EMM209 Ass2

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ASSESSMENT COVERSHEET Student Name: Marney Holt Student Number: 11757565 I, Marney Holt confirm that this work is my own. I have acknowledged the work or ideas of other authors within the assignment and that the work has not been submitted for any other assignments. Subject Code & Name: EMM418: Mathematics: Content & Pedagogy Assessment Number: 2 – S&P Practical Learning Experiences Date Submitted: 10/04/2024 Subject Coordinator: Dr Kylie Press 1

Assessment 2 Part A: Sequence of three (EMM209) or four (EMM418) practical learning activities: Title: Exploring Probability Stage: Stage 2 (S2) Year: 3 Statistics and Probability Outcome: MA1-CHAN-01 - Recognises and describes the element of chance in everyday events - Conduct chance experiments, identify and describe possible outcomes, and recognise variation in results (ACMSP067) Working Mathematically Outcome: MAO-WM-01 - Develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly Concept:  Identify possible outcomes from chance experiments Syllabus content: Chance A - Use the term outcome to describe any possible result of a chance experiment - Record all possible outcomes in a chance experiment where the outcomes are equally likely - Predict the number of times each outcome might occur in a chance experiment involving a set number of trials - Conduct experiments and compare the predicted and actual results where the outcomes are equally likely Activity 1: Exploring Outcomes in Chance Experiments Preparation:  Have the link to the interactive spinner open in tabs  Print out the worksheet, enough for each student  Floor space for students to sit in front of the interactive whiteboard  Desks set up in rows Resources:  Interactive spinner https://polypad.amplify.com/p#random  Worksheet https://www.twinkl.com.au/resource/au-n-432-order-the-likelihood-of-familiar-events-activity-sheet  Probability Posters https://topteacher.com.au/resource/chance-probability-posters-set/ In this activity, begin with direct instruction (Siemon et al., 2015), by introducing the concept of probability, including the idea of outcomes in chance experiments (MA1- 2

CHAN-01) (NSW Education Standards Authority [NESA], 2022). To begin as a whole class, allow students to predict how many times the spinner might stop on green. Following this each student will have a turn coming up to the whiteboard to spin the interactive spinner. Students may have limited experiences exploring probability and chance in stage 1, leading to misconceptions based on naïve, intuitive, or subjective thinking. For instance, students may incorrectly respond based on non-probability thinking: “I will predict that they spinner will land on green because it is my favourite colour”. Following this, compare the predictions with the actual outcomes and discuss any patterns or trends noticed (MAO-WM-01) (NESA, 2022), as a whole. Constructivism is evident throughout as students engage in constructing their understanding of probability through active participation in hands-on activities, such as the spinner experiment. By predicting outcomes, analysing data, students actively construct their knowledge of probability concepts based on their experiences (Siemon et al., 2015). After this, provide students with a worksheet to apply their understanding of the topic. In the worksheet, students will be required to think critically about the order of events in relation to their likelihood of occurring, whether they are certain, likely or unlikely. However, students may struggle to grasp the concept of uncertainty and may assume that all events are either certain to happen or impossible. Finally, reinforce the concept of outcomes and how they can vary in chance experiments and encourage students to consider how they can apply their understanding of outcomes in everyday situations (MAO-WM-01) (NESA, 2022). Differentiation: Additional support: Display chance and probability posters (Top Teacher n.d.) around the classroom. It could be beneficial to start with simpler spinners with fewer outcomes and gradually increase the complexity as students become more confident. During the learning process, you can also break down complex vocabulary into simpler terms to avoid confusion. For example, instead of using ‘outcomes,’ explain that it means ‘possible results’ (Hurrell, 2015). Also start by completing a few problems together as a class. Extension: Introduce advanced probability terminology and concepts to stretch students' understanding. Have them analyse the data collected from the spinner experiment and explore patterns or trends in the outcomes. Additionally, provide more challenging probability problems that require higher-level thinking and problem-solving skills. These problems could involve multiple steps, encouraging students to apply various probability concepts and strategies to solve them. Activity 2: Listing Outcomes with Marbles Preparation:  Print out the worksheet, enough for each student  Sort out 3x marble bags, each with 2x red, 3x green, 4x purple and 6x black marbles  Desks set up in workstations Resources:  Probability Experiment Worksheet (Appendix)  3x Marble Bags, 6x Red Marbles, 9x Green Marbles, 12x Purple Marbles, 18x Black Marbles  Monte Hall Problem Activity Sheet (Appendix) After completing the introductory activity on Exploring Outcomes in Chance Experiments, students will advance to Activity 2: Listing Outcomes with Marbles. This activity builds upon their initial exploration of probability concepts, providing a structured opportunity for students to deepen their understanding through hands-on experimentation with marbles. In this activity, students will randomly select three marbles from a bag (MA1-CHAN-01) (NESA, 2022). Students often overlook the fact that each event in a random situation is independent and unaffected by past occurrences. They may assume that if a certain outcome has occurred repeatedly in the past, it is more or less likely to 3

occur again in the future. This belief stems from disregarding the independence of successive events in a random situation or believing that previous patterns dictate future possibilities (Bryant & Nunes, 2009). Students will each receive a marble bag and a probability worksheet. Students will take turns, with their partner reaching into the bag without looking and recording the outcome of each draw on their worksheet. Once everyone has completed their worksheet, discuss the results as a class, by using r eflective questioning (Siemon et al., 2015). During the discussion, analyse how many marbles of each colour were drawn, identifying the most and least common colours. We will then ask the students questions such as: What does it mean if a colour has a high probability of being drawn? What does it mean if a colour has a low probability of being drawn? How can we calculate the probability of drawing a certain colour marble? (MAO-WM-01) (NESA, 2022). Differentiation: Additional support: Step-by-Step Guidance: Break down complex problems into smaller, more manageable steps, and guide the student through each step with clear explanations. Encourage them to ask questions and seek clarification as needed. For example, provide prompts and cues to help the student identify relevant information and apply appropriate strategies to solve probability problems. Give students visual aids, such as fraction sheets for reference if needed Extension: Introduce the student to classic probability paradoxes that challenge conventional thinking. For example, consider the Monte Hall Problem, where a student must choose between three doors, one of which hides a prize. After making a selection, ask the student “Why they decided to choose that door” Activity 3: A Roll of the Dice into Probability Preparation:  Have enough six-sided dice for 2x per pair, easily accessible for students  Have whiteboard markers and easers, easily accessible for students. Make sure there is enough whiteboard markers for each student and at LEAST one easer per pair  Print out Bingo Cards (Appendix) and laminate them, enough for each student  Have the link to the random generator open in tabs, with ALL student names inserted  Floor/desk space, allow for a diverse range of learning environments Resources:  Probability Bingo Cards (Appendix)  Six-sided dice, enough for 2x pair  Whiteboard markers and easers  Random Generator https://wheelofnames.com/ Following the exploration of probability with marbles, students will progress to this activity further extending their learning by engaging students in a practical application of probability concepts and allow them to recognise the variation in results that can occur in chance experiments (MA1-CHAN-01) (NESA, 2022). In probability bingo, each student will receive a bingo card, whiteboard marker, and eraser. The game will be played in pairs, and each pair will be given a set of dice. When the dice are rolled, the outcome will be considered one 'bingo call.' Pairs will be chosen randomly and the bingo cards will be distributed. During gameplay, each pair will take turns rolling the dice, adding the two numbers displayed together, and calling out the outcome. Students will mark off the corresponding squares on their bingo cards. The student who marks off all the squares on their card will win. Allow students to play as many rounds as possible within the allocated time and monitor the gameplay, providing assistance and guidance as needed (Siemon et al., 2015). Finally, u tilise reflective questioning by asking students to reflect on which events were more likely or less likely to occur and why. P rompt 4

students to critically analyse the outcomes and consider why certain results occurred. Guide them to reflect on the factors that may have influenced the results, such as sample size, randomness, or biases (Hurrell, 2015). Students may generalise outcomes based on limited data, assuming that patterns observed in a small sample size apply universally (Bar-Hillel, 1984). It is possible that students may not fully grasp that probabilities are based on mathematical calculations (theoretical), rather than based on the observed (experimental). Therefore, expand students' understanding of theoretical probability by encouraging them to share strategies for filling out their bingo cards and e ncourage students to consider alternative strategies to increase their chances of winning (MAO-WM-01) (NESA, 2022) . Differentiation: Additional support: Prior to students proceeding with their independent work, facilitate guided practice sessions to familiarise them with the game. This allows students to practice completing their bingo cards and comprehend the outcomes of the dice rolls in a relaxed setting. Additionally, pair students with differing levels of comprehension together, matching those who require assistance with peers who possess a stronger understanding of the concepts. This arrangement encourages peer support and modelling. Lastly, offer sentence starters or prompts to aid struggling students in expressing their thoughts and strategies during the game. For example, provide prompts such as "I predict that the next roll will be..." or "I observed that...". Extension: To add more complexity to the game, you can introduce variations that require complex probability calculations. For example, you can use multiple dice with varying numbers of sides. Another way to make the game more challenging is by introducing conditional probabilities. In this case, you can add conditions to the bingo squares, and students will have to calculate probabilities based on given conditions. For instance, they can determine the likelihood of rolling a prime number on one die given that the other die rolled an even number. 5

Assessment 2 Part B: Analytical Report: Analysis of key S&P concept(s): Students' prior knowledge starts with their recognition that events may or may not happen, and they begin to describe familiar events that involve chance. As their understanding of chance situations becomes more sophisticated, they become able to describe the outcomes of chance experiments, develop an understanding of randomness, recognise bias, make predictions, and explain why expected results may differ from the actual results of chance events (ACARA, 2020). Activity one serves as the foundational step in introducing students to basic probability concepts and terminology, establishing a common understanding of key terms and ideas. Throughout the activity, students are introduced to the concept of probability as the likelihood or chance of an event happening. They learn to accurately define probability, identify and describe events and outcomes in simple chance experiments, and use language to describe the likelihood of events using terms such as certain, likely, and unlikely (UnC1; UnC2, ACARA, 2020). Additionally, students develop skills in predicting the probability of simple events and documenting the outcomes of chance experiments in tables, demonstrating how they may differ from expected results. By engaging in these activities, students lay the foundation for deeper exploration of probability concepts in subsequent activities. Moreover, students are expected to demonstrate proficiency in this area of knowledge (MA2-1WM) (NESA, 2022), by using proper terminology to effectively communicate their understanding of possible outcomes of chance experiments. By the conclusion of activity one, students are equipped with a solid grasp of basic probability theory, providing them with a strong foundation for further learning and application of probability concepts. Activity two serves as an extension of activity one, building upon the foundational knowledge established therein. While students are expected to possess a basic understanding of probability concepts and terminology, activity two provides them with opportunities for further development in applying these concepts to practical situations. Throughout the activity, students actively engage in critical thinking, learning to make informed decisions and gaining confidence in their problem- solving abilities. They are challenged to select and employ appropriate mental or written strategies to calculate theoretical probabilities and conduct experiments to gather data for experimental probabilities (MA2-2WM) (NESA, 2022). By actively participating in these tasks, students showcase their understanding of experimental probabilities as they identify all possible outcomes of one-step experiments, record outcomes in tables, and explain the concept that probabilities of all chance events lie somewhere between impossible and certain to happen (UnC3, ACARA, 2020). Additionally, students are encouraged to express their probabilities as fractions, decimals or percentages (UnC4, ACARA, 2020). Through activity two, students not only deepen their conceptual understanding of probability but also refine their mathematical communication skills and problem-solving techniques, setting the stage for further exploration and application of probability concepts. Activity three consolidates students' understanding of probability through practical application. It encourages students to apply their knowledge in a meaningful context, reinforcing their understanding of probability concepts. Through collaborative problem-solving, students analyse outcomes and reflect on the likelihood of events, deepening their understanding of probability and its applications. This activity allows students to showcase their understanding of probability by explaining why some results have a higher probability than others (UnC5, ACARA, 2020). By actively participating in this activity students demonstrate their proficiency in probability concepts, problem-solving skills, and critical thinking abilities, which are essential for building a deeper conceptual understanding of probability. Additionally, the collaborative nature of the activity fosters peer interaction and discussion, promoting a supportive learning environment where students can learn from each other and reinforce their understanding through dialogue and collaboration. Students next logical knowledge following on from activity three, would be in relation to UnC6, where must recognise combinations of events and the impact they have on assigning probabilities and solve solves conditional probability problems informally using data in two-way tables and authentic contexts (ACARA, 2020). By solving these conditional probability problems in an authentic context, students would recognise combinations of events and understand how the occurrence of one event affects the likelihood of another event happening. Justification for: a) For the design of sequenced activities b) Targeting possible student misconceptions: In primary mathematics education, employing effective pedagogical models and practices is crucial for promoting active engagement, conceptual understanding, and mathematical proficiency among students (Miller, 2019). Among the various pedagogical approaches, constructivism is as an approach that emphasises active learning, where students construct their 6

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