1. 3/25/2017
Day 2 CCSS-M in the Classroom: Grades 3-5 Number and Operations Fractions Weaving Content and Standards for Mathematical Practices

2. Welcome Back Activity Success, challenge, breakthrough
Reflect on your experience using one of the assigned tasks in your classroom. Use separate post-it notes to capture your successes, challenges and/or breakthroughs Post on appropriate poster Read post-it notes on all posters and select one that resonates with you. Quick share of selected post-its.
First two lines of slide will be on the slide, allow 1-2 minutes, then click and the next two lines will come in together, allow 3-5 minutes, then do last line and share with remaining time.
Prepare a poster for each category: success, challenge, barrier, breakthrough
Encourage participants to label each post-it with the category for quick posting and accurate sharing.
Have participants put their post-its on the appropriate poster after a minute or two of reflecting & writing.
Allow 3-5 minutes for participants to read others ideas and select one to read aloud to the whole group.
Stand in a circle and Whip around to share each thought. Open up discussion, if time.

3. Reflection What is your current reality around classroom culture?
What can you do to enhance your current reality?
Handout

4. Outcomes: Day 2
Understand how to analyze student work with the Standards for Mathematical Practice and content standards.
Understand the SBAC Assessment Claims 3 & 4 for Mathematics.
Analyze and adapt a task with the integrity of the Common Core State Standards.

5. Looking at Student Work
Form small groups of 3 or 4
Each person selects 1 or 2 works samples to share with the group
Follow the Collaboration Protocol to review student work samples and record information
observations
Inferences
implications
Handouts for protocol and recording sheet – screen shots are on the next two pages

6. Collaboration Protocol-Looking at Student Work (55 minutes)
1. Individual review of student work samples (10 min)
All participants observe or read student work samples in silence, making brief notes on the form “Looking at Student Work”
2. Sharing observations (15 min)
The facilitator asks the group
What do students appear to understand based on evidence?
Which mathematical practices are evident in their work?
Each person takes a turn sharing their observations about student work without making interpretations, evaluations of the quality of the work, or statements of personal reference.
3. Discuss inferences -student understanding (15 min)
Participants, drawing on their observation of the student work, make suggestions about the problems or issues of student’s content misunderstandings or use of the mathematical practices.
Adapted from: Steps in the Collaborative Assessment Conference developed by
Steve Seidel and Project Zero Colleagues
Select one group member to be today’s facilitator to help move the group through the steps of the protocol.
Teachers bring student work samples with student names removed.
4. Discussing implications-teaching & learning (10 min)
The facilitator invites all participants to share any thoughts they have about their own teaching, students learning, or ways to support the students in the future.
How might this task be adapted to further elicit student’s use of Standards for Mathematical Practice or mathematical content.
5. Debrief collaborative process (5 min)
The group reflects together on their experiences using this protocol.
Please remind teachers that they will first focus on evidence for students use of the Mathematical Practices and understanding of content.
Example of Evidence:
Students drew pictures to represent the problem.
Students could …
Inference:
Students didn’t know how to find a common denominator

7. Looking at student work
This form is adapted from the book Looking Together at Student Work

8. Homework Debrief….
What instructional strategies did you use with this lesson?
What Standards for Mathematical Practices did you notice your students engaging in during this task?
Using the SMP Matrix how would you describe the students in your classroom

9. Applying your learning
From previous homework: you found/created or adapted to make a rich task.
Have you turned in your item for the Intel CCSS-M Item bank – digital is preferred, hard copy is better than not at all!!!!!

10. BREAK TIME!!! 15 minutes….Go

11. Assessment Claims for Mathematics
“Students can demonstrate progress toward college and career readiness in mathematics.”
Overall Claim (Gr. 3-8)
Overall Claim (High School)
“Students can demonstrate college and career readiness in mathematics.”
Claim 1
Concepts and Procedures
“Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.”
Claim 2
Problem Solving
“Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.”
Overall Claims (gr 3-8) refers to the content standards
Claim 3
Communicating Reasoning
“Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.”
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.”

12. Revisit the practices - The foundation for student learning is built using the CCSS for Mathematical Practices. It is important that we as educators provide students with opportunities to experience mathematics in a way that builds understanding. The 8 standards have been grouped into the following categories:
1 and 6 Over arching Habits of Mind of a productive mathematical thinker, Reasoning and Explaining, Modeling and using tools and seeing structure and generalizing.

14. Claim 2 – Problem Solving
Claim 2: Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.
Apply mathematics to solve well-posed problems arising in everyday life, society, and the workplace
Select and use tools strategically
Interpret results in the context of the situation
Identify important quantities in a practical situation and map their relationships.
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.”

15. What is the depth of knowledge of these tasks?
Looking at SBAC Tasks: Depth of Knowledge and Mathematical Practices, two lenses
What is the depth of knowledge of these tasks?
Which mathematical practices do they promote?
When we look at the SBAC sample items think about these two areas?
Use the top part of the deep Task Analysis template

16. 4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b
This comes back to the work on day one with unit fractions
MP 1?
DOK 2?

17. Claim 3 – Communicating Reason
Claim 3: Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.
Test propositions or conjectures with specific examples.
Construct, autonomously, chains of reasoning that justify or refute propositions or conjectures.
State logical assumptions being used.
Use the technique of breaking an argument into cases.
Distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in the argument—explain what it is.
Base arguments on concrete referents such as objects, drawings, diagrams, and actions.
Determine conditions under which an argument does and does not apply.
Claim 3 focuses on Communicating Reasoning. The purpose of this claim is to elicit evidence that students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Items and tasks written for this claim will ask students to explain his or her reasoning, justify a conjecture, and assess the validity of a claim.
The Claim 3 targets require students to:
{+}
Test propositions or conjectures;
Construct chains of reasoning that justify or refute propositions or conjectures;
State logical assumptions that are made;
Use techniques of breaking arguments into cases;
Distinguish correct logic and flawed reasoning and explain what it is
Base arguments on concrete referents
And determine conditions under which an argument does and does not apply.
Items and tasks written for Claim 3 will provide evidence for several of the Claim 3 assessment targets. Each target should not lead to a separate task. Tasks generating evidence for Claim 3 in a given grade will draw upon knowledge and skills articulated in the standards in that same grade, with strong emphasis on the major work of the grade.”

18. What is the depth of knowledge of these tasks?
Looking at SBAC Tasks: Depth of Knowledge and Mathematical Practices, two lenses
What is the depth of knowledge of these tasks?
Which mathematical practices do they promote?
When we look at the SBAC sample items think about these two areas?
Use the top part of the deep Task Analysis template

19. 3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size
MP 3 and 6
DOK 3

20. Focus and Coherence through the Major and Supporting Clusters of the CCSS-M
Discuss with a partner which:
Major, Supporting, and/or
Additional Clusters are
involved in the task
3rd Grade: Major Numbers and Operations with Fractions—developing understanding of fractions as numbers
Supporting Geometry—reasoning with shapes and their attributes

21. Which Fraction Pair is Greater?
Think like a fourth grader
Give one or more reasons for the comparison
Try not to use models or drawings
DO NOT USE cross multiplication
Elementary and Middle School Mathematics VandeWalle, p. 311
Alternate slides for Session 3 following in the same day as Session 2

22. Ways In Which To Compare
Same size whole (same denominator) - B&G
Same number of parts (same numerator) but different sized wholes – A,D,& H
More than/less than one-half or one – A,D,F,G, and H
Closeness to one-half or one – C,E,I,J,K, and L

23. Exploring Equivalent Fractions
atics.org/pages/fractions_pro gression
Facilitator discretion about the inclusion of these ideas

24. Task Analysis Protocol Sheet

25. Fraction Tracks Play Fraction Tracks (access online)
Video
Add protocol for video review
NCTM Illuminations has Fraction Tracks
Encourage games for practice of these skills
Fraction Tracks is available to play online.
Ann has cards to use for hard copies

26. How did this activity support the big idea of equivalent fractions
Reasoning about the size of a unit fraction based on the meaning of a unit fraction
Turn and talk with your partners and then we will share out with the group

28. Exploring Adding Fractions
atics.org/pages/fractions_pro gression
Watch video again??

29. Justify the truth of this statement in at least two different ways.
See if they solve it based on number sense i.e. 2/5 is less than ½ so ½+1/3 must be greater than 2/5
Strategies: Reasoning based on
Numerators
Denominators
Benchmark fractions
Reaso
The clock can be useful for this.—if they don’t come up with alternatives, we will return to it after string work, after the work ask them about how their responses have changed and what might have affected their understanding.

30. Exploring Multiplying Fractions
atics.org/pages/fractions_pro gression
Semi-concrete explanation
Multiplying part 2
Abstract explanation
Multiplying part 1
Facilitator discretion

31. Read a story and explore the math
Read Beasts of Burden from The Man who Counted by Tahan 1993
Rusty Bresser in Math and Literature describes three days of activities with fifth graders, based on this story

32. Paper folding
I planted half my garden with vegetables this summer. One third of the half that is vegetables is planted with green beans. What fraction of the whole garden is planted with green beans?
Math Matters- use a piece of paper to model this action.
Multiplication with whole numbers and a fraction follows the repeated addition using whole numbers
When two fractions are multiplied it is based on a fraction as an operator.
A fraction is operating on another number and changes the other number. The use of the word “of” to multiply is based on the operator interpretation of fraction: when we multiply ½ and 8 we are taking one half of eight.

34. Math Matters
When two fractions are multiplied it is based on a fraction as an operator.
A fraction is operating on another number and changes the other number. The use of the word “of” to multiply is based on the operator interpretation of fraction: when we multiply ½ and 8 we are taking one half of eight.
p. 125 Math Matters Chapin and Johnson

35. Finding a fraction of a fraction
Conceptually – with no subdivisions
Pictures – YES Algorithms - NO

36. Finding a fraction of a fraction

37. Exploring Dividing Fractions
atics.org/pages/fractions_pro gression
This three minutes covers a huge conceptual leap for most elementary teachers
Fraction as a division problem ie
Two whole numbers written as a fraction give the same number as the result of taking the numerator divided by the denominator
And why the invert and multiply rule works

38. Dividing by one –half Illustrative Math Example
Shauna buys a three-foot-long sandwich for a party. She then cuts the sandwich into pieces, with each piece being 1/ 2 foot long. How many pieces does she get?
Phil makes 3 quarts of soup for dinner. His family eats half of the soup for dinner. How many quarts of soup does Phil's family eat for dinner?
A pirate finds three pounds of gold. In order to protect his riches, he hides the gold in two treasure chests, with an equal amount of gold in each chest. How many pounds of gold are in each chest?
Leo used half of a bag of flour to make bread. If he used 3 cups of flour, how many cups were in the bag to start?
Illustrative Math Example

39. Video—Cathy Humphreys
Defending Reasonableness: Division of Fractions 1 ÷ 2/3 Use the Standards for Mathematical Practice Matrix to reflect on where this classroom falls on the continuum and be ready to discuss any activities you could use to move this classroom forward on the scale
Defending Reasonableness Division of Fractions Chapter 4 Connecting Mathematical Ideas by Jo Boaler and Cathy Humphreys
Can use Video Reflection Recording Sheet if desired

40. Standards for Mathematical Practice Matrix
Where are the opportunities for students in most of your classroom lessons

41. Claim 4 – Modeling and Data Analysis
Claim 4: Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.
Apply mathematics to solve problems arising in everyday life, society, and the workplace.
Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem.
State logical assumptions being used.
Interpret results in the context of a situation.
Analyze the adequacy of and make improvement to an existing model or develop a mathematical model of a real phenomenon.
Identify important quantities in a practical situation and map their relationships.
Identify, analyze, and synthesize relevant external resources to pose or solve problems.
Claim 4 focuses on modeling and data analysis. Claim four requires extended response items and performance tasks that elicit evidence that students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Items and tasks written for this claim will ask students to investigate problems that have more than one solution pathway, summarize the results within the context of the problem, and evaluate the solution within the context of the problem.
The assessment targets associated with Claim 4 require students to:
{+}
Apply mathematics to solve problems arising in everyday life;
Construct chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for complex problems;
State logical assumptions that are made;
Interpret results in the context of a situation;
Analyze the adequacy of and make improvements to an existing model or develop a mathematical model of a real phenomenon;
Identify important quantities in a practical situation and map their relationships;
And identify, analyze, and synthesize relevant resources to pose or solve problems.
Items and tasks written for Claim 4 will provide evidence for several of the Claim 4 assessment targets. Each target should not lead to a separate task. Tasks generating evidence for Claim 4 in a given grade will draw upon knowledge and skills articulated in the progression of standards up to that grade, with strong emphasis on the “major” work of the grades.
Now, let’s shift our focus from the claims and assessment targets specified in the Content Specifications to the additional information presented in the Item Specifications.

42. Looking at SBAC Performance Task: Depth of Knowledge and Mathematical Practices, three lenses
What are the specific content standards in this performance task?
What is the depth of knowledge of these tasks?
Which mathematical practices do they promote?
When we look at the SBAC sample items think about these two areas?
Use the top part of the deep Task Analysis template

43. Planting Tulips DOMAINS: Operations and Algebraic Thinking, Number and Operations—Fractions, Measurement and Data

44. Using your materials modify or create a rich task or one that is more conceptually focused for your students

45. Applying your learning
From previous homework: you found/created or adapted to make a rich task.
Have you turned in your item for the Intel CCSS-M Item bank – digital is preferred, hard copy is better than not at all!!!!!

46. Where to Look for Resources for Mathematics
RMC website:
OPSI website:
Sandy’s Math wiki:

47. Reflection What is your current reality around classroom culture?
What can you do to enhance your current reality?
Handout

48. Wrap up Activity Evaluations Thank You! 3/25/2017 Handout:
HS Evaluation
URL: