1. Secondary Math Progressions
Expressions and Equations
Lisa Ashe
Secondary Mathematics Consultant
NC DPI

2. Welcome Who’s in the Room?
Survey participants: first timers, math coaches, classroom teachers, central office, principals, etc…
Who’s in the Room?

3. NORMS Listen to others’ ideas Disagree with ideas, not people
Be respectful
Helping is not the same as giving answers
Confusion is part of learning
Say your becauses…

4. Goals of the PD
To develop a progressive understanding of the standards in the 6-8 Expressions & Equations domain.
To explore the Teaching Practices that facilitate how to make the Standards for Mathematical Practices come alive for students and teachers.
To analyze resources that can be used to teach within the Expressions and Equations domain for MS math with understanding.
To DO some MATH!

5. Common Core Key Instructional Shifts

6. Common Core Key Instructional Shifts

7. Common Core Key Instructional Shifts

8. FOCUS
Postsecondary instructors want deeper mastery of fewer things than high school instructors desire.
Focus…fewer standards at a deeper level.

9. RIGOR
Balance of teaching approaches and learning experiences for the student
Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. (CCSS, p. 4)

10. How do we get from here to there with coherence?
2003 SCOS
How do we get from here to there with coherence?
CCSS Standards
Mathematics for the work of teaching as a whole…the connections between and across conceptual categories.

11. Symbolic Standards for Mathematical Practice
Make sense of problems and persevere in solving them.
Reason abstractly and qualitatively.
Construct viable arguments and critique the reasoning of others.
Model with Mathematics.
Use appropriate tools strategically.
Attend to Precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Create a SYMBOL for the assigned SMP.

12. Eight High-Leverage Instructional Practices
Establish mathematics goals to focus learning
Implement tasks that promote reasoning and problem solving
Use and connect mathematical representations
Facilitate meaningful mathematical discourse
Pose purposeful questions
Build procedural fluency from conceptual understanding
Support productive struggle in learning mathematics
Elicit and use evidence of student thinking

13. Teaching Practices vs. Student Practices

14. Expressions and Equations Scavenger Hunt
How well do you know your standards
Using only each other as resource, determine what grade level each cluster heading represents?
Introduce yourself to at least 3 people that you have never met.

15. Domain Progressions
Use the resources on your table to organize the cluster headings in the most logical learning progression.

20. Establish mathematics goals to focus learning.
Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses goals to guide instructional decisions.

21. A little math humor

22. Standards and Learning Targets
How does a standard differ from a learning target?

23. Standards and Learning Targets
Standard: What we want students to be able to know and do at the end of any given time. Standards are provided by the state and derived from the National Standards.
Learning Targets: These are statements of intended learning based on the standards. Learning targets are in student friendly language and are specific to the lesson for the day and directly connected to assessment.

24. Let’s Practice! 5.OA.1 Write and interpret numerical expressions.
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
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.
Knowledge:
Vocabulary: expression, grouping symbols (parenthesis, bracket, braces, vinculum,…), operations (+, -, ×, ÷), order of operations
Skills:
Write numeric expressions based on a context
Evaluate numeric expressions
Interpret numeric expressions
Compare expressions without calculating
Recognize the structure of an expression

25. Your Turn! What mathematics is being learned? Why is this important?
How does it relate to what has already been learned?
Where are these mathematical ideas going?
Grade Level
Knowledge (Nouns)
Skills (Verbs)
Essential Understandings (Themes)
Learning goals (student friendly)

26. Principles to Action - page 16
Ask participants to list some specific strategies that they use to establish mathematics goals to focus learning in their classroom.
Examples
Begin every lesson/unit with an essential question that is continually revisited throughout the lesson/unit.
Mastery trackers---short quizzes that track a students mastery of particular concepts over time
Principles to Action - page 16

27. Reflection for Planning Learning
What kinds of expressions and equations do your students work with in your mathematics class?
How are expressions and equations in elementary school different from expressions and equations in middle school?

28. Progressions of Expressions
Numeric expressions in upper elementary and lower middle school mathematics; expressions with brackets
Numeric and algebraic expressions in middle school; expressions with whole number exponents; equivalent expressions leading to equations
Complex algebraic expressions in high school and beyond; connections to functions
We see this progression through a pathway of Expressions to Equations to Functions. Note that this is one possible progression. It can also be extended to more intensive work within Algebra.

29. From Numeric to Algebraic Expressions
K-8th Grades
HS Math
K-3rd
Numeric
Expressions
Numerical expressions begin in elementary school
Students in elementary grades use the order of operations to evaluate numeric expressions and write numeric expressions (without evaluating)
The work in middle school builds on the elementary experience working with the properties of operations with whole numbers, decimals and fractions to algebraic expressions.
Students have worked with variables to represent unknowns since 3rd grade, they begin to work with algebraic expressions in 6th grade
7th grade is where students begin to look at the structure of an algebraic expression and determine the form needed when solving problem; this is also where they start to simplify general linear expressions.
The work in 8th grades expands to expressions with exponents including the properties of integer exponents.
High school work focuses on simplifying, writing and evaluating more complex expressions…quadratic, exponential, trigonometric, rational, radical, etc.
3-6th Grade
Patterns and Sequences
5-7th Grade
Expressions
8th Grade
Expressions w/ exponents
High School
Complex Algebraic Expressions

30. Task: Comparing Expressions, Expressing Relations
Without any calculation, within each set, what do you know about the relationship between the products for the given expressions?
Set 1
4 × 15
8 × 15
4 × 30
Set 2
10 × 40
20 × 40
10 × 80
Set 3
20 × 150
40 × 150
20 × 300
Implementing the Common Core State Standards through Problem Solving Grades 3-5; Task 2.2 (Page 13)
Questions to encourage discussion:
What do you notice about these expressions?
Which product will be the smallest? Without telling us the product, how do you know?
Which product will be the greatest? Without telling us the product, how do you know?
What do we know about multiplication?
Implementing the Common Core State Standards through Problem Solving Grades 3-5; Task 2.2 (Page 13)

31. Task: Wacky Parentheses
Naima’s grandmother is going to double Naima’s money. Naima started with $4, and she then earned $6 more. Naima expected to have $20 total after her grandmother doubled her money. Her grandmother thought that Naima would have $14. How was Naima thinking about it? How was her grandmother thinking about it?
Implementing the Common Core State Standards through Problem Solving Grades 3-5; Task 2.9 (Page 29)
Questions to encourage discussion:
Who do you agree with, Naima or her grandmother? Why might Naima be right? Why might her grandmother be right?
How could you rewrite this expression to make sure Naima receives her money?
What do you think parentheses mean when we use them in math? Are they necessary?
Naima’s work
2 × = 20
Her grandmother’s work
2 × = 14
Implementing the Common Core State Standards through Problem Solving Grades 3-5; Task 2.9 (Page 29)

32. Task: Evaluating Expressions
For each of the following, a student evaluated the given expression. Your job is to determine whether the student is correct or not. If you think the student is wrong, explain the mistake and provide the correct answer.
Parts (a) through (e) – standard 6.EE.2c
Parts (f) and (g) – standards 6.EE.3 and 6.EE.4
(b) and (c) are correct

33. Expressions Task Progression
Look at the Expressions Task Progressions
To what learning goal(s) is each task aligned? At what grade level(s)?
How does the progression of the task support student understanding as they progress through middle grades?

34. Progression of Equations
Equations with an unknown in upper elementary school; informal language; relational definition of equivalence.
Equivalent expressions in middle school (formal language); symbolic transformations of expressions/equations; creating, solving and interpreting equations; simultaneous equations; functions.
Different types of equations in high school; more extensive work with functions; comparing function types.

35. Equivalent Expressions to Equations
6th-8th Grades
Math I-III
6th
One-step equations (one-variable)
7th- HS Mathematics
Multi-step Equations
(one-variable)
The work of equivalent expressions in upper elementary grades evolves into one-variable equations in middle school with one-step equations in 6th grade and eventually becoming multi-step equations in the remainder of middle school.
This will eventually lead to multiple two-variable equations/simultaneous equations in the 8th grade laying the foundation for systems of equations in High School and beyond.
In 8th grade students solve linear systems by graphing and substitution.
They will use other algebraic methods in high school and work with systems of different types of equations (combinations of linear, quadratic and exponential equations).
7th Grade
Ratios and
Proportions
8th Grade – HS Mathematics
Systems of Equations
Math II and III
Simple trig, absolute value and simple rational equations
Math III
Advanced Trigonometric and Logarithmic equations

36. Reasoning about Equations
Column A
Column B
48 × 67 × 6 = k
347 × 25 × 4 = p
746 × 398 ÷ 42 = t
398 × 746 ÷ 746 = d
= y
= j
475 × 2365 = 352 × w
8790 × 598 = 879 × n
Compare the equations in columns A and B.
Developing Essential Understanding of Algebraic Thinking, Grades 3-5, page 26

37. Task: Secret Number
Pick a number between 1 and 10. This is your secret number.
Using your secret number together with any other numbers you wish, write an expression that “changes” your secret number into 25. You may use any arithmetic operations you wish.
Write four more expressions that change your secret number into 25. Use different operations, symbols, and numbers, including parentheses, exponents, fractions or decimals, and negative numbers. You should have a list of five expressions that change your number into 25. Choose two of your expressions to write equality statements (called equations).
Next, get ready to exchange your equations that contain your equivalent expressions with a classmate, who will try to guess your secret number from the equation. Because you don’t want anyone to see what your number is, you should use a placeholder for your number. You can use a letter to represent your number.
Exchange your equation with someone at your table. See whether you can guess your friend’s secret number, and if he or she can guess yours. What do you need to do to figure it out? Write down your steps.

38. Task: Using Solved Problems
Consider the equation 3x – 8 = 8x + 12 and the three solutions below.
Method A
Method B
Method C
3x – 8 = 8x + 12
3x = 8x + 20
−5x = 20
x = −4
−5x = 12
3x – 8 – 8x – 12 = 0
3x – 8 – 8x = 12
3x – 8x = 20
In Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students (What Works Clearinghouse), it is recommended to use solved problems to engage students in analyzing algebraic reasoning and strategies.
Compare and contrast the solution methods and evaluate the efficiency of each method.

39. Equations Task Progression
Look at the Expressions Task Progressions
To what learning goal(s) is each task aligned? At what grade level(s)?
How does the progression of the task support student understanding as they progress through middle grades?

40. Elicit Evidence of Student Thinking
Preparation of each lesson needs to include intentional and systematic plans to elicit evidence that will provide “a constant stream of information about how student learning is evolving toward the desired goal.”
Principles to Actions pg. 53

41. Elicit Evidence of Student Thinking
“My Favorite No: Learning From Mistakes”
During the video;
Identify strategies the teacher uses to access, support, and extend student thinking.
How do these strategies allow for immediate re-teaching?
What student behaviors were associated with these instructional strategies?

42. Sorting Expressions Task
In your table groups, sort the expressions
Use sticky notes to label your categories.
Be prepared to compare and defend your strategy to others in the class.

43. Summary: Establish Mathematics Goals to Focus Learning – Points to Remember
Learning progressions or trajectories describe how students make transitions from prior knowledge to more sophisticated understandings
Both teachers and students need to be able to answer these crucial questions:
What mathematics is being learned?
Why is this important?
How does it relate to what has already been learned?
Where are these mathematical ideas going?
Situating learning goals within the mathematical landscape supports opportunities to:
Build explicit connections
See how ideas build and relate to one another
Develop a coherent and connected view of the discipline

44. Let’s Play!
student.desmos.com
Code: f2nv

45. Math Tasks
There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perception about what mathematics is than the selection or creation of the tasks with which the teacher engages students in shaping mathematics.
Principles to Actions, page

46. Intellectual Need
The tasks that teachers present, what issues are discussed, and the way in which student questions or alternative solutions are addressed all have a pronounced effect on where classroom activity falls in the problem-laden versus problem-free spectrum.

47. The Problem-FREE classroom:
The immediate goal is not understood by students.
The goal of the activity as a whole is unclear.
There is no intellectual necessity for the method of solution.
Students know what to do in advance, so there is no need for the problem to be considered carefully.

48. Implement Tasks That Promote Reasoning and Problem Solving
Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and that allow for multiple entry points and varied solution strategies.

49. Cognitive Levels of Demand
High
Low

50. Cognitive Demand Sort Find your group.
Read the handout describing the different cognitive demand task types. Come to a shared understanding of the demand types:
Memorization
Procedures without Connections
Procedures with Connections
Doing Mathematics
Use the contents of the envelope to sort the tasks by cognitive demand.

51. Your Turn! Look at the task(s) that you have brought:
To which learning goal does the task align?
Determine the level of cognitive demand for the task.
How can you increase the cognitive demand of the task?
What evidence of student thinking are you for in the student responses?

53. Implementation Matters
High
Low
High
Low
Moderate
Task
Implementation
Student Learning

54. Principles to Action - page 24

55. Implement tasks that promote reasoning and problem solving – Points to Ponder
Effective math teaching and learning uses carefully selected tasks as one way to motivate student learning and build new knowledge.
Research on math tasks over the past two decades has found:
Not all tasks provide the same opportunities for student thinking and learning.
Student learning is the greatest in classrooms where tasks consistently encourage high-level student thinking and the least in classrooms where tasks are routinely procedural in nature.
Tasks with high cognitive demands are the most difficult to implement well and are often transformed into less demanding tasks.
To ensure that students have the opportunity to engage in high- level thinking, teachers must regularly select and implement tasks the promote reasoning and problem solving.
Referenced from Page 39 of Strength in Numbers
“Well-chosen challenge problems serve as a good starting point for group work because they help students to get directly to the heart of mathematical issues.”
Presenting problems that map onto an example encourages students who work well with patterns quickly to take over that task; which ultimately lowers the cognitive demand of the task

56. Task: Hexagon Trains
Trains 1, 2, 3 and 4 are the first four trains in the hexagon pattern. The first train in this pattern consists of one regular hexagon. For each subsequent train, one additional hexagon is added.
Compute the perimeter for each of the first four trains.
Draw the fifth train and compute the perimeter of the train.
Determine the perimeter of the 25th train without constructing it.
Write a description that could be used to compute the perimeter of any train in the pattern.
Determine which train has a perimeter of 110.
Train 1
Train 2
Train 3
Train 4
This task is sample task. Use a task with multiple entry points and a variety of representations to model the next two teaching practices.
Monitor groups as they solve the task above. If you notice everyone using the same strategy, ask the group to approach the problem in at least two different ways.
Select and sequence: Decide which groups you want to share and in what order…the Boomerang task can be approached in at least 3 different ways (table, graph and algebraically). Begin with the graphical representation, then tabular and last algebraic.
Connect: Ask the students to compare/contrast the different approaches. Examine the connections between representations.
Additional Questions:
What was the presenter doing before, during and after the tasks?
Are there additional questions that could further student understanding?
How does conceptual understanding relate to these teaching strategies?

57. Facilitate meaningful mathematical discourse
Effective teaching of mathematics facilitates discourse among students in order to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments.
Table Talk: In what ways do you encourage mathematical discourse in your class?
Ask if anyone is willing to share…why does you think it is so difficult for our students/teachers? What are some ways that we can start this process?

58. “What students learn is intertwined with how they learn it
“What students learn is intertwined with how they learn it. And the stage is set for the how of learning by the nature of classroom-based interactions between and among teacher and students.”
(Smith & Stein, 2011)
Teachers need to develop a range of ways of interacting with and engaging students as they work on tasks and share their thinking with other students.

59. 5 Practices for Orchestrating Productive Mathematics Discussions
Anticipating
Monitoring
Selecting
Sequencing
Connecting
Group jigsaw
Reference page 30 in the Orchestrating Productive Mathematics Discussions book

60. Table Talk Think back to the Hexagon Trains Task
Review the student work samples.
As a table, determine the mathematical goal for the task.
Determine which group should present a solution, and in what order the solutions should be presented.
What questions should be asked to connect solutions?
Goal:
Order and Reasoning:
Connections:
Refer to example on pg 27 showing students moving through representations

62. Facilitate Meaningful Discourse – Points to Ponder
Effective mathematics teaching engages students in discourse to advance the mathematical learning of the whole class.
Smith and Stein (2011) describe five practices for effectively using student responses in class discussions:
Anticipating student responses prior to the lesson
Monitoring students’ work on engagement with tasks
Selecting particular students to present their mathematical work
Sequencing students’ responses in specific order for discussion
Connecting different students’ responses and connecting responses to key mathematical ideas
Students must have opportunities to talk with, respond to, and question one another as part of the discourse community, in ways that support the mathematics learning for all students in class.

63. Principles to Actions pg. 50
Teachers greatly influence how students perceive and approach struggle in the mathematics classroom. Even young students can learn to value struggle as an expected and natural part of learning.
Principles to Actions pg. 50

64. Supporting Productive Struggle

65. Comparing Instructional Tasks
The table of values below describes the perimeter of each figure in the pattern of blue tiles. The perimeter P is a function of the number of tiles t.
Choose a rule to describe the function in the table.
A. P = t + 3 B. P = 4t
C. P = 2t + 2 D. P = 6t – 2
How many tiles are in the figure if the perimeter is 20?
Graph the function. T 1 2 3 4 P 6 8 10
This slide is being used

66. Principles to Actions pg. 53

67. Start Small, Build Momentum, and Persevere
The process of creating a new cultural norm characterized by professional collaboration, openness of practice, and continual learning and improvement can begin with a single team of grade-level or subject-based mathematics teachers making the commitment to collaborate on a single lesson plan.

68. What questions do you have?

69. Follow Us! NC Mathematics www.facebook.com/NorthCarolinaMathematics

70. DPI Mathematics Section
Kitty Rutherford
Elementary Mathematics Consultant
Denise Schulz
Lisa Ashe
Secondary Mathematics Consultant
Vacant
Dr. Jennifer Curtis
K – 12 Mathematics Section Chief
Susan Hart
Mathematics Program Assistant 70

71. For all you do for our students!

72. References
Blanton, M., Levi, L., Crites, T., & Dougherty, B. J. (2011). Developing Essential Understanding of Algebraic Thinking Grades 3-5. Reston, Va: The National Council of Teachers of Mathematics, Inc. Foote, M. Q., Earnest, D., & Mukhopadhyay, S. (2014). Implementing the Common Core State Standards through Mathematical Problem Solving Grades 3-5. (F. R. Curcio, Ed.) Reston, VA: The National Council of Teachers of Mathematics, Inc. Gurl, T. J., Artzt, A. F., & Sultan, A. (2013). Implementing the Common Core State Standards through Mathematical Problem Solving Grades 6-8. (S. E. Frances R. Curcio, Ed.) Reston, VA: The National Council of Teachers of Mathematics, Inc. Gurl, T. J., Artzt, A. F., & Sultan, A. (2012). Implementing the Common Core State Standards through Mathematical Problem Solving High School. (S. E. Frances R. Curcio, Ed.) Reston: The National Council of Teachers of Mathematics. Keeley, P., & Rose, C. M. (2006). Mathematics Curriculum Topic Study. (R. Livsey, P. Cappello, D. S. Foster, & J. Tasch, Eds.) Thousand Oaks, CA: Corwin Press.

73. References, cont’d
 Leinwand, S., Brahier, D. J., Huinker, D., Berry III, R. Q., Dillon, F. L., Larson, M. R., et al. (2014). Principles to Actions Ensuring Mathematical Success for All. Reston, Va: The National Council of Teachers of Mathematics.
Lloyd, G., Herbel-Eisenmann, B., & Star, J. R. (2011). Developing Essential Understanding of Expressions, Equations, and Functions for Teaching Mathematics in Grades 6-8. (S. E. Rose Mary Zbiek, Ed.) Reston, Va: The National Council of Teachers of Mathematics, Inc.
Project (1993). Benchmarks for Science Literacy. New York, NY: Oxford University Press.
The Common Core Standards Writing Team. (2011, March 29). The University of Arizona, Institute for Mathematics and Education. Retrieved June 22, 2015, from Progressions Documents for the Common Core Mathematics Standards: