Rigor & Mathematical Practices in Grades 6-8

Rigor & Mathematical Practices in Grades 6-8
June 2017 Speaker’s Notes: This is the beginning of the second course of the week here at the Standards Institute. Materials for the day: Rigor Task Handout (digital) – 1 per participant; by grade Module Overview and Topic A (digital) – 1 per participant; by grade Mathematical Practice Video Content standards (from participants) Mathematical Practices (from participants) Lesson Plan (from participants) 3-column chart (printed) – 2 per participant; by grade

RIGOR IN GRADES 6-8 Welcome Back!
1min Speaker's Notes: Thank you for your time and attention yesterday! We are excited to dig into rigor today.

RIGOR IN GRADES 6-8 Thank You for Your Feedback!
+ ∆ 1min Speaker's Notes: Thank you for your feedback! I want to talk through some trends for the glows and grows and let you know what I’m doing for the grows within my control.

Take responsibility for yourself as a learner
RIGOR IN GRADES 6-8 Norms That Support Our Learning Take responsibility for yourself as a learner Honor timeframes (start, end, activity) Be an active and hands-on learner Use technology to enhance learning Strive for equity of voice Contribute to a learning environment in which it is “safe to not know” 1 min Speaker's Notes: Review norms.

RIGOR IN GRADES 6-8 This Week
Day Ideas Monday Focus and Within Grade Coherence Tuesday Rigor and the Mathematical Practices Wednesday Across Grade Coherence and Instructional Practice Thursday Adaptation and Curriculum Study Friday Adaptation and Practice “Do the math” Connect to our practice 2 min Speaker Notes: Here is what this week will look like. Our approach is to blend the conceptual with the practical: We work to understand the big ideas of the Shifts, how they look in practice, and how we can use them to meet the needs of our students. We will apply our understanding of the shifts most rigorously on Days 4 and 5 as we dive into curriculum. The two strands that run through all of our work are: digging deep into math content by “doing” the math connecting all of the ideas and principles we look at to our work back in our districts We will understand the principles that lie beneath curriculum, how to adapt curriculum, and how to interact with curriculum. This happens best when we understand the “load-bearing walls” of the curriculum – the big ideas that curriculum is based on.

Morning: Rigor in Grades 6-8
RIGOR IN GRADES 6-8 Today Morning: Rigor in Grades 6-8 Afternoon: Rigor and the Mathematical Practices in Grades 6-8 1 min Speaker's Notes: Here’s what we’re looking at today.

RIGOR IN GRADES 6-8 Morning Objectives
Participants will be able to define the procedural skill and fluency, conceptual understanding, and application aspects of mathematics understanding. Participants will be able to recognize signals of procedural skill and fluency, conceptual understanding, and application within language of standards. Participants will be able to identify characteristics of tasks that emphasize conceptual understanding, procedural skill and fluency, and modeling/application. 2 min Speaker's Notes: These are the objectives for the morning session.

RIGOR IN GRADES 6-8 Morning Agenda
What Is Rigor? Why Rigor? Deep Dive into Rigor: Conceptual Understanding Deep Dive into Rigor: Procedural Skills and Fluency Deep Dive into Rigor: Application Putting It All Together 1 min Speaker’s Notes: Here’s a look at our agenda for this session. Buckle your seatbelts! Materials for the session: Task Handout (digital) Module overview and Topic A (digital) Content standards (from participants)

RIGOR IN GRADES 6-8 What Is Rigor?
3 min Speaker's Notes: Ask participants to write and share how they have heard the word “rigor” used in the their school or district (e.g., by their colleagues, principals, or district leaders). Consider asking participants to share what they wrote. The emphasis here is on misconceptions; for example, some participants may share that people have used “rigor” to mean “hard” or “challenging.” Stop & Jot: How have you heard the word “rigor” used in your school or district?

“Rigor refers to deep, authentic command of mathematical concepts, not making math harder or introducing topics at earlier grades.” 2 min Speaker's Notes: A common misconception is that rigor just means “hard.” It doesn’t. “Rigor” has a specialized meaning in the context of Common Core math. From CoreStandards.org: “Rigor refers to deep, authentic command of mathematical concepts, not making math harder or introducing topics at earlier grades.” Yesterday we talked about what mathematical content is important and what connections exist between standards. Rigor has everything to do with how students engage with mathematical content: it implies a balance of conceptual understanding, procedural skill and fluency, and application.

RIGOR IN GRADES 6-8 Aspects of Rigor Conceptual Understanding: The standards call for conceptual understanding of key concepts, such as place value and ratios. Procedural Skills and Fluency: The standards call for speed and accuracy in calculation. Application: The standards call for students to use math in situations that require mathematical knowledge. 3 min Speaker's Notes: The Common Core names and emphasizes three aspects of rigor. Conceptual Understanding: Instead of teaching math as a series of mnemonics and discrete procedures, students should build a conceptual framework that supports students in accessing concepts from different perspectives. Conceptual understanding supports fluency and application. From the front matter of the Standards: “But what does mathematical understanding look like? One way for teachers to do that is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.” Procedural Skills and Fluency refers to both to “skills” and “fluency.” Students do need to have speed with calculations (e.g., single-digit facts) in order to access more complex concepts and procedures. Application is not “just doing a bunch of real-world problems,” but should genuinely require that students know which ideas to apply when. So, why is “rigor” emphasized in the standards?

RIGOR IN GRADES 6-8 Why Rigor?
2 min Speaker's Notes: Researchers in the late 1990s identified 5 components (or strands) of mathematical proficiency. This is from a report called “Adding It Up” that compiled research on math education from a variety of resources. Note these three in particular: conceptual understanding, procedural fluency, and strategic competence. These form the basis of what we call “rigor” in the standards and are a direct reflection of this research on mathematical understanding. Strategic Competence is the ability to formulate, represent, and solve mathematical problems and is reflected in the “application” shift. Question: “How well do you think our instruction is currently balanced among these three aspects of rigor?” Ask participants to share at their tables, then as a whole group.

From TIMSS Video Study RIGOR IN GRADES 6-8 3 min Speaker's Notes:
THE POINT of this slide is to focus on what the US is doing against the research not against other countries. As mentioned in the previous slide, the research calls for a balance of procedure, application, and conceptual understanding. The TIMSS video study of the late 1990s, which compared eighth grade math instruction in a variety of countries, looked at how students spent work time. While these data points are VERY OLD, they are some of the most recent data we have about classroom practice at the national AND international levels. It is important to acknowledge the longstanding tradition of mimicking procedures in US math education. Many of us grew up with this form of instruction—it is one of the trends that the Common Core Standards seek to change. Despite the recognition that a balance of the different strands of mathematical proficiency is necessary (Adding It Up), on average 75% of “private work time” is spent repeating procedures in the United States.

Deep Dive into Rigor: Conceptual Understanding
RIGOR IN GRADES 6-8 Deep Dive into Rigor: Conceptual Understanding Goals for This Activity: Do the math for each task. Identify the language/wording of the task that emphasizes conceptual understanding. Determine the standard(s) aligned with each task, and identify the language of the standard(s) that calls for conceptual understanding. Identify how the task aligns to the standard. Chart the characteristics of the tasks that exemplify the conceptual understanding. Protocol: time 10 min: Individual work time 10 min: Group collaboration 10 min: Each group share out 2 min Speaker's Notes: Rigor is baked into the standards. We’ll start to see what rigor looks like and how it is embodied in the standards, starting with conceptual understanding. You are going to be given 3 grade-level tasks. On your own, “do the math” for each task, and identify the language of the task that emphasizes conceptual understanding. Determine the standard(s) the task aligns to, and identify the language of the standard(s) that calls for conceptual understanding. Also, decide whether the standard is part of major work. At your table, share and come to agreement about the standard(s) aligned to each task. Discuss the language of the standard(s) that indicate conceptual understanding. As a group, record the characteristics of the tasks that exemplify the conceptual understanding aspect of rigor. But first, we are going to model the full protocol.

RIGOR IN GRADES 6-8 Sample Task
8.EE.B.5 The graphs below show the cost y of buying x pounds of fruit. One graph shows the cost of buying x pounds of peaches, and the other shows the cost of buying x pounds of plums. Which kind of fruit costs more per pound? Explain. Bananas cost less per pound than peaches or plums. Draw a line alongside the other graphs that might represent the cost y of buying x pounds of bananas. 7 min Speaker’s Notes: We will model one round of the protocol before you go off on your own. Yesterday, some of us did this 8th grade task. Take a minute to read the task and determine the alignment. What is the math the students are asked to do? (This task requires students to understand the slope of a line and the meaning of the unit rate graphically and how to represent a proportional relationship graphically give certain constraints.) What is the aligned standard? (8.EE.B.5) Is it part of the major work? (Yes.) Turn and talk – What characteristics of the task exemplify conceptual understanding? Highlight the following responses: The task requires students to interpret a graph without a scale. Students have to prove their answer through a visual.

RIGOR IN GRADES 6-8 Identifying Rigor in the Standards
Conceptual Understanding 8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. 2 min Speaker's Notes: Turn and talk – How does the language of 8.EE.B.5 embody the conceptual understanding aspect of rigor? Highlight the following response: 8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Conceptual Understanding
RIGOR IN GRADES 6-8 Conceptual Understanding Goals for This Activity: Do the math for each task. Identify the language/wording of the task that emphasizes conceptual understanding. Determine the standard(s) aligned with each task, and identify the language of the standard(s) that calls for conceptual understanding. Identify how the task aligns to the standard. Protocol: time 10 min: Individual work time 10 min: Group collaboration 10 min: Each group share out 10 min Materials Chart paper Markers Rigor tasks Speaker's Notes: Let’s begin with some individual work time. You will now work to solve your grade-level tasks, identify the language of the task that emphasizes conceptual understanding, and identify the standard(s) aligned to each task and whether it’s part of the major work of the grade. Tasks Sixth 6.RPA.3.A, Engage NY, Module 1, Topic B, Lesson 9, Example 1 6.RPA, Illustrative Math, Equivalent Ratios 1 used as CU task in MP section) 6.RPA.1, Many Ways to Say It Seventh 7.RPA.3 and 7.NS.A.2.b, Illustrative Math, Temperature Change 7.RPA.1, Engage NY, Module 4, Topic A, Lesson 1, opening exercise 7.EE.A.2, Illustrative Math, Ticket to Ride Eighth: 8.F.B.5, Illustrative Math, Bike Race 8.EE.A.,3 Illustrative Math, Orders of Magnitude 8.F.A.1, Illustrative Math, Function Rules

Transition to Group Time!
Speaker's Notes: It’s now time to transition to working in groups at your table. Transition to Group Time!

RIGOR IN GRADES 6-8 Conceptual Understanding Goals for This Activity: Determine the standard(s) aligned with each task, and identify the language of the standard(s) that calls for conceptual understanding. Identify how the task aligns to the standard. Chart the characteristics of the tasks that exemplify the conceptual understanding. Protocol: time 10 min: Individual work time 10 min: Group collaboration 10 min: Each group share out 10 min Speaker's Notes: In your grade-level group: Share and come to agreement about the standard(s) aligned to each task. Discuss the language of the standard(s) that indicate conceptual understanding. As a group, record the characteristics of the tasks that exemplify the conceptual understanding aspect of rigor.

Transition to Whole Group!
Speaker's Notes: Now let’s talk as a group.

RIGOR IN GRADES 6-8 Conceptual Understanding Protocol: time 10 min: Individual work time 10 min: Group collaboration 10 min: Each group share out Goals for This Activity: Chart the characteristics of the tasks that exemplify the conceptual understanding aspect of rigor. 10 min Speaker's Notes: Options: Grade-level groups hang their chart papers on the wall, and everyone walks around to read other groups’ charts. They go back and discuss what they see that’s different. One grade-level group shares out characteristics of rigor in the tasks. Subsequent groups add to or modify the previous groups but do not repeat suggestions.

Characteristics of Conceptual Understanding in Grades 6-8
RIGOR IN GRADES 6-8 Characteristics of Conceptual Understanding in Grades 6-8 Students explain their thinking with words, drawings, and/or equations. “Equivalent Ratios” Students understand the meaning of equivalence. “Cups of Water, Cups of Flour” “Matching” “Orders of Magnitude” Students use or interpret models and drawings. “Many Ways to Say It” “Bike Race” “Temperature Change” Students use and apply mathematical rules. “Function Rules” Students apply properties. 5 min Speaker's Notes: What are characteristics of tasks that embody conceptual understanding? What is the language in the standards that embodies the conceptual understanding aspect of rigor? Possible Answers Grade 6: 6.RPA.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. 6.RPA.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 6.RPA.3.A Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. Grade 7: 7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 7.RPA.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. 7.RPA.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units Grade 8: 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1 8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 8.EE.A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other

Procedural Skills and Fluency
RIGOR IN GRADES 6-8 Procedural Skills and Fluency Goals for This Activity: Do the math for each task. Identify the language/wording of the task that emphasizes procedural skills and fluency. Determine the standard(s) aligned with each task, and identify the language of the standard(s) that calls for procedural skills and fluency. Identify how the task aligns to the standard. Protocol: time 10 min: Individual work time 10 min: Group collaboration 10 min: Each group share out 10 min Speaker's Notes: Now we’ll take a look at some tasks that reflect procedural skills and fluency. The protocol will be the same. Let’s begin with some individual work time. You will now work to solve your grade-level tasks, identify the language of the task that emphasizes procedural skills and fluency, and identify the standard(s) aligned to each task and whether it’s part of the major work of the grade. Tasks 6th grade: 6.NS.B, Illustrative Math, Tenths of (and so on) 6.NS.A.1, Engage NY, Module 2, Topic A, Lesson 7, Problem Set #1-3 6.NS.B.3, Engage NY Fluency Support 6-8, Sprint, Multiplication of Decimals, pg 48 7th Grade: (Also used in PSF slide of practices section) 7.NS.A.1 and 7.NS.A.2, SAP Procedural Skills and Conceptual Understanding Mini-assessment, questions 1-8 7.RPA.3, Engage NY Fluency Support 6-8, Sprint- Fractions, Decimals and Percents, Pg 99 7.EE.A.2, Sprint- Generating Equivalent Expressions, Pg 89 8th Grade: 8.F.A, SAP Functions Mini-assessment, question 1 8.EE.C.7, Engage NY Fluency Support 6-8, Multistep equations, pg 125 8.EE.A.1, Engage NY Fluency Support 6-8, Properties of Integer Exponents, pg 115

RIGOR IN GRADES 6-8 Procedural Skills and Fluency Goals for This Activity: Determine the standard(s) aligned with each task, and identify the language of the standard(s) that calls for procedural skills and fluency. Identify how the task aligns to the standard. Chart the characteristics of the tasks that exemplify procedural skills and fluency. Protocol: time 10 min: Individual work time 10 min: Group collaboration 10 min: Each group share out 10 min Speaker's Notes: In your grade-level group: Share and come to agreement about the standard(s) aligned to each task. Discuss the language of the standard(s) that indicate procedural skills and fluency. As a group, record the characteristics of the tasks that exemplify the procedural skills and fluency aspect of rigor.

RIGOR IN GRADES 6-8 Procedural Skills and Fluency Protocol: time 10 min: Individual work time 10 min: Group collaboration 10 min: Each group share out Goals for This Activity: Chart the characteristics of the tasks that exemplify the procedural skills and fluency aspect of rigor. 10 min Speaker's Notes: Options: Grade-level groups hang their chart papers on the wall, and everyone walks around to read other groups’ charts. They go back and discuss what they see that’s different. One grade-level group shares out characteristics of rigor in the tasks. Subsequent groups add to or modify the previous groups but do not repeat suggestions.

Characteristics of Procedural Skills and Fluency in 6-8
RIGOR IN GRADES 6-8 Characteristics of Procedural Skills and Fluency in 6-8 Students perform procedures often with the expectation of speed and accuracy: Multiplying/Dividing Fractions and Decimals “Tenths of (and So On)” Rewriting Expressions 7th grade mini-assessment Solving for variables 8th grade mini-assessment Quick computation practice “Sprints” and “Rapid White Board Exchanges” 5 min Speaker's Notes: What are characteristics of tasks that embody procedural skills and fluency? What is the language in the standards that embodies the procedural skills and fluency aspect of rigor? Possible Answers: Grade 6: 6.NS.A.1 Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm. 6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 6.RPA.3.C Find a percent of a quantity as a rate per 100….. Grade 7: 7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. 7.NS.A.1Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7.NS.1.D Apply properties of operations as strategies to add and subtract rational numbers 7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 7.NS.A.2.C Apply properties of operations as strategies to multiply and divide rational numbers. 7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Grade 8: 8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. 8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

15 min Break

Application Goals for This Activity: Protocol: time
RIGOR IN GRADES 6-8 Application Goals for This Activity: Do the math for each task. Identify the language/wording of the task that emphasizes application. Determine the standard(s) aligned with each task, and identify the language of the standard(s) that calls for application. Identify how the task aligns to the standard. Protocol: time 10 min: Individual work time 10 min: Group collaboration 10 min: Each group share out 10 min Speaker's Notes: Now we’ll take a look at some tasks that reflect application. The protocol will be the same. Let’s begin with some individual work time. You will now work to solve your grade-level tasks, identify the language of the task that emphasizes application, and identify the standard(s) aligned to each task and whether it’s part of the major work of the grade. Tasks Grade 6: 6.RPA.3.B, Illustrative Math, Data Transfer 6.RPA.3.C, Engage NY, Problem Set (page 3) * 6.EE.A.6 and 7, Illustrative Math, Firefighter Allocation Grade7 : 7.RPA, Buying Bananas, Illustrative Math 7.RPA.3, Illustrative Math, Tax and Tip 7.NS.A.3, Illustrative Math, Sharing Prize Money Grade 8: 8.F.A.2, SAP Functions Mini-assessment, question 5 (Also used in application slide of practices section) 8.F.A.2, Illustrative Math, Battery Charging 8.F.B.4, Illustrative Math, High School Graduation

Application Goals for This Activity: Protocol: time
RIGOR IN GRADES 6-8 Application Goals for This Activity: Determine the standard(s) aligned with each task, and identify the language of the standard(s) that calls for application. Identify how the task aligns to the standard. Chart the characteristics of the tasks that exemplify application. Protocol: time 10 min: Individual work time 10 min: Group collaboration 10 min: Each group share out 10 min Speaker's Notes: In your grade-level group: Share and come to agreement about the standard(s) aligned to each task. Discuss the language of the standard(s) that indicate application. As a group, record the characteristics of the tasks that exemplify the application aspect of rigor.

Application Protocol: time 10 min: Individual work time
RIGOR IN GRADES 6-8 Application Protocol: time 10 min: Individual work time 10 min: Group collaboration 10 min: Each group share out Goals for This Activity: Chart the characteristics of the tasks that exemplify the application aspect of rigor. 10 min Speaker's Notes: Options: Grade-level groups hang their chart papers on the wall, and everyone walks around to read other groups’ charts. They go back and discuss what they see that’s different. One grade-level group shares out characteristics of rigor in the tasks. Subsequent groups add to or modify the previous groups but do not repeat suggestions.

Characteristics of Application in 6-8
RIGOR IN GRADES 6-8 Characteristics of Application in 6-8 Students ratio and rate reasoning to solve real-world math problems. “Buying Bananas” “Solving Percent Problems” “Data Transfer” Students apply understanding of concepts, rules, and equations (e.g., ratio, percent, functions) to solve word problems and mathematical problems. “Sharing Prize Money” “Firefighter Allocation” “High Jump Competition” “High School Graduations” 5 min Speaker's Notes: What are characteristics of tasks that embody application? What is the language in the standards that embodies the application aspect of rigor? Possible Answers: Grade 6 6.RPA. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Grade 7 7.RPA Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.EE.B.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. Grade 8 8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

RIGOR IN GRADES 6-8 Balance of Rigor
The three aspects of rigor are not always separate in materials and standards. Nor are the three aspects of rigor always together in materials and standards. “The Standards… set high expectations for all three components of rigor in the major work of each grade.” 5 min Speaker Notes: We have been looking at the 3 aspects of rigor separately. However, as you probably saw in some of the tasks, the aspects for rigor are not always neatly separated. A single standard often addresses more than one aspect. Search for a standard containing primarily one aspect of rigor (e.g., 6.NS.B.2 – procedural skills and fluency). Search for standards containing multiple aspects of rigor (6.EE.A.2.C – procedural skills and fluency and conceptual understanding). The point here is to understand and attend to the different aspects of rigor as dictated by the standards. The Publishers’ Criteria makes clear that what is important is a balance of rigor. The Standards themselves balance rigor, so it’s important that instructional materials and instruction align to the appropriate aspects of rigor called for by the Standards in order to preserve that balance. How do we pursue a “balance of rigor”? We must have a focused curriculum in order for teachers to be able to develop fluencies, conceptual understanding, and application.

RIGOR IN GRADES 6-8 Putting It All Together
What would you expect to see, in terms of rigor, in a unit on ratio or functions at your grade level? 5 min Speaker Notes: Now we are going to think about what rigor looks like with respect to a unit of instruction. What would you expect to see, in terms of rigor, in a unit on ratio or functions at your grade level? First, look at some of the standards in your grade that address ratios or functions. Then, think about the aspects of rigor associated with those standards. Thinking about the characteristics of rigor we listed earlier, what might lessons in a unit on expressions and equations contain to support these aspects of rigor? Grade 6: 6.RP.A.1 – Conceptual understanding Grade 7: 7.RP.A.1 – Procedurals skills and fluency Grade 8: 8.F.A.1 – Conceptual understanding We will repeat this exercise by looking at an actual module from EngageNY.

RIGOR IN GRADES 6-8 Why EngageNY?
1 min Speaker Notes: Standards Institute is focused on promoting and supporting open education resources (OER). EngageNY is the best available OER mathematics curriculum. EdReports found EngageNY (also known as Eureka Math) to be the only K-8 curriculum fully aligned to Gateway 1(focus and coherence) and Gateway 2 (rigor and the mathematical practices). The only curriculum rated fully aligned for Grades K–8, based on Gateways 1 and 2.

RIGOR IN GRADES 6-8 Curriculum Map
1 min Speaker's Notes: Here’s a quick overview about the basic structure of ENY. At the highest level, we have curriculum maps. For 6-8, this document is called “A Story of Ratios.”

RIGOR IN GRADES 6-8 Module + Topic Overviews
1 min Speaker's Notes: The year is broken into units called modules; each module has many topics within it. The modules and topics have overviews that explain the content, show standards alignment, introduce vocabulary, and give other helpful info.

RIGOR IN GRADES 6-8 Lessons + Assessments
1 min Speaker's Notes: Each module has a set of lessons, as well as a mid- and end-of-module assessment with answer keys and rubrics. We’ll look at these more tomorrow.

RIGOR IN GRADES 6-8 Rigor in the Modules Independently examine the standards for these modules: Grade 6: Module 1 Grade 7: Module 1 Grade 8: Module 5 1. What are the aspects of rigor associated with each standard? (There may be more than one!) 2. Predict the kinds of problems and activities you’d expect to see associated with each standard. 5 min Speaker's Notes: We’re going to be looking at Module 1 for Grades 6 and 7 and Module 5 for Grade 8. Examine the standards for the module, listed in the module overview. What are the aspects of rigor associated with each standard? Thinking about the characteristics of the different aspects of rigor we generated earlier, what kinds of problems and activities would you expect to see in this module, given the identified standards? [Facilitators may choose to have participants actually do highlighting (markers will be provided at tables) of the standards for this activity. This will give a visual picture of the rigor in the standards.]

Share aspects of rigor you found in the standards with a partner.
SESSION 1 (111M): WHAT IS RIGOR AT THIS GRADE LEVEL? – FORMULA FOR MATH SUCCESS K-2 Share Out Share aspects of rigor you found in the standards with a partner. 3 min Speaker's Notes: Ask participants to share with a neighbor and then highlight a few responses with the whole group. Answers are as follows: Grade 6, Module 1: 6.RP.A.1 emphasizes conceptual understanding (“Understand the concept of a ratio…”) and application (“use ratio language to describe relationships between quantities.”) Prediction—students will be asked to explain why they used a certain ratio to describe a real world situation 6.RP.A.3 emphasizes application (“solve real world and mathematical problems”) and conceptual understanding (“by reasoning…”) Prediction—students will solve a variety of real world problems using ratios Grade 7, Module 1: 7.RP.A.1 emphasizes procedures (“Compute unit rates…”) Prediction—students will have to perform a great deal of calculations with fractions in unit rate problems 7.EE.B.3 emphasizes application (“Represent quantities in a real world or mathematical problem…”) and conceptual understanding (“by reasoning about the quantities…”) Prediction—students will be asked to solve many real world problems and to explain their reasoning Grade 8, Module 5: 8.F.A.1 emphasizes conceptual understanding (“Understand that a function is a rule that assigns to each input exactly one output.”) Prediction—students will have to justify whether or not a given rule is a function. 8.G.C.9 emphasizes application (“use them to solve real-world and mathematical problems.”) Prediction—students will solve many real world problems using the volume formulas.

Rigor at This Grade Level
RIGOR IN GRADES 6-8 Rigor at This Grade Level 1 min Speaker's Notes: You are now going to take a closer look at module 1 for your grade level to find specific evidence for each aspect of rigor. Let’s get oriented with the layout of the module materials.

Teacher Version RIGOR IN GRADES 6-8 1 min Speaker’s Notes:
Every lesson has a teacher version and a student version. The teacher version is indicated with a “T” in the top right corner. The teacher version contains helpful information about lesson design and lesson reasoning.

Student Version RIGOR IN GRADES 6-8 1 min Speaker’s Notes:
The student version has all of the same work as the teacher version but without the narrative and answers.

Student Outcomes RIGOR IN GRADES 6-8 Speaker’s Notes:
These are the learning objectives for the lesson.

Examples RIGOR IN GRADES 6-8 1 min Speaker’s Notes:
Each lesson usually begins with examples. These form the basis for a discussion, exploration, or other ways of developing new ideas.

Problem Set RIGOR IN GRADES 6-8 1 min Speaker’s Notes:
Each lesson has a problem set for independent work for the students. This can be used strategically in class or as a homework assignment.

Exit Ticket RIGOR IN GRADES 6-8 1 min Speaker’s Notes:
The lesson assessment is an exit ticket tied to the outcomes of the lesson.

Exploratory Challenge
RIGOR IN GRADES 6-8 Exploratory Challenge 1 min Speaker’s Notes: One additional feature of the ENY lessons in the 6-8 grade band is the exploratory challenge. Building conceptual understanding, these challenges allow students to dive into (i.e., explore) a concept on their own for later classroom discussion.

RIGOR IN GRADES 6-8 Rigor at This Grade Level Examine the tasks and activities in the lessons and problem sets within Topic A. Find at least two tasks or activities that emphasize the aspect of rigor you would expect to see in a unit containing these standards. What evidence do you have? 25 min Speaker's Notes: Okay, now we will look in Topic A and find evidence of rigor. First, you will work individually. (10 min) Find at least two tasks or activities that emphasize the aspect of rigor you would expect to see in a unit containing these standards. Record the standard. Record the evidence that supports that aspect of rigor. Next, you will work with a partner. (10 min) Each partner shares 1 task for each aspect of rigor and the evidence. Record one task on chart paper. Label the standard and the aspect of rigor. Gallery walk (5 min) Participants do a gallery walk, looking at other pairs’/groups’ work. Let’s begin with some individual work time.

RIGOR IN GRADES 6-8 Rigor at This Grade Level Why spend so much time classifying aspects of rigor? Is there a hierarchy to the aspects of rigor? How does understanding the different aspects of rigor affect your instruction? 5 min Speaker's Notes: After the gallery walk, ask participants to share out answers to the following: Why spend so much time classifying aspects of rigor? One reason is to know all three exist and that it is important to attend to all of them, especially conceptual understanding. Most of us are comfortable teaching procedural skill and less comfortable teaching for conceptual understanding. It’s important that we do both. Is there a hierarchy to the aspects of rigor? One aspect of rigor is not more important than another, but it is important to recognize that conceptual understanding is often the foundation of the work with procedures and achieving fluency. Application is not relegated to happening after gaining fluency or facility with procedures. It should happen in tandem, providing context for math as well as part of supporting conceptual understanding. How does understanding the different aspects of rigor affect your instruction? One example is that understanding the different aspects of rigor can help you figure out how to address misconceptions in student thinking.

SESSION 1 (111M): Rigor– Calibrating Common Core (6 – 8)
BREAK Lunch

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Today
Morning: Rigor in Grades 6-8 Afternoon: Rigor and the Mathematical Practices in Grades 6-8 1 min Speaker's Notes: Here’s what we’re looking at today.

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Afternoon Objectives
Participants will be able to identify characteristics of tasks that require students to use the Mathematical Practices to reinforce the intended procedural skills and fluency, conceptual understanding, and/or applied understandings. Participants will be able to evaluate whether a lesson or task attends to conceptual understanding, procedural skills and fluency, and/or application/modeling. 2 min Speaker's Notes: These are the objectives for the afternoon session.

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Afternoon Agenda
The Mathematical Practices and Rigor The Mathematical Practices in Action Aligning Lessons for Rigor 1 min Speaker’s Notes: Here’s a look at our agenda for this session. Materials for the session: Mathematical practices (from participants) Module overview and Topic A (digital) Content standards (from participants) Lesson (from participants) 3-column chart (printed)

Make sense of problems and persevere in solving them.
RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 The Mathematical Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. 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. 15 min Speaker’s Notes: Most of us have heard of the eight mathematical practices. Today, we are going to look more closely at the practices to see how we can use the practices to support rigor: conceptual understanding, procedural skills and fluency, and application. First, let’s review the practices – pull up the practices on your device. What is the first practice? (read aloud) Read the description of the practice to yourself. As you read, think about the aspects of rigor and whether the practice reflects a particular aspect of rigor. Turn and talk: Can you associate this practice with an aspect of rigor? Which one(s)? Repeat for each practice.

Strands of Mathematical Proficiency
RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 The Practices and Rigor Process Standards Problem Solving Reasoning and Proof Communication Representation Connections -NCTM Strands of Mathematical Proficiency Adaptive Reasoning Strategic Competence Conceptual Understanding Procedural Fluency Productive Disposition -Adding it Up: National Research Council 3 min Speaker’s Notes: The practices are rooted in the NCTM Process standards and the Strands of Mathematical Proficiency we discussed earlier. As you can see, the shift of rigor is tightly connected to the mathematical practices. The practices are the embodiment of how we want our students to engage with the content. Teachers must be purposeful in creating opportunities for engagement in the practices.

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8
Conceptual Understanding 3. Construct viable arguments and critique the reasoning of others. …students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They justify their conclusions, communicate them to others, and respond to the arguments of others. 6. Attend to precision. …students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. In the elementary grades, students give carefully formulated explanations to each other. 6 min Speaker’s Notes: Now we will look more closely at how certain practices tend to embody certain aspects of rigor. While we will be drawing parallels between particular practices and aspects of rigor, the parallels are not exhaustive or exclusive. Let’s look at MP.3 and MP.6 together first. What aspect of rigor comes to mind when examining these practices together? (Give participants time to study and discuss at their tables, then reveal the title: Conceptual Understanding.) What is some of the language in MP.3 that reflects conceptual understanding? …students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They justify their conclusions, communicate them to others, and respond to the arguments of others. …students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. What is some of the language in MP.6 that reflects conceptual understanding? …students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. In the elementary grades, students give carefully formulated explanations to each other.

Procedural Skills and Fluency 7. Look for and make use of structure. …students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. 8. Look for and express regularity in repeated reasoning. …students notice if calculations are repeated, and look both for general methods and for shortcuts. 5 min Speaker’s Notes: Now we’ll look at two more practices, MP.7 and MP.8. What aspect of rigor comes to mind when examining these practices together? (Give participants time to study and discuss at their tables, then reveal the title: Procedural Skills and Fluency. Note that in some contexts, these practices also lend themselves well to building conceptual understanding. The excerpts chosen above from the full text of each practice are selected to highlight procedural work.) What is some of the language in MP.7 that reflects procedural skills and fluency? …students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. MP.8 also reflects procedural skills and fluency. What is some of the language in MP.8 that reflects procedural skills and fluency? …students notice if calculations are repeated, and look both for general methods and for shortcuts. …students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Application 2. Reason abstractly and quantitatively. …students make sense of quantities and their relationships in problem situations. …the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols… 4. Model with mathematics. …students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. …interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 6 min Speaker’s Notes: Finally, let’s look at MP.2 and MP.4 together. What aspect of rigor comes to mind when examining these practices together? (Give participants time to study and discuss at their tables, then reveal the title: Application.) What is some of the language in MP.2 that reflects application? …students make sense of quantities and their relationships in problem situations. …the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols …the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. MP.4 also reflects application. What is some of the language in MP.4 that reflects application? …students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. …students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Conceptual Understanding
5 min Speaker’s Notes: Now we are going to reexamine a task from this morning and look for evidence that supports student engagement with the practices. We can also think about how we might strengthen engagement with a practice. Here is a task related to conceptual understanding that some of us looked at this morning. What is the math the students need to do? What is the standard? (6.RPA.1) Which practice is most evident in this task? (MP.6) What is the evidence of the practice? Students have to attend to precision in language. Students have to examine the impact of careful communication. What ”teacher moves” would you use to help emphasize the Mathematical Practice Standard highlighted by the activity? Many Ways to Say It

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Procedural Skills and Fluency
5 min Speaker’s Notes: Here is a task related to procedural skills and fluency that some of us looked at this morning. What is the math the students need to do? What is the standard? (7.NS.A.1 and 7.NS.A.2) Which practice is most evident in this task? (MP.7) What is the evidence of the practice? Students look for patterns. Students use structure of numbers to engage pattern. What ”teacher moves” would you use to help emphasize the Mathematical Practice Standard highlighted by the activity?

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Application
Sam wants to take his MP3 player and his video game player on a car trip. An hour before they plan to leave, he realized that he forgot to charge the batteries last night. At that point, he plugged in both devices so they can charge as long as possible before they leave. Sam knows that his MP3 player has 40% of its battery life left and that the battery charges by an additional 12 percentage points every 15 minutes. His video game player is new, so Sam doesn’t know how fast it is charging, but he recorded the battery charge for the first 30 minutes after he plugged it in. 5 min Speaker’s Notes: Here is a task related to application that some of us looked at this morning. What is the math the students need to do? What is the standard? (8.F.A.2) Which practice is most evident in this task? (MP.4) What is the evidence of the practice? Students must apply the mathematics they know to solve problems, and analyze relationships to draw conclusions. What ”teacher moves” would you use to help emphasize the Mathematical Practice Standard highlighted by the activity? Battery Charging, If Sam’s family leaves as planned, what percent of the battery will be charged for each of the two devices when they leave? How much time would Sam need to charge the battery 100% on both devices?

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 The Mathematical Practices in Action
Can you make (and back up/support) a conjecture about any relationship that you see between these three growing patterns/functions. Remember that conjectures must hold true for similar patterns beyond the problem set that we are examining. 15 min Materials Video: Speaker’s Notes: Now we’re going to watch a video of an 8th grade classroom. What standard(s) does this task most closely align with? (Read standard language.) (8.F.B.4) How does this task relate to work in Grade 8? (uses conceptual understanding gained from initial exposure to functions, requires them to reason about and generalize in application) What is the aspect of rigor most evident in this task? (conceptual understanding) As you watch the video, think about the practice that is most evident. (MP.3) What do students do? (identify commonalities amongst patterns, identify rate that applies to all, construct conjecture using reasoning) What does the teacher do? (repeated questioning of students, reinforcing key content, supporting their ability to reason about content, has students explain their thinking) Which parts of the task emphasize engagement with MP.3? (making a conjecture)

Break 15 min Break

Read Lesson 1 and identify the intended standard(s).
RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Rigor and the Practices in Lessons Directions Read Lesson 1 and identify the intended standard(s). Identify the aspects of rigor reflected in the standard(s). Find evidence of how the the activities in the lesson align to the intended aspects of rigor. Identify the mathematical practice that is most evident in the lesson. How does it support the associated aspect of rigor? 3 min Materials 3-column chart Speaker’s Notes: Earlier we looked through Topic A to find examples of the different aspects of rigor. Now we’re going to take a closer look at a lesson to observe alignment for rigor. We’re going to study Lesson 1 in Topic A to determine if the lesson activities and problems as well as the teacher moves are appropriate for addressing the intended aspects of rigor. First, we’ll identify the intended standard(s). Then, we’ll identify aspects of rigor reflected in the standard Next, we’ll determine if the activities in the lesson align to the intended aspects of rigor. What does the teacher do to ensure alignment to rigor? Identify a practice in the lesson and how it supports the associated aspect of rigor. (For some lessons, participants may have to suggest a practice that could be added to a lesson.) You will record your findings on the 3-column chart.

Identify the intended standard(s).
RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Rigor and the Practices in Lessons Identify the intended standard(s). Identify the aspects of rigor reflected in the standard(s). Find evidence of how the the activities in the lesson align to the intended aspects of rigor. how the identified practice supports rigor. If you can’t find evidence, make suggestions for improving the alignment to rigor. 25 min Speaker’s Notes: Use the 3-column chart to record your evidence. At the end, have participants share out answers. Possible answers: 6th Grade: 6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. (emphasis shows aspect of conceptual understanding) 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations (emphasis shows aspect of application to problem solving) Activities: Define ratio, give context for use, apply context to class to generate ratios. 7th Grade: 7.RP.A.2 Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. (emphasis shows aspect of conceptual understanding and application) In the lesson, students are reintroduced to the meanings of value of a ratio, equivalent ratios, rate, and unit rate through a collaborative work task where they record their rates choosing an appropriate unit of rate measurement. 8th Grade: 8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (emphasis shows conceptual understanding)

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Aligning Lessons for Rigor and the Practices Directions Identify the intended standard(s). Identify the aspects of rigor reflected in the standard(s). Find evidence of how the the activities in the lesson align to the intended aspects of rigor. Identify the mathematical practice that is most evident in the lesson. How does it support the associated aspect of rigor? Make suggestions for better alignment of rigor. 1 Min Materials: 3-column chart Speaker’s Notes: Now you are going to align the lesson you brought with you for rigor.

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Aligning Lessons for Rigor and the Practices Identify the intended standard(s). Identify the aspects of rigor reflected in the standard(s). Find evidence of how the the activities in the lesson align to the intended aspects of rigor. how the identified practice supports rigor. If you can’t find evidence, make suggestions for improving the alignment to rigor. 35 min Speaker’s Notes: Fill in the chart using the lesson you brought with you. Identify objective standards. Identify the appropriate aspects of rigor for the standards. Find evidence of the appropriate aspects of rigor. Align for fidelity of rigor. Incorporate an appropriate practice or find evidence of a practice.

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Aligning Lessons for Rigor and the Practices
Share: What adjustments did you make to your lesson to better align it for rigor? 5 min Speaker’s Notes: Share out: Have participants share out how they will better align their lessons for rigor.

Feedback Please fill out the survey located here: Click “Summer 2017” on the top of the page. Click “Details” on the center of the page. 7 min Speaker's Notes: Please fill out the survey to help us improve!

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Reference List
Slide Source 10 12 13 15 36 39 – 41, 44 – 51 59 63 64 65 66

RIGOR AND THE MATHEMATICAL PRACTICES IN GRADES 6-8 Image References
Slide # Name and Photographer/Artist 2 "Welcome Mat" by Dru Bloomfield (Flickr) 9 “Notes” by Brady (Flickr) 10 “’Incomplete’ Bridge” by Ken Scarboro (Flickr) 18, 24, 31 "Transitions" by Arjan Almekinders (Flickr) 20, 26, 33 “208/365 - He's got the whole world in his hands.” by Courtney Carmody (Flickr) 29 “Latte Art Smile” by Brainy J (Wikimedia Commons) 37 “Function” by Vestman 43 “Microphone” by Alex Indigo (Flickr) 53 “Share” by GotCredit (Flickr) 54 “Coffee Break” by Sam Carpenter (Flickr) 67 “Snack Break” by IPlayHockey (Flickr) 72 “Adjustments” by _dali_ (Flickr)

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Rigor & Mathematical Practices in Grades 6-8

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