7SP-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 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 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 frequencies?// 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.6. Attend to precision. 7.MP.7. Look for and make use of structure. 7.MP.1. Make sense of problems and persevere in solving them. 7.MP.2. Reason abstractly and quantitatively. 7.MP.3. Construct viable arguments and critique the reasoning of others. 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.1. Make sense of problems and persevere in solving them. 7.MP.2. Reason abstractly and quantitatively. 7.MP.3. Construct viable arguments and critique the reasoning of others. 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.6. Attend to precision. 7.MP.7. Look for and make use of structure. 7.MP.8. Look for and express regularity in repeated reasoning. 7.MP.1. Make sense of problems and persevere in solving them. 7.MP.2. Reason abstractly and quantitatively. 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.7. Look for and make use of structure. 7.MP.8. Look for and express regularity in repeated reasoning. || ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Learning Target/Task Analysis**===
 * ===**Common Core Standards**===
 * 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.
 * 7.SP.8.**Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
 * a.** 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.
 * b.** 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.
 * c.** 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?// || ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Anchor Standard/Mathematical Practice(s)**===
 * 7SP5**
 * 7SP6**
 * 7SP7**
 * 7SP8**
 * ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Information Technology Standard**=== || ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Revised Bloom's Level of thinking**=== ||


 * 7.SP.5 This is students’ first formal introduction to probability.Students recognize that the probability of any single event can be can be expressed in terms such as impossible, unlikely, likely, or certain or as a number between 0 and 1, inclusive, as illustrated on the number line below.**
 * The closer the fraction is to 1, the greater the probability the event will occur. Larger numbers indicate greater likelihood. For example, if someone has 10 oranges and 3 apples, you have a greater likelihood of selecting an orange at random.**


 * Students recognize that all probabilities are between 0 and 1, inclusive, the sum of all possible outcomes is 1. For example, there are three choices of jellybeans – grape, cherry and orange. If the probability of getting a grape is 3/10 and the probability of getting cherry is 1/5, what is the probability of getting oranges? The probability of any single event can be recognized as a fraction. The closer the fraction is to 1, the greater the probability the event will occur. Larger numbers indicate greater likelihood. For example, if you have 10 oranges and 3 apples, you have a greater likelihood of getting an orange.**

There are three choices of jellybeans – grape, cherry and orange. If the probability of getting a grape is and the probability of getting cherry is, what is the probability of getting orange?
 * Example 1:**

Solution: The combined probabilities must equal 1. The combined probability of grape and cherry is. The probability of orange must equal to get a total of 1.


 * Example 2:** The container below contains 2 gray, 1 white, and 4 black marbles. Without looking, if Eric chooses a marble from the container, will the probability be closer to 0 or to 1 that Eric will select a white marble? A gray marble? A black marble? Justify each of your predictions.

Solution: White marble: Closer to 0 Gray marble: Closer to 0 Black marble: Closer to 1

Students can use simulations such as Marble Mania on AAAS or the Random Drawing Tool on NCTM’s Illuminations to generate data and examine patterns.

Marble Mania http://www.sciencenetlinks.com/interactives/marble/marblemania.html Random Drawing Tool - http://illuminations.nctm.org/activitydetail.aspx?id=67


 * 7.SP.6 Students collect data from a probability experiment, recognizing that as the number of trials increase, the experimental probability approaches the theoretical probability. The focus of this standard is relative frequency -- The relative frequency is the observed number of successful events for a finite sample of trials. Relative frequency is the observed proportion of successful event, expressed as the value calculated by dividing the number of times an event occurs by the total number of times an experiment is carried out.**

perform experiments multiple times, pool data with other groups, or increase the number of trials in a simulation to look at the long-run relative frequencies.
 * Example 1:** Suppose we toss a coin 50 times and have 27 heads and 23 tails. We define a head as a success. The relative frequency of heads is: 27/50 = 54%. The probability of a head is 50%. The difference between the relative frequency of 54% and the probability of 50% is due to small sample size. The probability of an event can be thought of as its long-run relative frequency when the experiment is carried out many times. Students can collect data using physical objects or graphing calculator or web-based simulations. Students can

conjectures about theoretical probabilities (How many green draws would are expected if 1000 pulls are conducted? 10,000 pulls?). Students create another scenario with a different ratio of marbles in the bag and make a conjecture about the outcome of 50 marble pulls with replacement. (An example would be 3 green marbles, 6 blue marbles, 3 blue marbles.) Students try the experiment and compare their predictions to the experimental outcomes to continue to explore and refine conjectures about theoretical probability.
 * Example 2:** Each group receives a bag that contains 4 green marbles, 6 red marbles, and 10 blue marbles. Each group performs 50 pulls, recording the color of marble drawn and replacing the marble into the bag before the next draw. Students compile their data as a group and then as a class. They summarize their data as experimental probabilities and make


 * Example 3:** A bag contains 100 marbles, some red and some purple. Suppose a student, without looking, chooses a marble out of the bag, records the color, and then places that marble back in the bag. The student has recorded 9 red marbles and 11 purple marbles. Using these results, predict the number of red marbles in the bag. (Adapted from SREB publication Getting Students Ready for Algebra I: What Middle Grades Students Need to Know and Be Able to Do)


 * 7.SP.7 Probabilities are useful for predicting what will happen over the long run. Using theoretical probability, students predict frequencies of outcomes. Students recognize an appropriate design to conduct an experiment with simple probability events, understanding that the experimental data give realistic estimates of the probability of an event but are affected by sample size.**

Students need multiple opportunities to perform probability experiments and compare these results to theoretical probabilities. Critical components of the experiment process are making predictions about the outcomes by applying the principles of theoretical probability, comparing the predictions to the outcomes of the experiments, and replicating the experiment to compare results. Experiments can be replicated by the same group or by compiling class data. Experiments can be conducted using various random generation devices including, but not limited to, bag pulls, spinners, number cubes, coin toss, and colored chips. Students can collect data using physical objects or graphing calculator or web-based simulations. Students can also develop models for geometric probability (i.e. a target).


 * Example 1:** If Mary chooses a point in the square, what is the probability that it is not in the circle?

Solution: The area of the square would be 12 x 12 or 144 units squared. The area of the circle would be 113.04 units squared. The probability that a point is not in the circle would be or 21.5%


 * Example 2:** Jason is tossing a fair coin. He tosses the coin ten times and it lands on heads eight times. If Jason tosses the coin an eleventh time, what is the probability that it will land on heads?

Solution: The probability would be. The result of the eleventh toss does not depend on the previous results.


 * Example 3:** Devise an experiment using a coin to determine whether a baby is a boy or a girl. Conduct the experiment ten times to determine the gender of ten births. How could a number cube be used to simulate whether a baby is a girl or a boy or girl?

• How many trials were conducted? • How many times did it land right side up? • How many times did it land upside down/ • How many times did it land on its side? • Determine the probability for each of the above results
 * Example 4:** Conduct an experiment using a Styrofoam cup by tossing the cup and recording how it lands.


 * 7.SP.8 Students use tree diagrams, frequency tables, and organized lists, and simulations to determine the probability of compound events. Probabilities are useful for predicting what will happen over the long run. Using theoretical probability, students predict frequencies of outcomes. Students recognize an appropriate design to conduct an experiment with simple probability events, understanding that the experimental data give realistic estimates of the probability of an event but are affected by sample size.**

Solution: Making an organized list will identify that there are 6 ways for the students to win a race A, B, C A, C, B B, C, A B, A, C C, A, B C, B, A
 * Example 1:** How many ways could the 3 students, Amy, Brenda, and Carla, come in 1st, 2nd and 3rd place?

drawing one blue marble followed by another blue marble.
 * Example 2:** Students conduct a bag pull experiment. A bag contains 5 marbles. There is one red marble, two blue marbles and two purple marbles. Students will draw one marble without replacement and then draw another. What is the sample space for this situation? Explain how the sample space was determined and how it is used to find the probability of


 * Example 3:** A fair coin will be tossed three times. What is the probability that two heads and one tail in any order will results? (Adapted from SREB publication Getting Students Ready for Algebra I: What Middle Grades Students Need to Know and Be Able to Do

Solution: HHT, HTH and THH so the probability would be .3/8

Solution: There are 24 possible arrangements (4 choices • 3 choices • 2 choices • 1 choice) The probability of drawing F-R-E-D in that order is .1/24 The probability that a “word” will have an F as the first letter is 6/24 or 1/4.
 * Example 4:** Show all possible arrangements of the letters in the word FRED using a tree diagram. If each of the letters is on a tile and drawn at random, what is the probability of drawing the letters F-R-E-D in that order? What is the probability that a “word” will have an F as the first letter?

‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**I can use random sampling to draw inferences about a population.**

 * **(7SP5) I can recognize all probabilities are between 0 and 1.**
 * **(7SP5) I can recognize the likelihood of an event occurring.**
 * **(7SP6) I can collect data from experimental probability events.**
 * **(7SP6) I can recognize that as the number of trials increases the closer the experimental probability approaches theoretical probability.**
 * **(7SP7) I can develop a probability model and use it to find the probabilities of events.**
 * **(7SP8) I can find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.**

‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Essential Vocabulary**

 * counting principle, sample space, compound event, independent event, dependent event, theoretical probability, experimental probability, variation/variability, distribution, measures of center, measures of variability**

‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Differentiation**

 * peer tutoring, guided notes, blogs, grouping, foldables, justification of answers, inquiry based learning to accelerate/remediate, conduct surveys and include the process of statistical measures to summarize data collected**

===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Instructional Resources**=== http://qta.quantiles.com/qtaxon/goal/29772/ http://qta.quantiles.com/qtaxon/goal/29773/ http://qta.quantiles.com/qtaxon/goal/29774/ http://qta.quantiles.com/qtaxon/goal/29775/ http://qta.quantiles.com/qtaxon/goal/29776/ http://qta.quantiles.com/qtaxon/goal/29777/ http://qta.quantiles.com/qtaxon/goal/29778/ http://www.montgomery.k12.nc.us/18032092913393583/site/default.asp
 * [|Activities with Probability]**
 * Math Partners resources**
 * http://www.kutasoftware.com,**
 * http://www.mathgoodies.com,**

[[file:7_Math_SPfinal.docx]]
===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Notes and Additional Information**===