1. Revisiting Sequence of Equations – Part 1

2. Reflecting Back On Our Work
Gather with your team.
Reflect on your work from yesterday. Create a coherent representation of your work so that others can understand the “gist” of your work.
Include in your discussion:
Size & Type of Number
Structure of Equation (Where is the equal sign?)
How are you planning to use this assessment?
Why did you choose the items? What will you learn about your students and what do you plan to do with that information?

3. Gallery Walk As you walk you should look for:
a progression of complexity from Grade to Grade.
Interesting problems you would like to modify for your grade level.
After you return to your table group, discuss the following:
Note the complexity from grade to grade.
To what degree do your items match grade level standards?
Before the gallery walk, what would you expect to see?

4. Equality and Relational Thinking: Abstracting from Computation
Common Core State Standards for Mathematics
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
Friday, July 25, 2014
Session Description:
Take a journey into the “Core” to inspect progressions of mathematical ideas and student learning,
to surface shifts from current practice, and to consider implications for instruction, curriculum, and assessment.

5. Learning Intentions & Success Criteria
We are learning to reason from the understanding that equality is a relationship that expresses the idea that two mathematical expressions hold the same value.
Success Criteria
We will be successful when we can recognize the difference between computational and relational thinking
We will be successful when we can use relational thinking to build, express, and justify mathematical relationships.

6. Shifting Perspective Session Description:
Take a journey into the “Core” to inspect progressions of mathematical ideas and student learning,
to surface shifts from current practice, and to consider implications for instruction, curriculum, and assessment.

7. Math Practice 7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure…(CCSSM). They can also step back for an overview and shift perspective. What shifts in perspective supported your thinking as you engaged in relational thinking during yesterday’s class?
Read MP 7 out of Think Math to add to their knowledge of the what it means to develop a habit of mind of looking for and making use of structure.
The CCSSM suggest mathematical habits of mind that are essential for mathematical proficiency – one of these is MP7.
Structures can be: place value, the properties of the operations, or generalizations about the behavior of the operations
***Developing Algebraic Habits of Mind by Mark Driscoll is very helpful to facilitating and structuring this discussion.

8. Supporting Relational Thinking
True or False? = 100 – – 28 = 86 – = What shifts in thinking did you have to make to view these true/false sentences through a relational lens?
Chart the shifts in thinking that participants made to think relationally
Structures can be: using place value, the properties of the operations, or generalizations about the behavior of the operations

9. Developing a Habit of Mind
Fill in the box to make these statements true: = x = 46 + ¾ + x = ¼ +
Meaning of the equal sign and thinking about the relationship between the quantities
Equation needs to balance
Structure of the base 10 system
Also MP 8.

10. Developing a Habit of Mind
Fill in the box to make these statements true: = x = 46 + ¾ + x = ¼ +
Meaning of the equal sign and thinking about the relationship between the quantities
Equation needs to balance
Structure of the base 10 system
Also MP 8.

11. What happens when…
How would students typically approach solving for x?
11 –
3x – 2
= 6 50
Read MP 7 out of Think Math to add to their knowledge of the what it means to develop a habit of mind of looking for and making use of structure.
The CCSSM suggest mathematical habits of mind that are essential for mathematical proficiency – one of these is MP7.
Structures can be: place value, the properties of the operations, or generalizations about the behavior of the operations
***Developing Algebraic Habits of Mind by Mark Driscoll is very helpful to facilitating and structuring this discussion.

12. What happens when…
How would students typically approach solving for x?
11 –
3x – 2
= 6 50
Read MP 7 out of Think Math to add to their knowledge of the what it means to develop a habit of mind of looking for and making use of structure.
The CCSSM suggest mathematical habits of mind that are essential for mathematical proficiency – one of these is MP7.
Structures can be: place value, the properties of the operations, or generalizations about the behavior of the operations
***Developing Algebraic Habits of Mind by Mark Driscoll is very helpful to facilitating and structuring this discussion.

13. 6.EE.2b
Apply and extend previous understandings of arithmetic to algebraic expressions.
2. Write, read, and evaluate expressions in which letters stand for numbers.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient);
view one or more parts of an expression as a single entity.
For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

14. When mathematical ways of thinking begin to become automatic – not just ways one can use, but ways one is likely to use – it is reasonable to call them habits: mathematical habits of mind. --EDC Transitions to Algebra

15. Transitioning to Algebra
Session Description:
Take a journey into the “Core” to inspect progressions of mathematical ideas and student learning,
to surface shifts from current practice, and to consider implications for instruction, curriculum, and assessment.

16. What Happens When You Add?
If you add 1/3 to a number, is the result larger than the number you started with, or smaller, or is it the same, or can’t you tell?
In a recent study, 87% of a cohort of community
college students enrolled in remedial mathematics
courses answered this question correctly.
Chart the shifts in thinking that participants made to think relationally
Structures can be: using place value, the properties of the operations, or generalizations about the behavior of the operations
Givven, K., Stigler, J. and Thompson, B. (2011)

17. What Happens Now?
If a + 1/3 = x, is x larger than a, or smaller, or is it the same, or can’t you tell?
30% of the community college students in the study answered “You can’t tell”. (Even though 78% of that 30% had answered the previous question correctly.)
Chart the shifts in thinking that participants made to think relationally
Structures can be: using place value, the properties of the operations, or generalizations about the behavior of the operations
Givven, K., Stigler, J. and Thompson, B. (2011)

18. Video Game Scores (Part 1)
Eric is playing a video game. At a certain point in the game , he has points. Then the following events happen, in order:
He earns 2450 additional points
He loses 3310 points
The game ends, and his score doubles.
Write an expression for the number of points Eric has at the end of the game. Do not evaluate the expression.
Where does the idea of understanding the meaning of the equal sign come up. Consider students in first grade and the expectation set as they are working to understand this fundamental idea for developing algebraic reasoning.
Illustrative Mathematics:

19. Video Game Scores (Part 2)
Eric’s sister Leila plays the same game. When she is finished playing, her score is given by the expression 3( ) – Describe a sequence of events that might have led to Leila earning this score.
Where does the idea of understanding the meaning of the equal sign come up. Consider students in first grade and the expectation set as they are working to understand this fundamental idea for developing algebraic reasoning.
Illustrative Mathematics:

20. 5.OA.2
Write and interpret numerical expressions. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × ( ) is three times as large as , without having to calculate the indicated sum or product.
Where does the idea of understanding the meaning of the equal sign come up. Consider students in first grade and the expectation set as they are working to understand this fundamental idea for developing algebraic reasoning.

21. Ms. Brown’s Cookies
Ms. Brown baked some cookies for her class. Each student ate 2 cookies, and there were 8 cookies left over. Write an expression that represents the number of cookies Ms. Brown baked. (Use s to represent the number of students in the class.) Write an equation that relates the number of students in the class and the number of cookies.
Where does the idea of understanding the meaning of the equal sign come up. Consider students in first grade and the expectation set as they are working to understand this fundamental idea for developing algebraic reasoning.

22. Ms. Brown’s Cookies Redux
The next day, Ms. Brown baked some more cookies for her class. Together with the 8 cookies left over from the first day, she had twice as many cookies as there were students in her class. Write an equation that relates the number of students in the class and the number of cookies Ms. Brown baked on the second day.
Where does the idea of understanding the meaning of the equal sign come up. Consider students in first grade and the expectation set as they are working to understand this fundamental idea for developing algebraic reasoning.

23. Back to the Handshake Problem
Everyone at a party shakes hands (once!) with everyone else. Write expressions for the total number of handshakes if there are: 4 people at the party; 5 people; 6 people; 20 people; k people; (n + 1) people.
Where does the idea of understanding the meaning of the equal sign come up. Consider students in first grade and the expectation set as they are working to understand this fundamental idea for developing algebraic reasoning.

24. 6.EE.2b
Apply and extend previous understandings of arithmetic to algebraic expressions.
2. Write, read, and evaluate expressions in which letters stand for numbers.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient);
view one or more parts of an expression as a single entity.
For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

25. 6.EE.6
Reason about and solve one variable equations and inequalities.
6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

26. A-SSE.1
Interpret the structure of expressions.
1. Interpret expressions that represent a quantity in terms of its context.

27. Transporting Sand
A construction company uses 2 trucks to transport sand to a building site. One truck makes x trips, and carries S tons of sand per trip. The second truck makes y trips, and carries T tons of sand per trip. What is the meaning of the following expressions? a. x + y b. S + T c. xS + yT
Where does the idea of understanding the meaning of the equal sign come up. Consider students in first grade and the expectation set as they are working to understand this fundamental idea for developing algebraic reasoning.
Illustrative Mathematics

28. Revisiting Sequence of Equations - Part 2

29. Revisiting Sequence of Equations
Revisit your equations and modify based on what you studied this morning. Complete your chart.
Please leave your chart with us today (unless you really want to work further on it).
Make sure your Grade and names are on it.
We will take photos or scan and share with everyone.

30. Curriculum Connections

31. Curriculum Connections
Work as a team and search out what we have studied so far in class in your curricular materials.
Use post-its, highlighters, etc. to identify specific connections in your book to what we have studied in class.
You will have 30 minutes for this task. Enjoy!

32. Curriculum Connections
As you reviewed your curricular materials, what were you noticing?

33. House Keeping

34. Housekeeping
Notebooks: Tab all PRRR, leave notebook on top of your pile of materials.
Turn in Sequence of Equations assignment.
Clean up (Thanks!)
Relax this weekend!!

35. Disclaimer Core Mathematics Partnership Project
University of Wisconsin-Milwaukee,
This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.