1. Standards and Instructional Strategies Module 4B
ESUHSD
June 2012
Welcome

2. Outcomes
Increase understanding of the Common Core State Standards (CCSS) in Mathematics by exploring and engaging in
Instructional Strategies that support all students’ learning
Formative Assessment Lessons
Number Talks
Discuss and Reflect on Next Steps
Review Outcomes

3. Agenda Welcome Back and Review CCSS Formative Assessment Lesson
Number Talk
Reflection and Next Steps
Share Agenda

4. Domains and Conceptual Categories Distribution
INTENT: To understand how domains and conceptual categories are distributed across the grade levels
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This diagram illustrates how the domains are distributed across the CCSS.
This diagram does not show how a domain may impact multiple domains in future grades. An example is K-5 Measurement and Data, which splits into Statistics and Probability and Geometry in grade 6. Likewise, Operations and Algebraic Thinking in K-5 provides a foundation for Ratios and Proportional Relationships, The Number System, Expressions and Equations, and Functions in grades 6-8.
Findell & Foughty (2011)
College and Career-Readiness through the Common Core State Standards for Mathematics
California’s Common Core State Standards: Toolkit | Overview

5. High School Mathematics
The CCSS high school standards are organized in 6 conceptual categories:
Number and Quantity
Algebra
Functions
Modeling (*)
Geometry
Statistics and Probability
California additions:
Advanced Placement Probability and Statistics
Calculus
Modeling standards are indicated by a (*) symbol.
Standards necessary to prepare for advanced courses in mathematics are indicated by a (+) symbol.
Intent: Participants will understand the high school content standards are organized by conceptual cluster.
Talking Points:
Let’s turn now to the high school standards.
The Common Core high school standards are organized in six conceptual categories, rather than courses. The categories are:
Number and Quantity
Algebra
Functions
Modeling
Geometry
Statistics and Probability
Modeling standards are indicated by a star symbol; those standards are included throughout the other categories.
When the SBE adopted the CCSS with California additions on August 2, 2010, they also included California’s standards for Advanced Placement Probability and Statistics and Calculus.
6-12 Mathematics Presentation CTA - CLAB: Developed by SCFIRD with support from ELCSD, SCALD, and AAD 5 5

6. High School Mathematics Standards
Conceptual Categories
Number & Quantity
Algebra
Functions
Modeling
Geometry
Statistics & Probability
Conceptual Categories
Number & Quantity
Algebra
Functions
Modeling
Geometry
Statistics & Probability
Modeling:
Links classroom mathematics and statistics to everyday life, work, and decision-making
Is the process of choosing and using appropriate mathematics and statistics
Uses technology to explore consequences and compare predictions with data
INTENT: To distinguish between modeling in the K-8 standards and modeling in the High School Standards
___________________________________________________________________________________
Recall that at the high school level, standards are organized by Conceptual Categories, and one of those categories is
[click] Modeling. Modeling:
[click] Links classroom…
[click] Is the process of choosing…
[click] Uses technology…
As you can see, Modeling describes the kinds of thinking and skills in Quadrants B and D of the Rigor/Relevance Framework.
“Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol.” (CCSS, p. 61)
At the high school level, we see an emphasis on modeling as we prepare students for college and career in the 21st century. So when we consider the Standards for Mathematical Practice at the high school level, we must also consider the Modeling Standards as described on p. 60 of the CCSS.
California’s Common Core State Standards: Toolkit | Instruction, | Mathematics

7. Standards for Mathematical Practice
Overarching habits of mind of a productive mathematical thinker
Reasoning and explaining
Modeling and using tools
Seeing structure and generalizing
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
1. Make sense of problems and persevere in solving them.
6. Attend to precision.
INTENT: To provide some background on the CCSS for Mathematics
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William McCallum, one of the writers of the CCSS for Mathematics, has paired the Standards for Mathematical Practice in the following way.
Review the slide.
When we look at the Standards for Mathematical Practice, we can see how these standards support the kinds of thinking and work described in Quadrants B, C, and D.
This organization of the standards helps us consider the implications the CCSS will have on instruction. In addition to teaching content, we need to plan instruction to help students make sense of problems and persevere in solving them—a challenge we’ve always been faced with in mathematics, but now it is a standard we need to help students achieve.
In our lesson plans, we need to create opportunities for students to engage in reasoning and explaining or using multiple representations and tools to model problems.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
adapted from McCallum (2011)
Standards for Mathematical Practice
California’s Common Core State Standards: Toolkit | Instruction, | Mathematics

8. Forming Quadratics Formative Assessment Lesson
INTENT: To transition to the FAL
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[Transition Slide]
California’s Common Core State Standards: Toolkit | Overview

9. Mathematical Goals
Understand the various algebraic forms of a quadratic function and what each reveals about the characteristics of its graphical representation.

10. Quadratic Functions
Read through the task and try to answer it as carefully as you can.
Show all you work so I can understand your reasoning.
It is important that students are allowed to answer the questions without your assistance, as far as possible.
Students should not worry too much if they cannot understand or do everything, because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to be able to answer questions such as these confidently. This is their goal.
Assessing students’ responses
Collect students’ responses to the task. Make some notes on what their work reveals about their current levels of understanding, and their different problem solving approaches.
We suggest that you do not score students’ work. The research shows that this will be counterproductive, as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics.
Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit.
We suggest that you write a list of your own questions, based on your students’ work, using the ideas that follow. You may choose to write questions on each student’s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the end of the lesson.

11. Graphs and Equations
What does an equation in standard form tell you about the graph?
What does an equation in completed square form tell you about the graph?

12. Key Features of a Quadratic Curve
Using graph paper draw the x-and y-axis and sketch two quadratic curves that look quite different from each other.
What makes your two graphs different?
What are the common features of your graphs?
Allow students to work for a few minutes and then ask them to show you their whiteboards.
Be selective as to which student you ask to explain his or her graphs. Look for two sets of curves in particular:
• one of which has a maximum point, the other a minimum;
• one of which one has two roots, the other one or none;
• that are not parabolas.

13. Key Features of a Graph of a Quadratic
Ask about turning points:
How many turning points does each of your graphs have? Is this turning point a maximum or
minimum?
Can the curve of a quadratic function have more than one turning point/no turning points?
If all students have drawn graphs with minimums, ask students to draw one with a maximum.
Ask about roots:
How many roots does each of your graphs have?
Where are these roots on your curve?
Does anyone have a graph with a different number of roots?
How many roots can a quadratic have?
If all students have drawn graphs with two roots, ask a student to draw one with one or no roots.
Ask about y-intercepts:
Has anyone drawn a graph with different y-intercepts?
Do all quadratic curves have a y-intercept?
Can a quadratic have more than one y-intercept?

14. Three Equations of Quadratic Functions
Standard Form
Factored Form
Completed Square Form
y = x2 – 10x + 24
y = (x – 4)(x – 6)
y = (x – 5)2 - 1
Here are the equations of three quadratic functions.
Without performing any algebraic manipulations, write the coordinates of a key feature of each of their
graphs.
For each equation, select a different key feature.
Explain to students they should use key features from the list on the board.
For example, students may answer:
Equation 1. The y-intercept is at the point (0,24). The graph has a minimum, because the coefficient of x is positive.
Equation 2. The graph has a minimum and has roots at (4,0) and (6,0).
Equation 3. The graph has a minimum turning point at (5, −1)
If students struggle to write anything about Equation 3, ask:
How can we obtain the coordinates of the minimum from Equation 3?
To obtain the minimum value for y, what must be the value of x? How do you know?
Equation 3 shows that the graph has a minimum when x = 5. This is because (x − 5)2 is always greater than or
equal to zero, and it takes a minimum value of 0 when x = 5.
What do the equations have in common? [They are different representations of the same function.]
Completed square form can also be referred to as vertex form.

15. Compare/Contrast y = - (x + 4)(x – 5) y = -2(x + 4)(x – 5)
What is the same and what is different about the graphs of these two equations? How do you know?
For example, students may answer:
• Both parabolas have roots at (−4,0) and (5,0).
• Both parabolas have a maximum turning point.
• Equation 2 will be steeper than Equation 1 (for the same x value Equation 2’s y value will be double
that of Equation 1).

16. Intro to Dominos Whole-class introduction to Dominos (10 minutes)
Organize the class into pairs. Give each pair of students cut-up “dominos” A, E, and H from Domino Cards 1
and Domino Cards 2.
Explain to the class that they are about to match graphs of quadratics with their equations, in the same way that
two dominoes are matched.
If students are unsure how to play dominos, spend a couple of minutes explaining the game.
The graph on one “domino” is linked to its equations, which is on another “domino.”
Place Card H on your desk. Figure out, which of the two remaining cards should be placed to the right
of card H and which should be placed to its left.
Encourage students to explain why each form of the equation matches the curve:
Dwaine, explain to me how you matched the cards.
Alex, please repeat Dwaine's explanation in your own words.
Which form of the function makes it easy to determine the coordinates of the roots/ y-intercept/ turning point of the parabola?
Are the three different forms of the function equivalent? How can you tell?
The parabola on Domino A is missing the coordinates of its minimum. The parabola on Domino H is missing the
coordinates of its y-intercept. Ask students to use the information in the equations to add these coordinates.
What are the coordinates of the minimum of the parabola on Card A? What equation did you use to work it out? [(4, −1)]
What are the coordinates of the y-intercept of the parabola on Card H? What equation did you use to work it out? [(0,16)]

17. Matching Dominos
Take turns at matching pairs of dominos that you think belong together.
Each time you do this, explain your thinking clearly and carefully to your partner.
It is important that you both understand the matches. If you don't agree or understand, ask your partner to explain their reasoning. You are both responsible for each other’s learning.
On some cards an equation or part of an equation is missing. Do not worry about this, as you can carry out this task without this information.
Which form of the function makes it easy to determine the coordinates of the roots/y-intercept/turning point of the parabola?
How many roots does this function have? How do you know? How are these shown on the graph?
Will this function be shaped like a hill or a valley? How do you know?

18. Sharing Work
One student from each group is to visit another group's work
If you are staying at your desk, be ready to explain the reasons for your group's matches.
If you are visiting another group:
Write your card matches on a piece of paper.
Go to another group's desk and check to see which matches are different from your own.
If there are differences, ask for an explanation. If you still don't agree, explain your own thinking.
When you return to your own desk, you need to consider as a pair whether to make any changes to your own work.

19. Mathematical “Big Ideas” in the Model Lesson
Students will understand
what the different algebraic forms of a quadratic function reveal about the properties of its graphical representation.
how the factored form of the function can identify a graph’s roots.
how the completed square form of the function can identify a graph’s maximum or minimum point.
how the standard form of the function can identify a graph’s intercept.

20. Misconception
Students may make incorrect assumptions about what the different forms of the quadratic equation reveal about the properties of its parabola.

21. Formative Assessment Lesson Structure
Curriculum Leadership Council
CLC December 6, 2011
Formative Assessment Lesson Structure
Students…
work on their own, completing an assessment task designed to reveal their current understandings.
participate in a whole-class interactive introduction
work in pairs on a collaborative discussion tasks (in this case, matching the dominoes).
return to their original task and try to improve their responses.
Intent: Participants will understand the big ideas of the lesson.
Talking Points: Share the big ideas of the lesson with participants.

22. To what extent are teachers using strategies modeled in the FAL?
FAL Walk Through
Standards
Instructional Strategies
Connections to Current Classroom Practice
To what extent are teachers using strategies modeled in the FAL?
Chart Instructional Strategies as participants call-out what they experienced.

23. Research on Formative Assessment
Guidelines issued by professions organizations (NRC, 2001)
Standards for Teacher Practice (AERA/APA/NCME, 1999)
Research on the effects of classroom assessment on student learning (Black & Wiliam, 1998; Brookhart, 2004, Shepard, 2001)
Share that the there is research on the effects of classroom assessment on student learning

24. It’s teachers that make the difference
Take a group of 50 teachers
Students taught by the best teacher learn twice as fast as average
Students taught by the worst teacher learn half as fast average
And in the classrooms of the best teachers
Students with behavioral difficulties learn as much as those without
Students from disadvantaged backgrounds do as well as those from advantaged backgrounds
Inside the black box-findings from the meta-analysis
(Black & Wiliam, 1998; Brookhart, 2004, Shepard, 2001)
Assessing for student learning-CSTPs

25. Impact of Interventions
Increase in speed
of learning
Improve teachers’ use of learning styles 0%
Make teachers do an MA in Education
<5%
Increase teacher content knowledge from weak to strong
10%
Minute-by-minute and day-by-day
assessment for learning
80%
Share findings from the meta analysis

26. Cost/Effect Comparisons
Intervention
Extra months of learning per year
Cost/class-room/yr
Class-size reduction (by 30%) 4
$30k
Increase teacher content knowledge from weak to strong 2 ?
Formative assessment/
Assessment for learning 8
$3k
(Black & Wiliam, 1998; Brookhart, 2004, Shepard, 2001)
Assessing for student learning-CSTPs

27. The way we just debriefed this question is a “Number Talk.”
Mental Math
What is 6% of 35?
The way we just debriefed this question is a “Number Talk.”

28. Number Talks
• A daily routine for whole‐class instruction • Number Sense (efficiency, accuracy & flexibility) • Generalized Arithmetic-conceptual understanding • Reasoning and Problem Solving • Mental Mathematics • 10 minutes per day • Preview-­Review-­Conceptual Understanding
Intent: Participants will review what benefits of number talks.
Talking Points:
Kathy Richardson…

29. How many dots do you see? How did you see them?
Number Talk with Dots
How many dots do you see? How did you see them?
Intent: Participants will actively participate in a number talk.
Talking Points: Participants will be shown a dot image for a few seconds. They will then be asked to identify how many dots they saw and how they saw the dots. Show participants the dot image again for a few seconds. Facilitator will then ask the whole group to share how many dots they saw and list their answers. There may be a range of results. Facilitator will then ask participants to share how they saw the dots (show dot image on screen). Facilitator will then script responses using pre-printed copies of the dot images; including a matching equation.
This slide is animated; dots will fly in according to the description above.
Give participants time to reflect on the Number Talk with Dots using the Number Talk Reflection Handout.

30. Number Talk
If 75% of the original price is $120, what is the original price?
Intent: Participants will actively participate in a number talk.
Talking Points: Ask participants to solve the problem stated above using mental math. Participants may struggle answering this problem using algorithms. If they do, share students’ use of bar models and landmark numbers such as finding 25% and building 100%. Solution is 160.
Give participants time to reflect on the Number Talk using the Number Talk Reflection Handout.

31. True/False Number Talk
− = − True or False? Why?

32. Kirsten says that 10xy-5xy + 4xy equals xy
Dilemma Number Talk
Kirsten says that 10xy-5xy + 4xy
equals xy
David says that 10xy-5xy + 4xy equals 9xy
10-5+4
*Introduce Dilemmas as a type of number talk
*What could come before and after this number talk. Give rationale.
Explain the mathematical reasoning that both David & Kirsten used to simplify the expression above. 32

33. Spatial Reasoning Math Talk
How many cubes? How do you see them? What is the surface area?

34. What’s My Rule? Math Talk
Input, Output, x-value, etc.
Output, Range, y-value, etc. 8 4 21 11

35. Questions Teachers Might Ask
Who would like to share their thinking?
Did someone solve it a different way?
Who else used this strategy to solve the problem?
How did you figure it out?
What did you do next?
What did you need to know?
Why did you do that? Tell me more.
Which strategies do you see being used?
Intent: Participants will review possible questions teachers may ask of their students during a number talk.
Talking Points:
These questions are just a few questions a teacher would ask.
The teacher could also have sentence stems that students use to share out their solutions and strategies. (Especially helpful for EL students.)

36. Get It Together Teams of four Distribute the clues
You may not look at anyone else’s clues
You may share your clue by telling others what’s on it, but you may not show it to anyone else!

37. Reflection FAL, Number Talk, Get It Together
Instructional Strategies connected to CCSSM and SMP
Questioning and Prompts
Collaboration
Oral Language Production
What are you currently doing in your classroom that exemplifies these strategies and what might you need to enhance?
Return to FAL Chart
Chart Number Talk and Get It Together
Chart Debrief