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Evaluating Student Learning: Preparing to Report: Unit 7 Data Analysis and Probability This unit provides an opportunity to report on the Statistics and Probability (Data Analysis) and Statistics and Probability (Chance and Uncertainty) strands. Master 7.4: Unit Summary: Data Analysis and Probability provides a comprehensive format for recording and summarizing evidence collected. Here is an example of a completed summary chart for this Unit: Strands: Statistics and Probability (Data Analysis), Statistics and Probability (Chance and Uncertainty) Conceptual Understanding Procedural Knowledge ProblemSolving Skills Communication Overall Ongoing Observations 3 3 3 4 3 Work samples or portfolios; conferences 2 3 3 3 3 Unit Test 2 3 2 3 2/3 Unit Problem Promoting Your Cereal 3 3 4 3 3 Achievement Level for reporting* 3 * Use locally or provincially approved levels, symbols, or numeric ratings. Recording How to Report Ongoing Observations Use Master 7.2 Ongoing Observations: Data Analysis and Probability to determine the most consistent level achieved in each category. Enter it in the chart. Choose to summarize by achievement category, or simply to enter an overall level. Observations from late in the unit should be most heavily weighted. Portfolios or collections of work samples; conferences or interviews Use Master 7.1 Unit Rubric: Data Analysis and Probability to guide evaluation of collections of work and information gathered in conferences. Teachers may choose to focus particular attention on the Assessment Focus questions. Work from later in the unit may be more heavily weighted. Teachers may choose to assign some or all of the questions in the Unit Review. Master 7.1 Unit Rubric: Data Analysis and Probability may be helpful in determining levels of achievement. See Assessment for Learning at the end of each lesson for specific data. Unit Test Master 7.1 Unit Rubric: Data Analysis and Probability may be helpful in determining levels of achievement. #2 provides evidence of Conceptual Understanding; #4 provides evidence of Procedural Knowledge; #3 provides evidence of Problem-Solving Skills; #5 provides evidence of Communication. Unit Problem Use Master 7.3 Performance Assessment Rubric: Promoting Your Cereal. The Unit Problem offers a snapshot of students’ achievement. In particular, it shows their ability to synthesize and apply what they have learned. Student Self-Assessment Note students’ perceptions of their own progress. This may take the form of an oral or written comment, or a self-rating. Use any of Master 7.5, PM 2, PM 3, PM 4, PM 5, PM 6, PM 7, and PM 8. Comments Analyse the pattern of achievement to identify strengths and needs. In some cases, specific actions may need to be planned to support the learner. Learning Skills Ongoing Records PM 9: Learning Skills Checklist PM 11: Observation Record 1 PM 12: Observation Record 2 PM 14: Work Sample Records Use to record and report evaluations of student achievement over clusters, a reporting period, or a school year. These can also be used in place of the Unit Summary. Use to record and report throughout a reporting period, rather than for each unit and/or strand. 54 Unit 7: Evaluating Student Learning Name Date Unit Rubric: Data Analysis and Probability Master 7.1 Not Yet Adequate Adequate Proficient Excellent Conceptual Understanding Demonstrates and explains: – strengths and limitations of various types of graphs – choice of graph for a given data set and context/ purpose – how format and formatting choices affect interpretation – conclusions that can (and cannot) be supported by a given data set or graph – a rule for determining the probability of independent events little understanding; may be unable to demonstrate or explain: – strengths and limitations of various types of graphs – choice of graph – effect of format and formatting choices – conclusions that can (and cannot) be supported by a given data set or graph – a rule for probability of independent events some understanding; partially able to demonstrate or explain: – strengths and limitations of various types of graphs – choice of graph – effect of format and formatting choices – conclusions that can (and cannot) be supported by a given data set or graph – a rule for probability of independent events shows understanding; able to demonstrate and explain: – strengths and limitations of various types of graphs – choice of graph – effect of format and formatting choices – conclusions that can (and cannot) be supported by a given data set or graph – a rule for probability of independent events shows depth of understanding; in various contexts, demonstrates and explains: – strengths and limitations of various types of graphs – choice of graph – effect of format and formatting choices – conclusions that can (and cannot) be supported by a given data set or graph – a rule for probability of independent events limited accuracy; often makes major errors/ omissions in: – comparing information in different graphs – identifying misrepresentations – identifying misinterpretations – determining probability partially accurate; makes frequent minor errors/omissions in: – comparing information in different graphs – identifying misrepresentations – identifying misinterpretations – determining probability generally accurate; makes few errors/ omissions in: – comparing information in different graphs – identifying misrepresentations – identifying misinterpretations – determining probability accurate and precise; rarely makes errors/ omissions in: – comparing information in different graphs – identifying misrepresentations – identifying misinterpretations – determining probability does not use appropriate strategies to solve probability problems uses some appropriate strategies with partial success to solve probability problems uses appropriate strategies to successfully solve probability problems consistently uses effective, and often innovative, strategies to solve probability problems does not record and explain reasoning and procedures clearly and completely records and explains reasoning and procedures with partial clarity; may be incomplete records and explains reasoning and procedures clearly and completely records and explains reasoning and procedures with precision and thoroughness Procedural Knowledge Accurately: – compares information provided by different graphs for the same data set – identifies misrepresentations and misinterpretations – determines and verifies the probability of two independent events Problem-Solving Skills Uses appropriate strategies to solve problems involving the probability of independent events Communication Records and explains reasoning and procedures clearly and completely, including appropriate terminology (e.g., scale of a graph; outcome) The right to reproduce or modify this page is restricted to purchasing schools. 55 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.2 Date Ongoing Observations: Data Analysis and Probability The behaviours described under each heading are examples; they are not intended to be an exhaustive list of all that might be observed. More detailed descriptions are provided in each lesson under Assessment for Learning. STUDENT ACHIEVEMENT: Data Analysis and Probability* Student Conceptual Understanding Procedural Knowledge Problem-Solving Skills Communication • Explains and demonstrates strengths and limitations, misrepresentation and misinterpretation of graphs • Compares graphs; identifies misrepresentations; misinterpretations. Determines probability of independent events. • Solves problems that involve the probability of independent events • Records and explains reasoning and procedures clearly and completely, including appropriate terminology * Use locally or provincially approved levels, symbols, or numeric ratings. 56 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.3 Date Performance Assessment Rubric: Promoting Your Cereal Not Yet Adequate Adequate Proficient Excellent Conceptual Understanding • Shows understanding of data presentation by choosing graphs that: – misrepresent a data set – accurately represent a data set Choice of graphs shows very limited understanding of data presentation Choice of graphs shows partial understanding of data presentation Choice of graphs shows understanding of data presentation Choice of graphs shows thorough understanding; may introduce complexity or subtleties (e.g., misrepresentation may be hard to detect) limited accuracy; major errors or omissions in: – constructing chosen graphs – representing and misrepresents data as required – determining probability of winning Partially accurate; some errors or omissions in: – constructing chosen graphs – representing and misrepresents data as required – determining probability of winning generally accurate; few errors or omissions in: – constructing chosen graphs – representing and misrepresents data as required – determining probability of winning accurate and precise; very few or no errors in: – constructing chosen graphs – representing and misrepresents data as required – determining probability of winning uses few effective strategies; does not construct the game successfully uses some appropriate strategies with partial success; game may have some flaws uses appropriate strategies to successfully construct a game to given specifications uses effective strategies to successfully constructs a relatively complex or innovative game does not present work and explanations clearly, uses few appropriate mathematical terms presents work and explanations with some clarity, using some appropriate mathematical terms presents work and explanations clearly, using appropriate mathematical terms presents work and explanations precisely, using a range of appropriate mathematical terms Procedural Knowledge • Accurately: – constructs chosen graphs – represents and misrepresents data as required – determines the probability of winning the game Problem-Solving Skills • Uses appropriate strategies to construct a game that involves the probability of two independent events Communication • Presents work and explanations clearly, using appropriate mathematical terminology The right to reproduce or modify this page is restricted to purchasing schools. 57 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.4 Date Unit Summary: Data Analysis and Probability Review assessment records to determine the most consistent achievement levels for the assessments conducted. Some cells may be blank. Overall achievement levels may be recorded in each row, rather than identifying levels for each achievement category. Most Consistent Level of Achievement* Strands: Statistics and Probability (Data Analysis), Statistics and Probability (Chance and Uncertainty) Conceptual Understanding Procedural Knowledge ProblemSolving Skills Communication Ongoing Observations Work samples or portfolios; conferences Unit Test Unit Problem Promoting Your Cereal Achievement Level for reporting *Use locally or provincially approved levels, symbols, or numeric ratings. Self-Assessment: Comments: (Strengths, Needs, Next Steps) 58 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Overall Name Master 7.5 1. Date Reflecting on Learning: Unit 7 Read each of the Learning Goals for data analysis and probability. Think about your learning. Rate your learning for each goal using these symbols. + = I can understand and do this very well. √ = I can understand and do this all right. – = I am still having trouble with this. × = This is a big problem for me. Learning goal Critique ways in which data are presented My rating In my own words, this means I am able to Solve problems that involve the probability of independent events In my own words, this means I am able to 2. Which of the following activities is usually most helpful to you in math: – listening to explanations – talking to your classmates or working in a group – working with concrete objects – using real-life examples – doing practice work (pencil and paper) Choose one and explain how it helps you learn. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ The right to reproduce or modify this page is restricted to purchasing schools. 59 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.6a Date Additional Activity 1: Scenario Graphs Work in groups of 3 or 4. You will need Scenario Cards (Master 7.6b), a die, 0.5-cm grid paper (PM 22), and percent circles (Master 7.10) or a compass and protractor. How to Play: Choose a scenario card. Fill in each blank on the card by rolling the die and recording the number shown. Display the data using the graph that best represents the data. Present your graph to the rest of the group and describe a scenario when your graph would be the best choice to display the data. Other members of the group may “challenge” you. To do this, they must choose a different type of graph that they think would better display the data for your scenario and explain their reasoning to the group. The group decides which type of graph is best or whether more than one graph is best. You win 5 points if your type of graph is best. Challengers win 2 points if their type of graph is best. Continue playing until each group member has used 2 scenario cards to create a graph. Take It Further: Create your own scenario cards. 60 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.6b Date Scenario Cards Ujarak collected these data about the colours of socks students in his class wore on Monday: No socks: ____ students White: ____ students Grey: ____ students Black: ____ students Other: ____ students The number of hours Maurice spent doing extracurricular activities after school are: Monday: ____ h Tuesday: ____ h Wednesday: ____ h Thursday: ____ h Friday: ____ h The final marks for the students in Mrs. Van Ee’s class were: A: ____ students B: ____ students C: ____ students D: ____ students F: ____ students The number of people in the families of the students in Nathan’s class are: 2: ____ students 3: ____ students 4: ____ students 5: ____ students 6 or more: ____ students A band purchases T-shirts for all of its members. The sizes it orders are: XS: ____ band members S: ____ band members M: ____ band members L: ____ band members XL: ____ band members The ocean temperature near Prince Rupert, BC on 5 consecutive days was: Day 1: ____°C Day 2: ____°C Day 3: ____°C Day 4: ____°C Day 5: ____°C The number of times Annika went to the movie theatre each month was: Jan.: ____ times Feb.: ____ times Mar.: ____ times Apr.: ____ times Haruna collected these data about how often her father napped for 4 weeks: Week 1: ____ times Week 2: ____ times Week 3: ____ times Week 4: ____ times The favourite seasons of the students in Mikayla’s class are: Autumn: ____ students Winter: ____ students Summer: ____ students Spring: ____ students Maurice recorded how often each of his sisters said “like” in an hour: Mandisa: ____ times Keira: ____ times Jayda : ____ times Latasha: ____ times Tony collected these data about the weekly allowance students in his class received: None: ____ students Less than $5: ____ students $5–$9.99: ____ students $10–$14.99: ____ students $15 or more: ____ students The number of hours Malik spent watching television each week was: Week 1: ____ h Week 2: ____ h Week 3: ____ h Week 4: ____ h The right to reproduce or modify this page is restricted to purchasing schools. 61 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.7 Date Additional Activity 2: Misleading Graphs Scavenger Hunt Work with a partner. You will need newspapers, magazines, or access to the Internet. How to Play: To complete this scavenger hunt, look through newspapers, magazines, or the Internet with your partner for different types of misleading graphs. Record where you found each graph in the reference column of the table below. Look for these misleading graphs: Reference Pictograph Bar Graph Line Graph Circle Graph Other: Other: Other: Other: Stop searching for graphs after 30 minutes. Compare graphs with another pair of students. Explain why each graph is misleading and why it may have been displayed in that way. Whichever pair finds the most types of misleading graphs wins. Take It Further: Draw an accurate representation of the data in each graph, if possible. 62 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.8a Date Additional Activity 3: Spinner Winner Play in pairs. You will need a copy of Spinners for Spinner Winner (Master 7.8b). How to Play: Choose one of the spinners. Colour or shade it to create different sectors. Suppose you spin the pointer once. Which events are possible? What is the probability of each event? Suppose you spin the pointer twice. Which events are possible? What is the probability of each event? Continue to find as many events and their probabilities as you can for up to 5 spins. For example, if you choose a spinner with congruent red and blue sectors, you might find the probabilities of these events: Landing on red or blue: 1 1 2 1 8 : Landing on red once : Landing on red, then blue, then blue 1 32 : Landing on blue 5 times in a row After 3 minutes, play stops. With your partner, compare the possible events you found. Cross out all the events you have in common. If one person has an event the other person does not have, the other player draws a probability tree or uses the rule to check that the probability of the event is correct. Each correct probability is worth 1 point. Whoever has more points wins. Take It Further: Use two or more spinners. The right to reproduce or modify this page is restricted to purchasing schools. 63 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.8b Date Spinners for Spinner Winner 64 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.9 Date Additional Activity 4: Probability Maze Work alone. How to Play: Find a path through this maze. Your path will go through several dots. Suppose an ant walks through the maze. At each circle, it has the option of going in 2 directions (it cannot go back the way it came). So, the probability of it going in either direction is 1 2 . Find the probability that the ant walks along your path. Take It Further: Find a path through the maze with probability less than 0.05%. The right to reproduce or modify this page is restricted to purchasing schools. 65 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.10 Date Percent Circles 66 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Date Step-by-Step 1 Master 7.11 Lesson 7.1, Question 8 Table A Table B Sizes of Shoes Sold in May Step 1 Yearly Sales Size Number of Pairs Sold Year Sales ($) 6 60 2000 579 000 7 239 2001 621 000 8 217 2002 598 000 9 156 2003 634 000 10 61 2004 657 000 11 43 2005 642 000 12 36 2006 675 000 A line graph displays data that change over ________________ Which table contains this type of data? ____________________ Step 2 Look at the data in the other table. Check off each characteristic that is true for the data. Characteristics The data are discrete The data change over time The data might be compared There is a clear part-to-whole relationship The data are large numbers The data are small numbers Which types of graph could you use to display the data? ______________________ Which type of graph would best display the data? Explain. ____________________ ___________________________________________________________________ The right to reproduce or modify this page is restricted to purchasing schools. 67 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.12 Date Step-by-Step 2 Lesson 7.2, Question 10 Step 1 When you first look at the graphs, does it seem like more boys or more girls participate in sports?________________________________________ Why do you think so? _______________________________________________ _________________________________________________________________ Step 2 Use the 2 graphs to complete the table. Swimming Soccer Baseball Cross-Country Running Total Number of Boys Number of Girls Do more boys or more girls participate in sports? _______________________ Step 3 Complete this chart. Graph A Each bar is Graph B squares wide The vertical scale starts at 1 square represents . participants How could the graphs be changed to present the data accurately?_____________ _________________________________________________________________ Step 4 What other graph could you use to accurately represent the data? double bar graph line graph circle graph pictograph 68 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.13 Date Step-by-Step 3 Lesson 7.3, Question 12 Step 1 A bag contains 6 red (R), 4 blue (B), 2 yellow (Y) marbles. The total number of marbles is: _____________ On the back of this page, draw a table to show all the possible outcomes of choosing a marble out of the bag 2 times. Step 2 Find each probability. P(red) = P(yellow) = P(red then yellow) = Step 3 Find each probability. P(blue) = P(blue 2 times in a row) = Step 4 Find each probability. P(not blue) = P(yellow) = P(not blue then yellow) = Step 5 Suppose the marble is not returned to the bag after the first draw. How many marbles are left? _________ Does removing one marble affect the possible outcomes of the next draw? __________ Explain. _______________________________________________________________ ______________________________________________________________________ Are the events independent? _________ The right to reproduce or modify this page is restricted to purchasing schools. 69 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.14 Date Step-by-Step 4 Lesson 7.4, Question 9 Step 1 In a deck of cards, there are: • an equal number of cards of each suit (♥, ♦, ♠, ♣) The number of suits are: _______ • an equal number of cards of each colour (red and black) The number of card colours are: _______ • an equal number of cards of each value (A, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K) The number of card values are: _______ Nadine (N), Joshua (J), and Shirley (S) each have a complete deck of cards. Step 2 Find each probability. P(N:♥) = P(J:♥) = P(S:♥) = P(J:♠) = P(S:red) = P(J:black) = P(S:A) = P(N:♥ and J:♥ and S:♥) = Step 3 Find each probability. P(N:♠) = P(N:♠ and J:♠ and S:red) = Step 4 Find each probability. P(N:not ♥) = P(N:not ♥ and J:black and S:A) = 70 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.15a 1. Date Unit Test: Unit 7 Data Analysis and Probability Each graph displays the number of students who were late for class at A.R.C. High School in one week. a) What is an advantage of each graph? b) What is a disadvantage of each graph? c) Which graph would you choose to show how the number of late students changed throughout the week? d) Which graph would you choose to show that the number of late students was about the same each day? The right to reproduce or modify this page is restricted to purchasing schools. 71 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.15b 2. Date Unit Test continued These graphs show the number of sport cards Robbie and Spencer have each collected. a) What impression does each graph give? b) Who do you think drew the graphs, Robbie or Spencer? Why? c) What would you change about the graphs to present the data accurately? 3. These tables show data about movie ticket sales at Clear View Theatre. Table A Table B Number of Tickets Sold Last Week Total Number of Tickets Sold Ticket Type Number of Tickets Week Number of Tickets Child 553 1 3558 Student 780 2 3459 Adult 1434 3 3010 Senior 312 4 2780 72 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.15c Date Unit Test continued a) Graph the data in a misleading way. b) What impression does your graph give? ________________________________________________________________________ ________________________________________________________________________ c) Describe how you created that impression. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ The right to reproduce or modify this page is restricted to purchasing schools. 73 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.15d 4. Date Unit Test continued A spinner has three congruent sectors coloured orange, green, and purple. a) Use the rule to find the probability of each event: i) Landing on orange, then landing on purple. ii) Landing on the same colour 2 times in a row. b) Use a tree diagram to verify your answers to part a. 5. A bag contains 2 red marbles, 1 white marble, and 3 blue marbles. A marble is removed without looking, its color is recorded, and it is returned to the bag. Find the probability of each event: a) Removing a red marble, then a white marble, then a blue marble. b) Removing a marble that is not red 3 times in a row. c) Removing a blue marble, then a black marble, then a red marble. d) Removing a blue marble, then a white marble, then 3 red marbles. 74 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2008 Pearson Education Canada Name Master 7.16 1. 2. 3. Date Unit 7 Test Sample Answers Answers may vary. For example: a) Line graph: Shows the actual number of students who are late, and the trend over time: more students were late just after or just before the weekend. Circle graph: Shows how the number of late students for each day compares to the total number of late students for the week. b) Line graph: Since the graph uses the zigzag symbol on the vertical axis, it is difficult to compare the total number of late students for different days. It is a bit difficult to read data values not on horizontal lines. Circle graph: The actual number of late students isn’t given, so you can’t tell whether there were about 100 late students each day or about 5. c) Line graph d) Circle graph b) Ticket sales are about the same—they are all low. c) I made the vertical scale very large and extended the graph a long way above the largest data value. 4. 1 × 3 1 = 3 ii) 1 × 1 9 = 3 1 3 b) First Spin Second Spin Possible Outcomes orange orange/orange green orange/green purple orange/purple orange green/orange green green/green purple green/purple orange purple/orange green purple/green purple purple/purple orange Answers may vary. For example: a) The left graph gives the impression that Spencer doesn’t have many sport cards. The right graph gives the impression that Robbie has a lot of hockey and basketball cards. b) Robbie; It looks like he has more cards. c) Graph the data on a double bar graph so I could make sure the vertical scales were the same. Make each bar have the same width. Answers may vary. For example: a) 1 a) i) green purple 5. a) 2 6 × 1 6 × 3 6 = b) 4 6 × 4 6 × 4 6 = 1 3 1 6 × 2 2 × 3 1 2 × × 3 = 2 = 3 1 36 8 27 c) 0 d) 3 6 × 1 6 × 2 6 × 2 6 = 1 2 × 1 6 × 1 3 × × 1 3 2 6 × 1 3 = 1 324 The right to reproduce or modify this page is restricted to purchasing schools. 75 This page may have been modified from its original. Copyright © 2008 Pearson Education Canada