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Patterns & Relations/Data & Probability Front Matter Content Page Contents 1 2 Math Place Components for the Patterns & Relations/ Data & Probability Kit 4 Overview



Contents 2 Math Place Components for the Patterns & Relations/ Data & Probability Kit 4 Overview 9 Getting Started with Patterns & Relations 13 1: Patterns and Relations 89 2: Equality and Inequality 151 Getting Started with Data & Probability 154 1: Data 208 2: Probability 240 References

Math Place Components for the Patterns & Relations/Data & Probability Kit Read Aloud Texts Four Read Aloud texts are included to set a whole-class focus for learning and to provide realistic contexts for the math and help students connect with it. Big Book The Patterns, Relations, Data, and Probability big book (with an accompanying 8 little book copies) is used to develop spatial reasoning and visualization and to create context for the math. Math Little Books Two math little books (8 copies of each) are used in guided math lessons with small groups for focused and differentiated instruction tailored to the needs of the students. They also offer opportunities to observe and assess students as they verbalize what they visualize, and apply math concepts in problem-solving situations. Teacher’s Website A variety of online projectable and printable resources are available to support instruction and students’ problem solving. Also included are modifiable Home Connections letters and Observational Assessment Tracking Sheets. 2 Patterns & Relations/Data & Probability

Teacher’s Guide A Teacher’s Guide supports teachers in building students’ conceptual understanding of math by providing hands-on learning experiences, using a variety of concrete materials and tools. This allows students to apply all of the core and curricular competencies as they solve problems. • Lessons include an About the Math section, which incorporates recent research to explain math concepts and why they are so critical to students’ current and future learning. • Detailed three-part lesson plans include rich problems for the students to solve and many opportunities for collaborative learning, communication of ideas, independent problem solving, and practice. The consolidating prompts and discussions are designed to connect students’ mathematical thinking and bring clarity to the big ideas. • The three-part lessons offer suggestions on how to differentiate the learning to meet the specific needs of all students. • Activities develop mental math strategies based on conceptual understanding and many ‘visualization’ activities are included to support and develop students’ spatial reasoning and visualization skills. • Lessons support assessment for learning by offering suggestions on how to assess through observations, conversations, and products. There are also ‘Teacher Look-Fors’ to further support assessment and evaluation, and to serve as a guide for co-constructing success criteria with your students. • Further Practice and Reinforcement activities offer students the opportunity to practise newly acquired skills. • Math Talks provide support for posing comments and questions that promote interactive talk. • Blackline Masters (BLMs), such as creating growing patterns and find the missing number, are included in the book of Reproducibles and can easily be used to prepare for lessons. There are also some graphic organizers which help students record and organize their observations and mathematical thinking. In addition, all BLMs are available digitally on the Teacher’s Website. Overview Guide A digital Overview Guide provides support for teaching all areas of math in Math Place, Grade Two. The guide offers background information about the math, the role of spatial reasoning and visualization in mathematics, balanced problem solving, assessment, First Peoples Perspectives, and strategies for meeting the needs of all students in your classroom. In addition, the Overview Guide outlines and explains the various high-impact instructional approaches used in the resource. 3

Overview What Are Patterns and Relations? Patterns are sequences that repeat, grow, or shrink according to certain rules. These rules represent the ‘relations’ or relationships between the elements in the pattern. Algebra is about relationships, including the relationships found in various patterns. Beatty and Bruce cite research about the importance of patterning and algebra in the early years. “Patterning activities are introduced in elementary school so that students can think about relationships between quantities early in their math education, which is intended to help them transition to formal algebra in middle school and high school” (Beatty & Bruce, 2012, p. 1). Students learn about algebraic relationships more formally in later grades. It is their experiences with patterns and relations in the primary grades that help to prepare them. Young children recognize patterns intuitively. In the primary grades students explore different representations of patterns (e.g., concrete materials, pictorial representations, numbers) and identify the relationships between them. As students develop a sense of number in the patterns, they begin to recognize the concept of equality, which supports their development in algebraic reasoning. They also investigate variables, or quantities that can change, and constants, or quantities that remain the same in equations and in everyday contexts. Grade two students investigate equality and inequality and use symbols (=, ≠) to represent the two concepts. This understanding is crucial to the study of algebra and algebraic relationships in later grades. Students also predict and solve for how quantities change in equations and expressions. The study of how quantities change involves working with variables, or unknowns, that are presented in different positions within an equation and reflect various contexts. What Are Data and Probability? The data and probability units offer many opportunities to explore real-world problems and make connections to students’ everyday lives. Students need to recognize that data can be represented in different ways and can be used to help answer questions. By learning to represent data and interpret the results, students will recognize the relationships between different sets of data. This knowledge will support their thinking in different curriculum areas (e.g., Science) as well as help them to develop critical thinking skills. According to Marian Small, probability “is the study of measures of likelihood for various events or situations” (Small, 2009, p. 544). Young students think about familiar events and the likelihood that they will occur. This requires learning the accompanying vocabulary so students can describe likelihood and make comparisons. In grade two, students use the words ‘certain,’ ‘uncertain,’ ‘more likely,’ ‘less likely,’ and ‘equally likely.’ Including Number Throughout the Strands Number is the foundation for all mathematical understanding and permeates all strands. Whenever possible, helping students make connections between the concepts they have explored with numeration and the strands in this unit will 4 Patterns & Relations/Data & Probability

reinforce their mathematical understanding and support flexibility in their thinking. Skip counting and operational sense, for example, relate directly to patterns and relations as students identify and perform the operations required to extend a number pattern or to determine equality. Marian Small notes that, “if we can connect a new idea being taught to related ideas that have been previously learned, it is more likely that the new knowledge will be assimilated” (Small, 2013, p. 18). Spatial Reasoning Spatial reasoning involves the way we move things and see things both mentally and physically. In order to build spatial reasoning skills in our students, we need to include activities that use tools for representation, involve concepts of space, and help them use the processes for reasoning. This resource offers teachers a variety of ways to help students develop their spatial reasoning skills. Patterning requires students to use both visual-spatial skills as well as number sense. The graphical representations that students work with in data management are highly visual and allow students to explore connections and relationships between space, pictorial representations, and the data. Importance of Multiple Representations Communicating and representing are curricular competencies that support students to make their thinking visible. The use of multiple representations is important to make connections between concepts across strands and to differentiate for students who may use different approaches to solving problems. Small notes that “the more flexible students are in recognizing alternative ways to represent mathematical ideas, the more likely they are to be successful in mathematics” (Small, 2013, pp. 24–25). Providing opportunities for students to share their thinking aloud, using a variety of representations, and having them expand their repertoire by experimenting with alternate models, will support their overall understanding. A Balanced Approach: Acquiring Conceptual Understanding, Basic Skills, Math Facts, and Mental Math Strategies A conceptual understanding of mathematics allows students to develop a deep understanding of math concepts which they can apply to a variety of real-world problems. Small cites research by Carpenter and Lehrer that explains conceptual understanding as “the development of understanding not only as the linking of new ideas to existing ones, but as the development of richer and more integrative knowledge structures” (Small, 2017, p. 3). Students can investigate math through problem solving and then develop conceptual understanding through meaningful math talk and consolidation, where alternative strategies are honoured and discussed. It is also important for students to develop basic skills and proficiency within the different strands, for example becoming proficient with the commutative property to help as they find the missing number in equations. Students also gain fluency in using various strategies in order to learn their addition and subtraction facts to 20. Practice plays a key role so students internalize the skills and can independently apply them in new situations. By using concrete materials and discussing their ideas during Math Talks, students develop mental math strategies that help them visualize the concepts and gain 5

automaticity of number facts and calculations. This balanced approach aligns with Indigenous teaching that emphasizes “experiential learning, modeling, collaborative activity and teaching for meaning” (Beatty & Blair, 2015, p. 5). Embedding First Peoples Perspectives Curricular Competencies • U nderstanding and solving: Engage in problem-solving experiences that are connected to place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures • C onnecting and reflecting: Incorporate First Peoples worldviews and perspectives to make connections to mathematical concepts First Peoples • Learning is holistic, reflexive, reflective, experiential, and relational (focused Principles of Learning on connectedness, on reciprocal relationships, and a sense of place) • Learning recognizes the role of Indigenous knowledge • Learning is embedded in memory, history, and story • Learning requires exploration of one’s identity When students are conceptually learning Big Ideas about patterns, data, and probability, working within a meaningful context is critical so they can connect their own experiences to the math and see its relevance. Throughout this resource, there are several opportunities to deepen understanding through other cultural lenses, including First Peoples perspectives. This approach reflects themes identified as Characteristics of Aboriginal Worldviews and Perspectives. These include: • E xperiential Learning: “Look for ways to incorporate hands-on learning experiences for students” (British Columbia Ministry of Education, 2015, p. 36). Find ways to incorporate natural objects from the outdoors as students create patterns using concrete materials. • T he Power of Story: Think of place-based stories that can represent the math and celebrate the cultural identities of all students. “Metaphor, analogy, example, allusion, humour, surprise, formulaic phrasing, etc. are storytelling devices that can be applied when explaining almost any non-fiction concept. Make an effort to use devices of this sort in all subject areas and to draw upon stories of the local Aboriginal community” (British Columbia Ministry of Education, 2015, p. 30). • Emphasis on Identity: “Embrace learner-centred teaching practice” (British Columbia Ministry of Education, 2015, p. 26). • C onnectedness and Relationships: “Look for ways to relate learning to students’ selves, to their families and communities, and to the other aspects of Aboriginal Worldviews and Perspectives” (British Columbia Ministry of Education, 2015, p. 16). Learning is a social process, not only in the classroom but within the family and community as well. • Local Focus: Look at how Indigenous people in this area would use patterns, data they collect, and probability—in what context, for what purpose (focus on local Indigenous history, experience, stories, imagery, ecology) (British Columbia Ministry of Education, 2015, p. 22). 6 Patterns & Relations/Data & Probability

• E ngagement with the Land, Nature, and the Outdoors: “Look for opportunities to get students interested and engaged with the natural world immediately available (place-based education in the area near your school). Illustrations using locally observable examples and phenomena, physical education activities, homework assignments, and student projects are examples of opportunities to promote this type of engagement” (British Columbia Ministry of Education, 2015, p. 24). In consultation with community members, think of place-based stories that can represent the math and celebrate the cultural identities of all students. Include activities that allow students to actively experience the learning. Find ways to incorporate natural objects from the outdoors as students create patterns using concrete materials. This inclusive approach allows all students to make connections between mathematics and their identities. mtoaMhftienaatruthehtleeeaTssbraseeldokgu5snirn–,imnn1oig5arnygwfrbhoeeereneuensvdeedr Math Talks the day. There are numerous Math Talks linked to the lessons in Patterns & Relations/ Data & Probability which support the understanding of math concepts through purposeful discussion, help to reinforce and extend the learning, and offer opportunities for further investigation. (For more on Math Talks, see the Overview Guide.) In order to maximize student participation and active listening, you can strategically integrate the following ‘math talk moves’ into all discussions (adapted from Chapin, O’Connor, & Canavan Anderson, 2009). Math Talk Moves Example Talk Move Description Wait Time Teacher waits after posing a question before – Wait at least 10 seconds after posing calling on a student so all students can think. a question. – If a student has trouble expressing, say “Take your time.” Repeating Teacher asks students to repeat or restate what “Who can say what said in another student has said so more people hear the their own words?” idea. It encourages active listening. Revoicing Teacher restates a student’s idea to clarify and “So you are saying…. Is that what you Adding On emphasize and then asks if the restatement is were saying?” Reasoning correct. This can be especially helpful for students who may need support with the language. “Can someone add on to what just said?” Teacher encourages students to expand upon a proposed idea. It encourages students to listen to peers. Teacher asks students to respond to other “Who agrees? Who disagrees?” .” students’ comments by contributing and justifying their own ideas. “You agree/disagree because (sentence starter) 7

aabeSiltpnsBneenitmrydsuucoaosidosemltkdotoeuteartpinnrnuhyawatstggrscsgo,wogherGwucinlkadteagrhehbosthnteehlw.omeondrbutoeitmehsttuavotgkeotuMahherbmskieeenterehiddssnes,ktare.esnt,sy Growth Mindsets and Positive Attitudes in Math Math Place offers many opportunities to build and reinforce habits of mind, growth mindsets, and positive attitudes towards math, beginning with an introductory lesson, Instilling a Growth Mindset (see the Overview Guide), which sets the tone for nurturing and developing growth mindsets in students. The lesson can be used at the beginning of the year to establish what a growth mindset is and to develop the criteria for building growth mindsets. The pertinent messages can be regularly reinforced throughout the year using the prompts and suggestions that are embedded in many of the lessons in this kit. For interview prompts and questions to build growth mindsets and positive attitudes, see the Overview Guide or the Teacher’s Website. 8 Patterns & Relations/Data & Probability