Grade Level: Unit:



Approximate Time Frame: 2-3 weeks

Connections to Previous Learning:

Students in Grade 6 learn the concepts of ratio and unit rate as well as the precise mathematical language used to describe these relationships. They learn to solve problems using ratio and rate reasoning using a variety of tools such as tables, tape diagrams, double number lines and equations.

Focus of this Unit:

In Grade 7 students develop a general understanding of the likelihood of events occurring by realizing that probabilities fall between 0 and 1. They gather data from simulations to estimate theoretical probability using the experimental probability. Students will make predictions about the relative frequency of an event by using simulations to collect, record, organize and analyze data. They will develop probability models to be used to find the probability of simple and compound events. Students will determine from each sample space the probability or fraction of each possible outcome.

Connections to Subsequent Learning:

The concepts developed at Grade 7 extend to the high school Conditional Probability and Rules of Probability Standards.

From the 6-8 Statistics and Probability Progression Document pp.7-8:

Chance processes and probability models In Grade 7, students build their understanding of probability on a relative frequency view of the subject, examining the proportion of “successes” in a chance process—one involving repeated observations of random outcomes of a given event, such as a series of coin tosses. “What is my chance of getting the correct answer to the next multiple choice question?” is not a probability question in the relative frequency sense. “What is my chance of getting the correct answer to the next multiple choice question if I make a random guess among the four choices?” is a probability question, because the student could set up an experiment of multiple trials to approximate the relative frequency of the outcome. And two students doing the same experiment will get nearly the same approximation. These important points are often overlooked in discussions of probability.

Students begin by relating probability to the long-run (more than five or ten trials) relative frequency of a chance event, using coins, number cubes, cards, spinners, bead bags, and so on. Hands-on activities with students collecting the data on probability experiments are critically important, but once the connection between observed relative frequency and theoretical probability is clear, they can move to simulating probability experiments via technology (graphing calculators or computers).

It must be understood that the connection between relative frequency and probability goes two ways. If you know the structure of the generating mechanism (e.g., a bag with known numbers of red and white chips), you can anticipate the relative frequencies of a series of random selections (with replacement) from the bag. If you do not know the structure (e.g., the bag has unknown numbers of red and white chips), you can approximate it by making a series of random selections and recording the relative frequencies. This simple idea, obvious to the experienced, is essential and not obvious at all to the novice. The first type of situation, in which the structure is known, leads to “probability”; the second, in which the structure is unknown, leads to “statistics.”

A probability model provides a probability for each possible non-overlapping outcome for a chance process so that the total probability over all such outcomes is unity. The collection of all possible individual outcomes is known as the sample space for the model. For example, the sample space for the toss of two coins (fair or not) is often written as {TT, HT, TH, HH}. The probabilities of the model can be either theoretical (based on the structure of the process and its outcomes) or empirical (based on observed data generated by the process). In the toss of two balanced coins, the four outcomes of the sample space are given equal theoretical probabilities of ¼ because of the symmetry of the process—because the coins are balanced, an outcome of heads is just as likely as an outcome of tails. Randomly selecting a name from a list of ten students also leads to equally likely outcomes with probability 0.10 that a given student’s name will be selected. If there are exactly four seventh graders on the list, the chance of selecting a seventh grader’s name is 0.40. On the other hand, the probability of a tossed thumbtack landing point up is not necessarily ½ just because there are two possible outcomes; these outcomes may not be equally likely and an empirical answer must be found by tossing the tack and collecting data.

The product rule for counting outcomes for chance events should be used in finite situations like tossing two or three coins or rolling two number cubes. There is no need to go to more formal rules for permutations and combinations at this level. Students should gain experience in the use of diagrams, especially trees and tables, as the basis for organized counting of possible outcomes from chance processes. For example, the 36 equally likely (theoretical probability) outcomes from the toss of a pair of number cubes are most easily listed on a two-way table. An archived table of census data can be used to approximate the (empirical) probability that a randomly selected Florida resident will be Hispanic.

After the basics of probability are understood, students should experience setting up a model and using simulation (by hand or with technology) to collect data and estimate probabilities for a real situation that is sufficiently complex that the theoretical probabilities are not obvious. For example, suppose, over many years of records, a river generates a spring flood about 40% of the time. Based on these records, what is the chance that it will flood for at least three years in a row sometime during the next five years?

|Desired Outcomes |

|Standard(s): |

|Investigate chance processes and develop, use and evaluate probability models. |

|7.SP.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an|

|unlikely event, a probability around ½ indicates an event that in neither unlikely nor likely, and a probability near 1 indicates a likely event. |

|7.SP.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the |

|probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. |

|7.SP.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. |

|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. |

|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 |

|7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. |

|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. |

|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. |

|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? |

|Supporting Standards: |

|7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. |

|Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. |

|Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive |

|inverses). Interpret sums of rational numbers by describing real‐world contexts. |

|Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply |

|this principle in real‐world contexts. |

|Apply properties of operations as strategies to add and subtract rational numbers . |

|7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. |

|Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products |

|such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real‐world contexts. |

|Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non‐zero divisor) is a rational number. If p and q are integers then – (p/q) = (-p)/q = p/(-q). |

|Interpret quotients of rational numbers by describing real‐world contexts. |

|Apply properties of operations as strategies to multiply and divide rational numbers. |

|Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats |

|7. NS.3 Solve real-world and mathematical problems involving the four operations with rational number. |

|7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a +0.05a = 1.05a means that “increase by 5% |

|is the same as multiply by 1.05.” |

|7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of |

|operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25|

|an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50 for a new salary of $27.50. If you want to place a towel bar 9 ¾ inches long in the center of a door that is 27 ½ inches |

|wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. |

|WIDA Standard: (English Language Learners) |

|English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. |

|English language learners will benefit from: |

|Explicit instruction with regard to understanding the contexts for probability models. |

|Explicit vocabulary instruction with regard to probability language. |

|Understandings: Students will understand that … |

|The probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. |

|The probability of a chance event is approximated by collecting data on the chance process that produces it, observing its long-run relative frequency, and predicting the approximate relative frequency given the |

|probability. |

|A probability model, which may or may not be uniform, is used to find probabilities of events. |

|Various tools are used to find probabilities of compound events. (Including organized lists, tables, tree diagrams, and simulations.) |

|Essential Questions: |

|How are probability and the likelihood of an occurrence related and represented? |

|How is probability approximated? |

|How is a probability model used? |

|How are probabilities of compound events determined? |

|Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) |

|1. Make sense of problems and persevere in solving them. Students make sense of probability situations by creating visual, tabular and symbolic models to represent the situations. They persevere through |

|approximating probabilities and refining approximations based upon data. |

|*2. Reason abstractly and quantitatively. Students’ reason about the numerical values used to represent probabilities as a value between 0 and 1. |

|*3. Construct viable arguments and critique the reasoning of others. Students approximate probabilities and create probability models and explain reasoning for their approximations. They also question each other |

|about the representations they create to represent probabilities. |

|*4. Model with mathematics. Students model real world populations using mathematical probability representations that are algebraic, tabular or graphic. |

|5. Use appropriate tools strategically. Students select and use technological, graphic or real-world contexts to model probabilities. |

|6. Attend to precision. Students use precise language and calculations to represent probabilities in mathematical and real-world contexts. |

|7. Look for and make use of structure. Students recognize that probability can be represented in tables, visual models, or as a rational number. |

|8. Look for express regularity in repeated reasoning. Students use repeated reasoning when approximating probabilities. They refine their approximations based upon experiences with data. |

| |

|Prerequisite Skills/Concepts: |Advanced Skills/Concepts: |

|Students should already be able to: |Some students may be ready to: |

|Understand statistical variability (6.SP.2) |Understand independence and conditional probability and use them to interpret data. (S-CP.1-5) |

|Display data in various ways (6.SP.4) |Use the rules of probability to compute probabilities of compound events in a uniform probability model. (S-CP.6-7) |

|Interpret and summarize data as a numerical set in relation to its context (6.SP.5) | |

|Knowledge: Students will know… |Skills: Students will be able to… |

|0 represents an event that is impossible. |Represent the probability of a chance event as a number between 0 and 1. (7.SP.5) |

|1 represents an event is certain. |Use the terms “likely”, “unlikely,” to describe the probability represented by the fractions used. (7.SP.5) |

|The closer to 1, the more likely an event is to occur. |Approximate the probability of a chance event by collecting data on the chance process that produces it and observing |

|The closer to 0, the less likely an event is to occur. |its long-run relative frequency. (7.SP.6) |

| |Predict the approximate relative frequency of a chance event given the probability. (7.SP.6) |

|All standards in this unit go beyond the knowledge level. |Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine |

| |probabilities of events (7.SP.7) |

| |Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance |

| |process. (7.SP.7) |

| |Compare probabilities from a model to observed frequencies. (7.SP.7) |

| |If the agreement between a model and observed frequencies is not good, explain possible sources of the discrepancy. |

| |(7.SP.7) |

| |Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation (7.SP.8) |

| |Represent the probability of a compound event as the fraction of outcomes in the sample space for which the compound |

| |event occurs. (7.SP.8) |

| |Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. (7.SP.8) |

| |For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space |

| |which compose the event. (7.SP.8) |

| |Design and use a simulation to generate frequencies for compound events. (7.SP.8) |

|Academic Vocabulary: |

| | |

|Critical Terms: |Supplemental Terms: |

|Simulation |Empirical probability |

|Compound event |Equally likely |

|Probability |More likely |

|Sample space |Less likely |

|Random sample |Fair |

|Outcome |Unfair |

|Theoretical probability |Simple event |

|Experimental probability |Fraction |

|Relative Frequency |Decimal |

|Tree diagram |Percent |

|Likelihood |Combination |

|Counting Principle |Permutation |

|Uniform probability model |Dependent Event |

| |Independent Event |

| |Complementary Event |

| |Relative frequency |

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