1. Module 6 PARTNERS for Mathematics Learning Grade 8 Partners 1

2. Algebra Talks  A class conversation about an algebraic2
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

3. Algebra Talks Rules  When the problem is put up, solve in your3
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

4. Algebra Talks 1-Step Equations4
Algebra Talks 1-Step Equations
 x+2=6
 3 = y + (-7)
 a – 3 = -2
 b – (-8) = -2
Partners
for Mathematics Learning

5. Algebra Talks 2-Step Equations5
Algebra Talks 2-Step Equations
 2x + 1 = 5
 3x – 1 = 5
 2 – 4x = 6
 13 = 7x - 1
Partners
for Mathematics Learning

6. Algebra Talks 1-Step Inequalities6
Algebra Talks 1-Step Inequalities
 x+3>4
 4–x<7
 2x > 14
 15 > -3x
Partners
for Mathematics Learning

7. What Do You Think?  How will you use Algebra Talks in your7
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

8. Focus: Algebra  Thinking about change is not simple8
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

9. Variables  What is changing?  Over what time period is the change9
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

10. Slope  What is the rate at which the change is10
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

11. Relations and Functions11
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

12. Let’s Play a Game! X Y  Can you “Guess My Rule”? 1 2 5 7 50 Partners12
Let’s Play a Game!
 Can you “Guess My Rule”? X Y 1 2 5 7 50
Partners
for Mathematics Learning

13. 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

14. Thinking Mathematically14
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

15. 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

16. 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

17. What Is Slope?  It is the rate of change in one quantity17
What Is Slope?
 It is the rate of change in one quantity
with respect to change
in another
quantity
Partners
for Mathematics Learning

18. Activity – Rising waters18
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

19. Filling A Hot Tub  What could a  What would be graph of the water19
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. 20
Water In A Hot Tub
Partners
for Mathematics Learning

21. Activity – Filling a Container21
Activity – Filling a Container
Work together at your
tables
Partners
for Mathematics Learning

22. Mystery Graph  When looking at the graph on the next22
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. 23
Mystery Graph
Partners
for Mathematics Learning

24. Qualitative Graphs Choose the graph that best represents each24
Qualitative Graphs
Choose the graph that best represents each
situation
Partners
for Mathematics Learning

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

26. Walk That Function  Walk the graphs using a CBR  What are common26
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

27. Distance-Time Graphs  “In order to analyze the graph of a set of27
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

28. Your Trip To School  You live 6 miles from school and your trip28
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

29. Slope as a Numerical Value29
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. 30
Water In A Hot Tub
Partners
for Mathematics Learning

31. He’s Back!  How much did the level of the  What numerical quantity31
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

32. Modeling Functions  Patterns and Functions  Rate of change32
Modeling Functions
 Patterns and Functions
 Rate of change
 Meaning of slope and y-intercept
 Relate a function to its graph
Partners
for Mathematics Learning

33. Olympic Data  Create a model of the data  How long would it take33
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

34. The NC Standards  Collect, organize, analyze, and display data34
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

35. What’s the Big Idea?  Bivariate data may be displayed and then35
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

36. 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

37. 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

38. PML Dissemination Consultants38
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

39. 2009 Writers Partners Staff Kathy Harris Rendy King Tery Gunter39
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

40. Module 6 PARTNERS for Mathematics Learning Grade 8 Partners 40