Probability Unit Grade 5 Overall Expectations: • represent as a fraction the probability that a specific outcome will occur in a simple probability experiment, using systematic lists and area models. Specific Expectations: – determine and represent all the possible outcomes in a simple probability experiment (e.g., when tossing a coin, the possible outcomes are heads and tails; when rolling a number cube, the possible outcomes are 1, 2, 3, 4, 5, and 6), using systematic lists and area models (e.g., a rectangle is divided into two equal areas to represent the outcomes of a coin toss experiment). – represent, using a common fraction, the probability that an event will occur in simple games and probability experiments (e.g., “My spinner has four equal sections and one of those sections is coloured red. The probability that I will land on red is ¼.”) – pose and solve simple probability problems, and solve them by conducting probability experiments and selecting appropriate methods of recording the results (e.g., tally chart, line plot, bar graph…?) Big Ideas: The likelihood an outcome will occur can be described as impossible, unlikely, likely, or certain. A fraction can describe the probability of an event occurring. The numerator is the number of outcomes favourable to the event; the denominator is the total number of possible outcomes. A tree diagram is an efficient way to find all possible combinations of outcomes of an event that consists of two or more simple events. The results of probability experiments often differ from the theoretical predictions. As we repeat an experiment, actual results tend to come closer to predicted probabilities. In probability situations, one can never be sure what might happen next. Sometimes a probability can be estimated by using an appropriate model and conducting an experiment. An experimental probability is based on past events, and a theoretical probability is based on analyzing what could happen. An experimental probability approaches a theoretical one when enough random samples are used. In this unit: Students use probability vocabulary (impossible, unlikely, likely, certain) to describe the likelihood of different outcomes in a variety of situations. Students find the number of possible outcomes of an event and describe the probability of a particular outcome as a fraction of all possible outcomes. They use tree diagrams, tables, area models, systematic lists and other graphic organizers to record and count all possible outcomes of an event. Students predict the probability that an outcome will occur. They conduct probability experiments and compare actual results to predicted results. (source: Math Makes Sense 5, Unit 11) Why Are These Concepts Important? Throughout their lives, students will make decisions in situations involving uncertainty. The abilities to understand, calculate, and predict probabilities are valuable life skills that refine and extend basic intuitions about chance events. The work students do in this unit will help prepare them for later studies of probability theory, data analysis, and statistics. (source: Math Makes Sense 5, Unit 11) Success Criteria and Misconceptions Success Criteria: I can use vocabulary to clearly talk about probability experiments (likelihood, outcome, possibility, impossible, possible, unlikely, likely, certain, probability, probable, improbable, odds, chances, outcomes, combinations, permutations) I can pose and solve simple probability problems, and solve them by conducting probability experiments I can figure out (determine) all of the possible outcomes of a probability experiment I can show (represent) the results in different ways (circle graph, table, tree diagram?, ) I can use fractions to show the probability that an event will occur in simple games and probability experiments Success Criteria for Problem Solving: I will highlight the important information in the word problem I can make a K/W/H chart to help me find out what the question is asking me to solve I will choose a strategy to help me solve the problem I will try my strategy (carry out the plan) I will check to see if my answer is reasonable I will use math language to justify my thinking/answer ESL Strategies Illustrate a poster. Next to each colour (or possible outcome) print the appropriate probability statement: This outcome is likely. This outcome is unlikely, and so on. (Adapted from MMS). Possible misconceptions: Some students may feel they have a “lucky” number or colour that is more likely to be chosen. How to Help: Encourage students to discuss the concept of chance. Remind them that events are not controlled by thoughts or feelings. (from MMS) Students may think outcomes are influenced by magic, luck, outside forces such as wind, or preference. Students may think: past events influence present probabilities, for example, if they flip heads three times, they may think they are due for tails. How to Help: Point out that the probability never changes. Some students may think middle values are more likely than the extremes. Students may think it is more likely to roll a 3 or 4 on a die than it is to roll a 1 or 6. In fact, when rolling one die, all six numbers are equally likely. (However, if you roll a pair of dice, middle values are most likely—for example, with a pair of dice, rolling a 7 is much more likely than rolling a 2 or 12.) How to Help: Use a systematic list to show the difference. Students may over-generalize from a small sample. For example, a student may flip a coin 10 times and get 3 heads. The student may predict that if you flip the same coin 100 times, it will come up heads about 30 times, rather than 50. How to Help: It is important to demonstrate that with larger sample sizes, experimental results tend to approach theoretical probability. (source: Explorelearning.com) One of the most common is that past events influence present probabilities. For example, if you are rolling a normal six-sided die and it’s been a long time since you’ve rolled a 1, you may think you are “due” for a 1. But the probability of rolling a 1 never changes; it is always 1 out of 6. Students may over-generalize from a small sample. For example, a student may flip a coin 10 times and get 3 heads. The student may predict that if you flip the same coin 100 times, it will come up heads about 30 times, rather than 50. It is important to demonstrate that with larger sample sizes, experimental results tend to approach theoretical probability. Note: “Probability is associated with games and gambling, but it also underlies many of the major decisions made by governments, companies and other decision-makers.” (Explorelearning.com) Be sensitive to the fact that some Christians, such as Baptists, prohibit playing games of chance. Lesson 1 – Diagnostic and Launch – Grade 5 Probability – Sandra Van Elslander Curriculum Expectations: Overall: represent as a fraction the probability that a specific outcome will occur in a simple probability experiment, using systematic lists and area models. Specific: – pose and solve simple probability problems, and solve them by conducting probability experiments and selecting appropriate methods of recording the results. – represent, using a common fraction, the probability that an event will occur in simple games and probability experiments. Task/Problem To pre-teach vocabulary. To solve a simple probability problem and express the answer as fraction, and in other ways using appropriate vocabulary. To identify prior knowledge and areas of need (diagnostic assessment). Learning Goal: Solve a simple probability problem. Use vocabulary from the book (including possible, probably, and impossible) to show the answer in different ways. Part 1 Before, Minds On or Student Success Criteria: Brainstorm words about probability: Question: When you think about probability, what words come to mind? (Activate prior knowledge/ assess knowledge of vocabulary, chart answers). I can use vocabulary to clearly talk about probability experiments. Read: That’s a Possibility: A Book About What Might Happen by Bruce Goldstein. I can use fractions to show the probability that an event will occur. Answer the question using fractions. I can pose and solve simple probability problems. Highlight vocabulary and add words and definitions to the chart as they occur in the text. Questions: From text: What’s a possibility? Could this ball knock down 12 pins in one roll? Why? If one of these fish swims under the bridge, what kind of fish will it be? Will this bee land on a white flower? Will this butterfly land on one of the purple flowers? What colour gumball will you probably get? What prize are you most likely to get? What other prizes are probable? Which prize is improbable? Part 2 – During, Work on It Stop at page 14. Ask students to answer the questions using probability language. Students work with a partner to record their answers in as many ways possible on white paper. “If the cat pounces on one ball of yarn, what color will it probably be?” What other color is possible? Can you think of a color that’s impossible for this cat to get? (There’s 3 blue and one yellow). With your Strategies: Make a picture Use the words from the chart Use a fraction Use a circle graph Use a table Use a pictograph Express the answer as a ratio Tools: Paper, pencils, markers Pre-printed vocabulary words and definitions, chart paper, tape partner, show the answer in as many ways you can. Can you show your answer using a fraction? What other ways can you also show the answer? Clipboard with class list for anecdotal comments and noting areas of need Questions: How many balls of yarn are there? How many of them are blue? How many are yellow? What do we know about fractions? What other ways can you show your answer? What probability words can you use to explain your thinking? Part 3 – After, Congress Misconceptions: Highlight 3 chosen pieces of work to show a variety of vocabulary words and a variety of ways of representing the answer, including fractions. Fractions: Invert the numerator and denominator. The denominator is not the whole (for example 1 yellow 3 blue expressed as 1/3). Make a chart of the ways students come up with (fraction, circle graph, table, tallies, etc.) Expressions and idioms in language apply to math, i.e. “nothing is impossible.” Congress Questions: How do you know what is probable, possible and impossible? Preference (i.e., the cat likes blue, red parrots talk more), magic, luck, or other outside factors, such as wind, influence outcomes. Can you tell me what (vocabulary word) means? What does the fraction tell us? How are these two ways of showing the answer alike? What does not affect the outcome, mathematically speaking? (address misconceptions – magic, luck, wind, etc.) Extension: There are three small balls of yarn and one big one. What other questions could you ask? Answer questions on page 15 (a dog and 9 various biscuits). Vocabulary: Spin the Big Wheel! Vocabulary Certain – definite. o If an outcome is certain, it will happen every time. Impossible – unable to occur. o If an outcome is impossible, it can never happen. Outcome – a possible result of a trial. Probability – a number between zero and one that states how likely an outcome is. o A probability of 0 means that the event is impossible. o A probability of 1 means that the event is certain. o A probability of Sample space – the set of all possible outcomes of a trial. o 2 means that the event will occur about 2 out of every 3 trials. 3 For example, if you are flipping a coin, the sample space has two outcomes: “heads” or “tails.” Trial – a test of something, an experiment. o For example, a trial could be flipping a coin once or spinning a wheel once. o In general, the more trials you do, the more reliable your results will be. Lesson 2 – Spin the Big Wheel – Grade 5 Probability – Sandra Van Elslander Curriculum Expectations: Overall: represent as a fraction the probability that a specific outcome will occur in a simple probability experiment, using systematic lists and area models. Specific: – determine and represent all the possible outcomes in a simple probability experiment – pose and solve simple probability problems, and solve them by conducting probability experiments and selecting appropriate methods of recording the results. – represent, using a common fraction, the probability that an event will occur in simple games and probability experiments. Task/Problem Solve a probability problem: Students spin the wheel to explore concepts o probability, certainty, and impossibility and design their own wheels. -Understand that experimental results will not exactly match theoretical probability. Learning Goal: I can say if an event will be certain, likely/ probable, unlikely/improbable or impossible. I can do an experiment and record the results in a table and circle graph and as a fraction. Part 1 Before, Minds On or Student Success Criteria: Student Exploration page 1– as a whole class on projector. Review vocabulary: certain, impossible, outcome, probability, sample space (set of all possible outcomes), trial I can use vocabulary to clearly talk about probability experiments. Which wheel gives you the best chances of winning? Why did you choose that wheel? Warm-up: How many sections in the wheel divided into? How many sections result in a small prize? How many sections result in a big prize? What fraction of the wheel says “big prize?” Spin the wheel. What did you win? Click clear. 10 kids are going to spin the wheel. How many do you think will win a big prize? Click 10 and press go. How many players won a small prize, a big prize, no prize? How do the results compare to what you thought would happen? I can pose and solve simple probability problems by doing probability experiments. I can use fractions to show the probability that an event will occur. -I can show the results in different ways (circle graph, table) Part 2 – During, Work on It Strategies: Page 2 – Activity A: Individually on computers in the library. Tools: (See attached Student Exploration sheets) Questions 4 and 5 are open – students design their own wheels and test the results. If time, students complete Activity B and on-line Use a circle graph Use a table Express the answer as a fraction Student exploration sheet, pencils Computers and username/password list Chart paper and markers Chart of learning goals Overhead projector in class assessment. Part 3 – After, Consolidation Misconceptions: In class – Whole group discussion, refer to learning goals. Fractions: Invert the numerator and denominator. The denominator is not the whole (for example 1 yellow 3 blue expressed as 1/3). Add to Student Success Criteria: -I can pose… by doing probability experiments -I can show the results in different ways (circle graph, table) Questions: How do you know if an event will be certain, likely/ probable, unlikely/improbable or impossible? What did the wheel you designed look like in 4A – It is certain you will win a big prize? 4B – It is impossible you will win a big prize? (A few students can come up and draw their wheels on the program using the overhead projector). 5. Describe the wheel you designed. Can you describe it using fractions? How did your predictions compare to your results? What do you think would happen if you did 100 more trials? 1000 more trials? Did you display the results in a table or circle graph? Why? Go over assessment questions as a whole class using the overhead projector. Ask: How did you know? If a few results are the same in a row, I am due for a different result. Name: ______________________________________ Date: ________________________ Student Exploration: Spin the Big Wheel! Vocabulary: certain, impossible, outcome, probability, sample space, trial Prior Knowledge Questions (Do these BEFORE using the Gizmo.) Walking through the fair, the carnies tempt you to try your luck at their booth. “Step right up, step right up. Spin the wheel and win a prize, there’s nothing to it!” 1. Which wheel gives you the best chance of winning? _______________________________ 2. Why did you choose that wheel? _______________________________________________ _________________________________________________________________________ Gizmo Warm-up The Spin the Big Wheel! Gizmo™ allows you to test your luck and try to win a prize. First, observe the wheel. 1. How many sections is the wheel divided into? _______ 2. How many sections result in a small prize? _______ How many sections result in a big prize? _______ 3. Spin the wheel by dragging it sideways. What did you win? __________________________ 4. Click Clear. Then, in the top right corner, next to Players, click 10 and press Go. How many players won a small prize? _______ Big prize? _______ No prize? _______ Activity A: What is the most likely outcome? Get the Gizmo ready: Be sure that Run the game is selected at the top. Make sure you still have the original wheel (4 sections – 2 No prize, 1 Big prize, 1 Small prize). If not, click Refresh or Reload in your browser. The carnies make it sound like everyone will win. But what is really the most likely outcome? 1. The sample space of an experiment is the set of all possible outcomes, or results. A. What is the sample space of the given wheel? ______________________________ ___________________________________________________________________ B. Which outcome do you think is most likely? ________________________________ 2. What do you think will happen if you spin the wheel once? __________________________ Spin the wheel. What happened? __________________ Is that what you predicted? _____ 3. Click Clear. Select 100 Players, and click Go. Each spin is called a trial. A. How many players won a small prize? ______ Big prize? ______ No prize? ______ B. Based on this, what is the most likely outcome? _____________________________ 4. Select Design the game. Here you can design your own wheel. First, set the number of Spinner sections with the up and down arrows. Then click any section to change the prize. A. An outcome is certain if it always happens. Design a wheel on which it is certain that you will win a big prize. Sketch this to the right, on Wheel A: B. An outcome is impossible if it cannot happen. Design a wheel on which it is impossible that you will win a big prize. Sketch this to the right, on Wheel B: 5. Design and sketch two wheels. Predict the most likely outcomes. Test each prediction with 100 spins, and record the most common outcomes. Wheel 1 prediction: ________ Actual: ________ Wheel 2 prediction: ________ Actual: ________ Activity B: Get the Gizmo ready: Probability of winning Click Clear. Select Design the game. The dancing monkey is on strike, so the fair needs a new booth. You are in charge of designing a new attraction called “Spin the Big Wheel!” 1. Design a wheel. It should have all 3 possible outcomes: No prize, Big prize, and Small prize. Draw your wheel to the right. (Label your sections B for Big prize, S for Small prize, or N for No prize.) 2. The probability of an outcome is a number between 0 and 1. If the probability is 0, the outcome is impossible. If the probability is 1, the outcome is certain. Look at your wheel. A. Which outcome do you think is most probable? _____________________________ B. Which outcome do you think is least probable? _____________________________ 3. Turn on Make your own sign. The first sign reads: “Probability of winning a prize: # / #.” Set the denominator of the fraction to the total number of sections. Set the numerator to the number of sections that will win a prize. Click Submit to show your sign to the inspector. What is the probability of winning? ______ 4. Select Run the game, and choose Circle graph to view the results. A. If 100 people spin your wheel, how many do you think will win something? ________ B. Select 100 players, and click Go. When they have finished spinning, add up the Small prize and Big prize winners. How many total winners were there? _________ C. How close was your prediction? _________________________________________ D. Look at the circle graph. How does the circle graph compare to the wheel? ________ ___________________________________________________________________ 5. Click Go until the Total players reaches 1,000. How does the circle graph compare to the wheel now? _______________________________________________________________ Activity C: Get the Gizmo ready: Making wheels Click Clear. Select Design the game. Check that Make your own sign is turned on. Your spinning wheel was so successful that fairs from all over the country are ordering them! Your job now is to design wheels to fulfill your orders and satisfy your customers. 1. The Main Street Fair wants all 3 outcomes – Big prize, Small prize, and No prize – to have the same probability. Sketch 3 different possible wheels below. (Use the Gizmo to help.) 2. Design 3 different signs that describe these wheels and click Submit. List them below. _________________________________________________________________________ _________________________________________________________________________ 3. Select Run the game. Test each of your wheels with 100 players. What were the results? Wheel A Wheel B Wheel C No Prize Small Prize Big Prize 4. Main St. Fair thinks your wheel is broken – the 3 outcomes (Big, Small, and No prize) are not coming out exactly equal. Your wheel maker says that’s normal – the numbers should be close to each other but probably not exactly equal. Who do you think is right? Explain. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Arwyn Carpenter - Lesson #3 * Colleen please mark this one Lost Socks Part 1 (from Guide to Effective Instruction- DM & P Grades 4-6) Curriculum Expectations: Grade 5 Probability Overall: Represent as a fraction the probability that a specific outcome will occur in a simple probability experiment, using systematic lists and area models. Specific: Represent, using a common fraction, the probability that an event will occur in simple games and probability experiments. Task: Consider the sock problem, estimate probability of getting a matching pair, carry out an experiment based on the problem, record results, revisit initial question with new insight. Learning Goal: understand how to use results of an experiment where there are two possible outcomes to determine if the probability of obtaining a desirable outcome will be likely, equally likely or unlikely. Part 1 Activate Prior Knowledge Show quote: "Chance has no memory", and have them discuss meaning. Review probability language. Student Success Criteria: - I can use probability words to make predictions and discuss this problem - I understand that recording results during this experiment will help me determine the probability - I chose a tool to help me record my results in an organized way - I communicated my thinking with my partner and made connections to his/her ideas Answer questions below with one of these words: certain, likely, equally likely, unlikely, impossible 1- When I flip a coin it will land heads up. 2- Our class will have Phys. Ed next week. 3- When I roll a die it will land on an even number. 4- When I roll a die it will land on a 3. 5- When I roll a die it will land on a 7. Part 2- Work on It - Present the problem: "Suppose you have two pairs of socks lying loose in a drawer, one blue pair and one green pair. You reach in without looking and pick two socks. Which outcome is more likely: the two socks will match or the two socks will not match?" Give students a minute to consider and tally on the board the students' predictions of whether they think it will be likely, equally likely or unlikely that they will pull out a matching pair. Give pairs of students a paper bag with two sets of coloured tiles, ask them to try the experiment 30 times and record their results on paper. Questions: "Do you think one of you will have a better probability than the other at getting matching pairs?" "Does your recording show the total number of matches, non-matches and total number of trials?" Part 3 – Consolidation Have Gallery Walk to compare methods of keeping track of results. How did people record their data? What categories did people choose? Were there Strategies: Use prior probability knowledge to make reasonable predictions Consider the best tool for recording collected data Carry out probability experiment to test prediction Use questioning strategy to consider reason for finding more mismatched pairs Tools: Paper/markers Paper bags Coloured tiles Misconceptions: - students will think that because there are equal numbers of blue and green socks, that a matching outcome will be equally likely. some that confused you? Which method most clearly showed the results? Ask, "Based on what you found, would you like to revise your prediction?" Ask students to explain their thinking using probability language. Congress Question: Why are we seeing more mismatched pairs than matched? - students will view experiment as a competition against each other, distracting them from the concept of probability - students will not understand that both two blues and two greens constitute matching pairs, and so do not need to be recorded in different categories Arwyn Carpenter - Lesson #4 Lost Socks Part 2 (from Guide to Effective Instruction- DM & P Grades 4-6) Curriculum Expectations: Grade 5 Probability Overall: Represent as a fraction the probability that a specific outcome will occur in a simple probability experiment, using systematic lists and area models. Specific: Determine and represent all the possible outcomes in a simple probability experiment using systematic lists Task/Problem Now that students see that it is more likely that the pair of socks will be mismatched, their task is to figure out why. Again considering the sock problem, students are asked to list all the possible combinations when two socks are picked without looking. Then they are asked to find the fraction that expresses the probability of finding a matching pair, given that the numerator represents the number of times they list a desirable outcome and the denominator represents the total number of possible outcomes. Learning Goal: Understand how to create a systematic list of all possible outcomes in a given problem and how to use this to determine probability as a fraction. Part 1 Activate Prior Knowledge Look again at outcome results from previous lessons sock experiment Student Success Criteria: - I used a think aloud strategy to share my ideas. - I apply my previous knowledge about fractions to help conceive my approach. - I created a clear method of organizing my data. - I made sense of my data and was able to explain it using probability words Questions: Why do you think most students got a result of more mismatched than matched pairs of socks? Understand that the fraction representing probability will be desirable outcomes over possible outcomes. Can you think of a way to record all the possible outcomes? Part 2- Work on It Sketch four socks on the board, two in one colour and two in another colour. Identify each sock by writing the numbers 1, 2, 3 or 4 beneath them. Say, "Can you think of a way to list all the possible outcomes?" Listen to their ideas and suggest they use pairs of numbers to list all the possible combinations when two socks are picked without looking. Model the process for two of the combinations (1-2; 1-3) then ask students to find all the other combinations. Questions: Strategies: * determine probability based on the experiment * use prior knowledge of fractions, remembering that the numerator is the part and the denominator is the whole. * remember from previous learning that the greater the denominator, the smaller the part. Tools: Paper/markers "What fraction represents the probability of finding a matching pair?" Part 3 – Consolidation Think/Pair/Share to discuss how they listed their possible outcomes. Ask them to share how they determined the probability as a fraction Misconceptions: * When organizing results students make categories for "likely", "unlikely" and "equally likely", rather than "matching" and "unmatching" Congress Questions: "How did you ensure that you had counted all the possible outcomes?" "How would you explain to a friend that when pulling out two socks without looking, it is likely that socks will be mismatched?" *When listing outcomes students don't realize that the matching pair 1-2 is the same as 2-1. * Students mistakenly see 1-1 or 2-2 for example as possible outcomes. * When it comes to making a fraction they confuse single outcome with total number of instances of desirable outcome. Assessment FOR Learning Observation & Interview Grade 5 (ext 4-6) Lost Socks Learning Goal: Use of Fractions to Express Probability Mathematics Task/Problem • • You have two pairs of socks loose in a drawer, one blue pair and one green pair. Without looking pick two socks. Which outcome is more likely: the two socks will match or the two socks will not match? Can you express the probability of finding a matching pair as a fraction? Seating Plan Mathematical Thinking A. Discussing socks combinations in terms of probability B. Understand they are gathering info to help determine predictability of likelihood C. Logical list of outcomes logical D. Re-strategizing based on results Meesha/Sandy Mohamed/ Mikhail Maxwell/Anya Michael/Keagan Kaelan/Aayana Joey/Kara Jamoree/Cory Max/Faisal Nicole/Sripiraba Byan/ Housam E. Conceptualizing possible outcomes with drawings or numbers F. Use previous understanding of fractions to find fraction to show probability Misconceptions Problem Solving Strategies 1. 2. 3. 4. Used "matching" "unmatching" for tally (organized list) Made systematic list for all possible outcomes Counted desirable outcomes Created fraction with desirable outcomes over possible outcomes a. Students will presume the two outcomes are equally likely. b. Students see problem as a competition to achieve more matches. c. Students cheat to achieve the results they want. d. Students pair a #1 with a #1, not realizing you don't put the sock back in after you've pulled it. e. Students pair 1-2 and then count 2-1 as a different possibility. Probability – Lesson 5 Name: John Hong Lesson Title: Two Dice Roll Date: April 23, 2014 Learning Goal (Curriculum Expectations) 1) Students will solve simple probability problems, and solve them by conducting probability experiments and selecting appropriate methods of recording the results (e.g., talyl chart, line plot, bar graph). Student Success Criteria 1) I can predict a probability outcome. 2) I can solve simple probability problems. 3) I can use probability language when speaking or writing about probability outcomes. 4) I can record my results using a line plot. Lesson Components Before 1. Open up the conversation and ask the students to name some things or situations in life that are probable (e.g., It’s going to get hot in the summer). 2. Discuss the meaning of probability terms such as certain, impossible, likely, unlikely, somewhat likely, most likely, equally likely, etc. 3. Ask the students: “If you roll two dice and add the two numbers on each roll, what sum will come up most often?” 4. Ask the students to make a prediction and write it down in their workbooks. Then start the experiment. 5. Hand out 2 regular dice along with paper, and two different coloured pencils for recording. Part 1: Instruct the students to record the sums of the rolls until one of the numbers is recorded 3 times (using one of the coloured pencils) in a line plot (Refer students to the line plot anchor chart). Part 2: Tell the students to continue the experiment (recording the results with the other coloured pencil) until one of the sums is rolled a total of 15 times. 6. Remind students to label their line plot where appropriate. During 1. Observe how students organize and label their line plot. Look for omissions made or for students who might be struggling with the activity. 2. Observe their addition skills (e.g., counting on, doubling, making ten facts). 3. Observe statements or realizations expressed by the students (e.g., “There are more ways to make the sum of 7”). 4. Challenge and discuss any misconceptions (e.g., “It’s all luck” or “My favourite number is 3”). After (Consolidation) 1. Have students compare the results of the experiment with their own prediction. 2. Have students compare the results of Part 1(first sum to be rolled 3 times) with Part 2 (first sum to be rolled 15 times). 3. Have students compare their results with other students. 4. Add up everyone’s results on chart paper and have students compare it with their own results (Part 1 and Part 2). 5. Have a class discussion about what they discovered. 6. Ask students “Did the results of Part 1 and Part 2 differ? Why or why not? Did the results of Part 1 and Part 2 differ from the total class result? Why or why not?” (Discuss the misconception: overgeneralization from a small sample). 7. Discuss any theories, strategies, surprises, misconceptions, or difficulties they might have had. These discussions should lead to figuring out what all the possible sums are when rolling two dice. 8. Figure out all the possible sums or outcomes with the students. 9. Once all the outcomes are written on chart paper, show or project the picture of all the possible outcomes. After (Highlights and Summary) 1. Students will reflect and write in their journal about what they have learned using probability language (e.g., likely, unlikely, highly likely, equally likely, etc.) 2. Ask them “Why do you think some of the sums rolled many more times than other sums?” or “Do you think all numbers had the same chance of being rolled?” Have them explain using words, images, or numbers. Other Activities 1) Ask students to record the probability of each sum being rolled in fractions, decimals, or percentages (Gr. 5/6). 2) Ask students to display their results using an appropriate graph. 3) Do the same experiment but with one 6-sided die and one octahedron die. Resources used: The Ontario Curriculum (Mathematics 2005), A Guide to Effective Instruction, Prabability Games (Creative Publishers) Probability- Lesson 6 Name: John Hong Lesson Title: One Die and Quarter Roll Date: April 23, 2014 Learning Goal (Curriculum Expectations) 1) Students will solve simple probability problems, and solve them by conducting probability experiments and selecting appropriate methods of recording the results (e.g., tall chart, line plot, bar graph). 2) Students will determine and represent all the possible outcomes in a simple probability experiment using systematic lists and area models (e.g., T-chart). Student Success Criteria 1) I can predict a probability outcome. 2) I can use probability language when speaking or writing about probability outcomes. 3) I can solve simple probability problems. 4) I can tally and record my results using a T-chart. 5) I can represent all possible outcomes using a rectangle (divided into parts). Lesson Components Before 1. As in the Two Dice Roll lesson, ask the students to make probability statements about real situations (e.g., It is certainly raining today, the Toronto Maple Leafs may win their game tonight). 2. Discuss probability terms such as certain, impossible, likely, unlikely, somewhat likely, most likely, equally likely, etc. 3. Ask the students: “What are the possible outcomes when you flip a coin? ( Heads or tails) Then ask: “What are the possible outcomes when you roll a die?” (1,2,3,4,5,6). 4. Ask: “If you roll a regular die and flip a quarter 48 times, what coin side and number will come up most often?” 4. Ask the students to make a prediction and write it down. 5. Hand out a regular die, a quarter, a T-chart and pencil for recording. 6. Organize students into pairs or into small groups. 7. Instruct students to organize and label their T-chart where appropriate (Ask them: “How will you record the results? What are the possible outcomes?”). 8. Start the experiment. During 1. Observe how students organize and label their T-chart. Look for omissions made or for students who might be struggling with the activity. Notice if students list all outcomes first in their T-chart or if they record the outcomes as they come up. 2. Observe statements or realizations expressed by the students (e.g., ”They all have an equal chance”). 3. Discuss misconceptions (e.g., “It’s all luck” or “Heads and 6 have the best chance”). 4. Check to see if students are flipping the coin and rolling the die properly and fairly. After (Consolidation) 1. Have students compare the results of the experiment with their own prediction. 2. Have students compare their results with other students. 3. Add up everyone’s results on chart paper and have students compare it with their own results. 4. Have a class discussion about what they discovered. 5. Ask students: Did your results differ from the total class result? Why or why not?” (Discuss the misconception: over-generalization from a small sample). 7. Discuss any theories, strategies, surprises, misconceptions, or difficulties they might have had. 8. Figure out all the possible outcomes with the students. After (Highlights and Summary) 1. Hand out grid paper and ask the students to represent all the possible outcomes in a rectangle and label each outcome. 2. Ask the students to reflect and write in their journal about what they have learned from the experiment. Encourage the use of probability language (e.g., likely, unlikely, equally likely, etc.). 3. Ask: “Did your results follow theoretical probability (i.e., 12 equal outcomes)? Explain.” Other Activities 1) Roll two regular dice, record how often even numbers come up in a chart. First make a prediction, roll the dice 36, then 72 times. Compare the results for 36 rolls with 72 rolls. Compare the results with own prediction and whole class result. 2) Roll an octahedron die 40 times, predict most often outcome, record results in a chart, show results using fractions. Resources used: The Ontario Curriculum (Mathematics 2005), A Guide to Effective Instruction, Prabability Games (Creative Publishers) Probability – Lesson 7 Adapted from: Math Makes Sense Unit 11, Lesson 4 Lesson Title: Tree Diagrams Name: Candace Minifie Date: April 26, 2014 Big Ideas 1. A tree diagram can be used to display the possible outcomes in an event that consists of two or more simple events. 2. A tree diagram can be used to count outcomes. Learning Goal (Curriculum Expectations) ✔ Students understand that calculating the probability of an event requires counting the total number of possible outcomes. ✔ Students use a tree diagram to count all possible outcomes in a situation with many different outcomes. Student Success Criteria I can use a tree diagram to count the total number of possible outcomes. I can calculate the probability of an event. Lesson Components Before 1. Show students a coin, a number cube, and a 2-colour counter. 2 Ask: • How many outcomes are possible when we toss this number cube? What are they? (6; 1, 2, 3, 4, 5, and 6) • How many outcomes are possible when we flip this coin? What are they? (2; heads and tails) • How many outcomes are possible when we toss this counter? What are they? (2; red and white) During 1. Ask a volunteer to read the Guiding Question. Materials: You will need a coin, a number cube labelled 1 to 6, and a 2-colour counter. Guiding Question: What are all the possible outcomes of rolling the number cube, tossing the coin, and tossing the counter? 2. Ask one student to roll the number cube, another to toss the coin, and a third to toss the counter. 3. Record the results. 4. Repeat the experiment 10 times. Ask: How many different outcomes did we find? Ask: How can we be sure that we have found all of the outcomes? Ongoing Assessment: Observe and Listen As students work, ask questions, such as: • What combinations of outcomes have you found? (heads, 5, red; tails, 4, red; heads, 2, white) • How are you keeping track of what you have found? (In a table. I list all the outcomes that have heads in one column and all the outcomes with tails in another.) • Have you found all the possible outcomes? (No, other combinations could happen.) Watch to see students sort their results and look for ways to determine other possible outcomes. Encourage any reasonably efficient way to organize the outcomes. Misconceptions Students launch into producing various possible outcomes without organizing their work. How to Help: Help students begin a tree diagram by focusing on one of the variables in the situation. Talk through the process of deciding what each branch should represent. After (Consolidation) Engage students in discussing the following questions: 1. Did everyone find the same outcomes? Explain. 2. Did everyone record their results in the same way? If not, what were the differences? Explain. 3. What patterns did you see in your results? 4. Select 2 or 3 students to present their work. Ask questions, such as: • How many different combinations are possible? (24) • How do you know you have found them all? (I used a pattern: first I listed heads and red with every number from 1 to 6, and then I listed heads and white, and so on.) If no one used a tree diagram, explain that there is another efficient way to figure out possible combinations. Model drawing a tree diagram on the board. Invite students to help you fill in the diagram once they get the hang of it. Ask: • How does a tree diagram help us list all possible outcomes? (It shows that we have included every choice for the combined outcome.) • How can you find the probability that the student has purple socks? (I can count all the combinations that include purple socks (4), and the total number of possible outcomes (12) to make a fraction that tells the probability: 4 12 .) Additional Activities / Extensions 1) Have students make up a simple restaurant menu and create a story problem about the possible combinations of meals, drinks, and desserts. They should pose their problem to other students who are finished early. 2) Omar’s class is painting pottery. Students can choose to paint a bowl, a plate, or a mug. They can use blue, green, yellow, or purple paint. a) b) c) Use a tree diagram to show all the different pieces of pottery Omar could make. What fraction of the choices are mugs? What is the probability that a student will paint a yellow mug? Probability Unit: Culminating Activity / Lesson 8 Lesson Title: Paper Bag Probability STEM Design Challenge Big Ideas Name: Candace Minifie Date: April 25, 2014 1) A fraction can describe the probability of an event occurring. The numerator is the number of outcomes favourable to the event; the denominator is the total number of possible outcomes. Learning Goal (Curriculum Expectations) 1) Students will collect and organize data and display the data using charts and graphs, including broken-line graphs. 2) Students will demonstrate an understanding that a fair game offers both players equal chances of winning. 3) Students will demonstrate the ability to use a list or table of outcomes to identify all possible outcomes. Student Success Criteria 1) I can collect data. 2) I can organize data into a chart or graph. 3) I understand the concept of a fair game. 4) I can identify all possible outcomes in an experiment. Lesson Components Before 1. Invite students to the carpet and ask them to bring their pencil with them. Ask them to turn to a partner and talk about how they think their pencils were made. Ask them to try to identify the step-by-step process that goes into making each of their pencils. Students will likely have a variety of pencil types which will make for a rich conversation with opportunities to compare design processes. 2. Write the title: Design Process on chart paper or on a white board. 3. Invite students to list all of the steps in the process of designing a pencil. 4. Introduce the book In The Bag! Margaret Knight Wraps it Up! by Monica Kulling 5. Let students know that In the Bag! is a book about an inventor. Ask them to listen carefully to the steps that the inventor went through to bring their idea to light. 6. After reading, revisit the step-by-step design process co-created earlier. Compare it to the process Margaret went through to design the machine that mass-produced paper bags. I hope that students would add in the patenting process. During Introduce the Paper Bag Probability design challenge. Design Challenge: Your class has been asked to host this year’s Probability Palooza. __________(teacher’s name) needs your help to prepare the probability games for the event. Criteria: Each probability game must: - be original - ensure that players have an equal chance of winning - identify all of the possible outcomes Constraints: You may only use the materials provided. Materials: brown paper bag variety of small items in a variety of colours (i.e., marbles, candies, counters, bingo chips, buttons, nuts and bolts – to go with the theme of the text) pencil and eraser markers and pencil crayons ruler Introduce the Steps for S.T.E.M. Success: Steps for S.T.E.M. Success 1. Ask: What have others done? What are the constraints? For this culminating activity, students can think about the experiments that they have completed within this unit. In this early stage we want to activate students’ prior knowledge and get them thinking about what they are being asked to do. 2. Imagine: What are some solutions? Brainstorm ideas Choose the best idea In step 2, students begin to generate ideas and ultimately select their best idea. In young children, I find this often translates as their “favourite” idea. 3. Plan: Draw a picture Make a list of materials For this design challenge, the picture that they will be creating will take the form of a tree diagram. They will use the tree diagram to determine all of the possible outcomes. 4. Create: Create your design Test your design Next, they actually create their probability game and test it out. They will want to record results in an organized manner. 5. Improve: Reflect and share findings Make changes to make it better In step 5, students can share the results of their experiment in small groups or with a partner, or have two or three students try out the game. Students can give feedback on the probability game. Retest! If students make any changes to their games, they will want to retest their game and record the results again. I suggest doing at least three trials so that they have data to compare. After (Consolidation) Probability Palooza! Divide the class into three groups. Have one group set up their games and the students from the other two groups are the players. Cycle through the groups until everyone has had a chance to play and have their games played. After (Highlights and Summary) Take Probability Palooza! on the road! Have your students take their games to another class or invite another class in to play the games. Performance Assessment Rubric: Name: Date: Level 1 Level 2 Level 3 Level 4 • with assistance, chooses and carries out a limited range of appropriate strategies, with little success • chooses and carries out some appropriate strategies, with partial success • chooses and successfully carries out appropriate strategies • chooses and successfully carries out effective strategies • uses • limited ability to probabilities to design a fair design a fair game game Understanding of concepts • shows • shows very understanding limited by providing understanding reasonable by giving explanations of: inappropriate explanations of: • some ability to design a fair game • successfully designs a fair game • effectively designs a fair game • shows limited understanding giving appropriate but incomplete explanations of: • shows understanding by giving appropriate explanations of: • shows thorough understanding by giving appropriate and complete explanations of: • differences between predicted probabilities and actual results • differences between predicted probabilities and actual results • differences between predicted probabilities and actual results • differences between predicted probabilities and actual results • how • how • how probabilities and probabilities and probabilities and number of number of number of outcomes can outcomes can outcomes can be determined be determined be determined Application of mathematical procedures • uses • limited • somewhat appropriate accuracy accurate procedures to accurately • major errors or • several minor calculate omissions in errors or probabilities calculating omissions in using fractions probabilities calculating probabilities Communication • provides a • presentation • presentation clear and discussion and discussion presentation and are unclear and are partially clear explanation of imprecise and precise results • how probabilities and number of outcomes can be determined • how probabilities and number of outcomes can be determined • generally accurate • accurate • few errors or omissions in calculating probabilities • very few or no errors or omissions in calculating probabilities • presentation and discussion are generally clear and precise • presentation and discussion are clear, precise, and confident • uses language of probability (e.g., likely, probable, outcome) • uses appropriate mathematical terms • uses the most appropriate mathematical terminology Problem Solving • chooses and carries out appropriate strategies, including tables and diagrams • differences between predicted probabilities and actual results • uses few appropriate mathematical terms • uses some appropriate mathematical terms Resource List A Guide to Effective Instruction Explorelearning.com Good Questions: Great Ways to Differentiate Mathematics Instruction, Second Edition Grades K-8 by Marian Small Canadian Edition of Elementary and Middle School Mathematics, by John A. Van de Walle and Sandra Folk. Big Ideas from Dr. Small - Grades K-3, 4-8, 9-12

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# Probability Unit Grade 5