Secondary Math ProgressionsExpressions and Equations

Lisa AsheSecondary Mathematics Consultant

NC DPI

Who’s in the Room?

Welcome

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…

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!

Common Core Key Instructional Shifts

Common Core Key Instructional Shifts

Common Core Key Instructional Shifts

FOCUS

Postsecondary instructors want

deeper mastery of fewer things than

high school instructors desire.

RIGOR

Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of

sufficient richness. (CCSS, p. 4)

COHERENCE2003 SCOS

CCSS Standards

How do we get from here to there with

coherence?

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.

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

Teaching Practices vs. Student Practices

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.

Domain Progressions• Use the resources

on your table to organize the cluster headings in the most logical learning progression.

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.

Standards and Learning Targets

How does a standard differ from a learning target?

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.

Let’s Practice!5.OA.1 Write and interpret numerical expressions.1. Use parentheses, brackets, or braces in numerical

expressions, and evaluate expressions with these symbols.

2. 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 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

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)

Principles to Action - page 16

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?

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

From Numeric to Algebraic Expressions

K-8th GradesK-3rd

NumericExpressions

3-6th Grade Patterns and Sequences

5-7th GradeExpressions

8th GradeExpressions w/

exponents

HS Math

High SchoolComplex Algebraic

Expressions

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)

Task: Wacky ParenthesesNaima’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?

Naima’s work2 × 4 + 6 = 20

Her grandmother’s work2 × 4 + 6 = 14

Implementing the Common Core State Standards through Problem Solving Grades 3-5; Task 2.9 (Page 29)

Task: Evaluating ExpressionsFor 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.

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?

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.

Equivalent Expressions to Equations

6th-8th Grades

6th

One-step equations

(one-variable)

7th GradeRatios and Proportions

8th Grade – HS Mathematics

Systems of Equations

7th- HS MathematicsMulti-step Equations

(one-variable)

Math I-III

Math II and III

Simple trig, absolute value and simple rational equations

Math IIIAdvanced

Trigonometric and

Logarithmic equations

Reasoning about Equations Column A Column B48 × 67 × 6 = k 347 × 25 × 4 = p

746 × 398 ÷ 42 = t 398 × 746 ÷ 746 = d

978 + 778 = 394 + y 378 + 794 = 778 + 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

Task: Secret Number① Pick a number between 1 and 10. This is your secret number.

② 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.

③ 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).

④ 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.

⑤ 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.

Task: Using Solved Problems

Consider the equation 3x – 8 = 8x + 12 and the three solutions below.

Method A Method B Method C3x – 8 = 8x + 12 3x = 8x + 20

−5x = 20 x = −4

3x – 8 = 8x + 12 −5x = 12 −5x = 20 x = −4

3x – 8 = 8x + 12 3x – 8 – 8x – 12 = 0 3x – 8 – 8x = 12 3x – 8x = 20 −5x = 20

x = −4

Compare and contrast the solution methods and evaluate the efficiency of each method.

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?

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

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?

“My Favorite No: Learning From Mistakes”

Elicit Evidence of Student Thinking

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.

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

Let’s Play!

student.desmos.com

Code: f2nv

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.

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.

http://math.ucsd.edu/~jrabin/publications/ProblemFreeActivity.pdf

The Problem-FREE classroom:

1. The immediate goal is not understood by students.

2. The goal of the activity as a whole is unclear.

3. There is no intellectual necessity for the method of solution.

4. Students know what to do in advance, so there is no need for the problem to be considered carefully.

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.

Cognitive Levels of Demand

High

Low

Cognitive Demand Sort

1. Find your group.

2. 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

3. Use the contents of the envelope to sort the tasks by cognitive demand.

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?

Implementation Matters

High HighHigh

Low LowLow

High LowModerate

Task Implementation Student Learning

Principles to Action - page 24

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.

Task: Hexagon TrainsTrains 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.

1. Compute the perimeter for each of the first four trains.

2. Draw the fifth train and compute the perimeter of the train.

3. Determine the perimeter of the 25th train without constructing it.

4. Write a description that could be used to compute the perimeter of any train in the pattern.

5. Determine which train has a perimeter of 110.

Train 1 Train 2 Train 3 Train 4

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.

“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)

5 Practices for Orchestrating Productive Mathematics Discussions

1. Anticipating

2. Monitoring

3. Selecting

4. Sequencing

5. Connecting

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:

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.

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

Supporting Productive Struggle

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.

1. 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

2. How many tiles are in the figure if the perimeter is 20?

3. Graph the function.

T 1 2 3 4P 4 6 8 10

Principles to Actions pg. 53

Start Small, Build Momentum,and PersevereThe 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.

What questions do you have?

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DPI Mathematics Section

Kitty RutherfordElementary Mathematics Consultant919-807-3841kitty.rutherford@dpi.nc.gov

Denise SchulzElementary Mathematics Consultant919-807-3839denise.schulz@dpi.nc.gov

Lisa AsheSecondary Mathematics Consultant919-807-3909lisa.ashe@dpi.nc.gov

VacantSecondary Mathematics Consultant919-807-3842

Dr. Jennifer CurtisK – 12 Mathematics Section Chief919-807-3838jennifer.curtis@dpi.nc.gov

Susan HartMathematics Program Assistant919-807-3846susan.hart@dpi.nc.gov

For all you do for our students!

ReferencesBlanton, 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. 

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 2061. (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: http://ime.math.arizona.edu/progressions/