p238-278_Gr2ONAlgebraDataunit2 Probability_3rd

Unit 2: Probability Lesson Content Page Let’s Talk About Math: Introducing Probability 238 244 1 Read Aloud: That’s a Possibility!: First Reading 249 253 2 That’s a Possibility!: Second Reading 254 257 3 and 4 Describing Probability and Likelihood 262 264 3 Describing Probability 268 271 4 That’s a Possibility!: Third Reading 273 275 5 to 8 Investigating Probability Through Games and Experiments 5 That’s a Possibility!: Fourth Reading 6 Probability With Coins 7 Probability With Dice 8 Probability With Spinners 9 Reinforcing Probability Concepts and Skills

Let’s Talk About Math Math Introducing Probability Curriculum Expectations Use the following Let’s Talk About Math lesson to introduce the Probability strand to students. Provided are a series of prompts and possible partner investigations that stimulate discussions about math and how it relates to students’ lives. For interview prompts and questions to help students self-assess and reflect on their learning, see the Overview Guide or the Teacher’s Website. Data • D2.1 use mathematical language, including the terms ‘impossible,’ ‘possible,’ and ‘certain,’ to describe the likelihood of complementary events happening, and use that likelihood to make predictions and informed decisions PMraotcheesmseast:ical About the Problem solving, crreeofamlesmcotnuininngigc,aactoninndgnpercotviningg,, According to Marian Small, “probability is the study of measures of likelihood for various events or situations” (Small, 2009, p. 544). Students in grade two describe the likelihood of complementary events happening using language that includes ‘impossible,’ ‘possible,’ and ‘certain.’ Impossible and certain events are examples of complementary events since they cannot occur at the same time. Possible events may or may not occur, even if there is a theoretically high likelihood because randomness or chance play a role. Van de Walle and Lovin emphasize that “during the early years, our goal should be to develop in students an intuitive understanding of chance that will be a firm foundation for the more precise ideas that will be developed in grades 4 through 6” (Van de Walle & Lovin, 2006, p. 331). Marian Small suggests that as students are introduced to probability concepts, they need opportunities to discuss real-life situations and activities they would experience in their own lives to help deepen their understanding of the likelihood of certain things happening (Small, 2017, p. 628). Marilyn Burns notes that probability is also helpful in developing students’ critical thinking skills as well as giving them opportunities to apply their number skills. Looking at situations or running experiments has students predicting and questioning, as well as justifying their thinking about what will happen as they objectively take part in the activity. The games and activities that students take part in often require arithmetic skills, helping them strengthen and apply their number sense in context (Burns, 2000, pp. 59–60). Students can also predict whether the likelihood of the mode in one 238 Algebra and Data

game, activity, or experiment (e.g., 6 was rolled more than any other number) will be the same as the mode if the activity is repeated, using the same number of trials. As students carry out experiments to determine the likelihood of events in real-life, the experiment is serving as a mathematical model that simulates the problem. Students can refine and test out their model and decide whether it is helping them to answer questions and make predictions. These experiences serve as good opportunites to reinforce the Mathematical Modelling process. Use an anchor chart of the four components to highlight how students are working back and forth and in between the components as they refine their models. About the Lesson This lesson is made up of several Math Talks based on the pictures in “What Are the Chances?” (pages 18–19 in the Algebra and Data big book). The purpose of using visual images in this Math Talk is to give students an opportunity to view images from their everyday lives that connect to concepts in probability, stimulate discussion, and allow them to ask questions. This can further evoke inquiry about mathematics and how it relates to their lives. It is important for students to see how this math is connected to the world around them and help them see the importance of the activities we do at school. Each picture, or set of pictures, can support a stand-alone Math Talk and investigation. The Math Talks can be used on progressive days, one Math Talk and partner investigation per day, to investigate and explore how probability is used in our everyday lives. Through the discussion, a new investigation or problem may emerge. The Math Talks can also serve as a review of previously learned concepts to keep the concepts fresh in students’ minds. For each picture there are: • several possible prompts from which to pick and choose, depending on what concept you are working on or what your lesson is about, and • possible inquiries or problems for students to explore, since Math Talks also serve as natural springboards for carrying out investigations. Throughout the discussion, integrate the math talk moves on page 7. For example, continually encourage students to expand upon their responses and explain their reasoning. Have students respectfully react and respond to what other students are saying continued on next page Probability 239

Materials: so they become active listeners. Have students repeat or paraphrase what their peers have said. Ask questions such as “Do you agree?” or “Can anyone add onto what she said?” Have students turn and talk to a partner before sharing with the group. Provide wait time so students can reflect on what is being asked. Below is one way in which the Math Talks may be structured. Math Talk (10 minutes) Based on your area of study and learning goal, select some of the prompts or design your own questions to create the framework for your Math Talk. Rather than following as they are written, allow the students’ responses to guide the flow of the discussion, keeping in mind the goal of the lesson. Partner Investigation (10 minutes) Have students work in partners to further explore one of the prompts or the sample inquiry problem provided. All students may work on the same inquiry, or some may work on different problems, depending on their interests and level of understanding. This is a good assessment opportunity to uncover what students know and what misconceptions they may have. Consolidation (5 minutes) Building Social-Emotional Learning Skills: Critical and Creative Thinking: Strategically choose some of the students’ findings or solutions to discuss as a class, and focus on how the math relates to their lives as well as how different sets of data compare to one another. Build and nurture positive attitudes by discussing how students feel about math and what they find interesting about it. By making connections and sparking curiosity throughout the discussion, students can develop a positive attitude towards math and be motivated to engage in and persevere at problem solving. “What Are the NOTE: Select the prompts that best meet the needs of your students. Chances?” (pages 18–19 in the Algebra Die and Data big book) • What are we looking at here? Where have you seen it before? Why do we use Time: 10–20 minutes per day dice? (discussing 2–3 images each day) • What kinds of things do you think we can use dice for? • What games have required you to use dice before? Why are they helpful? 240 Algebra and Data • What number do you usually hope for when you roll a die? Why that number? • If we are playing a board game, which number do you want to roll? Why?

• What number is impossible to roll if I have one die? What if I have two dice? • Is it possible to roll a number less than 4? Why is that? • Is it more or less likely to roll a number greater than 4? • It is more or less likely to roll a number less than 5? Possible Partner Investigation • Have students each take one die and roll it 10 times to see what number they are more likely to get. Collect class data to see what numbers had the greatest and least frequencies. Loonie • What do you see in this picture? What name do we give this coin? What do you know about this coin? Have you ever used one before? • What kinds of things do we use the coin for? • Have you ever played a game with the coin? How did the game work? • Have you ever heard of “Heads or Tails”? When have you used it? Why do we use it? • How do we know which side of the coin is heads? What about tails? • If I we toss (flip) the loonie, where do you think it will land? Why do you think that? • Do you think it’s more or less likely to get heads when you toss a coin? Why or why not? Possible Partner Investigation • Give each student a coin and have them each toss the coin 10 times. Have students keep track of their outcomes and decide if they think it is more likely to get heads or tails when tossing a coin. Paper Bag & Tiles • What are we looking at in this picture? How many tiles do you see? What colour of tiles do you see? How many red tiles? How many blue tiles? How many yellow tiles? • Close your eyes and visualize all the tiles in the bag. Thinking about pulling one out, what colour would you pull out? Why do you think you would pull out that colour of tile? • Which colour of tile do you think will most likely be pulled out? Which colour would be impossible to pull out? What colour(s) is certain that you will pull out? Possible Partner Investigation: • Have students recreate the image using concrete objects. Have them make predictions about which tile will be pulled out most often. Then have students reach into the bag and pull out a tile, record the colour, and then put the tile back in. Have students repeat this 10 times to see if their prediction is correct. Probability 241

Spinners • Look at these two circles. What are they called? • Have you ever used spinners before? Where have you used them? • Have you ever played a game with spinners before? Which spinners look most like the spinner in your game? • How are these two spinners the same? How are these two spinners different? • If I spin the purple and orange spinner, what colour will it land on? How do you know? Why do you think that? Is more likely to land on the purple or orange? Why? • Close your eyes and visualize the red, yellow, and blue spinner. In your mind, spin that spinner. What colour does the arrow land on? Why do you think it lands there? • On the red, yellow, and blue spinner, what colour will the arrow most likely land on if I spin that spinner? Why do you think that? Possible Partner Investigation • Let students choose and create one of the spinners on this page. Have them spin the spinner 10 times to see what colour they are least likely to land on. Schedule • What do you call this chart? Have you seen something like this before? Why do we use a chart like this? • How is a class schedule helpful to students? Helpful to parents? Helpful to teachers? Helpful to the principal or vice-principal? • Who do you think uses a class schedule most often? • What subject would you like to see most often on the schedule? • What is the possibility of having Literacy on any day of the week? • What is the possibility of having Art on Wednesday? • What is the possibility of having Science on Thursday? • What subject are we most likely to have on any day? • What subject are we least likely to have on any day? • What subject would be impossible for us to find on our schedule? Possible Partner Investigation • Have students create their own schedule where they would be most likely to have their favourite subject every day. • Have students use their own class schedule and figure out the following: – Which subject are they most likely to have in a day? – Which subject are they least likely to have in a day? – Questions to ask about their schedule (e.g., Is it possible that we will have Science on Wednesday?). 242 Algebra and Data

Probability Lesson Topic Page 1 Read Aloud: That’s a Possibility!: First Reading 244 249 2 That’s a Possibility!: Second Reading 253 254 3 and 4 Describing Probability and Likelihood 257 262 3 Describing Probability 264 268 4 That’s a Possibility!: Third Reading 271 273 5 to 8 Investigating Probability Through Games and Experiments 275 5 That’s a Possibility!: Fourth Reading 6 Probability With Coins 7 Probability With Dice 8 Probability With Spinners 9 Reinforcing Probability Concepts and Skills Probability 243

1Lesson T hat’s a Possibility!: First Reading Language Introduction to the Read Aloud Curriculum Expectations That’s a Possibility! by Bruce Goldstone is excellent for investigating how math is embedded into literacy and in students’ lives. Throughout the Read Alouds for this book, there are two areas of focus: – Students apply their literacy skills such as inferring, using prior knowledge, and synthesizing information to understand the context, which supports many of the science curriculum expectations dealing with seasonal changes. – Students apply the mathematical processes to discover and explore the probability concepts embedded in the context. Oral Communication • 1.4 demonstrate an understanding of the information and ideas in oral texts by retelling the story or restating the information, including the main idea and several interesting details • 1.5 use stated and implied information and ideas in oral texts to make simple inferences and reasonable predictions, and support the inferences with evidence from the text • 1.6 extend understanding of oral texts by connecting the ideas in them to their own knowledge and experience; to other familiar texts, including print and visual texts; and to the world around them Reading • 1.7 identify the main idea and some additional elements of texts • 2.3 identify some text features and explain how they help readers understand texts Building Social-Emotional Learning Skills: Critical and Creative Thinking: This book allows students to look at math in everyday activities and allows students to use their imaginations to answer questions. It encourages students to ask questions, visualize different scenarios, and have fun as they build their mathematical vocabulary, their confidence, and their mathematical abilities. 244 Algebra and Data

About the Lessons Since there are 14 different scenarios related to probability, which are too many for students to discuss and solve in one reading, the text has been broken up into four Read Aloud lessons throughout the unit. Each Read Aloud lesson has a link to the math lessons that are interspersed, sharing common concepts and vocabulary students will need. Each of the scenarios presented in the book takes place on two pages. During the reading of the first page, students will apply their comprehension strategies to understanding new terms in the context of the page. They will then be able to apply these terms to solve problems as they delve into the math concepts in the scenario on the second page. You may decide to further break each lesson into two sessions. In the first session, read both pages of the scenario and discuss the text to dig deep into an understanding of the vocabulary. In the second session, read over the pages, this time focusing on the math and the questions relating to probability. Math Data Curriculum Expectations • D2.1 use mathematical language, including the terms ‘impossible,’ ‘possible,’ Teacher and ‘certain,’ to describe the likelihood of complementary events happening, Look-Fors and use that likelihood to make predictions and informed decisions Math Focus: Students visualize as a mental strategy and use their understanding of vocabulary to solve problems. Possible Learning Goal • Applies visualizations and knowledge of key terms to solve problems involving probability • Predicts answers to questions based on knowledge of vocabulary • Explains their visualization and how it helped solve their problem Probability 245

PMraotcheesmseast:ical About the Problem solving, An essential component in mathematics is communication. It is ccoomnnmeucntiincga,tirnegf,lecting, important for students to learn how to clearly and accurately reasoning and communicate their ideas in writing as well as orally. Being able to representing proving, understand and use mathematical vocabulary plays a large role in students’ ability to communicate. In grade two, students are required Math Vocabulary: to know and are assessed on using the terms ‘impossible,’ ‘possible,’ vpiossusailbizieli,typ, rmedigichtt, and ‘certain’ to describe the likelihood of events. They may also use words such as ‘more likely’ or ‘unlikely’ to compare the likelihood of possible events, although these terms do not need to be assessed and are formally introduced in grade three. Teachers can also model the use of new vocabulary during small-group instruction as well as large-group discussions. They can also add the new words to the Word Wall so they can be regularly reinforced. Materials: Literacy Focus: Students apply their literacy skills to understand the context, which helps them understand the math in the everyday activities of living things. Text Features: Key probability terms are written in a different colours. Literacy Assessment Opportunities Observations: Note each student’s ability to: – Attend to and engage with visual text and images – Make connections to the text, using prior knowledge – Visualize – Solve words using illustrations and context Scenario “What’s a Read Aloud: That’s a Possibility! Possibility?” (pages 2–3 in That’s a Possibility!), Summary: The text describes the possibility of events that might occur in a chart paper variety of life activities. Within this context, it also engages students in mathematical problem solving as questions are posed about the accompanying Written by Bruce illustrations. The descriptive language and text features allow for rich Goldstone visualization and making inferences based on prior knowledge and synthesis of the text. Text Type: Non-Fiction: Description – Factual NOTE: There are more prompts than are feasible to use in the allotted time. Information Select the prompts that best meet the needs and interests of your students. Time: 20–30 minutes 246 Algebra and Data

Before Reading Inferring/predicting Activating and Building On Prior Knowledge Building on prior • Show students the front cover and ask the students what they see. Read the knowledge title and the author’s name. Ask students to predict what they think the book might be about (e.g., probability of events happening, what might happen) and have them explain their reasoning (e.g., the word ‘possibility,’ the text that says “A Book About What Might Happen”). • Ask students if they think this is a fiction or non-fiction story and how they know. • Ask students if they have used a gumball machine before and have them explain how it works. Have students look at the gumball machine closely. Ask them to tell you what they see. Read the question, “Will I Get the Blue Gumball?” and ask them what they think. Have students discuss whether it is possible or impossible to get the blue gumball. • Ask the students to share some other activities they do or things they see every day that involve probability. Explain that probability is the study of the likelihood of events occurring in everyday lives. • Begin a co-created list that keeps track of the everyday situations involving probability throughout this book and in everyday events throughout the unit. • Setting a Purpose: Tell the students, “Now that we have made our predictions, let’s keep reading to learn more about probability.” During Reading Making connections/ Inside Cover visualizing/ • Show students the inside cover and ask them what they see. Ask students if they Problem solving/ reasoning and proving ever had a gumball before. Have students close their eyes and picture all the gumballs and ask which colour of gumball they would prefer. Analysing/using prior knowledge/predicting • Ask students which colours are possible to pick out of the pile and which Problem solving/ colours are impossible to pick out of the pile and how they know. communicating Page 2 • Show students page 2 and cover up the text at the bottom of the page. Read the title and ask what they think is possible (e.g., something that might happen). Have students name some things they know are possible. (e.g., It could rain; we will go outside today.) • H ave students look at the picture and think about how this connects to the word ‘possible.’ • Read the text, “Will this mouse find the cheese?” Have students close their eyes and visualize the maze and decide if they think the mouse could find the cheese. Ask students if they think it is possible. Have them explain their choice. • Read the rest of the text on the page, “That’s a possibility!” and have students turn and talk with a partner about why this is true. Probability 247

Analysing/text features/ Page 3 predicting • H ave students look at the text on the page and explain what they notice about Connecting and communicating it. (e.g., ‘Possibility’ is written in colour; the word ‘POPS’ is big and in an orange bubble.) Ask students why the author might put the word ‘possibility’ in a different colour from the rest of the text. (e.g., makes you look at it, shows it is important) Have students decide and explain why ‘POPS’ is big and in an orange bubble. They can predict what the page might be about by looking at the image and using the word ‘POPS.’ • Read the text at the top of page 3 and have students explain if this is possible and how they know. Read the rest of the text on the page, including the question at the bottom. Ask students to close their eyes and visualize all the other possibilities (i.e., which other animal balloons could pop). Have students explain how they know. After Reading Synthesizing • Ask students what they think the word ‘possible’ and the phrase ‘That’s a Visualizing/inferring possibility’ mean. Co-create an anchor chart of related terms and other probability words that arose during the conversation. • Ask students how visualizing the image in their minds helped them answer the questions that were on the page. Problem solving/ Possible Partner Investigation communicating • In partners, have students think of other scenarios that would be considered possibilities. They can write out the scenarios using the patterned text in the book as a model. Have pairs share their scenario with the rest of the class and then have all students discuss the possibility of the event occurring. Math Assessment Opportunities Observations: Pay attention to students’ understanding of the word ‘possibility’ (e.g., Are they able to figure out if there is more than one possibility or if it is impossible?). Pay attention to how students are visualizing the images they see and whether they can use them to help answer the questions posed during the Read Aloud. 248 Algebra and Data

2Lesson T hat’s a Possibility!: Second Reading Math Data Curriculum Expectations • D2.1 use mathematical language, including the terms ‘impossible,’ Teacher ‘possible,’ and ‘certain,’ to describe the likelihood of complementary events Look-Fors happening, and use that likelihood to make predictions and informed decisions Math Focus: Describe the possibility of real-life events using mathematical vocabulary Possible Learning Goals • Reflects on the importance of math in real-life contexts • Applies visualization and understanding of key concepts to solve problems involving probability • Demonstrates an understanding of probability concepts (e.g., possible, impossible, certain) • Identifies events that are possible, impossible, and certain • Explain why events have different possibilities PMraotcheesmseast:ical About the crrPeeorfamolesbmcoletnuimninngigcs,aoactlonivnndingngpe,rcotviningg,, In grade two, probability rests on students’ understanding of key vocabulary terms: ‘possible,’ ‘impossible,’ and ‘certain.’ They also need to understand how the terms relate to each other. For example, impossible and certain are complementary events. If it is certain that you will draw a green tile out of a bag that only has green tiles, then it is impossible for the complementary event of not drawing a green tile to occur. Marian Small notes that “Students are sometimes confused by the incongruity between the way particular words are used in everyday language and the way these terms are used in mathematics” (Small, 2010, p. 132), and so it is important for students to have many opportunities to work with and discuss the key terms used in probability in order to develop their understanding. continued on next page Probability 249

Math Vocabulary: About the Lesson possible, impossible, certain In this activity, students are introduced to the terms ‘possible,’ ‘impossible,’ and ‘certain.’ They are challenged to identify, describe, and explain activities seen in everyday life using these terms. Using images encourages students to visualize the quantities they see in their minds as they determine the chance of something happening. Materials: Literacy Focus: Students apply their literacy skills to understand the context, which helps them understand the math in the everyday activities of living things. Text Features: Key probability terms are written in different colours. Images help students visualize the context. Literacy Assessment Opportunities Observations: Note each student’s ability to: – A ttend to and engage with visual text and images – M ake connections to the text, using prior knowledge – V isualize – Solve words using illustrations, context, and prior knowledge During Reading Scenario “That’s “That’s Impossible!” Impossible!” and “That’s Page 4 for Certain” (pages 4–7 in That’s a Possibility!) • W ithout showing them the page, read the title on page 4 and ask students what Time: 20–30 minutes the title means. Ask them to predict what this page is about. Analysing/predicting • Show students the picture and read the sentence underneath the title. • H ave students look at the picture and think about what might be impossible in this image. 250 Algebra and Data

Reasoning and proving/ • Read the rest of the text to the students, up until it asks, “Can you see why?” communicating Ask students to explain why it’s impossible to knock down 12 pins. Have them think of something else that would be impossible with the bowling ball and pins. Ask how the words ‘impossible’ and ‘possible’ differ. Ask why an impossible event and its related possible event (knocking down 12 pins) cannot occur at the same time. Analysing/using prior Page 5 knowledge/predicting • H ave students look at the images on this page. Ask if they recognize the images and if they have ever seen them before. Discuss what might be possible or impossible. • Read the text around each egg and have students answer the question and explain their thinking. Ask students to describe something else that would be impossible to hatch from the eggs. Problem solving/ • H ave students turn and talk with a partner about other events they know are communicating impossible. Have each pair share one impossible event and discuss how they know it is impossible. Analysing/using prior “That’s for Certain” knowledge Page 6 Problem solving/ • Read the title of page 6 and ask students what the word ‘certain’ means. Read communicating the definition of ‘certain’ that is under the title and ask for examples of Predicting/visualizing everyday events that are certain (e.g., the Sun will set today). Ask if it is certain that the Sun will set today and whether the Sun cannot set today. Explain that it is impossible for the Sun not to set if it is certain that it will. The two events cannot both happen. • W rite the words ‘possible’ and ‘certain’ on the whiteboard and ask students to turn and talk with a partner about how these words are similar and how they are different. Discuss these similarities and differences as a class. • Read the question and have students turn and talk with a partner about what kind of fish would swim under the bridge. • Read the rest of the text and discuss why it would certainly have to be a goldfish. • H ave students close their eyes and picture putting one goldfish and one angelfish in the water. Ask if it is still certain that the fish that swims under the bridge will be a goldfish. Have students explain how they know. Page 7 • Show students the picture and ask them to close their eyes and visualize the strings and the scissors. Read the text and have students explain what is certain to happen to the puppet. • H ave the students open their eyes and describe what they saw as they thought about the strings being cut. After Reading Synthesizing • Ask what the two new words ‘impossible’ and ‘certain’ mean and add them to the anchor chart. Probability 251

Problem solving/ • On the whiteboard, draw a line and remind students of the three words: communicating ‘impossible,’ ‘certain,’ and ‘possible.’ Explain that the line is like a number line that shows the probability of events happening from really not possible to very possible. Have students turn and talk with a partner about where they would place ‘certain’ and ‘impossible’ on the line. Place ‘impossible’ on the far left of the line and ask where you might place ‘certain’ and why (e.g., on the far right because impossible can’t happen and certain is the opposite). Discuss with students where you would put the word ‘possible.’ Ask students where they would place the word ‘possible’ on the line and why they think so. Explain that possible events can occur anywhere between impossible and certain events. Discuss whether possible events can differ in their likelihood of occurring. Explain that this concept can be further explored in future lessons. Possible Partner Investigation • In pairs or small groups, have students discuss where on the probability line the scenarios we have read about in the book so far would be. Have them draw their own probability line and add in the events (e.g., bowling, eggs hatching, etc.). Assessment Opportunities (for Math) Observations: Throughout the reading, the related problem solving, and the discussions, note which key terms are too difficult or too easy for students so that next steps can be planned and lessons can be differentiated to meet individual needs. Conversations: As students are using the probability line, discuss their reasoning behind the placement of their events on the line. Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: • Ask students to explain how visualizing images in their mind helped them answer some of the questions today. Explain that visualizing helps mathematicians represent the math in their mind so they can see what is happening and explain it to others. Explain that visualization takes practice, and they will have many more opportunities to imagine events in their mind. 252 Algebra and Data

3 4Lessonsand Describing Probability and Likelihood PMraotcheesmseast:ical About the Representing, As Marian Small explains “you can never be sure what will happen on a communicating, particular occasion, unless the event is either impossible or certain” preroabsloenmingsoalvnidngp,roving, (Small, 2009, p. 544). For example, even though you may have math selecting tools class at 11:00 every day, it is not certain, because there could be an strategies and assembly or a fire drill that could interfere with the planned event. MpproaostbhsaibbVlioelic,tayc,ebirumtlaapinroy,s:msiboldee, Students can engage in discussions about what ‘possible’ means and whether some events are more likely to occur than others. When using a probability line like they did in the previous lesson, students can place possible events between the extremes of ‘impossible’ and ‘certain.’ They may even order activities according to which are more or less likely and use comparative terms such as ‘likely’ and ‘unlikely.’ This is not necessary since grade two students are only assessed on their understanding of the words ‘impossible,’ ‘certain,’ and ‘possible.’ As students experiment with probability, they can also predict whether the mode in one situation will be the same if it is repeated in the same manner or in another same-sized population. The element of chance may play a role. About the Lessons In these lessons students discuss and apply key terms needed to continue their work throughout the probability unit and participate in activities to develop and extend their understanding of ‘possible,’ ‘impossible,’ and ‘certain.’ Probability 253

3Lesson Describing Probability Math Data Curriculum Expectations • D2.1 use mathematical language, including the terms ‘impossible,’ ‘possible,’ Teacher and ‘certain,’ to describe the likelihood of complementary events happening, Look-Fors and use that likelihood to make predictions and informed decisions Materials: • D2.2 make and test predictions about the likelihood that the mode(s) of a data Digital Slides 158–160, set from one population will be the same for data collected from a different paper bags, variety of population coloured concrete objects (e.g., beads, pattern Possible Learning Goal blocks, connecting cubes) • Identifies and describes the likelihood of events occurring as possible, Time: 55 minutes impossible, or certain 254 Algebra and Data • Accurately identifies probability of events and gives reasons for their responses • Accurately describes probability of events using ‘possible,’ ‘impossible,’ or ‘certain’ and describes how the terms are different • Gives examples of events in their lives that are impossible, probable, and certain Minds On (20 minutes) • Show students Digital Slide 158 and ask them to describe what they see. (e.g., A paper bag with 2 stars, 1 heart, and 3 squares on it). Ask, “If I pull one shape out, could I pull out a circle?” (e.g., No, there are no circles.) Ask, “Could we say it’s impossible to pull out a circle?” (e.g., Yes, because there are no circles.) Students create one statement with each of these prompts about the mystery paper bag: – It is impossible to… – It is certain that… – It is possible… • Ask if it is possible that the shape to be pulled out is a star. (e.g., Yes, there are stars in the bag.) Ask what other possibilities there are. (e.g., square, heart) Ask students if they are sure that they will pull out a shape. (e.g., Yes, since there are only shapes in the bag.) Ask if it is certain they will pull out a shape. (e.g., Yes, because certain means we are sure it will happen.) • Show students Digital Slide 159 and ask them what is certain, impossible, and impossible to be pulled out of the bag. Have them explain their reasoning. • Repeat this line of questioning with Digital Slide 160 if students need more practice with using the terminology.

Working On It (15 minutes) • Explain to students they are going to create their own mystery paper bag. Tell them they can choose some concrete objects to put in the bag and their job will be to create three sentences about the mystery bag: one sentence about what’s impossible, one about what’s certain, and one about what’s possible to be pulled out of their mystery bag. • Have students work in pairs to complete the activity. Give each pair a paper bag and allow them to choose their own concrete objects. Differentiation • For ELLs, complete the Working On It section in a small group to help consolidate the key words (possible, impossible, and certain). Show students different examples relating to each key word and place them underneath the appropriate word cards so students have a reference to use. Use gestures (e.g., thumbs-up for possible, thumbs-down for impossible) to help students as they learn these new words. Assessment Opportunities Observations: • Pay attention to how accurately students are able to identify events using key terms. Watch to ensure they use the correct mathematical language to describe the probability of the event. Conversations: • As students are creating their own mystery bags, use the following prompts to check in with different pairs on their understanding of the probability language: – What do you know that is certain about your mystery bag? Explain. – What do you know that is possible about your mystery bag? Explain. – What do you know that is impossible about your mystery bag? What else is impossible? Consolidation (20 minutes) • Group pairs together and have them share their mystery bags and sentences with each other. • Meet as a class. Use some of the students’ mystery bags and discuss their statements. Have students take turns pulling one object out of the bag and then return it. Repeat this 10 times. Make a tally of class results. Compare the results to the students’ statements to see if they matched the results. Ask what the mode is. • Have students predict whether the mode would be the same if they repeated the activity again. Repeat the activity and have students compare the modes. Discuss what the mode would most likely be on future trials and why they think so. Ask why it is important to pull out the objects the same number of times in each trial. Probability 255

Further Practice • Have students switch bags and create three more probability statements using the new mystery bag. • Reflecting in Math Journals: Have students use pictures, numbers, and/or words to give an example of each of the key words used in the lesson (e.g., possible, impossible, certain). • B uilding Social-Emotional Learning Skills: Critical and Creative Thinking: Ask students what they wonder about probability. Have them share what questions they have and what else they might like to learn about probability. Remind them that mathematicians wonder all the time and it’s curiosity that helps us learn new things. Ask students about which words that were used today were words from last year or words they have heard another time. Remind them that it is okay if they don’t remember the words YET because we will have many opportunities to work with and learn these words together. 256 Algebra and Data

4Lesson T hat’s a Possibility!: Third Reading Possible Learning Goals • Reflects on the importance of math in real-life contexts • Applies understanding of key concepts to solve problems involving probability Teacher • Identifies the likelihood of events as possible, impossible, and certain; Look-Fors unlikely, less likely, and more likely PMraotcheesmseast:ical • Explains or shows why events can have different possibilities Problem solving, • Connects real-life experiences to probability crreeofamlesmcotnuininngigc,aactoninndgnpercotviningg,, About the Lesson Math Vocabulary: pclioeksretslayi,binlu,en,pliirkmoebplayobs(olsepi,bticloehn,aanl)ce, In previous lessons, students were introduced to the probability of events in terms of possible, impossible, and certain. This lesson reviews these terms and offers students new situations in which to discuss the probability of events. Students are introduced to the word ‘likely’ as another way of describing the probability of events. Literacy Focus: Students apply their literacy skills to understand the context, which helps them understand the math in the everyday activities of living things. Science and Overall Expectation: Technolog y Curriculum • Demonstrate an understanding that animals grow and change and have Expectations distinct characteristics Growth and Changes in Animals • 3.1 identify and describe major physical characteristics of different types of animals • 3.2 describe an adaptation as a characteristic body part, shape, or behaviour that helps a plant or animal survive in its environment • 3 .3 identify ways in which animals are helpful to, and ways in which they meet the needs of, living things, including humans, to explain why humans should protect animals and the places where they live Probability 257

Materials: Text Features: Key probability terms are written in different colours. Images help students visualize the context. Literacy Assessment Opportunities Observations: Note each student’s ability to: – Attend to and engage with visual text and images – Make connections to the text, using prior knowledge – Solve a word’s meaning using illustrations and context During Reading Scenarios “Will It BEE Will It BEE Likely? Likely?,” “Possibilities Pages 8–9 on the Wing,” “A Chance for Change,” “Pet • Show students page 8 and the title. Discuss the word ‘BEE’ and ask students Possibilities,” “That Seems Likely” (pages why it might be in all capital letters. Ask them what they notice about the word 8–17 in That’s a and whether it is correctly spelled. Write the words ‘BE’ and ‘BEE’ on the Possibility!) whiteboard and talk about which one should be in the title. Discuss why the author might have used the word ‘BEE’ in this title instead. Analysing/making connections • D iscuss the word ‘likely’ and ask students what it means if something is likely Reasoning and proving/ to happen. Discuss some examples of things that are likely to happen (e.g., It’s communicating likely that I’ll watch TV when I go home.). Ask whether ‘likely’ means the same as ‘possible.’ • Ask students if they have gardens at home and if they have seen bees or butterflies in their gardens. Ask them why they think bees and butterflies are drawn to flowers. Discuss what kinds of flowers they are attracted to and why. • Talk about the endangerment of bees and how planting specific types of flowers in the garden can help with that. • Possible Partner Investigation: Have pairs of students research the importance of bees and how we can help them. Have students share their findings as a class. • Look at the image on page 8 and ask students what they see. (e.g., Lots of pink flowers, one white flower.) Ask them if it is possible that the bee will land on a white flower and have students explain their thinking. 258 Algebra and Data

• Look at page 9 together. Ask students if it is possible that the butterfly will land on a yellow flower. Ask students if it is more or less likely that the butterfly will land on a purple flower rather than the yellow flower. Have them turn to a partner and talk about why. Discuss why their predictions may not come true. Making connections/using Possibilities on the Wing prior knowledge Pages 10–11 Using text features • Show students the image on page 10 and ask what type of birds they are, Problem solving/reasoning whether they have seen these birds before, and what they know about them. and proving/ Discuss why the birds’ feathers are so colourful. communicating • Show students page 11. Ask them what the birds are sitting on and if they’ve ever seen birds do this. Ask why they think most of these birds look the same (e.g., travel together, same bird families). Ask students if they think it is important for birds to travel together or why they think birds travel together. • Look at the title of the page. Discuss with students why the author used the word ‘wing’ in the title. • Look at page 11 and ask students why the author chose to print the words in white. • Read the text on page 10 and have the students answer the question and explain their thinking. • H ave students estimate how many birds are on page 11 and explain how they decided on their estimate. • H ave students work through the prompts on page 11 with a partner and then share their answers and reasons with the class. Remind students to use the correct mathematical language in their answers. Connecting/using prior A Chance for Change knowledge Pages 12–13 Using text features • Show students pages 12 and 13. Ask them if they have seen/used one of these Problem solving/ machines before and where they have seen/used it. Talk about how people use reasoning and proving/ it and what is needed to make the machine work. Draw students’ attention to the two images. Discuss how the machines are similar and how they are communicating different. Ask students which machine they would want to use and why. • Together, look at the title. Ask students what they think the author means by the word ‘change’ and why he might have used it in this title. (e.g., You use change to get something out of the machine.) • Ask students what words are in different colours and why. Discuss the words and their meaning and add the word ‘improbable’ to the key term anchor chart. Ask whether they think the word ‘improbable’ means the same as ‘impossible.’ • U se the prompt on page 12 and have students explain which colour gumball is most possible to get using this machine and which colour is improbable to get. Ask students which colour would be impossible to get. • H ave students order the gumball colours in order, from the most possible to the least possible to get if they used this machine. Ask students how the image of the gumballs outside of the machine helps them with this question. Ask Probability 259

whether their predictions are certain to come true. Discuss how chance can play a part in what happens with events that are possible. • Possible Partner Investigation: Have students place the items on page 13 in order from the most possible to the least possible item they would get if they used this machine. Remind them to explain their reasoning. Connecting/using prior Pet Possibilities knowledge/predicting Pages 14–15 Problem solving/ • Show students pages 14 and 15. Ask them if they have pets at home and if their communicating/reasoning pets have favourite toys or treats. and proving • H ave them look at the image of the cat and predict what this page might be about. Ask what the page with the image of the dog might be about. • U sing the prompts on page 14, ask students which colour is possible for the cat to pounce on. Ask them which colour is most possible that the cat will pounce on. Ask which colour is least possible. • Read the second prompt to the students and have them turn and talk with a partner about what colours would be impossible for the cat to pounce on. • U sing the prompts on page 15, have students decide which shapes are most possible and least possible for the dog to eat. Have students explain their thinking to a partner and then have a few students share their reasons with the class. Connecting/using prior That Seems Likely knowledge/predicting Pages 16–17 Connecting/inferring • Read the title of the page and ask students what it means. Read the question on page 16 aloud and have students recall the meaning of ‘probably.’ Ask students if they have ever been swimming or if they have jumped into the water before. Have them close their eyes and visualize jumping into the water. Ask if they can predict what might happen by visualizing and making a connection to their experience. • O n page 17, read the question about what the ant will probably do with the leaf. Ask what clues they found in the photo to help them figure it out. • Ask students to visualize the size of an ant and the size of a leaf. Ask them if they think the leaf is heavy for the ant and what they can infer about the strength of an ant. Ask students to describe something they have carried that would be as heavy to them as the leaf is to the ant. Assessment Opportunities (for Math) Observations: Throughout the reading, the related problem solving, and discussions, note how students are using mathematical language of probability. Note students’ ability to do the following: – R ecognize possibilities – Accurately describe events using correct mathematical language (e.g., possible, impossible, certain) – Connect ideas in the text to their own real-life experiences 260 Algebra and Data

Materials: Math Talk: “Is It Possible?” (page Math Focus: Using mathematical language to describe the likelihood of 20 in the Algebra and different events occurring Data big book and little books) Let’s Talk Teaching Tip Select the prompts that best meet the needs of your students. Integrate the math • Show students page 20 in the big book and read the title to the students, or talk moves (see page 7) throughout have a student read the title. Judging from the title, what do you think you are Math Talks to going to do on this page? maximize student participation and • What words do you see at the bottom of the page? What does each word mean? active listening. Give a real-life example of each word. • Point to the image of an elephant in the classroom. What is happening in the image? (e.g., elephant sitting in a classroom) Is this possible? (e.g., No, it’s impossible.) How do you know it is impossible? (e.g., Elephants don’t come into the classroom.) How might we change this picture to make it certain? (e.g., change the elephant to a student) • Point to the image of a child opening the gift. What is happening in the image? (e.g., child opening gift and it’s a cat) Is this possible? (e.g., It’s improbable.) Why do you think so? (e.g., I already have a cat; a pet doesn’t come as a wrapped present.) Is it impossible? (e.g., No, you could wrap up a cat.) How might we change this picture to make it more possible? (e.g., put a book or a toy in the present) • Continue this line of questioning with the other images on the page. Partner Investigation • Have students work together and come up with other events they could put with each word on the page (i.e., possible, certain, impossible). Follow-Up Talk • H ave each pair of students share one of the events thought of and have the class decide on the likelihood of the event. • A sk the students what helps them decide the probability, or likelihood, of the events. Probability 261

5 8Lessonsto Investigating Probability Through Games and Experiments Math Data Curriculum Expectations • D2.1 use mathematical language, including the terms ‘impossible,’ ‘possible,’ and ‘certain,’ to describe the likelihood of complementary events happening, and use that likelihood to make predictions and informed decisions • D2.2 make and test predictions about the likelihood that the mode(s) of a data set from one population will be the same for data collected from a different population Previous Experience About the with Concepts: Students have explored Students in the primary grades begin to develop informal ideas about probability to describe chance and probability. Participating in simple games and experiments will the likelihood that help them explore these ideas, make predictions about the possible everyday events will outcomes, and then compare them to the actual results. Students can also occur, using determine the mode, predict whether it will be the same if the experiment mathematical language is repeated, and then carry out the experiment again to compare the such as ‘possible,’ modes, ensuring that the same number of trials occur each time. Many ‘certain,’ and ‘impossible.’ young children believe that they can control the fact that an event will happen because they want it to or because it is what happened the last PMraotcheesmseast:ical time (Van de Walle, 2001, p. 354). Probability games and experiments Representing, support students in examining their misconceptions about chance communicating, (Ontario Ministry of Education 2007a, pp. 31–34). preroabsloenmingsoalvnidngproving, Many people have misconceptions about probability and chance. For example, some people think that if the number 5 is rolled on a die six times in a row, there is less chance that a 5 will be rolled a seventh time. In reality, the probability that a 5 will be rolled is 1 in 6 and is the same probability each time the die is rolled. The rolls are independent events and the outcome of one does not affect the others. As students carry out simple experiments with rolling dice, spinning spinners, or flipping coins, they can figure out the mode and predict whether it will be the same if they repeat the experiment. Students need to understand that the same number of trials or the same-sized population must be used in order to make comparisons. 262 Algebra and Data

Math Vocabulary: Investigating the likelihood of different outcomes in games also reinforces cpgisormraepoimnpsibnsuoaen,lsbstehss,irlei,,ibetamydxled,pioe,sped,,coreidetsmairsctieilaebsni,,intlis,t,y, students’ understanding of concepts related to fairness. Students can learn that a fair game means that players have the same probability of winning in terms of the rules and the tools that are involved. Chance plays a role, when tools like dice and spinners are involved, and affects whether the next game will have a similar or different outcome. About the Lessons These lessons offer students experiences to help them explore the concept of probability based on playing games and conducting experiments. They can also predict whether the mode will be the same if the game or experiment is repeated, using the same number of trials. Probability 263

5Lesson T hat’s a Possibility!: Fourth Reading Teacher Possible Learning Goals Look-Fors • Reflects on the importance of math in real-life contexts PMraotcheesmseast:ical • Applies understanding of probability to solve problems crrPeeorfamolesbmcoletnuimninngigcs,aoactlonivnndingngpe,rcotviningg,, • Uses mathematical language (e.g., possible, certain, impossible) to describe Math Vocabulary: observations pcocoeuhsrtatscanoibicnmle,e,e,m,simpopidrpnoenob,seatsrbr,iilbafeall,sec,,e cards, suit • Identifies events that are more or less possible (e.g., landing on a certain colour when spinning a spinner or tossing a certain number with a standard die) • Explains why events have different possibilities (e.g., If there are more red than blue counters in a bag, pulling out a red one is more likely because there are more of them.) About the Lesson In previous lessons, students were introduced to the possibility of events using the terms ‘possible,’ ‘impossible,’ and ‘certain.’ This lesson reviews these terms and offers students new situations in which to discuss the possibilities within the context of a game or experiment and predict whether the results and/or the mode(s) will be the same if the activity is repeated using the same number of trials. Literacy Focus: Students apply their literacy skills to understand the context, which helps them understand the math in everyday activities. Text Features: Key probability terms are written in different colours. Images help students visualize the context. 264 Algebra and Data

Materials: Literacy Assessment Opportunities “Odds Aren’t Strange,” “Spin Observations: Note each student’s ability to: to Win,” “It’s in the Cards,” – Attend to and engage with visual text and images “On a Roll” (pages 18–25 in – Make connections to the text, using prior knowledge That’s a Possibility!); Digital – Visualize Slide 52; BLM 51: Which – Solve a word’s meanings using illustrations and context Colour Do You Choose?, BLM 52: Spin to Win; decks of During Reading playing cards, pencils and paper clips (to use as Building Social-Emotional Learning Skills: Healthy Relationship Skills: spinners if needed), plastic or Before beginning the reading and carrying out the following lessons, discuss actual coins what the purpose of games and experiments is when investigating probability. Explain that students will be working together to learn about the likelihood of Time: 20–30 minutes events happening. It is not about winning or losing. Co-create a list of how the students can respectfully interact with each other in the upcoming lessons and activities. Refer to it if students start getting competitive rather than being objective mathematicians. Making connections Odds Aren’t Strange Pages 18–19 Making connections • Ask students what they know about the word ‘odd.’ (e.g., it’s not even; unusual, Predicting strange, etc.) Read aloud the statement on page 18, “Odds are the chances that Teaching Tip something will happen.” Some of the theoretical • Ask students if they recognize the American coins in the photo. Show students probabilities (e.g., 50%, one out of two possibilities) in the Canadian equivalent of the coins shown, using Digital Slide 52: Coins. Ask the prompts on page 18 are students if they recall (e.g., financial literacy) the symbols represented on each beyond the curriculum coin and if they can identify how much each coin is worth. expectations for grade two probability. Use the • D iscuss the meaning of ‘heads’ and ‘tails’ and relate these words to the sides of examples shown as models to experiment with only. coins. Ask what the possibilities are if they flip the coin. Ask whether both outcomes could happen at the same time. Ask students to predict the side on which the coin will land if they toss it. Record students’ predictions. Have students flip the coin two or three times to test their predictions. Probability 265

Reasoning and proving Spin to Win Pages 20–21 Analysing/predicting Reasoning and proving • U se the prompts on page 20 to have students predict what will probably happen with each spinner. Say, “If you could choose a colour that the spinner will probably land on, which colour would you choose? Turn and talk with a partner to share why you chose that colour.” Provide BLM 51: Which Colour Do You Choose? to students so they can experiment and prove their predicitions. • Ask students what they notice about the game board on page 21. Pose some of the following prompts: What kinds of spaces are on the game board? Where is the start? Where is the finish? How many players are there? Who is winning? • Read the prompts on page 21. Students choose which spinner each of the players on the game board should use to get closer to the finish line and why. • Possible Partner Investigation: In small groups, have students play a similar version of the game board that is depicted on page 21 of That’s a Possibility! using BLM 52: Spin to Win. While students are playing, note when they use one spinner instead of the other one. (e.g., Are they considering the possibilities and at certain spots on the game board, does their choice of spinner change? For example, did they choose a different spinner to avoid a number that might make them go back to the start?) Making connections It’s In the Cards Pages 22–23 Connecting/ reasoning and proving • H old up a deck of cards and ask students if they recognize what it is. Give pairs Teaching Tip of students a deck of cards so they can get familiar with the 52 cards by sorting them. Say, “What do you notice? What do you wonder?” When using the prompts in the book, be sure to • Show the picture on page 22 and ask if they sorted their cards in the same way. emphasize more likely, less likely, and likely Pose some of the following prompts: How is your organization the same? How as possible answers for is it different? How many cards are in a deck? Which colours? Which numbers? their predictions. What are face cards? What is a suit (what symbols are used)? Making connections • D iscuss the meaning of the words ‘probable’ and ‘possible.’ Ask students the prompts on page 23 and in partners, have them answer each of the prompts (students should have access to their deck of cards to reference as they work). • Possible Partner Investigation: Have students create their own possible/ probable question for their deck of cards. On a Roll Page 24 • H old up a die (number cube) and ask students if they’ve ever played games that use dice. Hand out a die to each student. • Ask students what they notice and what they wonder. Have them share with a partner and then record their observations on an anchor chart. • Show the photo on page 24. Make connections between the anchor chart and the details in the photo. Ask students the prompts on page 24 and have them share their ideas. • Ask students where ‘impossible’ would apply and why. 266 Algebra and Data

• Possible Partner Investigation: Give pairs of students one die, and have them roll the die 20 times, 10 by each partner. Before beginning, ask them to create an organizer to record their results (e.g., tally table) and have them predict how many times they will toss each number (1, 2, 3, 4, 5, 6) during the experiment. Have students predict whether the mode (the most common number) will be the same if they repeat the experiment and explain why they think so. Students can repeat the experiment to see if the mode is the same. Assessment Opportunities (for Math) Observations: Throughout the reading, the related problem solving, and discussions, note how students are using mathematical language of probability. Note students’ ability to do the following: – Predict possibilities and explain their reasoning – Accurately describe events using correct mathematical language (e.g., likely, possible, impossible, certain) – Connect ideas in the text to their own real-life experiences Probability 267

6Lesson Probability With Coins Teacher Possible Learning Goals Look-Fors • Reflects on the importance of probability in real-life contexts (e.g., tossing a coin to decide who goes first) • Describes the probability of an event using mathematical language • Carries out an experiment after first predicting the outcome, identifies the mode(s), and repeats the experiment after first predicting whether the mode will be the same • Explains what the mode represents in some of their experiments and predicts whether it will be the same if they repeat the experiment • Describes the results of tossing coins using mathematical language (e.g., heads, tails, possible) • Predicts a result based on the possible outcomes and previous experiments • Creates and records experimental results using a tally table • Identifies the mode and predicts whether it would occur again if they repeat the experiment Materials: Minds On (15 minutes) Digital Slide 161, a variety • P roject Digital Slide 161. Ask the following questions to ensure that students of plastic or real coins, mini- whiteboards and markers understand which side of each coin represents heads and tails: What coins (sticky notes and pencils) are these? How do you know? What are the two sides of each coin called? Time: 45 minutes How do you know which one is which? • Explain that students are going to carry out an experiment to find out the results of flipping a coin several times. Ask students what the mode of an experiment would be if a coin was flipped 5 times and it landed on heads 2 times and tails 3 times. Ask whether they think the mode will be the same if they repeat the experiment. Flip a coin 5 times and compare the results to their predictions. Working On It (15 minutes) • H ave students work in pairs. Together, create a tally table that students can use to record their results. Coin Tosses Heads Tails 268 Algebra and Data

• Explain that students will take turns flipping the coin 10 times and recording the results in the tally table. They identify the mode, if one exists, and then predict whether the mode will be the same if they repeat the experiment. • Students repeat the experiment, recording their results in another tally table. Remind them that each experiment must have the same number of trials or ‘flips’ in order to make comparisons. Differentiation • Some students may need assistance setting up their tally table or having it on a BLM. • For students who need more of a challenge, have them repeat the experiment two or three times, predicting the mode each time. Assessment Opportunities Observations: • Pay attention to students’ ability to accurately record their results using a tally table. • Pay attention to students’ use of the language of probability. • Pay attention to partner conversations about the results they are getting. Conversations: • Is your prediction looking likely so far? • Did you and your partner have the same prediction? • If not, which of your two predictions is looking more likely so far? • If you could change your prediction, what would you change it to? Why? Consolidation (15 minutes) • Meet as a class. Discuss how students carried out their experiments in the same way so results could be compared. • Ask what students predicted before carrying out the first experiment and whether their predictions were true. • Ask students whether their mode for the first experiment was the same as the mode for the second experiment. Count the number of people who got the same results both times compared to the people who did not. • Have each pair of students identify the mode for their first experiment and record the results. Repeat this for the second experiment. In each case, compare the number of times that heads came up to the number of time tails came up. Ask students what the mode is for the compiled results of the first and second experiments. • Ask whether students think the results of the first experiment can be used to predict the results of the second experiment. Discuss the element of chance and the role it can play in the outcomes of experiments. Probability 269

Teaching Tip Further Practice You may choose to • P artner/Small Group Discussion: Have students share with a partner a share some examples to support student real-life situation in which someone might be asked to predict “heads or conversations (e.g., tails” when a coin was tossed. Ask, “What would you suggest to the person a referee tossing a who predicts which side of the coin will come up? Why would you suggest coin to start a sports that?” game) 270 Algebra and Data

7Lesson Probability With Dice Teacher Possible Learning Goals Look-Fors • Reflects on probability in real-life contexts (e.g., rolling dice to play a variety Materials: of board games) Digital Slide 162, BLM 46: Line Plot Template, • D escribes the probability that an event will occur using mathematical dice (one for each language pair of students), • Carries out an experiment after predicting the outcome and possible chart paper, paper, mode(s), and repeats the experiment to discover whether the mode will be pencils the same Time: 50 minutes • Describes the results of rolling dice using mathematical language Teaching Tip • Predicts a result based on the possible outcomes (e.g., 1, 2, 3, 4, 5, 6) • Identifies the mode in a set of data This slide uses a • Consistently repeats an experiment and identifies the mode line plot to record • Accurately records results using a tally table experiment results. • Compares the mode in the two experiments Make the connection between this Minds On (20 minutes) organizer and Data Management. • S how students a standard six-sided die (number cube). Ask students when they have used a die before and what role it plays in games. • Discuss what outcomes are possible when a die is rolled (1, 2, 3, 4, 5, 6) and what outcomes are impossible. Ask students whether any outcomes are certain and how they know. • P roject Digital Slide 162. Explain that the class is going to carry out an experiment using a die. Each student will roll the die once and the results will be recorded on the line plot. Ask what title and labels are necessary and what each ‘X’ will represent. Ask how many ‘X’s’ will be on the completed line plot. • H ave students take turns tossing the die, predicting the outcome before they do so. Assign some students to record the results on the line plots. (See BLM 46: Line Plot Template.) • D iscuss whether students were able to correctly predict what the outcome of their roll would be. • Analyse the results. Ask what the mode(s) is. Ask students whether they think the mode will be the same if they repeated the experiment and why they think so. Probability 271

Working On It (15 minutes) • Tell students that they are going to work in pairs and carry out the same experiment two times. For each experiment, they take turns rolling the die 10 times, predicting what number they think they will roll each time. • Together, create a tally table that students can use as an example when creating their own organizer. Students use a different tally table for each game. • Students can also keep a separate tally of the number of times per game when they correctly predicted the number. Differentiation • Some students may benefit from having a BLM of the tally table organizer so they do not need to create it themselves. Assessment Opportunities Observations: • Monitor students’ reactions when they compare the results of the rolls to the predictions. If students start to feel like they are ‘winning’ because they have more correct predictions, remind them that they are mathematicians carrying out an experiment and that they are not playing a game that involves winning or losing. Consolidation (10 minutes) • Meet as a class. Discuss whether students were able to accurately predict what numbers would be rolled and whether they improved as the experiment progressed. Ask students whether they were more successful at predicting the outcomes when they flipped the coins in the previous lesson. Discuss why they think this might be. (e.g., There were only two outcomes with the coins and there were six possible outcomes with the die.) • Have students analyse their data and identify the mode for the first experiment. Ask whether they had more than one mode. Compile the results for the class using a line plot. Repeat this for the second experiment. • Analyse the results. Ask students whether the mode for the experiments was the same. Discuss whether they think they can predict what the mode will be from experiment to experiment. Talk about the role that chance plays when rolling a die. 272 Algebra and Data

8Lesson Probability With Spinners Teacher Possible Learning Goals Look-Fors • Predicts results of an experiment using mathematical language, consistently Materials: carries out the experiment, and compares the mode of the results Digital Slide 163, whiteboard markers, • Repeats the experiment to determine whether the mode will be the same as “What Are the Chances?” (page 18 in the Algebra the mode in the first experiment and Data big book), BLM 53: My Spinners, • Offers reasons for the outcomes of their experiments BLM 54: My Spinner Tally, BLM 55: • Predicts a result based on the possible outcomes (e.g., I think I will spin Spinner yellow more often than blue and red because the yellow space is larger.) Template, crayons and • Consistently carries out the experiment two times, using the same number of pencil trials and conditions crayons, paper clips, • Identifies and compares the mode of two experiments pencils • Gives reasons for the outcomes of the experiments Time: 60 minutes Minds On (20 minutes) • Show “What Are the Chances?” (page 18 in the Algebra and Data big book.) • Draw students’ attention to the two spinners. Pose some of the following prompts: – Which colour do you think the mode would be on the purple and orange spinner if you spun it 10 times? Why? – Which colour do you think the mode would be on the yellow, red, and blue spinner if you spun it 10 times? Why? – What is different about these two spinners? What is the same? – What do you think the mode would be if you spun the orange? • Have students work in small groups, with each group sharing one of the little book versions of the big book. Each group uses a paper clip and pencil to spin on each of the spinners 10 times. They can record their results in two different tally tables. You may want to co-create the tally tables. (See Digital Slide 164: My Spinners Tally.) Encourage students to predict the mode for each spinner. • Meet as a class. Record the mode for each group. Discuss the results and why or why not each group found the same mode. Working On It (20 minutes) • Show students Digital Slide 163, which has a variety of spinners. Explain that they are going to work in pairs, design their own spinners, and test whether the mode will be the same or different if they repeat an experiment. Each student Probability 273

Teaching Tip can design his/her own spinner, using the templates on BLM 53: My Spinners or using the blank spinner on BLM 55: Spinner Template to create an original one. Use math talk moves to have students • O nce they have designed their spinners, students in the pairs take turns testing agree, disagree, or build on peers’ out the spinners. In each case, they predict what the mode will be, spin the comments during the spinner 10 times, and record their results in a tally table. Students repeat the discussion. experiment and record the results in a separate tally table. They compare the modes of the two trials of each experiment. Differentiation • You may decide to have students create their own tally tables, or you can use some of the examples on the BLM. (See BLM 54: My Spinners Tally.) • For students who need more of a challenge, have students design a spinner that will most likely have the same mode when the experiment is repeated, and one spinner that makes it difficult to predict what the mode will be. Assessment Opportunities Observations: • Pay attention to students’ predictions of what the mode will be in their experiments. Do they think the larger space will be the mode? Do they talk about the element of chance? Do they think that the results of one experiment will affect the results of the next experiment? Conversations: • Ask any of the following prompts as you circulate: – Did you accurately predict what the mode would be? – Did you get the same mode when you repeated the experiment? Why might it be hard to predict the mode? Consolidation (20 minutes) • Have students meet with another pair. They can take turns predicting what the mode is for their experiments and then comparing the predictions with the results. • Meet as a class. Discuss how students predicted the mode and whether they based their decision on the design of their spinner. • Discuss whether the mode was the same when the experiment was repeated. • Ask students which spinners made it easier to predict the mode. • Ask students whether they think the results of one experiment affect the results when the experiment is repeated. Discuss the role that chance can play when using a spinner. 274 Algebra and Data

9Lesson Reinforcing Probability Concepts and Skills Math Data Curriculum Expectations • D2.1 use mathematical language, including the terms ‘impossible,’ ‘possible,’ and ‘certain,’ to describe the likelihood of complementary events happening, and use that likelihood to make predictions and informed decisions • D 2.2 make and test predictions about the likelihood that the mode(s) of a data set from one population will be the same for data collected from a different population Possible Learning Goals • Describes probability in everyday situations and in simple games • Reflects on probability in real-life contexts • Describes the probability that an event will occur • Describes the outcome of a simple game or experiment using mathematical language and identifies the mode Teacher • Uses mathematical language (e.g., possible, likely, certain, impossible) to Look-Fors describe game and experimental results (e.g., heads, ‘5,’ blue, likely) Previous Experience • Predicts a result based on the possible outcomes (e.g., heads or tails) with Concepts: • Predicts what the mode will be in an experiment and identifies it from Students have had opportunities to test analysing the results probability situations with rolling dice, tossing coins, • Describes the role that chance can play in events that are possible spinning spinners, reflecting on the results they obtained and comparing them to others’ results to confirm or check their predictions. Probability 275

PMraotcheesmseast:ical About the Lesson Representing, arsacenonofdlldmveiscmnpttgrriuona,nvgtciei,ncogasgni,teenilsnpeegcrco,ttibinrnelgega,mstooonlisng This lesson takes place over three to four days and involves students rotating through a series of activity centres. If you have noticed Math Vocabulary: through prior lessons that you may need to be available to support leeqsusallilkyelliyk,emly,ore students with a particular activity centre, focus on that centre and approach it as a guided group, while students work through the other centres independently. You can set up activities through which students rotate over the course of a few days, or they can freely visit the centres. Select the way that best suits your class. The following schedule offers an example of how students may rotate from activity to activity. likely, likely, fair, unfair Suggested Rotation Schedule: Activity/Day Group A Group B Group C Group D Session 1 Centre 1 Centre 5 Centre 4 Centre 3 Session 2 Centre 2 Centre 1 Centre 5 Centre 4 Session 3 Centre 3 Centre 2 Centre 1 Centre 5 Session 4 Centre 4 Centre 3 Centre 2 Centre 1 Session 5 Centre 5 Centre 4 Centre 3 Centre 2 Materials: Set-Up of Centres deck of playing cards • If you decide to follow the rotation schedule (above) have students rotate through a series of five activities. Feel free to create your own and substitute them for the suggestions below. It is also beneficial if some centres are open exploration so students can work independently. Examples for Centres: • Games or Digital Games/Activities: Students can engage in digital math games and activities that are designed to test probability with a variety of materials. • Play Your Cards Right!: Provide a deck of playing cards, pre-shuffled. Attach a simple prompt such as, “How many ways can you sort these cards? What do you notice?” Have students complete the following sentence starter with an example using the deck: When we can’t see all of the cards, how likely is it that you can choose a ______________? (e.g., black card, red card, face card, king, queen, five, heart, etc.) We sorted the cards by ___________________. 276 Algebra and Data

Materials: • What Are the Chances?: Using copies of BLM 56: Class Schedule, BLM 56: Class Schedule, brown paper prepare one cut up version of an entire schedule in an envelope or brown bag or envelope, colour paper bag for students to choose from. Have students use a full copy of counters BLM 56 to reference as a game board and provide them with coloured counters to keep track of their “draws.” Students can play in pairs or in a Materials: small group. Have each student choose a different day of the week in the pencils, paper clips, schedule. One at a time, students pick one of the class pieces from the dice envelope. If it is on their day, they mark the spot with a colour counter. If not, the turn passes to the next person. Each time, the student replaces the Materials: class piece back into the envelope. Have students reflect on the results of brown paper bags, the game and why they chose the day of the week that they did. concrete objects Games: Materials: • Open Investigation: Put out a variety of board games that students may BLM 55: Spinner Template be familiar with that include an element of probability (e.g., they use a die, a spinner). Have students explore a game and discuss how fair they think the game is and why. • It’s in the Bag: Provide some brown paper bags and a variety of concrete objects for students to create a probability bag. Once the bag is created, students need to answer these questions: What concrete object is the most likely to be pulled from the bag? What concrete object is the least likely to be pulled from the bag? If you were going to play a game with your bag, which concrete object would you choose? Why? • It’s Just a Game: Provide copies of BLM 55: Spinner Template. Have students create their own game board and spinners for a game. Students can play their game and reflect on how it works. Preparing for the Next Day (20–30 minutes) • Introduce each activity centre so students clearly understand all instructions, where to access the necessary materials, and what to do when they are done. Have two or three students model each activity. Answer any questions that they may have. Investing this time will help to ensure that students can work independently and stay on task while you are working with another group on the guided math lesson. Following Days • Assign students to groups, and assign groups to their activity centres for the first rotation, which can last about 20–30 minutes. You can then carry out a second rotation on the same day, with the remaining rotations on other days. • B uilding Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: Review and discuss some of the activities students have done in this unit. – Which activity did you enjoy the most? What did you like about it? Probability 277

– What did you learn about probability that was the most interesting? – H ow have these probability activities helped you to become a better mathematician? – How might you be able to use probability in your life? Give some examples. Use the following reference chart to identify which centres allow you to assess the following overall and specific expectations related to probability: Describe probability in Describe the likelihood Describe the probability everyday games an event will occur an event will occur Digital Games 4 4 Play Your Cards Right! 4 What Are the Chances? 4 4 Open Investigation It’s in the Bag 4 It’s Just a Game 4 278 Algebra and Data

References Beatty, R. & Blair, D. (2015). Indigenous pedagogy for early mathematics: Algonquin looming in a grade 2 math classroom. The International Journal of Holistic Early Learning and Development, 1, 3–24. Beatty, R., & Bruce, C. (2012). From patterns to algebra: Lessons for exploring linear relationships. Toronto, ON: Nelson Education Ltd. Burns, M. (2000). About teaching mathematics: A K–8 resource. Sausalito, CA: Math Solutions. Caldwell, J.H., Kobett, B.M.C., & Karp, K.S. (2014). Putting essential understanding of addition and subtraction into practice in prekindergarten-grade 2. United States: National Council Of Teachers Of Mathematics. Chapin, S.H., O’Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn, K–6, Second Edition. Sausalito, CA: Math Solutions. Clements, D.H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge. Copley, J.V. (2000). The young child and mathematics. Washington, DC: National Association for the Education of Young Children. Fosnot, C.T. (2007). Bunk beds and apple boxes: Early number sense. In Contexts for learning mathematics: Investigating number sense, addition, and subtraction (K–3). Portsmouth, NH: Heinemann. Lawson, A. (2016). What to look for: Understanding and developing student thinking in early numeracy. Don Mills, ON: Pearson Canada. Lawson, A. (2015). What to look for: Understanding and developing student thinking in early numeracy. Don Mills, ON: Pearson Canada. Moss, J., Bruce, C.D., Caswell, B., Flynn, T., & Hawes, Z. (2016). Taking shape: Activities to develop geometric and spatial thinking, grades K–2. Don Mills, ON: Pearson, Canada. Newcombe, N.S. (2010). Picture this: Increasing math and science learning by improving spatial thinking. American Educator: Summer 2010, pp. 29–43. Ontario Ministry of Education (2020). The Ontario curriculum, grades 1–8: Mathematics context, 2020. Toronto, ON: Queen’s Printer for Ontario. Ontario Ministry of Education (2016). A guide to effective instruction in mathematics, grades 1–3: Geometry and spatial sense. Toronto, ON: Queen’s Printer for Ontario. References 279

Ontario Ministry of Education (2014). Paying attention to spatial reasoning, K–12. Toronto, ON: Queen’s Printer for Ontario. Ontario Ministry of Education (2013). Paying attention to algebraic reasoning, K–12. Toronto, ON: Queen’s Printer for Ontario. Ontario Ministry of Education (2007a). A guide to effective instruction in mathematics, grades K to 3: Data management and probability. Toronto, ON: Queen’s Printer for Ontario. Ontario Ministry of Education (2007b). A guide to effective instruction in mathematics, grades K to 3: Patterning and algebra. Toronto, ON: Queen’s Printer for Ontario. Small, M. (2017). Making math meaningful to Canadian students, K–8, Third Edition. Toronto, ON: Nelson Education Ltd. Small, M. (2014). Uncomplicating algebra to meet common core standards in math, K–8. Toronto, ON: Nelson Education Ltd. Small, M. (2013). Making math meaningful to Canadian students, K–8, Second Edition. Toronto, ON: Nelson Education Ltd. Small, M. (2010). Big ideas from Dr. Small: Creating a comfort zone for teaching mathematics, grades K–3. Toronto, ON: Nelson Education Ltd. Small, M. (2009). Making math meaningful to Canadian students, K–8. Toronto, ON: Nelson Education Ltd. Van de Walle, J.A. & Lovin, L.H. (2006). Teaching student-centered mathematics grades K–3, Volume One. Boston, MA: Pearson. Van de Walle, J.A. (2001). Elementary and middle school mathematics: Teaching developmentally, Fourth Edition. Toronto, ON: Pearson Education, Inc. 280 Algebra and Data



Algebra & Data Teacher’s Guide Part of Math Place Grade 2 Lead Author: Diane Stang, National Math Consultant for Scholastic Education Authors: Krista Clarke, Halton District School Board (retired) Kristin Marshall, Halton District School Board Math Reviewers: Carrie Byer, York Region District School Board Kim Lacelle, Ottawa Catholic School Board Brenda Lamb, Peel District School Board Kristin Methot, Thames Valley District School Board Indigenous Consultant: Mélanie Smits, Conseil scolaire catholique Franco-Nord Director of Publishing: Molly Falconer Project Manager: Jenny Armstrong Editors: Rachel Albanese, Sundus Butt Art Director: Kimberly Kimpton Designer: Kimberly Kimpton Production Specialist: Pauline Galkowski-Zileff Copyright © 2021 Scholastic Canada Ltd. 175 Hillmount Road, Markham, Ontario, Canada, L6C 1Z7 ISBN: 978-1-4430-6692-1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, scanning, recording or otherwise, without the prior written consent of the publisher or a license from The Canadian Copyright Licensing Agency (Access Copyright). For an Access Copyright license, call toll free to 1-800-893-5777.