Patterns and Pre-Algebra Upper Elementary Grades 4–6 1

Agenda What Is Early Algebra and Why Teach It? There Are Different Types of Patterns Algebraic Thinking Is Generalizing Relationships Your Questions Break Algebraic Thinking Is Understanding Change Pattern Rules Describe Relationships Patterns, Relations and Functions Can Be Modelled Your Questions Lunch Algebraic Symbols Describe Mathematical Situations—Equality Your Questions Break Algebraic Symbols Describe Mathematical Situations—Variables Your Questions and Feedback 2

Mathematical Metaphors When I think of algebra, I feel… Note: From the University of Alberta/Edmonton Catholic Separate School District No. 7 Collaborative Project,

Why Teach Algebraic Thinking? Working with patterns enables students to make connections both within and beyond mathematics. Through the study of patterns, students come to interpret their world mathematically and value mathematics as a useful tool. 4

Exploring patterns and pre-algebra in elementary school lays the foundation for the study of formal algebra. Rather than a new topic, algebra becomes a natural extension of the elementary curriculum and is often defined as generalized arithmetic and geometry. Why Teach Algebraic Thinking? By generalizing patterns, students develop strategies that can be used to solve a wide range of problems. Mathematics is seen as reasoning rather than solving one unrelated problem after another. 3 ÷ 4 = 3/4 5

Numbers Charts Tables Concrete Representations Visual Patterns Context Problems 6

Patterns can be repeating and made up of a core set of elements – a core unit that is iterated. Patterns can be increasing or decreasing and created by orderly change

Learning Task – Last One Standing No. 1 says “in” and remains standing. No. 2 says “out” and sits down. No. 3 says “in” and remains standing. No. 4 says “out” and sits down. The contest continues repeatedly around the circle. The last one standing wins the contest! 8 Note:Excerpted and reprinted with permission from Green, D. A. (2002). Last One Standing: Creative, Cooperative Problem Solving. Teaching Children Mathematics, 9(3), 134–139, copyright 2002 by the National Council of Teachers of Mathematics. All rights reserved.

Learning Task – Last One Standing What patterns were non-numerical? Numerical? What patterns repeated? How did the patterns grow? 9

Learning Tasks – Charts and Diagrams What are the Clues? page 166 Circle Patterns page 183 Multiple Patterns page 179 Expert Groups Home Sharing Groups 10

Learning Tasks – Charts and Diagrams  Explain the task.  Describe the patterns you found.  What connections are there to other mathematical big ideas?  At what grade level could this task be used?  Share any suggested tips or extensions to the task. What are the Clues? Circle Patterns Multiple Patterns 11

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Algebrafying the Curriculum – Solids 13 Note:From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (pp. 26–27), by K. Willson, L. Gibeau and R. McKay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the authors. Reprinted with permission.

Algebrafying the Curriculum – Solids If you know the shape of the base, how can you find the number of edges, faces and vertices without counting them all? What patterns are in each of the columns? How is the base related to the edges, faces and vertices? How many edges, faces and vertices on a 20-sided pyramid? 14 Shape of Base No. of Sides on Base EdgesFacesVertices Triangle Rectangle/Square Pentagon Hexagon

Algebrafying the Curriculum – Solids

Algebrafying the Curriculum - Combinations If you had 2 shirts and 3 pants, how many different outfits could you make? Algebrafied Version: What if you had 2 shirts and 4 pants? 5 pants? 6 pants? How many shirts and pants will make exactly 21 outfits? 16

How many possible outcomes are there when you toss two coins? Algebrafied Version: How many outcomes for 3 coins? 4 coins? 5 coins? 10 coins? heads–tails tails–tails tails–heads heads–heads Adding one coin doubles the number of possible outcomes. Algebrafying the Curriculum - Probability 17

Learning Task – Pool Tile What changes? What stays the same? 18

Learning Task – Iguanas 19 Note:Adapted from Lessons for Algebraic Thinking: Grades K–2, pp. 91–117, by Leyani von Rotz and Marilyn Burns. Copyright © 2002 by Math Solutions Publications. Note:Student samples: Edmonton Catholic Schools, 2005.

Learning Task – Iguanas Fish Caterpillars 20 Note:Adapted from Lessons for Algebraic Thinking: Grades K–2, pp. 2–11, by Leyani von Rotz and Marilyn Burns. Copyright © 2002 by Math Solutions Publications.

Learning Task – What Changes? What Stays the Same? Frame 1Frame 2Frame 3Frame 4 Frame number 1234 Number of Tiles

2, 4, 6, 8,10,… Learning Task – What Changes? What Stays the Same? 1, 3, 6, 10, 15,… 2, 4, 8, 16, 32,…20, 16, 12, 8, 4 Create a visual pattern for your number sequence. Can you find others who had the same sequence? 22

Understanding Change in Real-life Contexts Time in seconds ↔ Level of water Distance driven ↔ Gas in tank Heartbeat ↔ Age 23

4 +  Input 2 Output 6 24 Note:Adapted from Lessons for Algebraic Thinking: Grades K–2, pp. 91–117, by Leyani von Rotz and Marilyn Burns. Copyright © 2002 by Math Solutions Publications. Note:Student samples: Edmonton Catholic Schools, 2005.

 A pattern rule must account for all elements of a pattern, including the first one.  Pattern rules can describe recursive or functional relationships. start at 2 Recursive rule: from step to step or frame to frame

Functional rule: from step no. to step or frame no. to frame “x 2 + x” (1 × 1) + 1 (2 × 2) + 2 (3 × 3) + 3 (4 × 4) + 4  A pattern rule must account for all elements of a pattern, including the first one.  Pattern rules can describe recursive or functional relationships. 26

Looking at relationships down columns Recursive Looking at relationships across columns Functional Frame Number Frame

Learning Task – Twelve Days of Christmas Analyzing Recursive Patterns 28 Note:From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (p. 34), by K. Willson, L. Gibeau and R. Mckay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the authors. Reprinted with permission.

How many gifts were given on the 12 th day? Learning Task – Twelve Days of Christmas Analyzing Recursive Patterns 29 Note:From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (p. 36), by K. Willson, L. Gibeau and R. Mckay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the authors. Reprinted with permission.

Learning Task – Twelve Days of Christmas Analyzing Recursive Patterns How many gifts were given on the 12 th day? I can’t do this anymore! 30 Note:From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (p. 40), by K. Willson, L. Gibeau and R. Mckay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the authors. Reprinted with permission.

How many gifts were given on the 12 th day? Learning Task – Twelve Days of Christmas Analyzing Recursive Patterns Note:From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (pp. 37–39), by K. Willson, L. Gibeau and R. Mckay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the authors. Reprinted with permission. 31

What is the total amount of each gift given? Using multiplication sentences helped these students to discover a new pattern. As the number of days each gift was given decreases by one, the quantity of each gift increases by one. 12 × 1 12 days with 1 partridge 11 × 2 11 days with 2 turtle doves 10 × 3 10 days with 3 French hens 9 × 4 9 days with 4 calling birds 8 × 5 8 days with 5 golden rings 7 × 6 7 days with 6 geese Learning Task – Twelve Days of Christmas Analyzing Recursive Patterns Note:From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (p. 41), by K. Willson, L. Gibeau and R. Mckay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the authors. Reprinted with permission. 32

Learning Task – Patterns with Perimeters Analyzing Functional Patterns 33

Learning Task – Patterns with Perimeters Analyzing Functional Patterns Perimeter = number of sides x number of tiles – [2 × (number of tiles – 1)] (square): P = 4 × number of tiles – [2 × (number of tiles – 1)] (pentagon):P = 5 × number of tiles – [2 × (number of tiles – 1)] (hexagon):P = 6 × number of tiles – [2 × (number of tiles – 1)] Grade 4: add 2 to the frame number/number of tiles Grade 5: t + 2 or  + 2 where t &  are the number of triangles in the string Grade 6: p = t + 2 where p is perimeter and t is the number of triangles 3 × 5 = 15(3 × 5) – (2 x 4) = 7 34

Learning Task – Patterns with Perimeters Analyzing Functional Patterns 35 Number of tiles Perimeter Equilateral triangle SquarePentagonHexagon 13 units4 units5 units6 units 24 units6 units8 units10 units 35 units8 units11 units14 units 46 units10 units14 units18 units 57 units12 units17 units22 units 1012 units22 units32 units42 units 2022 units42 units62 units82 units units202 units302 units402 units Across the rows, the perimeter increases by 100 units 20 units 10 units 5 units 4 units 3 units 2 units 1 unit Triangles Perimeter increases by 1 unit each frame Squares Perimeter increases by 2 units each frame Pentagon Perimeter increases by 3 units each frame Hexagon Perimeter increases by 4 units each frame

Developing relational thinking 1.Build the first three frames and extend: How many in the next frame? 2.Keep extending the pattern, frame by frame: The next frame after that? 3.Identify a pattern: What do you notice? Why is this happening? 4.Push students to generalize: How many in the 10 th frame? 100 th frame? How can you find the answer for any given frame? Typical Pattern Lesson 36

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Learning Task – Handshake Problem Ten students arrive at a special gathering for students taking part in a Mathematics Fair. As the students are all from different schools, the teacher wants each to get to know the others. The teacher asks each student to shake hands with each of the other students and introduce themselves. How many handshakes took place? 38

Learning Task – Handshake Problem Note: Student samples: Edmonton Catholic Schools,

Learning Task – Handshake Problem Models become tools for thinking 40

What Is Early Algebra? Representing and analyzing mathematical problems using algebraic symbols Understanding the meaning of equality Understanding the conventions and multiple meanings of variables 41

What do elementary students think the equal sign means? 42

and 17 Other Grades 1 and 25%58%13%8%16% Grades 3 and 49%49%25%10%7% Grades 5 and 62%76%21%1%0% = + 5 Note:From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (p. 4), by T. P. Carpenter, M. L. Franke and L. Levi, 2003, Portsmouth, NH: Heinemann. Copyright 2003 by the authors. Reprinted with permission. The answer comes next: = Use all the numbers (overgeneralizing associative property): = Extending the problem: = = 17 43

Robin: Second-grade student = + 29 “Twenty-nine is 2 more than 27, so the number in the box has to be 2 less than 18 to make the 2 sides equal. So it’s 16.” 44 Note:From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (p. 4), by T. P. Carpenter, M. L. Franke and L. Levi, 2003, Portsmouth, NH: Heinemann. Copyright 2003 by the authors. Reprinted with permission.

Equality and inequality between quantities can be considered as: whole to whole relationships (5 = 5) part–part to whole relationships (3 + 5 = 8) whole to part–part relationships (8 = 5 + 3) part–part to part–part relationships (4 + 4 = 3 + 5). 45

Determine if these equations are true or false without calculating the actual sum or difference. Use relational thinking! = – 27 = 34 – – 382 = 474 – – 389 = 664 – – 529 = 83 – 29 Learning Tasks – True or False Mini-lessons 46

Learning Task – Virtual Pan Balance ~ Shapes 47 Note:Excerpted and reprinted with permission from National Council of Teachers of Mathematics. (2006). Pan Balance—Shapes. Illuminations. Retrieved November 29, 2006, from copyright 2006 by the National Council of Teachers of Mathematics. All rights reserved.

Learning Task – Virtual Pan Balance ~ Numbers Note:Excerpted and reprinted with permission from National Council of Teachers of Mathematics. (2006). Pan Balance—Numbers. Illuminations. Retrieved November 29, 2006, from copyright 2006 by the National Council of Teachers of Mathematics. All rights reserved. 48

Generalizing The Meaning of the Operations part whole Big Quantity Small Quantity Differenc e 49 Part-part-whole or Collection Whole Unknown Connie has 15 red marbles and 28 blue marbles. How many marbles does she have? Part Unknown Connie has 43 marbles. 15 are red and the rest are blue. How many blue marbles does Connie have? Comparison Difference Unknown Connie has 15 red marbles and 28 blue marbles. How many more blue marbles than red marbles does Connie have? Unknown Big Quantity Connie has 15 red marbles and some blue marbles. She has 13 more blue marbles than red ones. How many blue marbles does Connie have? Unknown Small Quantity Connie has 28 blue marbles. She has 13 more blue marbles than red ones. How many red marbles does Connie have?

× K G 50 Problem Type Multiplication Givens: number of groups and number of objects in each group Measurement Division Givens: total number of objects and the number of objects in each group Partitive Division Givens: total number of objects and the number of groups Grouping/ Partitioning Equal Groups Gene has 4 tomato plants. There are 6 tomatoes on each plant. How many tomatoes are there altogether? Gene has some tomato plants. There are 6 tomatoes on each plant. Altogether there are 24 tomatoes. How many tomato plants does Gene have? Gene has 4 tomato plants. There is the same number of tomatoes on each plant. Altogether there are 24 tomatoes. How many tomatoes are on each tomato plant? Multiplicative Comparison The giraffe in the zoo is 3 times as tall as the kangaroo. The kangaroo is 6 feet tall. How tall is the giraffe? The giraffe is 18 feet tall. The kangaroo is 6 feet tall. The giraffe is how many times taller than the kangaroo? The giraffe is 18 feet tall. She is 3 times as tall as the kangaroo. How tall is the kangaroo?

WcWsWvWrWt PcPsPvPrPt W P CSVRT 51 Symmetric Problems Area and Array A farmer plants a rectangular vegetable garden that measures 2 m along one side and 5 m along an adjacent side. How many m 2 of garden did the farmer plant? A baker has a pan of fudge that measures 8 inches on one side and 9 inches on another side. If the fudge is cut into square pieces 1 inch on a side, how many pieces of fudge does the pan hold? A farmer plants a rectangular garden. She has enough room to make the garden 5 m on one side. How long does she have to make the adjacent side in order to have 10 m 2 of garden? Combinations The Friendly Old Ice Cream Shop has 2 types of cones I(waffle and plain). They have 5 flavours of ice cream (chocolate, vanilla, strawberry, rainbow and tiger). How many one-scoop combinations of an ice cream flavour and cone type can you get at the Friendly Old Ice Cream Shop?

“There are twenty-four students for every teacher in Grade 5.” Write an algebraic equation to represent this situation: 52

“There are twenty-four students for every teacher in Grade 5.” Write an algebraic equation to represent this situation: In various studies, from 1/3 to 1/2 of adults answered this question incorrectly. 53

Common Student Misconceptions About Variables Variables Are Objects n + 3 = 12 x + 3 = = 12  + 3 = 12 have different answers Variables Always Have One Value a = a is always true, but c = a is never true x – x = 0 is a solvable equation 54

Variables as a Specific Unknown ×= 36 ×= ×= 9 Mathematician’s Rule a 55

Variables as Dynamic Quantities Pattern Rules: Expressions and Equations Formulas, Conversions and Rates p + 4 s = 24p A = l × w p = $2.10 × d cm = 2.54 × i e = s + c 56

Variables as Generalizers a + b = b + a (x + y) + z = x + (y + z) a + 0 = a a – b ≠ b – a Proof Conjectures a + b = c b + a = c c – a = b c – b = a a × b = c b × a = c c ÷ a = b c ÷ b = a 57

Learning Tasks – Variables a c b The Variable Machine Algebra Islands 58

Understanding the Meaning of Variables Mike Note:From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School, by T. P. Carpenter, L. M. Franke and L. Levi, 2003, Portsmouth, NH: Heinemann. Reproduced with permission. 59

Each problem that I solved became a rule which served afterwards to solve other problems. René Descartes 60