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From Patterns to Algebra
Transcript of From Patterns to Algebra
By Ashley Aseltine and Samantha Minden Help students move from additive to multiplicative strategy How many tiles would you need to build the 20th position? 100th position?
How do you know?
What if I wanted to build position 6 and I only have 12 tiles, can I create the pattern? Why or why not? Predictable Growth What would the 4th position look like? How many tiles would we need? 10th position? What is the pattern rule? number of tiles = position number x 5 Ask students to state the pattern rule: Linear growing patterns grow by the same amount at each position and are predictable "Gallery Walk" Repeat activity with new pattern rule. Build position 4 of a friend's growing pattern. Some students created tile arrangements that did not follow a predictable pattern but did follow the pattern rule (correct number of tiles). From Patterns to Algebra Exploring Linear Relationships Schools & Students: The Bishop Strachan School Upper Canada College Grade 4 girls
Whole class exploratory lesson Grade 3 boys
Small group explicit lesson Materials: Tiles
Grid chart paper
White board / smartboard Lesson Study Mathematics Curriculum Expectations Enduring Understanding
"Identify, extend, and create a repeating
pattern involving two attributes(e.g., size, colour, orientation, number), using a variety of tools (e.g., pattern blocks, attribute blocks, drawings)" "Describe, extend, and create a variety of numeric and geometric patterns, make predictions related to the patterns, and investigate repeating patterns involving reflections...relating the term and the term number in a numeric sequence; generating patterns that involve addition, subtraction, multiplication." Grade 3 The Ontario Curriculum, Ministry of Education Patterning and Algebra Grade 4 Students will learn that numeric rules for simple growing patterns can be represented visually.
Creating visual representations will help deepen understanding of multiplication and pattern rules. Key Concepts Knowledge Building Inquiry linear relationships visual representations of algebraic concepts predictable growth multiplicative rules growing patterns Background Research Van de Walle, 2011 Beatty & Bruce, 2012 Moss & Beaty, 2006 "With these sequences, students not only extend a pattern, but they also look for a generalization or mathematical relationship that will tell them what the pattern will be at any point along the way." It is important for students to see that the relationship exists in both forms, physical version and symbolic (algebraic rule). Patterns offer a tremendous opportunity for students to explore some fundamental algebraic concepts. Patterns support students' abilities to generalize. There is a need for instructional models that help students understand the connections among representations of linear relationships. The application of knowledge building practices and principles in mathematics can foster deep understanding and broad participation. "Democratization of knowledge" and "epistemic agency". Exploratory Lesson Lesson Study Conference Explicit Lesson Explicit Lesson How are students reasoning about growing patterns ? Do they see a connection between the number of tiles and the position number? Can they find the rule? Are they using addition or multiplication? Do they see a linear relationship? The Bishop Strachan School, grade 4 girls With a partner look at and copy this growing pattern. Write down what you notice. (10 min.) Teacher reveals position numbers.
What do you think the 10th position will look like? Explain your reasoning. Repeat with new, slightly more challenging pattern. Step 1 Step 2 Step 3 Lesson Study in action at The Bishop Strachan School Collaboration between teachers from grades 3 to 6. Research theme: patterning & algebraic reasoning
Discuss teacher and student learning goals
Plan lesson together for each grade
Teach & observe
Reconvene to report & revise (and document).
Repeat Lesson Study is seen as a bridge, connecting lessons and skills across the grade levels. Upper Canada College, grade 3 boys Activation, part 1 Model pattern What is happening in the pattern? How is it growing? How many tiles are at each position?
(write on white board) How many tiles would be in position 4? position 10? Knowledge Building Questions 1. 2. 3. 4. Collaborative group discussion Upper Canada College, grade 3 boys Activation , part 2 collaborative small group discussion Repeat activity, but with greater focus on the linear relationship. number of tiles = position number X ____ Provide students with a pattern rule and have them build the first 3 positions. Explicit Lesson Upper Canada College, grade 3 boys Development Students are asked to justify their pattern.
Can they show how it follows the rule?
Is their visual representation linear? Consolidation Challenges Next Steps Explicit Lesson Upper Canada College, grade 3 boys Analyze and contribute solutions as a group. Together look at one particularly challenging growing pattern and come up with the 4th position. Knowledge Building Reconvene to discuss number of tiles = position number x 6 Students rotate places. Knowledge Building Patterns they thought were interesting?
Patterns they thought were easy or difficult to find the next position?
Patterns that might not be linear growing patterns? If students think a pattern is incorrect, then allow pattern builder to justify thinking. Some students continually relied on additive reasoning which was a challenge when thinking of far off positions, i.e 20th, 100th. 1. 2. However, through knowledge building practices in the development and consolidation periods, overall improvement was found. Continuation of lesson study conferences at The Bishop Strachan School. Conference with Cathy Bruce! We will present our findings from Upper Canada College. Dec. 6 Dec. 10-13 Clinical Interviews with students about their understanding of patterning and algebra. Feb. Public lesson study.
Invite teachers from other schools to attend. Discuss findings, collaborate to make improvements and learn! From Patterns to Algebra Questions?