YOUNGSTOWN CITY SCHOOLS ACCELERATED MATH: GRADE 7 UNIT # 7: PROBABILITY AND STATISTICS – (5 WEEKS) Synopsis: In this unit, students work with statistics and probability. They learn about measures of center and measures of variability as well as sampling techniques and how to represent data graphically. Probability is the second major part of the unit, and students play games of chance with spinners, number cubes, etc. to determine and explain theoretical and experimental probability. Students work with a probability model to find probabilities of events. Students work with a game situation and determine if it is fair and how to make it fair.
STANDARDS 7. SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
7. SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates ro predictions. (Estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be)
7. SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variablities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. ( the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable).
7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations (decide whether the words in a chapter of seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book)
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 is 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 ( when rolling a number cube 600 times, but probably not exactly 200 times.)
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 1 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 discrepancy. a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. (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.) (7.SP.7a) b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. (find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-ended down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?) (7.SP.7b)
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. (7.SP.8a) 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. (7.SP.8b) c. Design and use a simulation to generate frequencies for compound events. (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?) (7.SP.8c)
8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 8.SP.A.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. 8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 2 data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? (See SCATTERPLOTTING papers) MATH PRACTICES 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning
LITERACY STANDARDS L-1 Learn to read mathematical text (including textbooks, articles, problems, problem explanations ) L-2 Communicate using correct mathematical terminology L-3 Read, discuss, and apply the mathematics found in literature, including looking at the author’s purpose L-4 Listen to and critique peer explanations of reasoning L-5 Justify orally and in writing mathematical reasoning
MOTIVATION TEACHER NOTES
1. Teacher has students average grades for a fake set of data for a class: using these data, show mean, median, and mode. Then have students use their own grades and show how to calculate the mean for their own grade. Ask them “what if you wanted to get a certain grade, what would you have to get on the next 1 or 2 tests/grades to get the overall grade you want?”
2. Do something with sports (e.g., basketball or baseball) as another example for calculating mean.
3. Students establish academic and personal goals for the unit.
4. Teacher previews the Authentic Assessment for the end of the unit.
TEACHING-LEARNING TEACHER NOTES VOCABULARY Measures of Measures of Stem-and-Leaf Bar Graph 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 3 TEACHING-LEARNING TEACHER NOTES Center Spread Plots Measures of Measures of Box-and-Whisker Correlation Central Variability Plots Tendency Mean Variation Line Plot Outliers Median Range Circle Graph Probability Mode Random Sample Histogram Theoretical Probability Experimental Probability
Do survey on class to collect a variety of data: type of music they like, type of T.V. shows, sports, Sports Teams, Type of Tennis Shoes (sneakers), cars, minutes pent playing video games, minutes spent doing homework, etc.
1. Measures of Center or Measures of Central Tendency: refer back to the Motivation activities and discuss mean, median, and mode; have students describe what each measure tells about the data set. Reference Glencoe pages 237-242 (Section 5-8) for ideas or tips. (7.SP.1; 7.SP.4)
2. Measures of Spread – Variation – Variability – Range: Discuss when measures of variation are important. Indicate that we will use these when we create certain data displays, such as box-and-whisker plots and stem-and- leaf plots; histograms, circle graphs; and bar graphs. Discuss outliers here to have students see how they impact the data set. (7.SP.4) a. do survey with data collected from class and have students determine and create an appropriate graphic for the data set.
b. have students make comparative inferences from two data sets.
c. Chapter 12 Glencoe, pages 606-628; do not do every page or every item; pick and choose. Page 620 has an activity on comparing average gas mileage for different cars with a box-and-whisker plot. Box and Whisker Plots (use day # of birth make human box-and-whisker plot; for example all of the students born on the 10th of any month, would stand together)
3. Random Sampling: page 634 Glencoe on specific things that influence your sample responses a. skewed sample b. voluntary sample c. biased sample (include different ways that surveys and data sets can be biased) http://www.learner.org/courses/learningmath/data/session9/index.html http://www.learner.org/courses/learningmath/data/session1/part_d/index.html
“Something Fishy Activity” from Model Curriculum; attached to Unit Plan on pages 10-14 The following is from the Model Curriculum and has activities that can be used with students:
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 4 TEACHING-LEARNING TEACHER NOTES From the National Council of Teachers of Mathematics, Illuminations - Capture Recapture: http://illuminations.nctm.org/LessonDetail.aspx?id=L721 In this lesson, students experience an application of proportion that scientists use to solve real-life problems. Students estimate the size of a total population by taking samples and using proportions
4. Misrepresenting Data: Glenco Pre-algebra
5. Probability: put numberline 0-1 on board and ask students how likely something is to happen (e.g., teacher fly across the room, chance of everyone wearing yellow on the same day, chance of getting a head / tail when flipping a coin, etc.). Discuss weather with a 30% chance of rain - - what does this mean? Emphasize the closer to 1 the more likely something is to happen. Discuss lottery and the chances of winning. Focus on Theoretical and Experimental Probability: Theoretical being what one might predict should happen - - chances of getting a head or tail are 50%. For Experimental Probability show students that this is the result one gets when doing repeated trials - - flipping a coin, you will not get exactly half heads and half tails, so the experimental probability is what you get when you actually flip the coin multiple times. Some sources to help with this section: Ready: Common Core Instruction- 7 by Curriculum Associated; Scott Foresmann Book 3, pages 646-655. (7.SP.6)
Boxing Up: In this lesson students explore the relationship between theoretical and experimental probabilities. http://illuminations.nctm.org/LessonDetail.aspx? ID=L448
Probability Using Dice ORC #9737 This activity explores the probabilities of rolling various sums with two dice. http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery- Probability_Using_Dice.pdf
Do probability tests with students using colored marbles, spinners, coins, number cubes, cards, marshmallow toss (top/bottom): do test and have students record results together to show that the more experiments the closer one will be to theoretical.
5. Compound Events (7.SP.8) Teacher discusses sample space, compound events, and simple events. Do this with flipping a coin and rolling a number cube: what are the possible outcomes? Students should show that they could get a HEAD with the number 1, 2, 3, 4, 5, or 6 or they could get a TAIL with the number 1, 2, 3, 4, 5, or 6. Have students design a skateboard with a choice from 3 deck models and 2 wheel choices. Have students do number of outfits from given set of a number of shirts and pants styles.
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 5 TRADITIONAL ASSESSMENT TEACHER NOTES 1. Multiple-Choice Unit Test counts toward the 20% of the grade with other Unit Tests for the Quarter
TEACHER-MADE ASSESSMENT TEACHER NOTES 1. Teacher Classroom Assessments: 50% comes from grades the teacher takes. 2. 2- point and 4- point Constructed Response items
AUTHENTIC ASSESSMENT ( the combined Authentic TEACHER NOTES Assessments for the Quarter are worth 30% of the student’s grade.) 1. Have students evaluate their goals and give evidence
2. How to Fix and Unfair Game ORC #9718 This activity explores a fair game and “How to Fix an Unfair Game.” http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery- How_to_Fix_an_Unfair_Game.pdf
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 6 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 7 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 8 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 9 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 10 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 11 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 12 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 13 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 14 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 15 1. 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 16 SCATTERPLOTING
Name: ______What sort of trend is shown in the scatter plot?
a. No trend b. Negative trend c. Constant trend d. Positive trend
Solution:
In the scatter plot, weight of a person is increasing along with his age.
Positive trend is the one in which, if one set of values increases, the other set also tends to increase.
So, the scatter plot follows a positive trend.
Correct answer : (4) 2. A wholesale dealer of stationery products plots the scatter diagram of his income through the years 1985 to 2000. In which year was his income the highest?
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 17 a. 1990 b. 1985 c. 2000 d. 1995
Solution:
The height of each point in the graph indicates the income (in 1000)ofthedealerintheparticularyear.Fromthegraph,theincomesofthedealerin1985,1990,1995a nd2000are 40000, 45000, 35000, and 50000.Theincomeofthedealerin2000,whichis 50000 was the maximum.
Correct answer : (3) 3. The number of boys and girls in different grades of a school are given in the table. Pick the appropriate scatter plot for the data.
Grades Boys Girls 8th 20 25 9th 30 28 10th 40 35 11th 60 45 12th 80 65
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 18 a. Plot 1 b. Plot 2 c. Plot 3 d. Plot 4
Solution:
The height of each point represents the number of students of a particular gender in a particular grade.
The yellow colored points in the scatter plots represent the number of boys and the blue colored points represent the number of girls in each grade.
Observe each of the scatter plots and the heights of the points and match them with the values in the table.
It is observed that the scatter plot given in Plot 1 corresponds with the values given in the table.
So, Plot 1 is the appropriate scatter plot graph for the data.
Correct answer : (1) 4. The scatter plot shown gives the relationship between the demand and price of consumer goods. What trend does the graph follow?
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 19 a. No trend b. Negative trend c. Positive trend
Solution:
From the graph, as the demand of an item rises, there is a rise in the price of the item and as the demand reduces there is a decrease in the price of an item.
If one set of data tends to increase with an increase in the other set, the plot is said to have a positive trend.
So, the scatter plot shows a positive trend.
Correct answer : (3) 5. The graph depicts the relation between the price and the supply of an item. What trend does the graph follow?
a. no trend
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 20 b. data is insufficient to decide the kind of trend c. positive trend d. negative trend
Solution:
From the graph, as the supply of an item increases, the price of the item decreases and as the supply decreases, the price of the item increases.
If one set of data values tends to decrease with an increase in the other set of data, the plot is said to follow a negative trend.
So, the scatter plot shows a negative trend. [Price decreases as supply increases.]
Correct answer : (4) 6. Rose measured the depth of water in a bathtub at two-minute intervals after the tap was turned on. The table shows the data prepared by her. Pick the appropriate scatter plot for the data.
Time (in minutes) Depth (in cm) 2 7 4 8 6 13 8 19 10 20 12 24 14 32 16 37 18 38 20 41 22 47
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 21 a. Graph 2 b. Graph 3 c. Graph 1 d. Graph 4
Answer: (d)
Correct answer : (4) 7. The table shows the data collected by a Consumer Products Group on the relationship between the weight of a car and its average gas mileage.
Car Type Weight (pounds) miles per gallon A 2750 29 B 3125 23 C 2100 33 D 4082 18 E 2690 20 F 3640 21 G 4380 14 H 2241 25
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 22 I 2895 31 J 3659 17
Pick the appropriate scatter plot for the data.
a. Graph 4 b. Graph 1 c. Graph 3 d. Graph 2
Answer: (c)
Correct answer : (3) 8.
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 23 Which scatter plot shows no relationship between the test scores of Samuel and the temperature of the classroom?
a. Graph 2 b. Graph 4 c. Graph 3 d. Graph 1
Answer: (c)
Correct answer : (3) 9. The table shows the height and foot length of students in a class. Select the appropriate scatter plot for the data.
Height (cm) Foot Length (cm) 102 15.5 103 15.6 109 16 114 16.2 118 16.5 120 16.8 122 18 124 18.5 6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 24 126 18.8 130 19.3
a. Graph 2 b. Graph 3 c. Graph 4 d. Graph 1
Answer: (d)
Correct answer : (4) 10.
6/21/2013 YCS Grade 7 MATH UNIT 8: STATISTICS AND PROBABILITY 20 13-2014 25 Which scatter plot shows the positive relationship between the weight and the number of seeds in papaya?
a. Graph 3 b. Graph 2 c. Graph 1 d. Graph 4
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