1. Complex Math Tasks Lead to Accountable Talk: Evidence of Mathematical Practice Standards
Read; Introduce the module by discussing the point listed on the slide.

2. Student-Centered Mathematics
“Instruction involves posing tasks that will engage students in the mathematics they are expected to learn. Then, by allowing students to interact with and struggle with the mathematics using THEIR ideas and THEIR strategies – a student-centered approach – the mathematics they learn will be integrated with their ideas; it will make sense to them, be understood, and be enjoyed.”
“Teaching Student-Centered Mathematics”; John Van de Walle, 2006

3. Essential Question
What kinds of learning tasks and Accountable Talk strategies allow students the opportunity to demonstrate, both orally and in writing, their mathematical proficiency?
Read; Explain to participants that in this section of the workshop, they will discover the answers to the question on this slide.

4. Learning Objectives
Identify the attributes of a rich, instructional, problem-based approach and how it can support access to the Standards for Mathematical Practice
Use Accountable Talk and questioning strategies to promote mathematical discourse
Read; Explain to the participants the objective for this section of the workshop.

5. Attributes of Learning Tasks – Quick Write
What are the attributes of learning tasks that…
promote opportunities for mathematical proficiency?
give evidence of the mathematical practices?
Ask participants to think about the attributes of the workshop tasks that promote mathematical proficiency and give evidence of the mathematical practices.
Have each participant share an idea in a round-robin activity. Tell participants that they can pass if they have nothing to add. After all participants have shared once, anyone can add further comments.
Summarize relevant ideas on a chart that you will keep posted. This chart will be revisited at the end of this section.

6. Attributes of Learning Tasks
Encourage productive struggle by using challenging problems.
Connect the current problems to students’ prior learning.
Highlight important mathematics in the teacher’s verbal introductions to the lesson and summarization at the end.
Ask questions that emphasize the important mathematical concepts.
Provide coherence between current learning and future learning.
add any attributes that are missing to the chart.
“Worth the Doing
Choose high-level learning tasks.
Consider the language and the context of the tasks.
Encourage students to work independently of the teacher, either individually or cooperatively in groups.
Encourage students to analyze situations and pose higher-order questions.
Focus on student approaches and help them focus on important mathematics.” (Pearson Education, Inc. 2011, 30)
Do these task features change with different populations of students (for example, with ELL or Special Education Students)?
The tasks may look different because what is challenging for these students is different, but the key features are still important.

7. Designing and Selecting Problem-Based Tasks
A problem or task for learning mathematics is any task for which the students have no prescribed or memorized rules, nor is there any perception that there is a specific route to the solution.
Three points to remember:
What is problematic in the task must be the mathematics.
Tasks must be accessible to your students.
Tasks must require justifications and explanations for answers or methods.
A RICH TASK:
Asks students to think, plan and execute an approach to the problem. In textbooks, frequently the students simply need to “follow directions”.
Has MANY standards embedded in it and is therefore LOADED with mathematics and processes. That is ,students may need to problem solve, formulate an approach, calculate, justify, draw, connect, represent, etc. And above all, THINK!
Is interesting to the students.
Is naturally differentiated… that is, students of all ability and knowledge levels can engage in the task on some level.

8. There are 125 sheep and 5 dogs in a flock. How old is the shepherd?
Ask participants what students would do with this problem?

9. A Student’s Response
There are 125 sheep and 5 dogs in a flock.
How old is the shepherd?
125 x 5 = 625 extremely big
= 130 too big
= 120 still big
125  5 = 25 That works!

10. 1 A Rich Task? Put these two fractions on the number line below.5 4
and 1
Which fraction is closer to 1?
Why is it closer?
Ask participants to think about the questions. What are they asking?
Ask participants to think about which content and practice standards this problem uses.
Have a few participants present their thinking process as they solved the problem. Ideally a variety of approaches that vary in their mathematical maturity will be presented.
Which content and practice standards does this
task address?

11. DRAFT (Internal Use ONLY)
Connecting the Number Line Fractions Mini–Lesson to the Content and Practice Standards
Attend to precision: Distinguish between 4/5 and 5/4.
Grade 4 Number and Operations—Fractions: comparing fractions with different numerators and different denominators.
Construct viable arguments and critique the reasoning of others: Explain answers and check reasoning.
Ask questions of participants to make sure everyone understands the mathematical thinking involved in each approach.
Connecting to the Standards:
This is where we believe the problem best fits in the standards – 4th grade Number Operations Fractions. Discuss other places that it could be appropriate.
These are the 2 major practices seen in the problem – however, there are justifications that could be made for all 8. Be sure if they think that another one is included, that there is EVIDENCE and WHY it is so!
DRAFT (Internal Use ONLY)

12. Functions as Tables, Graphs, Words, and Formulas
Ask participants to think about which content and practice standards this problem uses
What is the rule for this table?
What does the horizontal axis show?
What does the vertical axis show?
What is the output when we input 2?
What is the output when we input 6?
A Rich Task?
(America’s Choice 2006, 91–100)

13. DRAFT (Internal Use ONLY)
Connecting the Number Line Fractions Mini–Lesson to the Content and Practice Standards
Look for and make use of structure: Distinguish between x/2 and 2x and notice pattern where graphs cross y-axis.
Reason abstractly and quantitatively: Contextualize as needed to understand referents.
Grade 8 Functions: “translate among representations and partial representations of functions.”
Construct viable arguments and critique the reasoning of others: Explain answers and check reasoning.
Connecting to Content and Practice Standards: .
DRAFT (Internal Use ONLY)

14. A RICH TASK?
My seven friends and I are going to share two pizzas for lunch. The pizzas are divided into eight slices each. How much pizza will each person get? (Draw a picture to represent your solution.)
How can you tell the difference between a good and poor task or problem?
Remember, RICH TASKS ARE NATURALLY DIFFERENTIATED! That is, all students can engage in the task at some level…Some will complete the basics or foundation of the problem, others will take it to big heights.

15. Revised Pizza Task – A Rich Task?
You are invited to go out for pizza with several friends. When you get there, you find your friends sitting at two different tables.
You can join either group. If you join the first one, there will be a total of 4 people in the group and all of you will be sharing 6 small pizzas.
If you join the other group, there will be 6 people in the group and all of you will be sharing 8 small pizzas.
If your goal was to get as much pizza as possible, which group would you join?
Explain your thinking and show your math.
Which content and practice standards does this
task address?

16. Model Lesson: The Tile Problem
Your summer job is to repair tiles in the corners of roofs. The tiles you will use are 1 foot by 1 foot squares. Jobs are identified as follows:

17. Middle School Questions: The Tile Problem
Draw jobs #3 and #6 below.
What do you need to know to draw the figures?
After laying the tile, you need to apply some waterproof caulking around the outside of the pattern. The caulking is sold by the foot. Job #2 requires 8 feet of caulking. Job #5 needs 20 feet of caulking.
Your supervisor assigns you to do job #12.
How much caulking will you need to complete the job? Explain how you figured this out.
Write a rule that would allow you to determine how much caulk you would need for any job. Explain why your rule should always work.
Which content and practice standards does this
task address?

18. High School Questions: The Tile Problem
Draw jobs #3 and #6 below.
What do you need to know to draw the figures?
After laying the tile, you need to apply some waterproof caulking around the outside of the pattern. The caulking is sold by the foot. Job #2 requires 8 feet of caulking. Job #5 needs 20 feet of caulking.
Your supervisor assigns you to do job #12.
How may tiles and how much caulking will you need to complete the job? Explain how you figured this out.
Write a rule that would allow you to determine how many tiles and how much caulk you would need for any job. Can you identify any patterns between the two rules?
Which content and practice standards does this
task address?

19. Do the math What are the benefits? A RICH MATH TASK RITUAL Work solo
Share with a partner
Share with the group
What are the benefits?
Remind participants how you used the Opening, Work Time, and Closing routine for the Rituals and Routines session.
Work time:
Solo - think about what works well, write it down.
Talk with partner or table.
Closing - share and discuss with whole group.

20. Questions That Connect to Practices Accountable Talk
The habit of connecting my talk to the talk of others models the mental habit of connecting the idea on my mind right now to ideas previously thought.
Phil Daro
In other words, this habit of talk teaches me how to link my own ideas together into longer chains linked by logic and analogy. I can make complicated ideas in mathematics just like the ones smart people make.

21. Aligning Questions to the Common Core State Standards:
Mathematical Practices

22. Standards for Mathematical Practice
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.
Give out handout

23. Practice #1 – Making sense of problems and Persevere in solving them:
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Huh: What Are You Thinking?
Does this make sense?
What does this term mean?
What do you want to know? Put it in a sentence.
Can you break the problem into simpler problems (multi-step)?

24. Practice #2 – Reason abstractly and quantitatively:
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Quantities:
What quantities are in the problem?
What are the relationships among the quantities in the situation?
How can we label the quantities?
What inferences can we draw?

25. Practice #3 – Construct viable arguments and critique reasoning of others
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Is it true?
Is it always true?
Is it never true?
When is it true? Under what conditions?
Is there a counter example?
A contradiction?
Is there another way to prove that this statement is true or not true?

26. Practice #4 – Model with mathematics
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Can you represent the idea in words, tables, diagrams, formulas, or graphs and explain the relationships between them?
Can you create your own visual representation of this situation?
Can you solve the problem in more that one way?

27. Practice #5 – Use appropriate tools strategically
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Why did you choose to use this model (manipulative) to help you understand the task?
How does your model compare to someone else’s?
Is there an additional representation for this concept?
Was this tool the most efficient?

28. Practice #6 – Attend to Precision
5/16/2018
How would you explain this to someone who didn’t understand?
How does your statement link to what others have said?
How does what you say add to what Anna just said?
Is it a justification? A special case? A generalization?
Evidence? A supporting argument?
A logical extension? A contradiction?
A counterexample?

29. Practice #7 – Look for and make use of structure
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Can you identify the basic component in this structure?
Can you break the problem into smaller components?
Is there a pattern?
Can you simplify the situation?

30. Practice #8 – Look for and express regularity in reasoning
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Do you notice a pattern?
Is there anything in this pattern that is repeating?
Is it possible to make a generalization? A rule?

31. Language Strategies for Math Accountable Talk
5/16/2018
“When students are challenged to think and reason about mathematics and to communicate the results of their thinking to others verbally or in writing, they are faced with the task of stating their ideas clearly and convincingly to an audience.”
From: Principles and Standards, p.85

32. Understanding Mathematics
One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a+b)(x+y) and a student who can explain where the mnemonic comes from.