Module 6 PARTNERS for Mathematics Learning Grade 8 Partners 1

Algebra Talks  A class conversation about an algebraic 2 Algebra Talks  A class conversation about an algebraic equation, in which students discuss and critique various strategies for solving the problem  The work is done mentally, though some writing may be offered by a student or by a teacher when a strategy is explained Partners for Mathematics Learning

Algebra Talks Rules  When the problem is put up, solve in your 3 Algebra Talks Rules  When the problem is put up, solve in your head  When you have solved, put your thumb up in front of your chest  Try to solve in a different way  For each different way you solve, put up another finger Partners for Mathematics Learning

Algebra Talks 1-Step Equations 4 Algebra Talks 1-Step Equations  x+2=6  3 = y + (-7)  a – 3 = -2  b – (-8) = -2 Partners for Mathematics Learning

Algebra Talks 2-Step Equations 5 Algebra Talks 2-Step Equations  2x + 1 = 5  3x – 1 = 5  2 – 4x = 6  13 = 7x - 1 Partners for Mathematics Learning

Algebra Talks 1-Step Inequalities 6 Algebra Talks 1-Step Inequalities  x+3>4  4–x<7  2x > 14  15 > -3x Partners for Mathematics Learning

What Do You Think?  How will you use Algebra Talks in your 7 What Do You Think?  How will you use Algebra Talks in your classroom?  How do rules of operations with integers fit within these discussions?  How do standard algorithms fit within these discussions? Partners for Mathematics Learning

Focus: Algebra  Thinking about change is not simple 8 Focus: Algebra  Thinking about change is not simple  It requires thinking about several things at once National Council of Teachers of Mathematics. (2001) Navigating through Algebra in Grades 6-8 , Reston, VA: author Partners for Mathematics Learning

Variables  What is changing?  Over what time period is the change 9 Variables  What is changing?  Over what time period is the change occurring? National Council of Teachers of Mathematics. (2001) Navigating through Algebra in Grades 6-8 , Reston, VA: author Partners for Mathematics Learning

Slope  What is the rate at which the change is 10 Slope  What is the rate at which the change is occurring? National Council of Teachers of Mathematics. (2001) Navigating through Algebra in Grades 6-8 , Reston, VA: author Partners for Mathematics Learning

Relations and Functions 11 Relations and Functions  Is the change following a pattern? National Council of Teachers of Mathematics. (2001) Navigating through Algebra in Grades 6-8 , Reston, VA: author Partners for Mathematics Learning

Let’s Play a Game! X Y  Can you “Guess My Rule”? 1 2 5 7 50 Partners 12 Let’s Play a Game!  Can you “Guess My Rule”? X Y 1 2 5 7 50 Partners for Mathematics Learning

Can You “Guess my Rule”?  What is changing? 13 Can You “Guess my Rule”?  What is changing?  Over what time period is the change occurring?  What is the rate at which the change is  Is the change following a pattern? Partners for Mathematics Learning

Thinking Mathematically 14 Thinking Mathematically  Middle-grades students need to learn to think about these questions mathematically National Council of Teachers of Mathematics. (2001) Navigating through Algebra in Grades 6-8 , Reston, VA: author Partners for Mathematics Learning

What Do Students Think of Slope? 15 What Do Students Think of Slope?  m = slope y = mx + b slope = rise/run m = (y 2 – y 1 )/(x 2 – x 1 ) Partners for Mathematics Learning

What Does Slope Tell Us?  It quantifies the steepness of a line! 16 What Does Slope Tell Us?  It quantifies the steepness of a line! Partners for Mathematics Learning

What Is Slope?  It is the rate of change in one quantity 17 What Is Slope?  It is the rate of change in one quantity with respect to change in another quantity Partners for Mathematics Learning

Activity – Rising waters 18 Activity – Rising waters Aesop’s fable: The Crow and the Pitcher  Look at your graph  What if the shape of the container changed? Partners for Mathematics Learning

Filling A Hot Tub  What could a  What would be graph of the water 19 Filling A Hot Tub  What could a graph of the water level look like?  What would be the independent and dependent variables? Partners for Mathematics Learning

20 Water In A Hot Tub Partners for Mathematics Learning

Activity – Filling a Container 21 Activity – Filling a Container Work together at your tables Partners for Mathematics Learning

Mystery Graph  When looking at the graph on the next 22 Mystery Graph  When looking at the graph on the next slide, think about  What common household appliance does the graph represent?  Why doesn’t the graph temperature start at zero?  Why isn’t the temperature constant at 350°? Partners for Mathematics Learning

23 Mystery Graph Partners for Mathematics Learning

Qualitative Graphs Choose the graph that best represents each 24 Qualitative Graphs Choose the graph that best represents each situation Partners for Mathematics Learning

Mathematical Models What is a mathematical model? Partners 25 for Mathematics Learning

Walk That Function  Walk the graphs using a CBR  What are common 26 Walk That Function  Walk the graphs using a CBR  What are common misconceptions students have about distance-time graphs such as these? Partners for Mathematics Learning

Distance-Time Graphs  “In order to analyze the graph of a set of 27 Distance-Time Graphs  “In order to analyze the graph of a set of distance-time data, the student must be able to interpret the rates of change in distance, velocity, and acceleration that are observable in a graph.” NCTM – Navigating through Algebra, 2001 Partners for Mathematics Learning

Your Trip To School  You live 6 miles from school and your trip 28 Your Trip To School  You live 6 miles from school and your trip takes you twenty minutes  There is one stop sign along the way and you must come to a complete stop at the stop sign Partners for Mathematics Learning

Slope as a Numerical Value 29 Slope as a Numerical Value  Describe in words what these slope values mean  2  3/2  1.5  -4  -2.5/3 Partners for Mathematics Learning

30 Water In A Hot Tub Partners for Mathematics Learning

He’s Back!  How much did the level of the  What numerical quantity 31 He’s Back!  How much did the level of the water rise for each marble that was added?  What numerical quantity describes this change?  What was the height of the water in the container before any marbles were added? Partners for Mathematics Learning

Modeling Functions  Patterns and Functions  Rate of change 32 Modeling Functions  Patterns and Functions  Rate of change  Meaning of slope and y-intercept  Relate a function to its graph Partners for Mathematics Learning

Olympic Data  Create a model of the data  How long would it take 33 Olympic Data  Create a model of the data  How long would it take Michael Phelps to swim 400 meters?  How realistic is your prediction? Partners for Mathematics Learning

The NC Standards  Collect, organize, analyze, and display data 34 The NC Standards  Collect, organize, analyze, and display data including scatter plots to solve problems  Approximate a line of best fit for a given scatterplot; explain the meaning of the line as it relates to the problem and make predictions  Develop an understanding an understanding of functions Partners for Mathematics Learning

What’s the Big Idea?  Bivariate data may be displayed and then 35 What’s the Big Idea?  Bivariate data may be displayed and then analyzed within the rectangular coordinate plane, where a linear equation may be a good model for the relationship between the two attributes Partners for Mathematics Learning

Learning Is …  Learning is not the result of development; 36 Learning Is …  Learning is not the result of development; learning is development. It requires invention and self-organization on the part of the learner. Thus teachers need to allow learners to raise their own questions, generate their won hypotheses and models as possibilities, and test them for validity. (Fosnot) From Van de Walle, J. A (2004). Elementary and Middle School Mathematics: Teaching Developmentally . Pearson Learning Inc. Partners for Mathematics Learning

Renee Cunningham Kitty Rutherford Robin Barbour Mary H. Russell 37 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 38 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 39 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

Module 6 PARTNERS for Mathematics Learning Grade 8 Partners 40