p1-8_Gr3BC_Patterns-Data_frontmatter

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



Contents 2 Math Place Components for the Patterns & Relations/ Data & Probability Kit 4 Patterns and Relations/Data and Probability Overview 9 Getting Started with Patterns and Relations 10 Inviting Patterns and Relations into the Classroom 13 1: Patterns and Relations 73 2: Equality and Inequality 106 Getting Started with Data and Probability 107 Inviting Data and Probability into the Classroom 109 1: Data 174 2: Probability 219 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, to help promote a growth mindset in math, and to provide realistic contexts for the math concepts. Big Book The Patterns, Relations, Data, and Probability big book (with an accompanying 8 little book copies) is used to develop spatial reasoning 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. 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 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 skills. • Lessons support assessment for learning by offering suggestions on how to assess through observations and conversations. There are also ‘Teacher Look-Fors’ to further support assessment, 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 dot configurations, are included in the Reproducibles guide 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. Teacher’s Website A variety of online resources, including Digital Slides, are available to support instruction and students’ problem solving. Also included are the digital big book, modifiable Home Connections letters and Observational Assessment Tracking Sheets. Overview Guide A digital Overview Guide provides support for teaching all areas of math in Math Place, Grade Three. The guide offers background information including the role of spatial reasoning in mathematics, balanced problem solving, assessment, and strategies for meeting the needs of all students in your classroom. In addition, the Overview Guide outlines and explains the various instructional approaches used in the resource, including three-part problem- solving lessons, whole-group lessons, guided math lessons, and Math Talks. 3

Patterns and Relations/ Data and Probability 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 patterns. Beatty and Bruce explain that “patterning activities are introduced in elementary school so 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 grade four. It is their experiences with patterns and relations in the primary grades that help to prepare them. In grade three, students create and represent increasing and decreasing patterns in many ways, as well as generalize to make pattern rules and describe them using words and/or numbers. They can also compare representations of patterns (e.g., concrete, pictorial, and numerical) and make connections between them. Experiences with number patterns are valuable since they extend students’ understanding of number relationships among numbers to 1000 as they recognize patterns inherent in our base ten system. In previous grades, students investigated the concepts of equality and inequality and the symbols that represent them (=, ≠). This understanding is crucial to the study of algebra and algebraic relationships in later grades. In grade three, students apply this knowledge to solve for an unknown number in one-step addition and subtraction equations. The unknown quantity can be located in various positions within the equation (e.g., start unknown, change unknown, and result unknown). What Are Data and Probability? The Data unit offers many opportunities to explore real-world problems and make connections to students’ everyday lives since people are continually exposed to data through advertising, news, polls, and social media. Students need to recognize that data can be represented in different ways and serves a real-life purpose in making informed decisions. In grade three, students collect and organize data in various ways, display sets of data using one-to- one correspondence in pictographs, bar graphs, charts, and tables, and analyze different data sets in order to make observations, comparisons, inferences, and then draw conclusions within a context. According to Marian Small, probability “is the study of measure 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 4 Patterns & Relations/Data & Probability

the accompanying vocabulary so students can describe likelihood and make comparisons. Exploring probability allows students to make predictions and to develop critical thinking skills that will support their thinking in different curriculum areas (e.g., Science, English Language Arts). Grade three students also learn about chance as they investigate simulated events using materials such as coins, spinners, and dice. Including Number Throughout the Strands Number is the foundation for all mathematical understanding and permeates all strands. Helping students make connections between the concepts they have explored in patterns, data, and in other strands will 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. 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 locations of objects, their shapes, their relations to each other, the paths they take as they move” (Newcombe, 2010, p. 30). It plays an integral role in all mathematical learning. For example, students can develop a strong understanding of equality if they can see the equivalence represented with concrete objects. This helps students develop the mental images that support their ability to visualize mathematical concepts in a meaningful way. The patterns and data strands offer 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 investigate in data are highly visual and allow students to make comparisons and draw conclusions. Importance of Multiple Representations The mathematical process of representing is evident as students make their thinking visible. Using multiple representations allows students to make connections between concepts and offers differentiation 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 represent their thinking in many ways and to verbally explain their thoughts to peers allows all students to expand their repertoire and experiment with alternate models. “The more ways that children are given to think about and test out an emerging idea, the better chance it has of being integrated into a rich web of ideas and relational understanding” (Van de Walle, 2001, p. 34). 5

First Peoples A Balanced Approach: Acquiring Conceptual Understanding, Principles of Basic Skills, Math Facts, and Mental Math Strategies Learning 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. Marian Small cites research by Carpenter and Lehrer (1999) 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, in which alternative strategies are honoured and discussed. It is also important for them to develop basic skills and proficiency within the different strands. Practice plays a key role so students internalize the skills and can independently apply them in new situations. By using concrete objects and discussing their ideas during Math Talks, students develop mental math strategies that help them visualize the concepts and gain computational fluency with number facts and calculations. Marian Small also suggests using “rich tasks embedded in real-life experiences of children, and… rich discourse about mathematical ideas” (Small, 2017, p. 3), which align with Indigenous teaching that emphasizes “experiential learning, modeling, collaborative activity and teaching for meaning” (Beatty & Blair, 2015, p. 5). Embedding First Peoples Perspectives As you plan and adapt the lessons in this resource, consider how you can integrate First Peoples knowledge, stories, perspectives, and worldviews into the context of the lessons. Finding math stories outside in nature, using natural materials gathered outdoors for concrete materials, and reading stories that involve local or place-based animals and plants, help students with Indigenous ancestry see their culture reflected in their school life and classroom. Curricular Competencies • Understanding 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 • Connecting and reflecting: Incorporate First Peoples worldviews and perspectives to make connections to mathematical concepts • Learning is holistic, reflexive, reflective, experiential, and relational (focused 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. • Learning ultimately supports the well-being of the self, the family, the community, the land, the spirits, and the ancestors. • Learning involves patience and time. • Learning involves recognizing the consequences of one’s action 6 Patterns & Relations/Data & Probability

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: • Experiential 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. • The 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). • Connectedness 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). • Engagement 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. 7

mtoaMhftienaatruthehlteeeaTssbraseeldokg5unsinr–,imn1noig5ranywgtfhrbhoeeeerendeueansvyde.edr Math Talks There are numerous Math Talks linked to the lessons in Algebra and Data 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 Chart Talk Move Description Example 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 “Who can say what       said in their own what another student has said so more people words?” hear the idea. It encourages active listening. Revoicing Teacher restates a student’s idea to “So you are saying…. Is that what you were clarify and emphasize and then asks if the saying?” restatement is correct. This can be especially helpful for ELLs. Adding On Teacher encourages students to expand upon “Can someone add on to what       a proposed idea. It encourages students to just said?” listen to peers. Reasoning 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) aabeSilapnsBneentnrydsuucao oysidosmltktdotoeuteiaripnmnrunaywasttggcessgo,wgheGrwctinlaktdhearhheborstnteeolhw.moendubrtoeigtmehstauvhotokeMugaoebmrshkuieneettrehiddstsnehs,ktree.set,s Building Habits of Mind, Growth Mindsets, and Positive Attitudes Toward Math Math Place offers many opportunities to build and reinforce habits of mind and growth mindsets, beginning with an introductory lesson, which sets the tone for nurturing and developing these important skills and attitudes. (The introductory lesson, “Thinking Like A Mathematician,” can be found in the Overview Guide. The lesson can be used at the beginning of the year to establish what the skills are and to develop the criteria for building them. The pertinent messages can be regularly reinforced throughout the year using the prompts and suggestions that are embedded in many of the Patterns, Relations, and Data and Probability lessons. For interview prompts and questions to build social-emotional learning skills and positive attitudes, see the Overview Guide or the Teacher’s Website. 8 Patterns & Relations/Data & Probability