Class Session – Day 1

Longwood University Professional Development Seminar Algebra, Number Sense, and Mathematical Connections in Grades 3-5 Community of Learners Complete X notecard: Name Email School name and location Grade you teach Number of years teaching & grade levels Why are you interested in this workshop? Workshop Introductions Course Materials Distribute Course Texts and Packets Your grant committment: 3-day summer workshop, 1- day fall and 1-day spring workshops, a lesson plan due by end of 2012-2013 school year, & a sampling of classroom observations New 2009 VA SOL Patterns, Functions, and Algebra Number and Number Sense Computation and Estimation Complete Workshop Pre-Assessment Resources Blanton, Maria, et al Developing Essential Understanding of Algebraic Thinking: Grades 3-5 Reston, Va.: National Council of Teachers of Mathematics, 2011 Jacob, Bill, and Catherine Twomey Fosnot The California Frog-Jumping Contest: Algebra Portsmouth, NH: Heinemann, 2007 Russell, Susan Jo, Deborah Schifter, & Virginia Bastable Connecting Arithmetic to Algebra Portsmouth, NH: Heinemann, 2011 Resources Fosnot & Jacob Young mathematicians at work: Constructing Algebra Portsmouth, NH: Heinemann, 2010 Blanton, Maria Algebra and the Elementary Classroom: Transforming Thinking, Transforming Practice Portsmouth, NH: Heinemann, 2008 Fosnot, C.T., and Dolk, M Young mathematicians at work: Constructing number sense, addition, and subtraction Portsmouth, NH: Heinemann, 2001 Goals of the Workshop To know more about algebraic thinking than you expect your students to know and learn An awareness of different models and representations to enhance algebraic thinking To become familiar with the connections between number and operation concepts and algebra concepts Goals of the Workshop To know what mathematics to emphasize and why in planning & implementing lessons To anticipate, recognize, and dispel students’ misconceptions about algebra Build on prior grades’ algebra ideas and know later-grade connections  vertical alignment From NCTM Principles and Standards: The Teaching Principle “Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well… (p.16) Teaching about algebraic thinking requires knowledge that goes “beyond what most teachers experience in standard preservice mathematics courses” (p.17) - NCTM (2000) Principles and Standards for School Mathematics Reston, VA: Author Class Structure and Norms  What you need to to contribute to the learning of the group?  What you need to to make sure this is a learning experience for yourself? Think-Pair-Share What is algebra? Make a list of what you think might be ‘essential components’ of algebra in grades 3-5 and grades 68 How are your lists the same and different? What classroom practices can teachers use to support students’ development of algebraic thinking? Activity: Commutative Property of Addition “Consider the commutative property of addition for yourself How many ways can you explain why + = + 2? Use number lines, story contexts, drawings, and a variety of representations.” -Russell, Schifter, & Bastable (2011), p 51 Show Me… Convince Me The Associative Property of Addition and The Associative Property of Multiplication “What if Ms Raymond or Ms Perez comes in the room and says she doesn’t understand, or she doesn’t even trust what you are saying How could you convince her that this equality is true? How can you prove it is true? You can use drawings, cubes, or a story to convince her.” -Russell, Schifter, & Bastable (2011), p 18 Generalizing patterns with Even/Odd Develop a set of addition problems that helps children think about what happens when they add two even numbers Is the result even or odd? How you know? How could you show your result symbolically using algebra? Develop a set of subtraction problems that helps children think about what happens when they subtract one odd number from another odd number Is the result even or odd? How you know? How could you show your result symbolically using algebra? Helping Children to Generalize Arithmetic Sample Teacher Questions 12 + 18 = What you notice about the addends? (They are even) 22 + 18 = What you notice about your solutions? (They are even) 28 + 14 = What can you conjecture about the sum of two even numbers? (An even number plus an even number is always even) 34 + 26 Do you think this is always true? Why? -Blanton, Maria (2008) (p 16) What if a conjecture is false?  A counterexample is a case where a conjecture does not hold true For example, if a student conjectured that the sum of an even number and an odd number is even; the single counterexample, + = 9, would be sufficient to prove the conjecture is false It is usually not possible to test all possible cases so there is always some sense of uncertainty about a conjecture Building on Generalizations Use the generalization to reason instead of needing to calculate Will the following sum be even or odd? Why? 1895 + 198 + 2073 + 5999 When students reason from structure instead of calculation they are reasoning algebraically -Blanton, Maria (2008) (p 17-18) Investigating Relationships with the Hundred Chart Students can analyze the structure of a series of moves Commutative of addition: (down row and right one column is the same as right one column and down row) 15 + 10 + = 15 + + 10 Addition and Subtraction are Inverse operations: Adding a number and then subtracting that number results in a change of zero(down row and then up row) 15 + 10 – 10 = 15 Why we need parentheses? Shane had a $50 bill He bought a t-shirt for $10 and a pair of jeans for $25 How much change should Shane get back when he pays? (We are assuming the prices already include sales tax.) First we need to find out how much he spent…  10 + 25 = 35 Then we need to find out how much he gets in change  50 – 35 = 15 He will receive $15 in change How could we combine these two operations in one math sentence? If we were to write our processes in words in one sentence we could say… Money paid – Total price = Change received I could write 50 – (10 +25) = 15 Parentheses help us write one math sentence that has more than one operation When we have parentheses the order of operations says that we must the calculation inside the parentheses first More practice with using parentheses Suzanne bought a pair of gloves for $12 and a scarf for $32 How many glove and scarf pairs can Suzanne buy for $220? Describe your processes in words…what we need to to solve this problem? Find the cost of one scarf-glove pair  12 + 32 = 44 How many $44 sets can be bought with $220?  220 ÷44 = I can buy scarf-glove pairs How can you use one math sentence to describe these processes?  220 ÷ (12+32) Parentheses and the Distributive Property Hannah loves lollipops She has bunches of cherry lollipops with lollipops in each bunch She has bunches of orange lollipops with lollipops in each bunch How many lollipops does Hannah have? Describe in words your process Cherry lollipops + Orange lollipops = Total lollipops  Find the cherry lollipops x = 15  Find the orange lollipops x = 18  Find the total lollipops 15 + 18 = 33  Written using one math expression x + x = 33  3(5 + 6) = 33  Why is x + x = 3(5 + 6) Set model Cherry Can you see sets of and sets of 6? Can you see sets of + 6? Orange Why is x + x = 3(5 + 6) Array or area model Cherry Orange Can you see rows of and rows of 6? Can you see rows of + 6? Thinking Strategies… Develop Algebraic Thinking “If students early work with arithmetic consists solely of practicing standard algorithms and memorizing facts to use in those algorithms, they are unlikely to engage in algebraic thinking during this study However, when students’ learning of arithmetic focuses on reasoning about mathematical relationships, they naturally engage in a great deal of algebraic thinking while thinking about arithmetic” (p 23) -Blanton, et al., 2011 Read: Homework Blanton et al (2011): Foreword, Preface, Introduction, and beginning part of Chapter (pp 1-24) Jacob & Fosnot (2007): Day One, (pp 15-20) Do: Develop a set of subtraction problems that helps children think about what happens when they add two odd numbers Is the result even or odd? How you know? How could you show your result symbolically using algebra? ... Packets Your grant committment: 3 -day summer workshop, 1- day fall and 1 -day spring workshops, a lesson plan due by end of 2 012 -2 013 school year, & a sampling of classroom observations New 2009... add and subtract 1; 29 + = 30 + Symbolic Generalization: a + b = (a + 1) + (b – 1) ‘Shift in the same direction’ – add the SAME amount to both numbers: 12 8 -19 ‘is the same as’ 12 9-20; use an... role: 1st step – observe and notice 2nd step – help students share what they notice 3rd step – planning opportunities for generalizing -Russell, Schifter, & Bastable, (2 011 ), pp 7, 9, & 16 Three