for Mathematics Learning PARTNERS for Mathematics Learning Grade Four Module 2 Partners for Mathematics Learning

Mathematics Learning… 2 Mathematics Learning…  Mathematics learning is about making sense of mathematics  Mathematics learning is about acquiring skills and insights to solve problems NCTM Standards 2000 Philosophy Partners for Mathematics Learning

Problem Solving  Solving problems is not only a goal of 3 Problem Solving  Solving problems is not only a goal of learning mathematics but also a major means of learning mathematics  Choose problems that engage students  Create environment that encourages exploration, risk-taking, sharing, and questioning – developing confidence in students engaged in problem-solving activities NCTM Standards 2000 Partners for Mathematics Learning

Children with Good Number Sense 4 Children with Good Number Sense  Have well-understood number meanings  Understand multiple interpretations and representations of numbers  Recognize the relative and absolute magnitude of numbers  Appreciate the effect of operations on numbers  Have developed a system of personal benchmarks NCTM Curriculum and Evaluation Standards , 1989 Partners for Mathematics Learning

Developing Number Sense . . . 5 Developing Number Sense . . .  Number Lines (Relative Magnitude)  What numbers do points C and D represent?  How far apart are A and B?  Why do you think so? A B C D E F G 75 200 Partners for Mathematics Learning

+5 Number Lines n 128 153 What number is n? Label the number line 123 6 Number Lines 123 +5 128 n 153 What number is n? Label the number line Partners for Mathematics Learning

Number Relationships  Greater than/ less than/ equal 7 Number Relationships  Greater than/ less than/ equal  Relative magnitude  Different ways to name the same number  Decomposed into a combination of other numbers  Composed with other numbers to name a new number  Multiples and factors Partners for Mathematics Learning

Number Relationships  1026 916 1176  Which two numbers are closest? 8 Number Relationships  1026 916 1176  Which two numbers are closest?  Which number is closest to 950? to 1100?  Name a number between 1026 and 1179  Name a multiple of 25 between 1026 and 1179  About how far apart are 1026 and 1179? Partners for Mathematics Learning

Number Relationships  1026 916 1176  1026 916 1176  If these are “big” numbers, what are some small numbers?  …numbers that make these seem small? Partners for Mathematics Learning

Games Provide Fun Practice 10 Games Provide Fun Practice  Order Up! • Need dice or number cards  Go Digit! • Children create number cards to use  Largest or Smallest • Use dice with 0-9  What’s My Number?  Target 100 (or choose other target) Partners for Mathematics Learning

 Name two odd numbers whose sum is 16 11 Mystery Number Pairs  Name two odd numbers whose sum is 16  Name two numbers whose sum is 20 and whose difference is 4  Name two numbers whose product is 24 and whose sum is 11  List 3 pairs of numbers with a difference of 54 Partners for Mathematics Learning

? Mystery Numbers  I am not an even number  I am between 20 and 50 12 ? Mystery Numbers  I am not an even number  I am between 20 and 50  The sum of my digits is 7 Partners for Mathematics Learning

? ? Mystery Numbers  I am not an even number  I am between 20 and 50 13 ? Mystery Numbers  I am not an even number  I am between 20 and 50  The sum of my digits is 7  I am a multiple of 5  What number am I? ? Partners for Mathematics Learning

Operation Sense  A complete and flexible understanding of 14 Operation Sense  A complete and flexible understanding of the operations  Understand various meanings of multiplication and division [and addition and subtraction]  Understand the effects of multiplying and dividing whole numbers [and of adding and subtracting]  Identify and use relationships between operations to solve problems  Understand and use properties of operations Partners for Mathematics Learning

Mastering Basic Facts  Components of Fact Mastery Development  Development of a strong understanding of number relationships related to the operations  Development of efficient strategies for fact retrieval  Then (and only then) drill in use and selection of these strategies Partners for Mathematics Learning

Mastering Multiplication Facts 16 Mastering Multiplication Facts  Relate new facts to existing facts  Understand and use commutative property  Children should use strategies for remembering that make sense to them  Drill activities should be individualized to promote different strategies and address different collections of facts Partners for Mathematics Learning

 Groups of…  4 groups of 3 4 x 3 = 12  3 groups of 4 3 x 4 = 12 17 Multiplication Models  Groups of…  4 groups of 3 4 x 3 = 12  3 groups of 4 3 x 4 = 12 Partners for Mathematics Learning

Multiplication Models 18 Multiplication Models  Arrays 3 x 4 = 12 6 x 2 = 12  Are any other arrays possible for factors of 12? Partners for Mathematics Learning

Multiplication and Division 19 Multiplication and Division  Equivalent sentences fit this array: xxxxxxxxxxxx  Multiplication: 3 x 12 = 36 36 = 3 x 12  Division: 36 ÷ 3 = 12 12 = 36 ÷ 3 12 x 3 = 36 36 = 12 x 3 36 ÷ 12 = 3 3 = 36 ÷ 12 Partners for Mathematics Learning

Practice Can Be Fun  Four's a Winner  Practices facts 20 Practice Can Be Fun  Four's a Winner  Practices facts  Uses strategy  Promotes mathematical conversation Partners for Mathematics Learning

Multiplication  A thirsty camel drinks about 20 gallons of 21 Multiplication  A thirsty camel drinks about 20 gallons of water  How many gallons will 80 thirsty camels drink? 20 x 80 = Partners for Mathematics Learning

Multiplication: The Array Model 22 Multiplication: The Array Model 10 10 10 10 10 10 10 10 10 20 × 80 (2 × 10) × (8 × 10) (2 × 8) × (10 × 10) 16 × 100 1600 Partners for Mathematics Learning

23 Do These Work? Partners for Mathematics Learning

24 Do These Work? How?

Thinking About Multiplication 25 Thinking About Multiplication  Beth and Sahil finished working a giant jigsaw puzzle. They saw that the pieces fit into rows of 45 pieces and that there were 29 rows in all. How many pieces did the puzzle have? 45 x 29 = Partners for Mathematics Learning

Multiplication: The Array Model 26 Multiplication: The Array Model Partners for Mathematics Learning

Multiplication: The Array Model 27 Multiplication: The Array Model Connecting to the compact algorithm: Partners for Mathematics Learning

“Deep Mathematics”  “Depth” means …that students know a lot 28 “Deep Mathematics”  “Depth” means …that students know a lot about multiplication before they deal with an algorithm for performing multiplication.  “Depth” does not mean making all students master arithmetic procedures earlier or with more digits…. Focusing on more arithmetic procedures or more digits at the expense of deeper explorations and problem solving is not the same as raising our expectations for all students. Cathy Seeley, President, NCTM, 2004, in Teaching children Mathematics , October 2004 Partners for Mathematics Learning

Thinking About Computation 29 Thinking About Computation  "The depressing thing about arithmetic badly taught is that it destroys a child’s intellect and, to some extent, his integrity. Before they are taught arithmetic, children will not give their assent to utter nonsense; afterwards they will. Instead of looking at things and thinking about them, they will make wild guesses in the hopes of pleasing a teacher…. Partners for Mathematics Learning

Thinking about Computation 30 Thinking about Computation  The essential quality for a mathematician is the habit of thinking things out for oneself. That habit is usually acquired in childhood. It is hard to acquire it later." From Mathematician’s Delight, W. W. Sawyer, University of Toronto, 1943 Partners for Mathematics Learning

Learning Through Problems 31 Learning Through Problems Jungle Problem # 4  Some kings found nine melons  They shared them equally  Then they found six more melons and shared them equally  How many kings were there and how many melons did each king get?  Show your work and explain your thinking Maharajah’s Tasks Partners for Mathematics Learning

Assessment: Scoring Rubric 32 Assessment: Scoring Rubric Score Indicator Noanswer,or Wronganswerbasedoninappropriateplan 1 Incorrectanswerbutusesanappropriatestrategy,or Onlycompletesonestepoftheproblem,or Correctanswerwithinaccurateexplanation, notrelatedtotheproblem 2 Correctanswerbutincompleteorunclearexplanation,or Correctanswer,butminorerrorsinwork 3 Correctanswerwithappropriatestrategyused,and Correctsolutionclearlystated,and Clearandcorrectwrittenexplanationofprocess Partners for Mathematics Learning

Division  Freda has a booth at the state fair 33 Division  Freda has a booth at the state fair Her tiger’s-eye marbles are a big seller She has 92 marbles to put equally into 4 bags How many marbles can she put into each bag? 92 ÷ 4 = Partners for Mathematics Learning

Division  First estimate the solution to this equation: 92 ÷ 4 = 34 Division  First estimate the solution to this equation: 92 ÷ 4 =  How did you make your estimate?  Without using the traditional algorithm, how many ways can you solve this problem? Partners for Mathematics Learning

Division – Equivalent Sentences 35 Division – Equivalent Sentences 92 ÷ 4 = 4x = 92 Partners for Mathematics Learning

Division – Equivalent Sentences 36 Division – Equivalent Sentences 92 ÷ 4 = 4x = 92 4 x 20 = 80 (92-80 =12) Partners for Mathematics Learning

Division – Equivalent Sentences 37 Division – Equivalent Sentences Partners for Mathematics Learning

Division – Equivalent Sentences 38 Division – Equivalent Sentences  Try using this method to solve these:  376 ÷ 8 =  2520 ÷ 40 = Partners for Mathematics Learning

Division - Chunking - 720 36 x 20 Answer: 27 36 )972 252 36 x 7 -252 39 Division - Chunking 36 )972 - 720 252 -252 36 x 20 36 x 7 Answer: 27 Partners for Mathematics Learning

Division – Mental Strategies 40 Division – Mental Strategies  How would you solve this problem without the traditional algorithm? 168 ÷ 21 Partners for Mathematics Learning

Division – Mental Strategies 41 Division – Mental Strategies 168 ÷ 21  Fourth graders who did not know the algorithm solved it with mental strategies  Sixth graders "couldn't do it" because they hadn't yet learned to "do it with 2 digits" From research by Julia Anghileri, University of Cambridge, shared at ICME-10, July, 2004 Partners for Mathematics Learning

Division – Mental Strategies 42 Division – Mental Strategies  Try this division problem without using the traditional algorithm 648 ÷ 3  How many different ways can you find to solve this problem? Partners for Mathematics Learning

Calculating with Number Sense 43 Calculating with Number Sense  "Calculating with number sense means that one should look at the numbers first and then decide on a strategy that is fitting – and efficient. Developing number sense takes time; algorithms taught too early work against the development of good number sense. Children who learn to think, rather than to apply the same procedures by rote regardless of the numbers, will be empowered. " From Fosnot and Dolk, Young Mathematicians at Work: Constructing Multiplication and Division, Heinemann, 2001 Partners for Mathematics Learning

What’s the Story?  Think about this equation: 47 ÷ 5 = 44 What’s the Story?  Think about this equation: 47 ÷ 5 =  Give a context in which each one of these answers would be the correct solution for the division of 47÷5: 9 , 9 r 2 , 10 , 9 2/5 , 9 or 10 Partners for Mathematics Learning

Problem-Based Lessons 45 Problem-Based Lessons  Provide problems that require thinking, not just following procedures and rules For example:  How many stamps are not on the border of a 10 by 10 sheet of stamps? Partners for Mathematics Learning

Avoiding Meaningless Math 46 Avoiding Meaningless Math  When students reflect on their own strategies, they are developing an understanding of concepts out of empirical experiences  “You can’t tell students what a concept is, but you can give them experiences that help them make abstractions for themselves.” -Michael Mitchelmore, Sydney, Australia, quoted at ICME-10, Copenhagen, Denmark, July, 2004 Partners for Mathematics Learning

Discovering Order of Operations 47 Discovering Order of Operations  Using the calculators provided by the leader, solve these equations  How did we get different answers to the same problems?  What answers does your personal calculator give? Partners for Mathematics Learning  4x7+3= 4+8x3= 10 – 6 ÷ 2 = 12 – 3 + 8 = 7+2x5–3= 8 x 27 ÷ 9 = 40 ÷ 8 – 2 =

“Good Manners of Mathematics” 48 “Good Manners of Mathematics” What is happening here?  4 x (7 + 3) = 40  (4 + 8) x 3 = 36  (10 – 6) ÷ 2 = 2  7 + 2 x (5 – 3) = 11 How do the parentheses impact the solution? Partners for Mathematics Learning

R emembering O rder of O perations 107 R emembering O rder of O perations How is the triangle a better model than the traditional P E M D A P E M A D S S for remembering the correct order? Partners for Mathematics Learning

Connections  Where in this module did you see algebra 50 Connections  Where in this module did you see algebra embedded in the number strand?  Which of the process standards did we use?  Problem Solving  Reasoning and Proof  Communication  Connections  Representation Partners for Mathematics Learning

DPI Mathematics Staff Everly Broadway, Chief Consultant Renee Cunningham Kitty Rutherford Robin Barbour Mary H. Russell Carmella Fair Johannah Maynor Amy Smith Partners for Mathematics Learning is a Mathematics-Science Partnership Project funded by the NC Department of Public Instruction. Permission is granted for the use of these materials in professional development in North Carolina Partners school districts. Partners for Mathematics Learning

PML Dissemination Consultants Susan Allman Julia Cazin Cara Gordon Tery Gunter Shana Runge Yolanda Sawyer Ruafika Cobb Anna Corbett Gail Cotton Jeanette Cox Leanne Daughtry Lisa Davis Ryan Dougherty Shakila Faqih Patricia Essick Donna Godley Barbara Hardy Penny Shockley Kathy Harris Julie Kolb Renee Matney Tina McSwain Marilyn Michue Amanda Northrup Kayonna Pitchford Ron Powell Susan Riddle Judith Rucker Partners for Mathematics Learning Pat Sickles Nancy Teague Michelle Tucker Kaneka Turner Bob Vorbroker Jan Wessell Daniel Wicks Carol Williams Stacy Wozny

2009 Writers Jeane Joyner, Co-PI and Project Director Partners Staff Kathy Harris Rendy King Tery Gunter Judy Rucker Penny Shockley Nancy Teague Jan Wessell Stacy Wozny Amanda Baucom Julie Kolb Partners Staff Freda Ballard, Webmaster Anita Bowman, Outside Evaluator Ana Floyd, Reviewer Meghan Griffith, Administrative Assistant Tim Hendrix, Co-PI and Higher Ed Ben Klein , Higher Education Katie Mawhinney, Co-PI and Higher Ed Wendy Rich, Reviewer Catherine Stein, Higher Education Please give appropriate credit to the Partners for Mathematics Learning project when using the materials. Jeane Joyner, Co-PI and Project Director Partners for Mathematics Learning

for Mathematics Learning PARTNERS for Mathematics Learning Grade Four Module 2 Partners for Mathematics Learning