Longwood University Professional Development Seminar Algebra, Number Sense, and Mathematical Connections in Grades 3-5 Community of Learners Complete 3 X 5 notecard: Name Email School.
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Longwood University Professional Development Seminar Algebra, Number Sense, and Mathematical Connections in Grades 3-5 Community of Learners Complete 3 X 5 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 1. To know more about algebraic thinking than you expect your students to know and learn. 2. An awareness of different models and representations to enhance algebraic thinking. 3. To become familiar with the connections between number and operation concepts and algebra concepts. Goals of the Workshop 4. To know what mathematics to emphasize and why in planning & implementing lessons. 5. To anticipate, recognize, and dispel students’ misconceptions about algebra. 6. 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 do you need to do to contribute to the learning of the group? What do you need to do 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 6-8. How are your lists the same and different? What classroom practices can teachers use to support students’ development of algebraic thinking? Algebraic Reasoning is… The study of relationships among quantities. The study of structures. A set of rules and procedures. Generalized arithmetic. --- Bassarear, T. (2005). Mathematics for Teachers. Algebraic Thinking “When children are given the chance to structure number and operation in their own way, they see themselves as mathematicians and their understanding deepens. They can make sense of algebra not as a funny set of rules that mixes up letters and numbers handed down by the authority of thousands but as a language for describing the structure and relationships they uncover” (p. 169). ---Fosnot & Jacob (2010). Something to Consider “What would you think about literacy instruction if the goal was primarily for students to learn to spell? Would you agree that simply teaching children to spell provides them with the necessary skills to write in various literary genres or read and critique the work of others? In a similar way, mathematics instruction that focuses only on arithmetic facts, skills, and procedures – while these are important – leaves a big void in developing children’s mathematical understanding” (p. 83). --Blanton, 2008. Classroom Practices… 5 Process Standards: Problem Solving Reasoning and Proof Communication Connections Representation ---Principles and Standards for School Mathematics (2000) and Virginia Standards of Learning (2009) Virginia Department of Education Instructional Goals that foster Algebraic Thinking “Represent: Provide multiple ways for children to systematically represent algebraic situations. Question: Ask questions that encourage children to think algebraically. Listen: Listen to and build on children’s thinking. Generalize: Help children develop and justify their own generalizations” (p. 94). --Blanton, 2008. Generalizing… “Children should view building a generalization as an important mathematical activity. It is at the heart of algebraic thinking” (p. 116). --Blanton, 2008. “Generalizing is the process by which we identify structure and relationships in mathematical situations.” “Generalizations may be expressed in a number of ways—through natural language, through algebraic notations using letters as variables… or, even through tables and graphs.” ---Blanton, et al., 2011, p.9. Task: Chair and Leg Problem Using at least two different strategies, show all your work and solve the following problem: Suppose that you have some chairs, and each chair has 4 legs. How would you describe the relationship between the number of chairs and the corresponding number of chair legs? --Blanton, et al., 2011, p.9 Discuss with a partner how you solved the problem. Generalizing the Chair and Leg Problem Language: “The number of chair legs is four times the number of chairs.” Algebraic notation: l = 4c, where c represents the number of chairs and l represents the number of legs. Function table: see p.10, Blanton et al. Graph: see p.10, Blanton et al. ---Blanton, et al., 2011, pp. 9-10 Using Word Problems: The Role of Context The potential to model the situation is built in the problem (Fruedenthal, 1973). The situation allows learners to realize what they are doing. The situation prompts learners to ask questions, notice patterns, wonder, ask why and what if. (p. 43) ---Fosnot, C.T., and Dolk, M. (2001). Hunter’s model train is set up on a circular. Six telephone poles are spaced evenly around the track. The engine of Hunter’s train takes 10 seconds to go from the first pole to the third pole. How long will it take the engine to go all the way around the track once? NCTM and VA Process Standard: Making Connections “Understanding algebraic thinking deeply requires you not only to know important mathematical ideas but also to recognize how these ideas relate to one another.” ---Blanton, et al., 2011, p. 3. Our Focus: 5 Big Ideas 1. The properties of arithmetic used with the four operations are the same for algebra. 2. An equation uses an ‘equal sign’ to show that two quantities are equivalent. 3. A variable is a tool that describes relationships in many different ways. 4. Quantitative reasoning extends relationships between and among quantities. 5. Functional relationships are expressed in words, symbols, tables, or graphs, and reasoning occurs between these representations. ---Blanton, et al., 2011, pp. 12-14. Teaching Responsibilities “We have a critical, very important role to play as teachers. We walk the edge between the structures of mathematics and the development of the child. This means we have to understand thoroughly the development of the mathematics by considering the progression of strategies, the big ideas involved, and the emergent models that potentially can become powerful forms of representations with which to think.” --- Fosnot and Jacob, 2010, p. 25. Big Idea 1 Addition, subtraction, multiplication, and division operate under the same properties in algebra as they do in arithmetic. ---Blanton, et al., 2011, p. 15. Essential Understanding 1a The fundamental properties of number and operations govern how operations behave and relate to one another. --Blanton et al., p. 20 What properties are used in real number computations? If you add two real numbers in either order you have the same result. If you multiply two numbers and then multiply by a third, or if you multiply the first by the product of the second and third, you have the same result. What properties are used in real number computations? If you multiply a real number by the sum of two real numbers, or if you add the products of the first real number with each of the second and third, you have the same result. What properties do zero and one have with respect to all other real numbers? Fundamental Properties of Number and Operations Task 1 For each operation, make a list naming the properties students need to learn and know for grades 3-5. Provide an example of each using real numbers. Use variables to represent the properties. Where are the different properties stated in the Virginia Standards of Learning? Task 2: Compare your lists of properties to those stated in the VA SOL found in your packet. In which grade do students focus on each property? Task 3: Compare your lists of properties to those stated on page 16 in the Blanton et al. course text. Task 4: For each of the properties on page 16 in the Blanton et al. course text develop a series of arithmetic tasks that would help children generalize that property. How would children express this generalization in words and in symbols? Why are these properties missing for subtraction and division? Justify your thinking! Frog Jump Training Camp http://www.youtube.com/watch?v=3ctSME2uEIg Make your own origami frog! With a partner, use an open number line to demonstrate 3 jumps and 2 jumps is the same as 2 jumps and 3 jumps. Can you think of another example using the commutative property? Share. Demonstrate the associative property of addition with an open number line using your origami frog. Essential Understandings The fundamental properties are essential to computation. (Essential understanding 1b) --Blanton et al., p. 17 Simplifying algebraic expressions entails decomposing quantities in insightful ways. (Essential understanding 1d) As teachers, our goal is to use ‘relational thinking’ and understand: How properties are used in computations How to explicitly identify them --Blanton et al., p. 20 Think ‘mental math’ strategies Task: Using Mental Mathematics 1. First, pencils or pens down, solve the given problems mentally in at least 2 different ways. 2. Second, write down and record your thinking – the strategies you used. 3. Share and discuss your strategies with a partner. 4. Now that we have shared our strategies, are any of the strategies generalizable? If so, write the generalizations in words and/or symbols. Children’s Invented Strategies Analyze each child’s strategy used to solve: 28 + 35 What is the logic behind each strategy? Are each of these useful strategies? Why or why not? Analyzing Student Work “The general ideas you notice in your students’ work become the starting points for engagement in generalizing about the operations.” Teacher’s role: 1st step – observe and notice 2nd step – help students share what they notice 3rd step – planning opportunities for generalizing ---Russell, Schifter, & Bastable, (2011), pp. 7, 9, & 16. Three Standards for Assessing Students’ Algorithms Efficient enough to be used regularly without considerable loss of time and frustration due to the number of recorded steps required Mathematically valid http://www.youtube.com/watch?v=PgjGXHZXCuY Generalizable Generalizable Strategies for Addition and Subtraction Compensation: making user-friendly numbers; add an amount to one addend and subtract the same amount from the other addend; making an easier equivalent expression; 29 + 6 add 1 and subtract 1; 29 + 6 = 30 + 5 Symbolic Generalization: a + b = (a + 1) + (b – 1) ‘Shift in the same direction’ – add the SAME amount to both numbers: 128-19 ‘is the same as’ 129-20; use an ‘open number line’ model. Symbolic Generalization: a – b = (a + 1) – (b + 1) A Focus on Reasoning “Gradually, students learn to develop and justify mathematical arguments based on images and models. Asking questions that require reasoning about problem without computing requires students to think more deeply about the operations.” ---Russell, Schifter, & Bastable (2011), p. 19. 2.3 Other Addition Thinking Strategies 1. Splitting – partial sums; decomposing numbers using place value (i.e., expanded form); also called “1010” method in the Dutch/ Netherlands curriculum 2. Making Jumps of Ten – called the “N10” method, split or decompose only one number (use hundred chart or open number line) ---Fosnot, C.T., and Dolk, M. (2001). Partial Sums for Addition Algorithm 25 + 34 9 50 59 20 + 30 = 50 5+ 4= 9 59 Model with base 10 blocks… Draw an open numberline… Partial Sums for Addition Algorithm This works for larger numbers too! 305 + 627 900 20 12 932 300 + 00 + 5 600 + 20 + 7 900 + 20 + 12 = 932 Model with base 10 blocks… Draw an open numberline… Other Subtraction Thinking Strategies 1. Splitting – partial differences; decomposing numbers using place value (i.e., expanded form); also called “N10” method 2. Making Jumps of Ten – split or decompose only one number; called the “N10” method in the Dutch/ Netherlands curriculum. 3. Count-back from the minuend. (use hundreds chart) 4. Count-up from the subtrahend – ‘adding up.’ 5. Use four-fact families: relate the inverse operations of addition and subtraction. ---Fosnot, C.T., and Dolk, M. (2001). Splitting or Partial Differences for Subtraction Algorithm 548 - 233 300 10 5 315 Model with base 10 blocks… Draw an open numberline… Splitting or Partial Differences for Subtraction Algorithm 642 - 293 Model with base 10 blocks… Draw an open numberline… 500 + 130 + 12 - 200 - 90 - 3 300 + 40 ________ +9 = 349 ‘Adding Up’ Subtraction Algorithm (Context: think making change with money) 50 - 23 Model with base 10 blocks… Use hundreds chart… Draw an open numberline… What do we need to add to 23 to get 50? Count-on from 23 until we reach 50. +2 + 10 + 10 + 5 27 Activity: A Multiplication Strategy What Do You Know About ________ “Asking what students know about an expression can be a good way to start the year because it invites a range of student observations and provides a window into the understandings and experiences students bring to the classroom. It is also useful throughout the year because you can choose expressions that connect to whatever ideas students are currently studying.” ---Russell, Schifter, & Bastable (2011), p. 16. More Generalizations “Generalizations in arithmetic can be derived from the fundamental properties” (p. 21). --Blanton et al., 2011. “Generalizing arithmetic includes helping children see, describe, and justify patterns and regularities in operations on and properties of real numbers” (p. 12). ---Blanton, Maria (2008). 4 Approaches to Justifying the ‘Truth of a Mathematical Statement’ 1. Accepting the claim on authority. 2. Trying it out with examples. 3. Applying mathematical reasoning based on a visual representation or story context. 4. Proving using algebraic notation and the laws of arithmetic. ---Russell, Schifter, & Bastable (2011), pp. 52-53. Arithmetic leads to Generalization 1. Choose accessible numbers when asking students to investigate general ideas about the operations. 2. Ask students to think about an expression or a set of related expressions without carrying out the computation. 3. Ask students to show their ideas using cubes, number lines, arrays, story contexts, or other representations. 4. Keep a list of ideas in progress. ---Russell, Schifter, & Bastable (2011), pp. 19 – 21. Arithmetic leads to Generalization 5. The teacher can then ask students to pose a conjecture, a statement about the pattern or regularity the students are noticing that can be either true or false. 6. These conjectures can be investigated to determine if they are always true, or sometimes true and need to be revised. 7. These conjectures can then be written as generalizations with words and/or symbols once they are found to be true. Forming a Generalization Are the following statements true? 3+0=3 0+4=4 100 + 0 = 100 How do you know these are true? Can you write a conjecture in words? Is there a way you can show that your conjecture is true or false? How? If your conjecture was true, write it as a generalization in words and in symbols. Another Generalization Are the following statements true? 14 x 1 = 14 1 x 359 = 359 1 x 2,346 = 2,346 How do you know these are true? Can you write a conjecture in words? Is there a way you can show that your conjecture is true or false? How? If your conjecture was true, write it as a generalization in words and in symbols. Activity: Commutative Property of Addition “Consider the commutative property of addition for yourself. How many ways can you explain why 2 + 5 = 5 + 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 do 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 do you know? How could you show your result symbolically using algebra? Helping Children to Generalize Arithmetic Sample Teacher Questions 12 + 18 = What do you notice about the addends? (They are even). 22 + 18 = What do 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, 4 + 5 = 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 7 + 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 1 row and right one column is the same as right one column and down 1 row) 15 + 10 + 1 = 15 + 1 + 10 Addition and Subtraction are Inverse operations: Adding a number and then subtracting that number results in a change of zero(down 1 row and then up 1 row) 15 + 10 – 10 = 15 Why do 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 do 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 do we need to do 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 = 5 I can buy 5 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 3 bunches of cherry lollipops with 5 lollipops in each bunch. She has 3 bunches of orange lollipops with 6 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 3 x 5 = 15 Find the orange lollipops 3 x 6 = 18 Find the total lollipops 15 + 18 = 33 Written using one math expression 3 x 5 + 3 x 6 = 33 3(5 + 6) = 33 Why is 3 x 5 + 3 x 6 = 3(5 + 6) Set model Cherry Can you see 3 sets of 5 and 3 sets of 6? Can you see 3 sets of 5 + 6? Orange Why is 3 x 5 + 3 x 6 = 3(5 + 6) Array or area model Cherry Orange Can you see 3 rows of 5 and 3 rows of 6? Can you see 3 rows of 5 + 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 Homework Read: Blanton et al. (2011): Foreword, Preface, Introduction, and beginning part of Chapter 1 (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 do you know? How could you show your result symbolically using algebra?