the student's measurement? percent error =
cross-product algorithm: This algorithm is a strategy for determining a missing value in a proportion. The proportion can be set up using either as a within ratios proportion or as a between ratios proportion, as shown in the diagrams below.
factor of change algorithm: This algorithm is a strategy for determining a missing value in a proportion. A unit-rate approach is used, whereby the factor of change (rate of a single unit) is established first from the given values in one ratio. Then, the missing value from the other ratio is computed by multiplying the known value in the other ratio by the factor of change.Example using the within ratios format: Three candies cost a total of $2.40. At that same price, how much would 10 candies cost? The within ratios proportion is:. To determine the factor of change (unit cost) for one candy, divide $2.40 by 3. The factor of change is (or $0.80 for one candy), meaning that as the number of candies increases by one, the total cost increases by $0.80. So, 10 candies ($0.80 x 10) would cost $8.00.
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