Standards and Instructional Strategies Module 4B

ESUHSDJune 2012

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

Agenda

• Welcome Back and Review• CCSS Formative Assessment Lesson• Number Talk • Reflection and Next Steps

Domains and Conceptual Categories

Distribution

Findell & Foughty (2011)College and Career-Readiness through the Common Core State Standards for Mathematics

High School MathematicsThe 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.

High SchoolMathematics Standards

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

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Conceptual Categories• Number & Quantity• Algebra• Functions• Modeling• Geometry• Statistics & Probability

Standards for

Mathematical PracticeOverarching habits of mind of a productive mathematical thinker

Reasoning and explaining

Modeling and using tools

Seeing structure and generalizing

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2. Reason abstractly and quantitatively.3. Construct viable arguments and

critique the reasoning of others.

4. Model with mathematics.5. Use appropriate tools strategically.

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

Forming QuadraticsFormative Assessment Lesson

Mathematical Goals

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

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.

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?

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?

Key Features of a Graph of a Quadratic

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

Compare/Contrasty = - (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?

Intro to Dominos

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.

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.

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.

Misconception

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

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.

http://map.mathshell.org/materials/

FAL Walk Through

• Standards• Instructional Strategies• Connections to Current Classroom Practice

To what extent are teachers using strategies modeled in the FAL?

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)

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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

Impact of InterventionsIntervention 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-dayassessment for learning

80%

Cost/Effect ComparisonsIntervention 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

Mental Math

What is 6% of 35?

The way we just debriefed this question is a “Number Talk.”

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

Number Talk with DotsHow many dots do you see?

How did you see them?

Number Talk

• If 75% of the original price is $120, what is the original price?

True/False Number Talk

True or False?Why?

Dilemma Number Talk

10-5+4 Kirsten says that 10xy-5xy + 4xy equals xy

David says that 10xy-5xy + 4xy equals 9xy

Explain the mathematical reasoning that both David & Kirsten used to simplify the expression above.

Spatial Reasoning Math Talk

How many cubes? How do you see them?

What is the surface area?

What’s My Rule? Math TalkInput, Output, x-value, etc.

Output, Range, y-value, etc.

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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?

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!

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?