for Mathematics Learning PARTNERS for Mathematics Learning Grade Two Module 3 Partners for Mathematics Learning

Equipartitioning  Equipartitioning means What experiences 2 Equipartitioning  Equipartitioning means “Sharing Fairly” What experiences have students had with sharing fairly? Partners for Mathematics Learning

What Happens in K and 1 st Grades? 3 What Happens in K and 1 st Grades?  Share collections of objects fairly and reassemble  Kindergarten: share between 2 people  1 st grade: share between 2-6 people  Share rectangles and circles among two or four people and reassemble  1 st grade Partners for Mathematics Learning

What Happens in 2nd Grade? 4 What Happens in 2nd Grade?  Look at the Essential Standards  Fairly share collections and reassemble them  Fairly share a rectangle or circle among two, four, eight; three and six people and reassemble it  Name the share as “1/nth of the whole”  Predict that the size of a fair share decreases as the number of shares increases, and vice versa Partners for Mathematics Learning

Equipartitioning  Parts of a whole  Continuous or area models 5 Equipartitioning  Parts of a whole Poland France  Continuous or area models  Equal shares by sub-dividing a whole Partners for Mathematics Learning

Equipartitioning  Parts of a group  Discrete or set models 6 Equipartitioning  Parts of a group  Discrete or set models  Dividing into subsets of equal size (equal numbers of elements)

Equipartitioning, Division, Fractions 7 Equipartitioning, Division, Fractions  All focus on equal portions Understanding the “whole” is important Often referred to as “fair shares” Models may look different depending on the context Partners for Mathematics Learning

Equipartitioning, Division, Fractions 8 Equipartitioning, Division, Fractions  How could 2 people fairly share the cake? The gumballs?  How could you divide the cake and the gumballs into 2 equal parts?  Show 1/2 of the cake and 1/2 of the gumballs Partners for Mathematics Learning

Division as Equipartitioning 9 Division as Equipartitioning  Get at least 15 two-color counters  Use them to solve these problems:  Larry and 3 friends are sharing 12 cupcakes. How many cupcakes will each child get if each gets a fair share?  Larry, Mary, Devin and Martez each have 3 brownies. How many brownies did Larry’s mom bake? Partners for Mathematics Learning

Student Thinking  Tom, Renee, and Sara collected some 10 Student Thinking  Tom, Renee, and Sara collected some apples. Each person got five apples. What was the total number of apples collected?  What conceptual understanding related to equipartitioning do students have?  How do we start to build understanding of the relationships between the whole and the equal parts? Partners for Mathematics Learning

Relationships : Wholes and Parts 11 Relationships : Wholes and Parts  Needs to be made explicit  Whole was 15 apples  5 apples represented 1/3 of the whole  5 represented a fair share for 3 people  Bringing together each of the three thirds makes the whole  If 5 people shared the same whole, the fair share would be smaller Partners for Mathematics Learning

Classroom Problems  Write several problems for 2 nd graders that 12 Classroom Problems  Write several problems for 2 nd graders that address division as equipartitioning  Ideas to include  Equal sized groups  Determine initial size of a group given the equal parts  Describe what happens when collections are divided fairly and what happens when reassembled Partners for Mathematics Learning

Considering Set Models 13 Considering Set Models  The whole is the total set of objects and subsets of the whole are fractional parts  Number of objects not size is important  Examples: counters, people, M & Ms, any discrete objects Partners for Mathematics Learning

Fair Shares: Rectangles 14 Fair Shares: Rectangles  Fold a square in half  Cut your square  How do you know each half is equal? Talk with a neighbor  Is your half equivalent to your neighbor’s?  What key ideas do you want 2 nd graders to understand in this activity? Partners for Mathematics Learning

Key Ideas  Each part is one-half of the whole square 15 Key Ideas  Each part is one-half of the whole square The two parts make the square Why are these not halves (non-examples)? Why may some students think they are?  If we started with different size squares, would the one-halves be equivalent? Partners for Mathematics Learning

16 One Half? How Many Ways? Partners for Mathematics Learning

17 One Half? How Many Ways? Partners for Mathematics Learning

Rectangles and Circles 18 Rectangles and Circles  2nd graders share rectangles and circles among 2, 4, 8, 3, and 6 people  What experiences do we need to provide students?  Context  Models Partners for Mathematics Learning

19 People Fractions Partners for Mathematics Learning

20 Goofy Fractions Partners for Mathematics Learning

Context  Maria’s mom baked  How can Maria share the brownies with 3 22 Context  Maria’s mom baked a pan of brownies  How can Maria share the brownies with 3 friends and herself? Partners for Mathematics Learning

Get to 30 Play the game What mathematics is addressed by the game? Partners for Mathematics Learning

Contextual Problems  With a partner brainstorm problems that 23 Contextual Problems  With a partner brainstorm problems that match the Essential Standard: “Explain the division of rectangles and circles to accommodate different numbers of people”  Share your ideas Partners for Mathematics Learning

Making Sense of Fractions 24 Making Sense of Fractions  Students need:  Multiple models  Experience with multiple contexts  Time  Language Partners for Mathematics Learning

 Content is new (proportional vs. additive 25 Fractions: Reasons for Difficulties  Content is new (proportional vs. additive reasoning) and notation is different  Conceptual-development experiences are limited; symbols are often emphasized  Instruction is  Too abstract  Too procedural  Without connections  With limited models  Without meaningful contexts

Fraction Research Students’ difficulties in understanding 26 Fraction Research Students’ difficulties in understanding fractions stem from learning fractions by using rote memorization of procedures without a connection with informal ways of solving problems involving fractions. Steffe and Olive, 2002 Partners for Mathematics Learning

No Symbols for Fractions 27 In 2nd Grade: No Symbols for Fractions 1 2 Partners for Mathematics Learning

Math & Literature  What happens to each child’s share 28 Math & Literature  What happens to each child’s share as the doorbell rings?  What happens to each child’s share when Grandma arrives? Partners for Mathematics Learning

Compensatory Principle 29 Compensatory Principle  The principle states that the bigger the unit, the smaller the number of that unit needed  1/8 is fewer than 1/4; 1/4 is few cookies than 1/2 of the cookies  What experiences might lead students in developing an understanding of this idea? Partners for Mathematics Learning

When Concepts Are Not Well-Developed…  NAEP Question: 30 When Concepts Are Not Well-Developed…  NAEP Question: Estimate the sum of 12/13 and 7/8 a. 1 b. 2 c. 19 d. 21  How did you decide your estimate?  Which answer did most 13 year old students choose? Partners for Mathematics Learning

Well-Developed…NAEP Question 31 When Concepts Are Not Well-Developed…NAEP Question  Estimate the sum of 12/13 and 7/8 a. 1 b. 2 c. 19 d. 21  Students look at fractions as though they represent two separate whole numbers  This leads to misinterpretation and an inability to access the reasonableness of results Partners for Mathematics Learning

Chinese Proverb “ Tell me and I'll forget; show me and I may remember; 32 Chinese Proverb “ Tell me and I'll forget; show me and I may remember; involve me and I'll understand. ” Partners for Mathematics Learning

What is Measurement?  A count of how many units are needed to 33 What is Measurement?  A count of how many units are needed to fill, cover, or match the attribute of the object being measured  A fundamental mathematical process interwoven throughout all strands of mathematics Partners for Mathematics Learning

Why Measurement?  An essential link between 34 Why Measurement?  An essential link between mathematics and other disciplines such as science, art, music, and social studies  It makes mathematics real and tangible for students giving them a handle on their world  What are the implications for teaching measurement?

Measurement: Big Ideas 35 Measurement: Big Ideas  An object can be described and categorized in multiple ways (attributes)  The measurement of a specific numerical attribute tells the number of units  The process of measurement is similar for all attributes, but the measurement system and tool vary according to the attribute  Measurements are accurate to the extent that the appropriate unit/tool is used properly

Measurement Standards 36 Measurement Standards  Look at the Essential Standards for Measurement  Look at the Clarifying Objectives Partners for Mathematics Learning

Attributes  What are the attributes of the present? 37 Attributes  What are the attributes of the present?  Which of these are measurable? Partners for Mathematics Learning

Process of Measurement 38 Process of Measurement  Determine the attribute to be measured  Choose an appropriate unit that has the same attribute  Choose the tool  Determine how many of that unit is needed by filling, covering, or matching the object Partners for Mathematics Learning

A Cloak for a Dreamer Covering space with nonstandard units 39 A Cloak for a Dreamer Covering space with nonstandard units leaving no gaps or overlaps Partners for Mathematics Learning

Measuring a Square  Use the pattern blocks to cover the square 40 Measuring a Square  Use the pattern blocks to cover the square  Only use one type of pattern block Partners for Mathematics Learning

Computer Activities www.arcytech.org/java/patterns/patterns_j.shtml 41 Computer Activities www.arcytech.org/java/patterns/patterns_j.shtml Partners for Mathematics Learning

Measurement Components 42 Measurement Components  Conservation Transitivity Partitioning Unit Iteration Compensatory Principle Partners for Mathematics Learning

Components of Measurement 43 Components of Measurement  Conservation: an object maintains the same size if it is rearranged, transformed, or divided in various ways Partners for Mathematics Learning

Conservation Examples 44 Conservation Examples  Students explore conservation  Take a clay ball, cut it in half. Then roll one half into a ball and other into a snake. Do they have the same mass?  Pour the same amount of liquid from one container to another. How tall is the liquid in the container? Does the amount stay the same?

Components of Measurement 45 Components of Measurement  Transitivity : two objects can be compared in terms of a measurable attribute using a third object If A = B and B = C, then A = C If A < B and B < C, then A < C If A > B and B > C, then A > C

Partitioning  Larger units can be subdivided into 46 Partitioning  Larger units can be subdivided into equivalent units…mentally subdividing  Choose a place in the room (table, wall)  Mentally divide it in half  Choose a unit of measure (marker, handspan)  Determine the measurement and double it Partners for Mathematics Learning

Components of Measurement 47 Components of Measurement  Units : the attribute being measured dictates the type of unit used; the unit must have the same attribute as the attribute to be measured Partners for Mathematics Learning

Components of Measurement 48 Components of Measurement  Unit iteration : units must be repeated in order to determine the measure Partners for Mathematics Learning

Iteration Experiences 49 Iteration Experiences  What questions might you ask to encourage these students to focus on the placement of units?  How can you provide opportunities for students to use iteration? Partners for Mathematics Learning

Compensatory Principle 50 Compensatory Principle  The compensatory principle states that the bigger the unit, the smaller the number of that unit is needed  Turn to your partner and give an example of what this means  What happens when the unit used to measure is smaller? Do you need more or fewer units? Partners for Mathematics Learning

Compensatory Principle in Action 51 Compensatory Principle in Action  Students measure the top of the cabinet and it is 60 cubes long  Then they measure it in craft sticks and find out that it is 5 sticks long  It takes more of the smaller units (cubes) to measure the cabinet than the larger unit (sticks) Partners for Mathematics Learning

Measurement Estimate  Choose two spots on a surface  Mark each spot 52 Measurement Estimate  Choose two spots on a surface  Mark each spot  Estimate the halfway point between the two spots and mark it  Cut a piece of string to represent the distance between the two spots  Fold it in half and use it to measure the halfway point  How close was your estimate? Partners for Mathematics Learning

Connecting Data & Measurement 53 Connecting Data & Measurement  What took place in the Giant Step activity?  How could iteration have been highlighted during this activity?  How could the compensatory principle have been more explicit during this activity? Partners for Mathematics Learning

Capacity and Volume Capacity is the focus for primary children 54 Capacity and Volume  Capacity refers to the amount a container will hold  Pitchers and bottles  Scoops, cups, quarts measure liquids  Volume is often though of as the amount of space an object takes up  Non-Standard units such wooden cubes, marbles, tennis balls teddy bear counters Capacity is the focus for primary children Partners for Mathematics Learning

Nonstandard Units of Capacity 55 Nonstandard Units of Capacity  Containers or scoops: small containers that are filled and poured repeatedly into the container being measured; the number of scoops provides the measure  Solid units: small congruent objects that fill of small objects provides the measure Partners for Mathematics Learning

C apacity: C ompensatory P rinciple 56 C apacity: C ompensatory P rinciple  How can the compensatory principle be developed through the concept of capacity in second grade? Partners for Mathematics Learning

Mass  Mass is the amount of 57 Mass  Mass is the amount of matter in an object  Weight is a measure of the pull or force of gravity on an object  On the moon there is less gravity so an object weighs less  The mass of an object is identical on the moon and on Earth Partners for Mathematics Learning

 Children can use their hands to estimate 58 Exploring Mass  Children can use their hands to estimate which of two objects is heavier  Children place marbles on one side of a balance scale to find the mass of a book using marbles as a non-standard measure Partners for Mathematics Learning

Classroom Experiences 59 Classroom Experiences  Look at the measurement handouts  Each group will work through the Scavenger Hunt and the Measuring Rectangles  As you work discuss how the activity addresses the Essential Standards Partners for Mathematics Learning

Measuring Rectangles  How would students 60 Measuring Rectangles  How would students order the rectangles?  What attributes do they use?  Do they realize that when they consider different attributes, the rectangles may be ordered differently?  Can they cover it with no gaps or overlaps?  Other ideas? Partners for Mathematics Learning

Scavenger Hunt  How could you extend the activity? Partners 61 for Mathematics Learning

Measurement Jigsaw  Roll This Weigh Side by Side Guess and Count 62 Measurement Jigsaw  Roll This Weigh Side by Side Guess and Count Paper Clip Measurement Which Takes Longer? Partners for Mathematics Learning

 2nd Grade Essential Understanding 63 Marker board Paths chair bookcase  2nd Grade Essential Understanding  Describe relationships of objects using proximity, position, direction, and turns  Look at the handout, “Paths Lesson” Partners for Mathematics Learning

NCTM Applets http://illuminations.nctm.org/ 64 NCTM Applets http://illuminations.nctm.org/  Hiding Ladybug Adventures  Ladybug Mazes  Turtle Pond Partners for Mathematics Learning

Measurement  Is one of the most widely used applications 65 Measurement  Is one of the most widely used applications of mathematics  Connects to geometry, data, number, and problem solving  Experience provide opportunities to learn and understand Partners for Mathematics Learning

Personal Reflection  Opportunities to learn and understand in an 66 Personal Reflection  Opportunities to learn and understand in an environment that is interesting, challenging, and enjoyable provide students with strong foundations upon which to build  What strengths does your instructional program currently have?  What three challenges will you take from this module for your classroom? Partners for Mathematics Learning

Renee Cunningham Kitty Rutherford Robin Barbour Mary H. Russell 67 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 68 PML Dissemination Consultants Susan Allman Julia Cazin Ruafika Cobb Anna Corbett Gail Cotton Jeanette Cox Leanne Daughtry Lisa Davis Ryan Dougherty Shakila Faqih Patricia Essick Donna Godley Cara Gordon Tery Gunter Barbara Hardy Kathy Harris Julie Kolb Renee Matney Tina McSwain Marilyn Michue Amanda Northrup Kayonna Pitchford Ron Powell Susan Riddle Judith Rucker Shana Runge Yolanda Sawyer Penny Shockley Pat Sickles Nancy Teague Michelle Tucker Kaneka Turner Bob Vorbroker Jan Wessell Daniel Wicks Carol Williams Stacy Wozny Partners for Mathematics Learning

2009 Writers Partners Staff Kathy Harris Rendy King Tery Gunter 69 2009 Writers Partners Staff Kathy Harris Rendy King Tery Gunter Judy Rucker Penny Shockley Nancy Teague Jan Wessell Stacy Wozny Amanda Baucom Julie Kolb 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 Two Module 3 Partners for Mathematics Learning