7.4.3 Probability & Proportionality
Use random numbers generated by a calculator or a spreadsheet or taken from a table to simulate situations involving randomness, make a histogram to display the results, and compare the results to known probabilities.
For example: Use a spreadsheet function such as RANDBETWEEN(1, 10) to generate random whole numbers from 1 to 10, and display the results in a histogram.
Calculate probability as a fraction of sample space or as a fraction of area. Express probabilities as percents, decimals and fractions.
For example: Determine probabilities for different outcomes in game spinners by finding fractions of the area of the spinner.
Use proportional reasoning to draw conclusions about and predict relative frequencies of outcomes based on probabilities.
For example: When rolling a number cube 600 times, one would predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Overview
Standard 7.4.3 Essential Understandings
The focus in 7th grade is on connecting probability to proportional reasoning. In previous grades, students have learned basic probability vocabulary, conducted experiments, calculated relative frequency and compared results with known probabilities. This 7th grade standard extends student knowledge to include using proportional reasoning to predict relative frequencies based on known probabilities. Students will also demonstrate understanding of relative frequency (experimental results) by constructing relative frequency histograms of the data. It is essential that students make the connection that relative frequency is another term for proportion, such that it is the value calculated by dividing the numbers of times an event occurs by the total number of times an experiment is carried out. Students also should understand that the probability of that event can be thought of as the relative frequency that happens when the experiment is carried out many times.
Also prior to 7th grade, students have determined sample space, setting up game-like simulations to determine experimental probability and have used different strategies to find probability and compare what was expected to the relative frequency of a situation. In 7th grade, students will simulate experiments by using random numbers gathered from a spreadsheet, calculator, or a table. Students will then use a histogram to represent the data gathered and will compare to known probabilities. Students will also calculate probability as a fraction of a sample space or as a fraction of area.
MN Standard Benchmarks
7.4.3.1 Use random numbers generated by a calculator or a spreadsheet or taken from a table to simulate situations involving randomness, make a histogram to display the results, and compare the results to known probabilities. For example, use a spreadsheet function such as RANDBETWEEN(1, 10) to generate random whole numbers from 1 to 10, and display the results in a histogram.
7.4.3.2 Calculate probability as a fraction of sample space or as a fraction of area. Express probabilities as percents, decimals and fractions. For example, determine probabilities for different outcomes in game spinners by finding fractions of the area of the spinner.
7.4.3.3 Use proportional reasoning to draw conclusions about and predict relative frequencies of outcomes based on probabilities. For example, when rolling a number cube 600 times, one would predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Find probability of a given a set of data;
- Express probabilities in the three different ways: percent, decimal and fraction;
- Predict the number of times an event can occur based on probability of the event;
- Scale up and down;
- Set up ratios and proportions then solve for missing values;
- Simulate situations;
- Make histograms of data collected or from a table, including histograms of relative frequencies;
- Compare results to known and previously collected data;
- Use the term 'relative frequency' when performing experiments as a way to compare the results as relative frequencies to a known probability;
- Be able to generate random numbers in various ways (graphing calculator, drawing numbers out of a bucket, rolling dice, etc.);
- Tally data from a random number generator.
Work from previous grades that supports this new learning includes:
- Write decimals, fractions and percents;
- Know basic one stage probability;
- Write ratios;
- Write probabilities as percents, decimals and fractions;
- Write equivalent fractions;
- Know decimal, fractional and percent equivalencies;
- Know and use relative frequency (experimental results);
- Determine the set of all possible outcomes (sample space) for an experiment;
- Represent the set of possible outcomes using a variety of strategies, such as tree diagrams, organized lists, tables and pictures;
- Use tree diagrams, tables and pictures to determine the size of the sample space (number of possible outcomes);
- Use representations of a sample space to determine which members are related to certain events;
- Determine, know and use sample space;
- Determine probability of an event using the ratio of size of event and size of sample space;
- Make predictions based on gathered data.
NCTM Standards
Data Analysis and Probability Standard for Grades 6-8
- Understand and apply basic concepts of probability:
- Use proportionality and basic understanding of probability to make and test conjectures about the results of experiments and simulations;
- Formulate questions, design studies and collect data about a characteristic shared by two populations or different characteristics within one population.
- Understand and use appropriate terminology to describe complementary and mutually exclusive events.
- 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.
See this page.
Common Core State Standards (CCSS)
- 7.SP (Statistics and Probability) Investigate chance processes and develop, use, and evaluate probability models.
- 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.8a. 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.
- 7.SP.8c. 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?
Misconceptions
- Students think the odds of an event occurring is the same as probability.
- Students mistakenly believe probability can be a number larger than 1.
- Students might get confused when a probability problem involves a fraction of area.
- When interpreting histograms, students think they are simply bar graphs that are condensed.
- Students think that all situations have an expected probability; in some cases this cannot be known and needs to be based on data collected. For example, the probability that a refrigerator would need to be fixed would be determined by previous data gathered about the same make and model of refrigerator.
- Students sometimes do not make the connection that histograms of experimental data should be proportional in size to another histogram from the same experiment with a different sample size.
Vignette
Vignette 1
This vignette helps students make graphs to determine the probabilities of multiple events, using the chances of shooting free throws in a basketball game as an example.
Teacher: You are in a two shot free throw situation. That means you get to shoot both shots no matter if you made the first one or not. You are a 60% free throw shooter. How can you determine how many points you should make?
Mike: Make a tree diagram?
Jan: Actually shoot the ball?
Teacher: True. But we don't have a basket in our classroom.
Sara: List all the outcomes?
Teacher: Well, all those ideas are good suggestions, but I don't know if they will help us. What do we know about probability?
Nick: That it can be represented by a fraction.
Teacher: Yes! So, based on that, we could find the fraction of the shots. This will be a two-step process.
Sara: So how will we do that?
Teacher: Let's all start with a 10x10 grid. How many squares does that have?
Nick: 100. So if we're 60% free throw shooter, we will make 60 points.
Teacher: You are partially right. Can someone tell me why that answer is only partially right?
Mike: Because we have to shoot twice. So then wouldn't we make 120 because 60 + 60 = 120?
Teacher: Well, unfortunately, it's a little more complicated. You are correct in saying we have to add points scored, though. OK, here we go. Now, if you make 60% of your shots, and we shoot 100 shots, how many will we make?
Sara: 60, just like Nick said.
Teacher: You're right, Sara, but we get to shoot them all again. Using your 10x10 grid, let's color in what we are doing. We are making an area model here to determine the probability of making one shot, making two shots and making zero shots. OK, to do this, the first thing we need to do is count across six columns. That would be how many squares?
Jan: 60, because there are 10 squares in each of the six columns.
Teacher: Right. Does everyone have those six columns vertically colored in? You don't have to color it in perfectly, just so you can tell they are shaded in. Those columns represent the first shots we made. OK, pick up your second color, and let's again shade in 60% of the shots, but this time we are going to color in six rows (going across). So count down six, and color all the way across. Do you see how there is some overlap of the colors? What do you suppose that means in terms of the free throw problem?
Jake: Two shots?
Teacher: Yes. Those boxes that have two colors mean that you made both of the shots. How many of those boxes are there?
Sara: 36.
Teacher: So, what do you think the boxes with only one color mean?
Mike: The times we made one shot?
Teacher: Bingo! You are right. Notice there are two sections of squares that have one color in each of the boxes. How many are in each section? Maybe count the rows and columns to help you count.
Sara: I count 24 in one section.
Teacher: How many are in the other section with one color?
Sara: 24 in that one, too.
Teacher: So that means we will make one shot (or one point) 24+24 or 48 times.
Jan: So we scored 48 points. And we scored two points 36 times, so 2 x 36 = 72. So far we have scored 120 points.
Teacher: There is one section left on your grid. There are no colors in that section, so what do you think that means?
Jake: The times we scored zero points.
Teacher: OK. Now let's look at our results from this lesson. We are going to use that data and predict how many times we would make zero, one and two shots if we shot 150 times instead of 100. How could we do that?
Tina: Well, for the 100 shots, we predicted we would make two shots 36 times. The number of times we make two shots out of 150 should be proportional to that since we are still a 60% free throw shooter. So, 36/100 would be equivalent to about 54 times out of 150. Because 36/100 = x/150.
Teacher: Nice. Then how many times would we make zero and one shot?
Jake: We would make zero points 16 times out of 100, so that means we would make zero points 16 x 1.5 times which is 24 times out of 150. 150 is 1.5 times as much as 100, so the number of times out of 150 would be 1.5 x the number out of 100.
Teacher: That is another nice way to do this. Good. What about the number of times we will make one shot?
Mike: I set up a proportion to do this. If 48/100 = x/150, then the number of times making one shot would be 72 times.
Teacher: Wow. We used area models, probability, relative frequency and proportions all in this one problem. Nice work today!
Vignette 2
This vignette presents a probability activity in which students make predictions about drawing different colored tiles from bags of mixed tiles. It will take at least two class periods - one to do the bags experiment and record data, and the next to compile the results and start the graphing.
Teacher: We are going to do a fun activity today. At the front of the room, you will see eight different paper bags, each of them numbered. Inside each bag are colored tiles. This is what they look like. (Hold up a tile.) Your task is to reach in, without looking, and pull out a tile. Each pair of students will have a different bag. You will record the color that you draw out, replace it and draw again. You will need to do this 30 times, recording each color as you go, and make sure you put the tile back after you record it. Are there any questions?
Seeing that there are no questions, the teacher walks around the room as the students proceed with the lesson. She hears several groups making predictions about what is in the bags.
Student: I don't think there are any greens in our bag.
Student: I think the tiles are all the same number in each bag.
Student: I think we have the most red in our bag.
Teacher: Now, make sure you don't peak in your bags when you are making your predictions for the sheets of paper. I will direct you when you can peak in your bag. Do NOT change your predictions of what fraction of each color is in your bag! That's what a prediction is; it is done BEFORE we know the results.
Teacher: We all have made our predictions, right? Let's compile our results on the board in a table. I'd like one from each group to come record your predictions on the board for each color. You are recording what fraction of your bag is each color. Why can't we just write our numbers on the board, instead of fractions?
Student: Because we don't know how many tiles there were, and we may have grabbed the same tile more than once.
Teacher: Correct, and if we say ½ are green, don't we know how many that would be if we knew the total? So it is a way we can compare values where the totals may be different, or when we don't know the total.
A student from each group records the group's predictions in the class data. Some have simply written the exact values they got. For example, one group had 11 blue, so their fraction they wrote was 11/30. Another group simply wrote ⅓ for blue.
Teacher: Could these two groups have had the same distribution of colors in their bags?
Student: I think so, because 11/30 is about 10/30, which could be simplified to ⅓.
Student: Not necessarily. We had no way of keeping track of which tiles we used, so they may have kept grabbing the same tile. This is our experimental data, so it is not necessarily what IS in the bag.
Teacher: Nice way of explaining that. You are correct: This is our experimental data. We don't know for sure if these two bags had the same number of tiles or not. OK, now let's look at our results and compare them to the actual numbers of colors of each tile. Were there any groups whose predictions were very close to the actual numbers?
A couple groups raise their hands.
Teacher: So the rest of you did it wrong then, right?
Student: No. We recorded what we pulled and it wasn't what was actually in bag.
Teacher: So why are the answers so different?
Student: Because it was by chance. It was completely random what we selected out of the bag.
Teacher: OK, let's work on our graphs now. You are to graph your experimental probabilities (as percents) on the graph paper. Use different colors, like the guidelines suggest. When you are done with your experimental graphs, then, on the SAME graph, graph the known probabilities.
Student: Do we still use the different colors?
Teacher: Yes. That way you can compare your answers.
Walk around as the students complete their graphs. If time doesn't allow, have them complete this task for the next day.
Teacher: Now let's compare our graphs. What I want you to look at is the same colors on two different graphs. Someone explain what they see.
Student: Well, our graph started out pretty high for our results, because four out of our first five were blue, but then it went down and by 30 draws, it was close to the percent that was actually in the bag.
Teacher: Why do you think that is?
Student: Because we did more trials, so it got closer?
Teacher: Anyone have a guess WHY that is?
Student: If you only do it a few times, then it won't be as accurate as if you do it a lot of times.
Teacher: Correct! The more trials we do, the closer the experimental results will get to what it should ACTUALLY be. So, if I would have had you all do 50 pulls, what do you think would've happened to our different graphs of the experimental and expected data?
Student: They would have ended up the same, or almost the same line.
Teacher: So, in summary, the more trials you do of an experiment, the closer your results will be to the relative probabilities.
Students should record their results on the following handouts:
(Sheet 1)
Graphing Probabilities
Use your data from the Mystery Bags lesson to calculate the experimental probabilities of drawing each color after five trials, 10 trials, 15 trials and so on, up to 30 trials. Graph this data on one coordinate plane by using a different color to represent the experimental probabilities of each color tile. The number of trials is the independent variable and is graphed on the x-axis, and experimental probability is the dependent variable and is graphed on the y-axis.
(Sheet 1)
Data table for graphing probabilities
Trials | Trials | Trials | Trials | Trials | Trials |
1. | 6. | 11. | 16. | 21. | 26. |
2. | 7. | 12. | 17 | 22. | 27. |
3. | 8. | 13. | 18. | 23. | 28. |
4. | 9. | 14. | 19. | 24. | 29. |
5. | 10. | 15. | 20 | 25. | 30. |
Write the probabilities as fractions and convert to percents.
Probability after 5 trials | Probability after 10 trials | Probability after 15 trials | Probability after 20 trials | Probability after 25 trials | Probability after 30 trials |
P(B): | P(B): | P(B): | P(B): | P(B): | P(B): |
P(Y): | P(Y): | P(Y): | P(Y): | P(Y): | P(Y): |
P(R):
| P(R):
| P(R):
| P(R):
| P(R):
| P(R):
|
P(G): | P(G): | P(G): | P(G): | P(G): | P(G): |
(Sheet 2)
Mystery Bags
Your teacher has prepared bags with color tiles. Some of the bags have identical color combinations. One at a time, you and your partner will select a tile from the bag, record the color of the tile and return the tile to the bag. Repeat the procedure for 30 trials, recording the pulled tile after each trial. Do NOT look in the bag until told to do so!
A. How many tiles drawn by your group were:
blue?________________
yellow? __________________
red? ________________
green? _____________
Which color do you think there are the greatest number of in the bag? __________
Which color tile do you think there are the least number of? _________
Explain.
B. Based on your experimental data, what fraction of the tiles in the bag do you think are:
blue? ______________
yellow? ___________
red?________
green? ____________
Record your findings on a class chart. Using the class information, determine which groups have the same color combination in their bags. Explain your decision. (You can use approximate fractions, such as ⅓ or ⅕. The fractions don't necessarily have to add to 1,as these are predictions).
C. Is each tile equally likely to be selected from the bag? ______________
Explain.
Is each color equally likely to be selected from the bag? ____________
Explain.
NOW you may look in your bag. Record your ACTUAL results:
blue ______________
yellow ___________
red________
green ____________
D. What is the probability of drawing a white tile from the bag? _______________
How many tiles would add to your bag so that the probability of drawing a green tile would be ½? (If it already is ½, you must add tiles and have it remain ½). Be specific.
Resources
Teacher Notes
The State of MN 2007 Math Standards do not include the word "theoretical" probability. Instead of experimental probability, the standards document references relative frequency of experimental data.
When writing probabilities, make sure the students are writing the total number of outcomes as the denominator (the 'out of' number).
Students should have numerous opportunities to make predictions and test their conjectures.
In comparing probabilities, students sometimes may not see that if the number of outcomes is doubled, the probabilities actually stay the same. For example, there are 12 marbles in a bucket, and five of them are red. If we double the number of marbles in the bucket (all colors are doubled), what happens to the probability? The probability will actually stay the same because the number of red marbles will become 10, the total number in the bucket will be 24, so the probability is now 10/24. (5/12 is equivalent to 10/24 by scaling up, i.e., multiplying both numbers by 2.)
Benchmark 7.4.3.1 references the use of technology with random number generators. There are multiple ways to achieve this. Students may use RANDBETWEEN in a spreadsheet, RANDINT on a graphing calculator, or a table of random numbers. Students can then use those "trials" to make displays of the data.
Histograms of experimental data need to be proportional in size to other histograms of the same data because all of the data should be in proportion to one another, no matter how big the sample space is, and also proportional to the relative frequency histogram.
"The study of probability helps us figure out the likelihood of something happening. For instance, when you roll a pair of dice, you might ask how likely you are to roll a seven. In math, we call the 'something happening' an 'event.' The probability of the occurrence of an event can be expressed as a fraction or a decimal from 0 to 1. Events that are unlikely will have a probability near 0, and events that are likely to happen have probabilities near 1*." See this page.
Expose students to problems that involve probability given as a fraction of a shaded area. In this diagram, the area of the blue shaded section is ¼ of the whole circle, so its probability is also ¼.
Ask students: When you keep having a shot at something, such as rolling a six-sided die and hoping for a four, what proportion of your shots end up being successful? If it's a symmetrical die, the answer is obviously 1 in 6. That proportion is known as probability. We usually write the proportion or probability as p. In this example, p = 1/6 = 0.1666 = 0.17 (to two decimal places).
Probability is a number between 0 and 1. When it's 0, there's no way you'll be successful, and when it's 1, you'll win every time. You can't have negative probability.
We can represent probability in several other ways. In the above example, we can talk about 1 chance in 6, 17 times in 100, 17%, a likelihood of 0.17, or 17% likely. See this page.
Remember the equation:
This equation will lead to an estimate of the probability of a particular outcome. The more trials performed, the more accurate the estimate of the probability becomes. Students can also create a histogram of the relative frequencies and use that to make conclusions and predictions about the situation. When making this histogram, the height of each bar is proportional to the frequency of that value that occurred in the experiment.
Help students understand that there are some instances when an expected probability (like 0.5 for getting heads when flipping a coin) is not known. Some probabilities cannot be calculated just by looking at a situation. For example, if students want to calculate the probability of the home team winning the football game this Friday, they cannot work out the probability of winning by assuming win, lose, or tie are equally likely. But students could look at previous results in similar games and use those results to estimate the probability that the home team will win. If past results show the home team winning 35 out of 50, the fraction 35/50 = 7/10 is not technically the probability of the home team winning, but is an estimate of the probability that they will win. It can be said the relative frequency of the home team winning is 7/10.
Students have not had much previous experience with histograms. Have them brainstorm similarities and differences between a histogram and a bar graph. Students in this standard are to be making histograms of data collected in an experiment. Students can start by making a frequency histogram of the data. Then have students make a table of the relative frequencies from the experiment. Note: the two histograms should be similar to each other in that the bars of the histograms will be proportional. This is due to the fact that the only difference between a frequency histogram and a relative frequency histogram is that the vertical axis uses relative or proportional frequency instead of simple frequency.
The relative frequency histogram could be exactly the same as this one, with the scale on the vertical axis changed to 1/14, 2/14, 3/14... because there are 14 scores in the data set and 1 happens two times, therefore the relative frequency is 2/14.
Spinner
This site allows students to change the number of sectors and increase or decrease their size to create any type of spinner. Then, the students conduct a probability experiment by spinning the spinner many times and then comparing the experimental probability with the theoretical probability.
Fire
Students choose a starting place for a wildfire and enter the probability that it will spread; then, they watch the results as the fire weaves through the forest or burns itself out.
Random Drawing Tool - Individual Trials
This applet simulates drawing tickets from a box; each ticket has a number written on it. After the student decides which tickets to place in the box, the applet chooses tickets at random. The relative frequency of each number is displayed in a frequency distribution at the bottom of the applet.
What is the Name of this Game?
This article reinforces the importance of using games in instruction of students.
Additional Instructional Resources
Random number generation - spreadsheet
This website explains how to generate random numbers in a spreadsheet.
Random number generation - calculator
This website explains how to generate random numbers using a Texas Instruments graphing calculator.
Simulated probability experiments
In this classroom activity, students use the random integer (randInt) command to simulate probability experiments on a TI graphing calculator. They also graph the number of trials and corresponding probabilities to observe the Law of Large Numbers. Simulated experiments involve tossing a coin, spinning a spinner and observing the gender of children in a family.
Probability and relative frequency
This interactive online lesson is about probability and relative frequency.
Pintozzi, C. (2001). Chapter 5: Ratios, Probability, Proportions, & Scale Drawings. In Mathematics Review (pp. 64-73).Woodstock, GA: American Book.
Jones, G.A. (2005). Exploring Probability in School: Challenges for Teaching and Learning. New York: Springer.
Bright, G.W., Frierson, Jr., D., Tarr, J. E. and Thomas, C. (2003). Navigating Through Data Analysis, Statistics and Probability. Reston, VA: National Council of Teachers of Mathematics, Inc.
This book offers the teacher ideas on how to make the abstract concepts of probability and statistics understandable to middle grades students. It introduces students to the basics: sample space and the use of tree diagrams and geometric regions to represent sample spaces. Many activities explore probability in the context of the fairness of games. Engaging problems deal with population samples, prediction over the long term, and the law of large numbers. A supplemental CD-ROM features activity pages for students, interactive electronic activities and additional readings for teachers.
In this online workshop, probability can be investigated by exploring games of chance, probability models, tree diagrams, random events and even the binomial probability model. Included in this session are video clips of teachers as they work through the problems, hands-on activities and an online simulation for exploring these abstract concepts. Instruction on each idea is clearly set out through hands-on problems as well as diagrams and text explanation. Practice problems with solutions are included. and links to online articles written for teachers. This is one session in the free online course: Learning Math: Data Analysis, Statistics and Probability. In this session, some basic ideas about probability are explored - a subject that has important applications to statistics.
Ask Dr. Math: Introduction to Probability
This website explains the basics of probability including vocabulary.
The Probability Web
This website has links to a variety of resources that include online tutorials, interactive demonstrations and general resources regarding probability.
This site has a unit on probability developed by a teacher (Toni Smith, Meade Middle School, Anne Arundel County, MD), with many worksheets and activities, including a computer lesson.
Probability theory is introduced in this unit. Experiments, outcomes, sample spaces, events and conditional probability theory are covered. Interactive spinners and die rolls are truly random.
"The probable is what usually happens." - Aristotle
New Vocabulary
- random: typically describes a set of single digit numbers that are generated so that each number has an equal chance of occurring each time.
- equally likely: two or more possible outcomes of a given situation that have the same probability. For example, when picking at random among 50 people, each person has an equal chance of being picked. Similarly, when flipping a coin, the chances that the coin lands heads up or tails up are equally likely.
- histogram: a histogram is a type of bar graph in which the bars are used to represent the frequency of numerical data that have been organized into equal intervals.
.
- simulate: to create a representation or model.
- area model: a model used to represent the probability of an event; the fraction of the area represented by each part is the probability that a given event might occur.
"It is a truth very certain that when it is not in our power to determine what is true we ought to follow what is most probable." - Descartes
- How was technology used to deepen students' understanding of probability?
- Can students use proportional reasoning to make predictions of outcomes and if they can't what other techniques can be used to help with their thinking?
- How did instruction and student tasks further develop students' understanding of probability?
- What is the evidence that students understand the similarities and differences between relative frequency and probability?
Assessment
1. Make a list of the topics and ideas that come to mind when you think of probability, including both everyday uses of probability and mathematical or school uses. See this page.
2.
3. Which of the following is an impossible event?
A. choosing an odd number from 1 to 10.
B. getting an even number after rolling a single 6-sided die.
C. choosing a white marble from a jab of 25 green marbles.
D. none of the above
See this page.
Answer: C
4.
A jar contains five red, three green, two purple and four yellow marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a purple and a red marble? A. $\frac{5}{98}$ B. $\frac{1}{2}$ C. $\frac{3}{98}$ D. $\frac{2}{49}$ Answer: A See this page. |
5.
At Pacific Middle School, 3 out of 5 students make honor roll. What is the probability that a student does not make honor roll? A. 65% B. 40% C. 60% D. None of the above |
Answer: B
6. A quality inspector examines a sample of 25 strings of lights and finds that 6 strings of lights are defective. What is the best prediction of the number of defective strings in a delivery of 1000 strings of lights?
A. 6 lights
B. 25 lights
C. 24 lights
D. 240 lights
Answer: D
Taken from MCA 2011 review data and probability
7.
Answer: D
Taken from MN MCA 7th grade item sampler
8.
Correct answer: B
Taken from MN MCA 7th grade item sampler
Differentiation
Struggling Learners
Provide visual aids for struggling learners.
Use concise and precise ways to do the process.
Help students earn the concepts: certain, probable, unlikely, impossible.
This site provides online practice problems, lower level problems and good review games.
Probability Fair - An Online Game
This site let students calculate probabilities based upon visual diagrams.
Probability games
On this site, students can try probability problems about simple independent events, word problems, spinners and dice. For the additional activities using graphing calculators, be sure to write all steps on the board for students, or give them the step-by-step instructions with diagrams so they can do this activity. It could also be done as a 'class' activity with the students taking turns entering the information into the graphing calculator. Using an overhead calculator or SmartBoard would assist with this option.
Let ELL students use manipulatives when learning probability.
This site provides online practice problems, lower level problems, good review games and nice graphics.
Review the meaning of relative frequency - the frequency of the event/value divided by the total number of data points. ELL students may struggle with this because of the word 'relative.' Provide an example: "In other words...If you picked 12 marbles out of a bag, and 9 of them were green, the frequency of green marbles would be 9... but the relative frequency would be that number (the frequency) divided by the total number of marbles... so the relative frequency would be 9/12 or 3/4."
Provide a writing prompt by asking: What is a random event? Give an example of something that happens randomly and something that does not.
Probability in Advertising
Ask students to look at newspapers and magazines for examples of how numbers are used in advertisements. For example, it is not unusual to see something like "two-thirds less fat than the other leading brand" or "four out of five dentists recommend Brand T gum for their patients who chew gum." Ask students why advertisers use numbers like these, what information are they trying to convey, whether the numbers give accurate information about a product, and why or why not.
They Said What?
Ask students to look at newspapers or magazines for examples of how politicians, educators, environmentalists, or others use data, statistics and probability. Then have them analyze the use of the information they find. Ask students: Why did the person use data? What points were effectively made? Were the data useful? Did the data strengthen the argument? Have students provide evidence to support their ideas.
Collect All Ten to Win!
Students simulate a contest that requires obtaining 10 tokens found in the bottom of specially marked cereal boxes. The students should determine the average number of boxes to be bought in order to collect all ten tokens.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) | Teachers are: (descriptive list) |
using organized lists to find outcomes. | modeling ways of listing outcomes. |
finding probability correctly in all three ways: fractions, decimals and percents. | modeling strategies for finding probabilities. |
comparing relative frequencies and probabilities. | posing thought-provoking questions based on answers derived from probability situations. |
understanding some basic ideas of probability and some of the relationships between probability and statistics. | giving students opportunities to make predictions and test their conjectures.
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comparing probabilities using multiple strategies. | modeling probability situations. |
| exposing students to problems set up as a fraction of area when talking about probability. |
Parent Resources
This probability site has interactive questions for students to review/practice.