Year 2 – Facilitator Notes for grade 6-8: Ratio and Proportion & Expressions and Equations
Goals:
- Reflect on teaching practices that support the shifts in the Common Core State Standards for Mathematics.
- Understand how to analyze student work with the Standards for Mathematical Practice and content standards.
- Analyze, adapt, and implement a task with the integrity of the Common Core State Standards.
- Deepen understanding of the progression of learning around Expressions and Equations.
Participants Bring: Standards Booklets and progression documents for notes & highlighting.
Time / Activity (Session 1) / Materials8:30-8:40
Slide
1-2
Norms Slide? / Welcome/Norms/Overall Outcomes
Norms: Leaving early, how to get everyone's attention, our intentions for the group, not a "team day." Its okay to work on computers, but only during appropriate times and for appropriate purposes, sidebar conversations,
Have norms for establishing a productive classroom culture
-have a reflection regularly about their classroom culture throughout the work / Participants Bring:
Standards Document
Major/Supporting clusters
8:40-8:55
Slides
3-9
Slide 3
Slide 7
Slide 8
8:55-9:40
Slides 10-41
Slide 16
Slide 17
Slide 19
Slide 25
Slide 27
Slide 30
Slide 34
Slide 35
Slide 38
Slide 41
9:40-9:55
Slides 42-53
Slide 43
Slide 44
Slide 50 / Classroom Culture/Day 1 Outcomes
Begin session with research-based best practices…we would like to encourage you to engage in personal challenge throughout the sessions, collaborate with your colleagues, welcome disequilibrium and wrong answers, and be open to questions…take them as a challenge to your ideas not to you personally.
If this is what needs to occur in the classroom, we are going to try to model this in the workshop. Offering your suggestions and techniques to establish a classroom culture that is conducive to learning mathematics. You will notice that these best practices encompass the Standards for Mathematical practice.
Could link with the shifts in classroom practice (7) if you are using that in your session.
Teachers Development Group-Best Practices Workshop
Donovan and Bransford, eds., 2005; Weiss et al, 2003;
Kilpatrick et al, 2001; Glenn et al, 2000;
Current Reality
Go through Greta’s slides with our grade level examples woven in
This is a reminder of the three shifts that are required by the Common Core
State Standards for Mathematics.
There are two levels of focus.
The first level is the focus of what is in versus what is out; what is being taught at each grade level compared to what is not. It is because of this level of focus that teachers will have the time to go deeper with the math that is most important. Compared to the typical state standards of the past (which in some cases were literally volumes of standards that would have taken years to “cover,” even one grade’s worth of math), the Common Core State Standards for Mathematics have fewer standards which are manageable and it is clear what is expected of the teachers and students at each grade level. That is the 1st level of focus.
The other level of focus is the shape of the content that is in each grade or course. What that means is that if you look at the “focused” list, say for Kindergarten, you can see the list in terms of shades. There are things that are really sharp and focused in the middle, that are the major content for that grade. The other topics are there in a supporting way and help to support that major work. So, even within the list that exists, there is focus. That is the 2nd level of focus.
(It is recommended that middle school teachers first explore 3-5 and then 6-8. If the facilitator chooses to only go over 6-8, please use the outline of notes on slides #11 and #12 to assist with this and the next page).
Let’s take sometime to closely review 6-8 grade band. After reviewing, please discuss with your groups the questions on this page.
Coherence is about math making sense.
Just like there are two levels of focus, there are two types of coherence found in the math standards. One is the coherence of topics across grades and the other is the coherence within a grade.
You will see coherence across grades as the Standards direct us to have students apply learning from a previous grade to learn a new topic. You will see coherence across grades as you see the thoughtfully laid out progressions of mathematics that are meaningful and make sense.
You will see coherence within a grade in the Standards as they direct us to have students reinforce a major topic in a grade by utilizing a supporting topic. You will see coherence within a grade as you see the meaningful introduction to topics in the same grade that complement each other.
Here are some examples of where coherence within a grade can be found.
•In every case in 1st – 5th grade, Represent and Interpret Data is a supporting cluster. It is clear that the intent for 1st – 5th is that representing and interpreting data needs to be done in a way that supports and deepens the understanding of the major work of the grade.
•Teachers do not have to search for the coherence in the Standards - in many cases it is very clear such as in 3rd and 5th grade when teachers are asked to relate area and volume to multiplication and addition. Here the teacher is shown that area and volume are not to be taught in isolation but as a topic that relates to something else. Again, this is about math making sense.
•In 6th grade, students are graphing in all 4 quadrants for the first time. It shouldn’t be a surprise that that is the first year students are introduced to rational numbers.
•In 8th grade, cluster 8.EE.B demands that students actually understand the connections between proportional relationships, lines, and linear equations. This is not left up to chance, but is written in the standards.
There are many meaningful progressions of mathematics in the CCSSM.
We are going to look at some ways the coherence across grades is evident in the Standards. In the activity we are going to do next, you will see how themes, topics, and language build across grades in the CCSSM.
Rigor, as defined here, does not mean hard problems. It doesn’t mean more difficult.
Rigor, here, means something very specific. We are talking about the balance of these components of conceptual understanding, fluency, and application.
We are going tolook at a set of problems; some assess fluency, some require conceptual understanding, and some are examples of application. By working through these problems, we can start seeing what thislooks like.
Here are a few frequently asked questions about the idea of rigor and balance.
How can we assess fluency other than giving a timed test?
When you talk about assessment items that get at fluency, it is not always only going to be weekly timed tests. That may be an instructional practice that works for certain people, but it is not amandate that everyone across the country will be doing timed tests. There are lots of different ways to get to this idea of fluency and there are lots of different ways to know if a student is fluent in something. On a side note, it is important to remember that assessing fluency is not necessarily the same as practicing fluency. Often schools will do timed tests 3 days a week and think they are fulfilling the “fluency requirement.” Students need opportunities to practice fluency, as well.
Is it really possible to assess conceptual understanding? What does it look like?
So, how do you take the math that we want students to know out of context and how do we ask them something that just by doing the problem will tell us that they have conceptual understanding?
We will be doing a set of sample problems in a moment. Spend some time really thinking about the problems under conceptual understanding and see what is different. Asking students to show work and explain can be informative, but it isn’t the only way to assess conceptual understanding and can become tiring for students.
Lastly, are the Common Core Standards in Math all about application and meaningful tasks?
It is a common misconception that CCSSM is only about rich application tasks. That is only one piece of the puzzle. You cannot just do these tasks and think that everything else is going to happen. This is not a criticism of the tasks that are out there.
The tasks are often good. Performance tasks are great, but where in those performance tasks are you getting at procedural skill and fluency? Where are you getting at conceptual understanding?
We cannot just expect those things to happen if we only do those tasks. Remember that rigor in the Standards is a balance of time spent on conceptual understanding, procedural skill and fluency, and application. The Standards themselves will typically guide the reader to what aspect of rigor is expected.
Not individually addressed….culture of the classroom
Assessment
SBAC Assessment System Components
System Highlights
-Smarter Balanced items and tasks will elicit evidence that students have the ability to integrate knowledge and skills across multiple assessment targets and are ready to meet the challenges of college and careers.
-Items and tasks must be constructed at various levels of cognitive rigor. Smarter Balanced has defined four levels of depth of knowledge.
-The first level focuses on recall and reproduction of facts and other types of information.
-The second level focuses on basic skills and concepts that require cognitive processes that extend beyond the recall of information.
-The third level focuses on strategic thinking and reasoning.
-The fourth and final level requires extended thinking that includes complex reasoning, planning, development, and cognition that occurs over an extended period of time.
Let’s take a look at a sample item for each of the four levels of depth of knowledge. / Handout –
Classroom Culture Template
Handout – Classroom Culture Template Document.
Handout (Slide 43)
SBAC Assess Visual
Handout (Slide 44)
System Highlights Visual
Handout-
Front and Back (Webbs DOK w/Visual)
Copy of Slide 50
9:55-10:10 / Break
10:10-10:55
Slides
54-58
Slide 55
Slide 57
Slide 58 / Standards for Mathematical Practice
Not individually addressed….culture of the classroom
We are going to give you time to dig deeper into these standards for mathematical practice.
Divide teachers into groups of 2 to 4. Use the same standard used for the Frayer model.
They fill out their own 4 quadrant chart and then match up with 1 or 2 other teachers and share. Summarize their discussion and have each group prepare a poster to share with the rest of the group.
Sharing posters, can be done in several ways. Each group can share out orally or you can give them post it notes and proceed through a gallery walk and add notes and comments to the posters. Teachers will often collect ideas from each other so they might like to have time to make some personal notes about what they learned from each other.
You can have them talk with an elbow partner and share what they learned during the share out. / Handout
Digging Deeper
Post it Notes
Poster Paper
Markers
10:55-11:45
slides
59-63
Slide 60
Slide 61
Slide 62 / The Claims – Claim 1
The number of Claim 1 assessment targets varies across grade levels. The assessment targets for each grade level are presented in detail in the Content Specifications and are explored in greater detail in the Grade Level Considerations training modules. For now, note that careful thought went into examining the progression of mathematical knowledge and skills as students progress from early elementary grades through high school and as students develop college and career readiness. This progression informed the development of each grade level assessment target. Also note that for high school, assessment targets are established for grade 11 only and reflect the skills and knowledge students are expected to demonstrate in order to be college and career ready.
To help use assessment targets to inform the development and review of items and tasks, let’s take a closer look at the structure of an assessment target.
The number of assessment targets refers to the actual number of clusters in each grade level. Each of the clusters will be assessed, but not necessarily each standards within each clusters. 75% will focus on major clusters and 25% on the rest.
Expressions and Equations 2
Grade: 7
Claim 1: Concepts and Procedures
Target: 1C
CCSS: 7.EE.2
DOK Level: 2
SMP: 1, 6, and 2
Shift: Rigor
This item would likely have a high difficulty level. High difficulty items are essential for measurement precision of top-performing students. Work on equivalent expressions in grades 6 and 7 supports students' ability to solve increasingly complex equations in grade 8 and high school.
Expressions and Equations 3
Grade: 8
Claim 1: Concepts and Procedures
Target: 1D
CCSS: 8.EE.7a
DOK Level: 2 or 3
SMP: 1, 6, and 2
Shift: Coherence
In grade 6 students generate equivalent algebraic expressions, in grade 7 these are expanded to include expressions with rational coefficients, and in grade 8 students use earlier strategies to solve increasingly complex equations. / Handout
Task 43053
Handout
Task 43056
11:45-12:45 / Lunch
12:45-1:15
Slides 65-67
Slide 65
Slide 66 / The Claims – Claim 2
Claim 2 focuses on Problem Solving. The purpose of this claim is to elicit evidence that students can solve a range of complex well-posed problems in pure and applied mathematics and can make productive use of knowledge and problem solving strategies. Items and tasks written to assessment targets for this claim will ask students to:
-Apply mathematics to solve well-posed problems arising in everyday life;
-Select and use tools strategically;
-Interpret results in the context of a situation;
-And identify important quantities in practical situations and to map their relationships.
Items and tasks written for Claim 2 will provide evidence for several of the Claim 2 assessment targets. Each target should not lead to a separate task: it is in using content from different areas, including work studied in earlier grades, that students demonstrate their problem solving proficiency.”
Interact with this item on the webpage.
Calculator
Grade:6
Claim 2: Problem Solving
Target:2A, 2B,
CCSS:6.RP.3
DOK Level: 2 or 3
SMP: 1, 2, 5, and 6
Shift: Coherence
Online calculators and other applications are common in the 21st century. Today's students should understand that the mathematics they learn supports many of the tools designed to make our lives easier. More sophisticated calculator simulations will be used to measure some of the high school standards. / Internet Access
SBAC site
1:15-3:00
Slides 68-79
Slide 69
Slide 71
Slide 72
Slide 73
Slide 74
Slide 75
Slide 76
Slide 77 / Standards for Mathematical Practice – Classroom Video
Talk through each of the 6 questions – provide them with a handout
Discussion around mathematical endeavors
Should address more than one standard
Potential to broaden skills and content knowledge
Potential for revealing patterns or lead to generalizations
Involves the learner in testing, proving, explaining, justifying, reflecting and interpreting
Handout Task Analysis
5th Grade Task
CCSS-M: Intro to Ratios and Proportional Relationships Standards (page 42 )
6.RP.1 - Understand the concept of a ratio and use ratio language
6.RP.3 – Use ratio to solve real world problems
Operations and Algebraic Thinking Standards (page 35 )
5.OA.3 – Identify apparent relationships between corresponding terms.
DOK Level: 2
SMP: 1, 3, 5, and 6
Shift: Coherence
Video
Mix up groups by counting off 1-8 so that those that hit on this standard earlier are spread out and can serve as experts.
Table discussion on similarities and differences from your observations of the Lesson Video.
Video
After the video, if we really need to attend to the three shifts, what are the implications reflected in your reality of classroom culture. Answer reflections questions and discuss. / Handout
What Makes a Rich Task?
Handout
Task Analysis Sheet
Bead Task Video
Handout
SMP with Explanations
Handout
Classroom Culture Template
3:00-3:30
Slides 78-81
Slide 78
Slide 79
Slide 80 / Preparation for Homework
Give participants the entire SBAC Task but emphasize they only need to bring student samples back for Part A.
Grade: 06
Primary Claim:Claim 4: Modeling and Data Analysis Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.
Secondary Claim(s): Claim 1: Concepts and Procedures Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency
DOK: 3
Standard(s): 6.RP.1, 6.RP.2, 6.RP.3, 6.EE.7, 6.EE.9, 6.NS.3, 5.NBT.4
Mathematical Practice(s): 1, 3, 4, 5
Consider how you might design the lesson help facilitate this.