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: Possible misconceptions:

 I can use vocabulary to clearly talk Some students may feel they have a “lucky” number or colour about probability experiments that is more likely to be chosen. How to Help: Encourage (likelihood, outcome, possibility, students to discuss the concept of chance. Remind them that impossible, possible, unlikely, likely, events are not controlled by thoughts or feelings. (from MMS) certain, probability, probable, improbable, odds, chances, outcomes, Students may think outcomes are influenced by magic, luck, combinations, permutations) outside forces such as wind, or preference.  I can pose and solve simple probability Students may think: past events influence present problems, and solve them by probabilities, for example, if they flip heads three times, they conducting probability experiments may think they are due for tails. How to Help: Point out that  I can figure out (determine) all of the the probability never changes. possible outcomes of a probability experiment Some students may think middle values are more likely than  I can show (represent) the results in the extremes. Students may think it is more likely to roll a 3 or different ways (circle graph, table, tree 4 on a die than it is to roll a 1 or 6. In fact, when rolling one die, diagram?, ) 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  I can use fractions to show the dice, rolling a 7 is much more likely than rolling a 2 or 12.) probability that an event will occur in How to Help: Use a systematic list to show the difference. simple games and probability experiments Students may over-generalize from a small sample. For example, a student may flip a coin 10 times and get 3 heads. Success Criteria for Problem Solving: The student may predict that if you flip the same coin 100 . I will highlight the important information times, it will come up heads about 30 times, rather than 50. in the word problem How to Help: It is important to demonstrate that with larger . I can make a K/W/H chart to help me sample sizes, experimental results tend to approach find out what the question is asking me theoretical probability. (source: Explorelearning.com) to solve . I will choose a strategy to help me solve One of the most common is that past events influence present the problem probabilities. For example, if you are rolling a normal six-sided . I will try my strategy (carry out the plan) die and it’s been a long time since you’ve rolled a 1, you may . I will check to see if my answer is think you are “due” for a 1. But the probability of rolling a 1 reasonable never changes; it is always 1 out of 6. I will use math language to justify my . Students may over-generalize from a small sample. For thinking/answer 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. 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).

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 Learning Goal: To pre-teach vocabulary. Solve a simple probability problem. Use vocabulary from To solve a simple probability problem and express the the book (including possible, probably, and impossible) to answer as fraction, and in other ways using appropriate show the answer in different ways. vocabulary. To identify prior knowledge and areas of need (diagnostic Answer the question using fractions. assessment).

Part 1 Before, Minds On or Student Success Criteria:

Brainstorm words about probability: Question: When you I can use vocabulary to clearly talk about probability think about probability, what words come to mind? experiments. (Activate prior knowledge/ assess knowledge of vocabulary, chart answers). I can pose and solve simple probability problems.

Read: That’s a Possibility: A Book About What Might I can use fractions to show the probability that an event Happen by Bruce Goldstein. will occur.

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 Strategies:  Make a picture Stop at page 14. Ask students to answer the questions  Use the words from the chart using probability language. Students work with a partner  Use a fraction to record their answers in as many ways possible on white paper.  Use a circle graph  Use a table  Use a pictograph  Express the answer as a ratio “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? Tools: (There’s 3 blue and one yellow). With your partner, show  Paper, pencils, markers the answer in as many ways you can. Can you show  Pre-printed vocabulary words and definitions, your answer using a fraction? What other ways can you chart paper, tape also show the answer?  Clipboard with class list for anecdotal comments Questions: and noting areas of need

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 Fractions: Invert the numerator and denominator. vocabulary words and a variety of ways of representing The denominator is not the whole (for example 1 yellow 3 the answer, including fractions. blue expressed as 1/3).

Make a chart of the ways students come up with Expressions and idioms in language apply to math, i.e. (fraction, circle graph, table, tallies, etc.) “nothing is impossible.”

Congress Questions: Preference (i.e., the cat likes blue, red parrots talk more), How do you know what is probable, possible and magic, luck, or other outside factors, such as wind, impossible? 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 means that the event will occur about 2 out of every 3 trials.

 Sample space – the set of all possible outcomes of a trial. o 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 Learning Goal: Solve a probability problem: Students spin the wheel to I can say if an event will be certain, likely/ probable, explore concepts o probability, certainty, and impossibility unlikely/improbable or impossible. and design their own wheels. I can do an experiment and record the results in a table -Understand that experimental results will not exactly and circle graph and as a fraction. match theoretical probability.

Part 1 Before, Minds On or Student Success Criteria:

Student Exploration page 1– as a whole class on I can use vocabulary to clearly talk about probability projector. Review vocabulary: certain, impossible, experiments. outcome, probability, sample space (set of all possible outcomes), trial I can pose and solve simple probability problems by doing probability experiments. Which wheel gives you the best chances of winning? Why did you choose that wheel? I can use fractions to show the probability that an event will occur. Warm-up: How many sections in the wheel divided into? -I can show the results in different ways (circle graph, How many sections result in a small prize? How many sections result in a big prize? table) 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?

Part 2 – During, Work on It Strategies:

Activity A: Individually on computers in the  Use a circle graph library.  Use a table (See attached Student Exploration sheets) Questions 4  Express the answer as a fraction and 5 are open – students design their own wheels and Tools: test the results.  Student exploration sheet, pencils  Computers and username/password list If time, students complete Activity B and on-line  Chart paper and markers assessment.  Chart of learning goals  Overhead projector in class Part 3 – After, Consolidation Misconceptions:

In class – Whole group discussion, refer to learning Fractions: Invert the numerator and denominator. goals. The denominator is not the whole (for example 1 yellow 3 blue expressed as 1/3). Add to Student Success Criteria: If a few results are the same in a row, I am due for a -I can pose… by doing probability experiments different result.

-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? 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: Get the Gizmo ready:

 Be sure that Run the game is selected at the top. What is the most  Make sure you still have the original wheel (4 sections – 2 No prize, 1 Big prize, 1 Small prize). If not, click likely outcome? 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:

 Click Clear. Probability of  Select Design the game. winning

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:

 Click Clear.  Select Design the game. Making wheels  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 Learning Goal: understand how to use results of probability of getting a matching pair, carry out an an experiment where there are two possible experiment based on the problem, record results, outcomes to determine if the probability of revisit initial question with new insight. obtaining a desirable outcome will be likely, equally likely or unlikely.

Part 1 Activate Prior Knowledge Student Success Criteria: Show quote: "Chance has no memory", and have - I can use probability words to make predictions them discuss meaning. Review probability and discuss this problem language. - I understand that recording results during this experiment will help me determine the probability Answer questions below with one of these words: - I chose a tool to help me record my results in an certain, likely, equally likely, unlikely, impossible organized way 1- When I flip a coin it will land heads up. - I communicated my thinking with my partner and 2- Our class will have Phys. Ed next week. made connections to his/her ideas 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: Strategies: "Suppose you have two pairs of socks lying loose in  Use prior probability knowledge to make a drawer, one blue pair and one green pair. You reasonable predictions reach in without looking and pick two socks.  Consider the best tool for recording Which outcome is more likely: the two socks will collected data match or the two socks will not match?"  Carry out probability experiment to test prediction Give students a minute to consider and tally on the  Use questioning strategy to consider board the students' predictions of whether they reason for finding more mismatched pairs think it will be likely, equally likely or unlikely that Tools: they will pull out a matching pair.  Paper/markers Give pairs of students a paper bag with two sets of  Paper bags coloured tiles, ask them to try the experiment 30  Coloured tiles 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 Misconceptions: Have Gallery Walk to compare methods of keeping - students will think that because there are equal track of results. How did people record their data? numbers of blue and green socks, that a matching What categories did people choose? Were there outcome will be equally likely. some that confused you? Which method most - students will view experiment as a competition clearly showed the results? against each other, distracting them from the concept of probability Ask, "Based on what you found, would you like to - students will not understand that both two blues revise your prediction?" Ask students to explain and two greens constitute matching pairs, and so their thinking using probability language. do not need to be recorded in different categories

Congress Question: Why are we seeing more mismatched pairs than matched? 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 Learning Goal: Understand how to create a more likely that the pair of socks will be systematic list of all possible outcomes in a given mismatched, their task is to figure out why. problem and how to use this to determine Again considering the sock problem, students are probability as a fraction. asked to list all the possible combinations when two socks are picked without looking. Then they Understand that the fraction representing are asked to find the fraction that expresses the probability will be desirable outcomes over probability of finding a matching pair, given that possible outcomes. the numerator represents the number of times they list a desirable outcome and the denominator represents the total number of possible outcomes.

Part 1 Activate Prior Knowledge Student Success Criteria: Look again at outcome results from previous - I used a think aloud strategy to share my ideas. lessons sock experiment - I apply my previous knowledge about fractions to help conceive my approach. Questions: - I created a clear method of organizing my data. Why do you think most students got a result of - I made sense of my data and was able to explain more mismatched than matched pairs of socks? it using probability words

Can you think of a way to record all the possible outcomes?

Part 2- Work on It Strategies: * determine probability based on the experiment Sketch four socks on the board, two in one colour * use prior knowledge of fractions, remembering and two in another colour. Identify each sock by that the numerator is the part and the writing the numbers 1, 2, 3 or 4 beneath them. denominator is the whole. Say, "Can you think of a way to list all the possible * remember from previous learning that the outcomes?" Listen to their ideas and suggest they greater the denominator, the smaller the part. use pairs of numbers to list all the possible combinations when two socks are picked without Tools: looking. Model the process for two of the  Paper/markers combinations (1-2; 1-3) then ask students to find all the other combinations. Questions:

"What fraction represents the probability of finding a matching pair?"

Part 3 – Consolidation Misconceptions: Think/Pair/Share to discuss how they listed their * When organizing results students make possible outcomes. Ask them to share how they categories for "likely", "unlikely" and "equally determined the probability as a fraction likely", rather than "matching" and "unmatching" *When listing outcomes students don't realize that Congress Questions: the matching pair 1-2 is the same as 2-1. * Students mistakenly see 1-1 or 2-2 for example "How did you ensure that you had counted all the as possible outcomes. possible outcomes?" * When it comes to making a fraction they confuse single outcome with total number of instances of "How would you explain to a friend that when desirable outcome. pulling out two socks without looking, it is likely that socks will be mismatched?" Assessment FOR Learning Observation & Interview

Grade 5 (ext 4-6) Lost Socks Learning Goal: Use of Fractions to Express Probability

Mathematics Task/Problem Mathematical Thinking

A. Discussing socks combinations in terms of probability • 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 B. Understand they are gathering more likely: the two socks will match or the two socks will not info to help determine match? predictability of likelihood • Can you express the probability of finding a matching pair as a fraction? C. Logical list of outcomes logical Seating Plan D. Re-strategizing based on results Meesha/Sandy Maxwell/Anya E. Conceptualizing possible Mohamed/ Mikhail Michael/Keagan outcomes with drawings or numbers Kaelan/Aayana Joey/Kara F. Use previous understanding of Jamoree/Cory Max/Faisal fractions to find fraction to show probability Nicole/Sripiraba Byan/ Housam Misconceptions

Problem Solving Strategies

1. Used "matching" "unmatching" for tally (organized list) 2. Made systematic list for all possible outcomes 3. Counted desirable outcomes 4. Created fraction with desirable outcomes over possible outcomes 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: over-generalization 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

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).

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.

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 Name: Candace Minifie Adapted from: Math Makes Sense Unit 11, Lesson 4 Lesson Title: Tree Diagrams 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.

✔ 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.

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.

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.

• 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: .)

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) Use a tree diagram to show all the different pieces of pottery Omar could make. b) What fraction of the choices are mugs? c) What is the probability that a student will paint a yellow mug? Probability Unit: Culminating Activity / Lesson 8 Name: Candace Minifie Lesson Title: Paper Bag Probability STEM Design Challenge Date: April 25, 2014 Big Ideas

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.

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.

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.

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 Problem Solving • chooses and • with assistance, • chooses and • chooses and • chooses and carries out chooses and carries out some successfully successfully appropriate carries out a appropriate carries out carries out strategies, limited range of strategies, with appropriate effective strategies including tables appropriate partial success strategies and diagrams strategies, with little success • effectively • uses • limited ability to • some ability to • successfully designs a fair probabilities to design a fair game design a fair game designs a fair game design a fair game game Understanding of concepts • shows • shows very • shows limited • shows • shows thorough understanding by limited understanding understanding by understanding by providing understanding by giving appropriate giving appropriate giving appropriate reasonable giving but incomplete explanations of: and complete explanations of: inappropriate explanations of: explanations of: explanations of: • differences • differences • differences • differences • differences between predicted between predicted between predicted between predicted between predicted probabilities and probabilities and probabilities and probabilities and probabilities and actual results actual results actual results actual results actual results • how probabilities • how probabilities • how probabilities • how probabilities • how probabilities and number of and number of and number of and number of and number of outcomes can be outcomes can be outcomes can be outcomes can be outcomes can be determined determined determined determined determined Application of mathematical procedures • uses appropriate • limited accuracy • somewhat • generally • accurate procedures to accurate accurate accurately • major errors or calculate omissions in • several minor • few errors or • very few or no probabilities using calculating errors or omissions in errors or omissions fractions probabilities omissions in calculating in calculating calculating probabilities probabilities probabilities Communication • provides a clear • presentation and • presentation and • presentation and • presentation and presentation and discussion are discussion are discussion are discussion are explanation of unclear and partially clear and generally clear and clear, precise, and results imprecise precise precise confident

• uses language • uses appropriate • uses the most of probability (e.g., • uses few • uses some mathematical appropriate likely, probable, appropriate appropriate terms mathematical outcome) mathematical mathematical terminology terms 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