time activity who (high effect strategy)education.ucf.edu/rtp3/docs/ccss_mathematics_july.pdf ·...

Exploring the Depth of the Common Core State Standards Presenter: Edward C. Nolan Dates: July 19 and 20, 2013 Time: 9:00 am to 3:00 pm

Location: TA 203 on July 19th; TA 130 on July 20th Goals:

• To teach mathematics for conceptual understanding • To understanding the Common Core State Standards for Mathematics and the Standards

for Mathematical Practice and what they mean for teaching • To translate high level of STEM content knowledge into the classroom • To use high effect instructional strategies

Time Activity Who (High Effect Strategy) 9:00-9:30 Introductions Leaders Warm-up Overview of the Common Core State Standards (CCSS) (previewing new content, processing new information, identifying critical information) 9:30-10:30 Standards for Mathematical Practice (SMP) activity Cross Grade Level Understanding the actions and predicting Groups (MS/HS) behavior (using questioning and discussion techniques, organizing to interact with new knowledge) 10:30-11:30 Learning Progression of a Topic Common Groups Explore learning progression across grades (MS or HS) (metacognition, demonstrating knowledge of content, making connections across content, teacher reflection) 11:30-12:00 Matching activity Common Groups Problem solving and SMP (critical thinking, creating and testing hypotheses) 12:00-12:30 Lunch (with continued conversations)

Time Activity Who (High Effect Strategy) 12:30-1:30 Adapting to the vision of the Common Core Common Groups Exploration of current instructional sequence and SMP rubric (adapting lessons, using effective questioning strategies, establishing a culture for learning) 1:30-2:15 Dan Meyer video and discussion Cross Grade Level “Math Class Needs a Makeover” Groups What impact does this have for instruction? (establishing a culture for learning, engaging students in learning, demonstrating enthusiasm) 2:15-2:45 What is the role of STEM in the mathematics Cross Grade Level classroom? Groups What are the key aspects of effective STEM education? (making connections across contents, engaging students in learning, designing coherent instruction) 2:45-3:00 Summary and Closure Whole Group (teacher reflection, demonstrating knowledge of content and pedagogy, identifying critical information) Resources:

• Presentation powerpoint • Copy of the full Common Core State Standards • Activity handouts

o Warm-up o Academic Language in Mathematics Teaching o Standards for Mathematical Practice (individually and full set) o Notice and Wonder problems o Cellular Growth sample item o Learning Progression examples o Matching SMP activity o Sample Algebra 1 Lesson Plan o Critical Thinking Selected Response Items o Capture sheet for STEM discussion

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Exploring the Depth of theCommon Core State Standards

July 2013

Edward C. Nolan

[email protected]

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Sample Assessment Item

Lo, Jane‐Jane and Feng‐Chiu Tsai. “Taiwanese Arithmetic and Algebra,”

Mathematics Teaching in the Middle School, March 2011. Page 424.

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Sample Assessment Item

“Review” for Next Year’s Test

If (31x – 23)(7x – 29) – (7x – 29)(11x – 13)

can be factored into (mx + n) (20x + p) and

m, n, and p are all integers,

then m + n + p = ?

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• To teach mathematics for conceptual understanding

• To understand the Common Core State Standards for Mathematics and the Standards for Mathematical Practice and what they mean for teaching

• To translate high level of STEM content knowledge into the classroom

• To use high effect instructional strategies

4

Goals

Agenda

Introduction and Overview of the Common Core

Activity on the SMP

Learning Progression of a Topic

Problem Solving and the Practices

Adapting to the Common Core Vision

STEM in the Mathematics Classroom

Summary and Closure

9:00am

9:30am

2:45pm

12:00pm

12:30pm

Lunch

THECOMMON CORESTATE STANDARDS

(CCSS)

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• CCSS adopted by 45 states and DC

• Written in 2010, led by the National Governors Association and Council of Chief State School Officers

• Content and practices

• Thinking, reasoning, problem‐solving

• Focus, coherence, and rigor

• Challenge and application

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CCSS: Current Status

Overview of the Structure of the CCSS

• Ratios and Proportional Relationships

• Functions

• The Number System

• Expressions and Equations

• Geometry

• Statistics and Probability

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CCSS Middle School Domains

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• Number and Quantity

• Algebra

• Functions

• Modeling

• Geometry

• Statistics and Probability

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CCSS High School Domains

• 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.

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Standards for Mathematical Practice

• Use appropriate tools strategically.

• Attend to precision.

• Look for and make use of structure.

• Look for and express regularity in repeated reasoning.

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2011‐2012 Item and Task Development

2012‐2013 Development/Field Testing

2013‐2014 Development/Field Testing

2014‐2015 Summative Assessment In Use

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PARCC Timeline

• Component 1 (Optional): Early Assessments

• Component 2 (Optional): Mid‐Year Assessments

• Component 3 (Required, summative): Performance‐Based, End of Year

• Component 4 (Required, summative):Computer Based, Machine Scorable ,End of year

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PARCC Design

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Cellular Growth (high school)

Source: ccsstoolbox.com

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STANDARDS FOR MATHEMATICAL PRACTICE (SMP)

ACTIVITY

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Standards for Mathematical Practice Wordlewww.wordle.net

• Read the assigned Standard for Mathematical Practice (1‐8).

• Underline all of the verbs in the assigned standard.

• Individually: think about the behaviors that students who are proficient with your SMP might display.

• Partner: Share your thoughts with your neighbor.

• Group: Share ideas at your table.

• Whole group: Report one interesting idea to the whole group.

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SMP Analysis Activity

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• What words are at the beginning of each of the eight SMPs?

• What do the SMPs describe: student actions or teacher actions?

• Why do you think that the Common Core SMP are written this way?

• What is the goal of the SMP?

• How will we get there?

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• Create a question (or two, or more) for each of the “Notice and Wonder” pictures.

• Share your question with a neighbor and talk about how you would answer the question(s).

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• With your partner, reflect on:

• The strategies that were used to answer the question.

• Are there other strategies that could be used to answer the question?

• How are the Standards for Mathematical Practice reflected in this activity?

• Were there SMP that were less evident?

• Share with your table.

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The eight Standards of Mathematical Practice are:

• Habits of Mind

• Fostered through a variety of experiences

• Embedded in everyday lessons

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LEARNING PROGRESSIONOF A TOPIC

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• Working with a partner, discuss what students should know before starting to learn the targeted standard . Next, determine what knowledge the targeted standard will help students learn in later courses.

• Locate the targeted standard in the CCSS document.

• Search the CCSS document for standards that are the precursor to the targeted standard and for which the targeted standard is a precursor.

• Record the standards and the reasoning on your paper.

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Grade 6Example

6.SP.3

• The Common Core is based on research‐based learning trajectories/progressions. These progressions are built around the development of student mathematical conceptual understanding and procedural fluency.

• When teachers explore and examine learning progressions, they learn about how student understanding builds upon itself. Examining these progressions will help teachers to develop pre‐assessments and offer enrichment.

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MATCHING ACTIVITY

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• Select either the seventh grade problems or the algebra problems.

• With a partner, think about and discuss how you would solve each of the problems.

• Think about where the Standards for Mathematical Practice match to the ways that you solved the problems.

• Based on the approach you used, match the primary SMP to each problem.

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Matching Activity

• What are your current thoughts about the:

• Common Core State Standards?

• Standards for Mathematical Practice?

• Learning progressions?

• Planning a mathematics lesson?

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Reflection So Far

LUNCH

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EXAMINATION OF ALESSON THROUGH THE LENS OF THESTANDARDS OF

MATHEMATICAL PRACTICE

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• Modeling with Algebra Tiles

• Which SMP is this?

• What is the value of each tile?

• How would I represent 3x + 5? 6x – 3? 4 – 5x?

• What about x2 + 5x + 6? x2 + x – 6?

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A Sample Algebra Lesson

• With a partner, work through the sample lesson on under‐standing algebraic factoring [DON’T RUSH, take your time and think like a student].

• Once we are close to everyone finishing the lesson, we will consider the representation of the Standards for Mathematical Practices in this lesson.

• Consider which Practice(s) (remember, we don’t expect to see them all) are the focus of this activity.

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A Sample Algebra Lesson

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Adjusting Based on the Practices

Institute for Advanced Study/Park City Mathematics Institute Summer 2011 Secondary School Teachers Program/Visualizing Functions

SMP5Exemplary Practice – when Students take Ownership

• Consider each of the three selected response items on the handout. Select one to solve with a member at your table. [Note: warm‐up was another example from this assessment.]

• Share your solution strategy with your partner.

• Discuss how this problem might be used within a lesson that targets both content and practice standards.

• Identify which SMP is the focus of your problem and how using this problem might help students to become more proficient with the Standards for Mathematical Practices.

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Exploring Integrating the Practices

“MATH CLASS NEEDSA MAKEOVER”

DAN MEYER

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Go To Video

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Dan Meyer Video

• With your table group:

• Share an idea that resonated with you.

• How might this idea (or one you heard from a tablemate) affect the way your students approach problem solving?

• How does Dan Meyer talk about getting students involved in mathematics? How does this relate to the Standards for Mathematical Practice?

• How might this video impact the way that you think about teaching?

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Video Reflection

WHAT IS THE ROLE OFSTEM IN THE

MATHEMATICS CLASSROOM?

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• STEM refers to the integration of the subjects of Science‐Technology‐Engineering‐Mathematics.

• Consider connections among these topics and jot down real world, meaningful, and relevant connections. It is important that the connections be natural and appropriate, seamless, rigorous, and based in logic and inquiry.

• Share your ideas at your table.

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What is STEM? How do we integrate STEM?

• What are you walking away with?

• What questions to you still have?

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Summary and Closure

GOALS• To teach mathematics for conceptual understanding• To understand the Common Core State Standards

for Mathematics and the Standards for Mathe‐matical Practice and what they mean for teaching

• To translate high level of STEM content know‐ledge into the classroom

• To use high effect instructional strategies

Hope you had a wonderful day!

July 2013

Edward C. Nolan

[email protected]

Warm-up

Source: Lo, Jane-Jane and Feng-Chiu Tsai. “Taiwanese Arithmetic and Algebra,” Mathematics Teaching in the Middle School, March 2011. Page 424.

Exploring the Depth of the Common Core State Standards

Academic Language in Mathematics Teaching Common Core State Standards: The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy. (from http://corestandards.org/) Standards for Mathematical Practice: The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). (from http://corestandards.org/the-standards/mathematics/introduction/standards-for-mathematical-practice/) STEM: STEM is an acronym that stands for Science, Technology, Engineering, and Math – usually in reference to talent. STEM can cut across the learning pipeline – from early childhood education through lifelong learning. It can be referenced in efforts focused on talent development, but also retention and recruitment or retraining/recertification efforts. Any career enabled by science, technology, engineering, or math may be a STEM career; it is the technician working with algebra in her water resources field work, just as it’s the doctoral-level bio-tech researcher. Just as the STEM acronym may not be readily recognized by many employers and workers, STEM-career awareness is key to ensuring the right leaders and partners are both engaged and benefiting. (from http://www.stemflorida.net/PressKit)

http://corestandards.org/

http://corestandards.org/the-standards/mathematics/introduction/standards-for-mathematical-practice/

http://corestandards.org/the-standards/mathematics/introduction/standards-for-mathematical-practice/

http://www.stemflorida.net/PressKit

High effect instructional strategies: Research-based strategies that have been demonstrated to lead to student achievement. Two leading proponents of the use of these strategies in the observation and evaluation of teaching are Charlotte Danielson (http://www.danielsongroup.org) and Dr. Robert Marzano (http://www.marzanoevaluation.com). Types of high effect instructional strategies that we may touch on in our workshop include (there are others): • Danielson

o Demonstrating knowledge of content and pedagogy

o Demonstrating knowledge of students

o Setting instructional outcomes o Demonstrating knowledge of

resources o Designing coherent instruction o Creating an environment of

respect and rapport

o Establishing a culture for learning

o Communicating with students o Using questioning and discussion

techniques o Engaging students in learning o Demonstrating flexibility and

responsiveness o Reflecting on Teaching

• Marzano o Identifying Critical Information o Organizing Students to Interact

with New Knowledge o Previewing New Content o Chunking Content into

“Digestible Bites” o Processing of New Information o Elaborating on New Information o Recording and Representing

Knowledge o Reflecting on Learning o Reviewing Content o Organizing Students to Practice

and Deepen Knowledge o Practicing Skills, Strategies, and

Processes o Revising Knowledge o Noticing When Students are Not

Engaged o Using Academic Games o Managing Response Rates

o Using Physical Movement o Maintaining a Lively Pace o Demonstrating Intensity and

Enthusiasm o Using Friendly Controversy o Understanding Students’

Interests and Background o Effective Scaffolding of

Information with Lessons o Lessons within Units o Attention to Established Content

Standards o Use of Available Traditional

Resources o Use of Available Technology o Needs of English Language

Learners o Needs of Students Receiving

Special Education o Needs of Students Who Lack

Support for Schooling

http://www.danielsongroup.org/

http://www.marzanoevaluation.com/

Cellular Growth

Part a. In a cellular regeneration experiment, Jaydon Laboratory found that for cells put in containers

with a particular growth medium, the number of cells at the end of each week was double the number

of cells at the end of the previous week.

The data for the first 6 weeks of the experiment are shown in the table. Fill in the blanks to complete

the table for weeks 7-10.

Part b. Assume that as the experiment continues, the number of cells

at the end of each week continues to be double the number of cells at

the end of the previous week. Let wn represent the number of cells in

the growth medium in week n. Drag the tiles to write a recursive

definition for the sequence that represents the number of cells in the

growth medium at the end of each week.

Part c. Let wn represent the number of cells in the growth medium at

the end of week n. Which of these statements are true about the

explicit formula for wn?

Select all that apply.

Part d. Write your answers to the following problem in your answer booklet.

Consider the table of data about the cellular regeneration experiment.

a. If the number of cells continues to grow according to the pattern shown in the table, at what

week number will the number of cells exceed one billion?

b. Explain how the process you used to find the week number relates to either the recursive

model or the explicit model you constructed in the previous questions.


Standards for Mathematical Practice Analysis

1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning

of a problem and looking for entry points to its solution. They analyze givens,

constraints, relationships, and goals. They make conjectures about the form and

meaning of the solution and plan a solution pathway rather than simply jumping

into a solution attempt. They consider analogous problems, and try special cases

and simpler forms of the original problem in order to gain insight into its solution.

They monitor and evaluate their progress and change course if necessary. Older

students might, depending on the context of the problem, transform algebraic

expressions or change the viewing window on their graphing calculator to get the

information they need. Mathematically proficient students can explain

correspondences between equations, verbal descriptions, tables, and graphs or

draw diagrams of important features and relationships, graph data, and search for

regularity or trends. Younger students might rely on using concrete objects or

pictures to help conceptualize and solve a problem. Mathematically proficient

students check their answers to problems using a different method, and they

continually ask themselves, “Does this make sense?” They can understand the

approaches of others to solving complex problems and identify correspondences

between different approaches.



2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships

in problem situations. They bring two complementary abilities to bear on problems

involving quantitative relationships: the ability to decontextualize—to abstract a

given situation and represent it symbolically and manipulate the representing

symbols as if they have a life of their own, without necessarily attending to their

referents—and the ability to contextualize, to pause as needed during the

manipulation process in order to probe into the referents for the symbols involved.

Quantitative reasoning entails habits of creating a coherent representation of the

problem at hand; considering the units involved; attending to the meaning of

quantities, not just how to compute them; and knowing and flexibly using different

properties of operations and objects.



3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions,

definitions, and previously established results in constructing arguments. They

make conjectures and build a logical progression of statements to explore the truth

of their conjectures. They are able to analyze situations by breaking them into

cases, and can recognize and use counterexamples. They justify their conclusions,

communicate them to others, and respond to the arguments of others. They reason

inductively about data, making plausible arguments that take into account the

context from which the data arose. Mathematically proficient students are also able

to compare the effectiveness of two plausible arguments, distinguish correct logic

or reasoning from that which is flawed, and—if there is a flaw in an argument—

explain what it is. Elementary students can construct arguments using concrete

referents such as objects, drawings, diagrams, and actions. Such arguments can

make sense and be correct, even though they are not generalized or made formal

until later grades. Later, students learn to determine domains to which an argument

applies. Students at all grades can listen or read the arguments of others, decide

whether they make sense, and ask useful questions to clarify or improve the

arguments.



4. Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve

problems arising in everyday life, society, and the workplace. In early grades, this

might be as simple as writing an addition equation to describe a situation. In

middle grades, a student might apply proportional reasoning to plan a school event

or analyze a problem in the community. By high school, a student might use

geometry to solve a design problem or use a function to describe how one quantity

of interest depends on another. Mathematically proficient students who can apply

what they know are comfortable making assumptions and approximations to

simplify a complicated situation, realizing that these may need revision later. They

are able to identify important quantities in a practical situation and map their

relationships using such tools as diagrams, two-way tables, graphs, flowcharts and

formulas. They can analyze those relationships mathematically to draw

conclusions. They routinely interpret their mathematical results in the context of

the situation and reflect on whether the results make sense, possibly improving the

model if it has not served its purpose.



5. Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a

mathematical problem. These tools might include pencil and paper, concrete

models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra

system, a statistical package, or dynamic geometry software. Proficient students

are sufficiently familiar with tools appropriate for their grade or course to make

sound decisions about when each of these tools might be helpful, recognizing both

the insight to be gained and their limitations. For example, mathematically

proficient high school students analyze graphs of functions and solutions generated

using a graphing calculator. They detect possible errors by strategically using

estimation and other mathematical knowledge. When making mathematical

models, they know that technology can enable them to visualize the results of

varying assumptions, explore consequences, and compare predictions with data.

Mathematically proficient students at various grade levels are able to identify

relevant external mathematical resources, such as digital content located on a

website, and use them to pose or solve problems. They are able to use

technological tools to explore and deepen their understanding of concepts.



6. Attend to precision.

Mathematically proficient students try to communicate precisely to others. They

try to use clear definitions in discussion with others and in their own reasoning.

They state the meaning of the symbols they choose, including using the equal sign

consistently and appropriately. They are careful about specifying units of measure,

and labeling axes to clarify the correspondence with quantities in a problem. They

calculate accurately and efficiently, express numerical answers with a degree of

precision appropriate for the problem context. In the elementary grades, students

give carefully formulated explanations to each other. By the time they reach high

school they have learned to examine claims and make explicit use of definitions.



7. Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure.

Young students, for example, might notice that three and seven more is the same

amount as seven and three more, or they may sort a collection of shapes according

to how many sides the shapes have. Later, students will see 7 × 8 equals the well

remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive

property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and

the 9 as 2 + 7. They recognize the significance of an existing line in a geometric

figure and can use the strategy of drawing an auxiliary line for solving problems.

They also can step back for an overview and shift perspective. They can see

complicated things, such as some algebraic expressions, as single objects or as

being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5

minus a positive number times a square and use that to realize that its value cannot

be more than 5 for any real numbers x and y.



8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look

both for general methods and for shortcuts. Upper elementary students might

notice when dividing 25 by 11 that they are repeating the same calculations over

and over again, and conclude they have a repeating decimal. By paying attention to

the calculation of slope as they repeatedly check whether points are on the line

through (1, 2) with slope 3, middle school students might abstract the equation (y –

2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x –

1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the

general formula for the sum of a geometric series. As they work to solve a

problem, mathematically proficient students maintain oversight of the process,

while attending to the details. They continually evaluate the reasonableness of their

intermediate results.


Notice and Wonder

What question(s) do you think of when looking at each of the following pictures?

http://www.101qs.com/images/fullsize/591-cd-burner.png


Learning Progression of a Topic Consider the learning progression of a topic that precedes and follows the listed Common Core State Standard. Explore the CCSS document to find standards that support students preparing to address the given standard, as well as standards that follow the assigned standard. Enter the standard as well as the reasoning for your choice. Grade Standard Rationale

4

5

6

7

Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.2 Recognize and represent proportional relationships between quantities.

a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

8

HS

HS

HS



4

5

6

7

8

Understand the connections between proportional relationships, lines, and linear equations. 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

HS

HS

HS



4

5

6

7

8

HS

Create equations that describe numbers or relationships. A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

HS

HS



4

5

6

7

8

HS

Prove theorems involving similarity. G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

HS

HS

Exploring the Depth of the Common Core State Standards Common Core Sample Problems – Grade 7

Problem 1. A student is making trail mix. Create a graph to determine if the quantities of nuts and fruit are proportional for each serving size listed in the table. If the quantities are proportional, what is the constant of proportionality or unit rate that defines the relationship? Explain how you determined the constant of proportionality and how it relates to both the table and graph. Problem 2. Use a number line to illustrate:

o p - q o p + (- q) o Is this equation true p – q = p + (-q)

Problem 3. Jamie and Ted both get paid an equal hourly wage of $9 per hour. This week, Ted made an additional $27 dollars in overtime. Write an expression that represents the weekly wages of both if J = the number of hours that Jamie worked this week and T = the number of hours Ted worked this week? Can you write the expression in another way?

Problem 4. Julie showed you the scale drawing of her room. If each 2 cm on the scale drawing equals 5 ft, what are the actual dimensions of Julie’s room? Reproduce the drawing at 3 times its current size.

Problem 5. The container below contains 2 gray, 1 white, and 4 black marbles. Without looking, if you choose a marble from the container, will the probability be closer to 0 or to 1 that you will select a white marble? A gray marble? A black marble? Justify each of your predictions.

From http://www.azed.gov/standards-practices/mathematics-standards/

http://www.azed.gov/standards-practices/mathematics-standards/

Exploring the Depth of the Common Core State Standards Common Core Sample Problems – Algebra

Problem 1. Express 2(x3 – 3x2 + x – 6) – (x – 3)(x + 4) in factored form and use your answer to say for what values of x the expression is zero. Problem 2. In February, the Bezanson family starts saving for a trip to Australia in September. The Bezanson’s expect their vacation to cost $5375. They start with $525. Each month they plan to deposit 20% more than the previous month. Will they have enough money for their trip? Problem 3. Use the formula for the sum of two fractions to explain why the sum of two rational expressions is another rational expression. Problem 4. Are the roots of 2x2 + 5 = 2x real or complex? How many roots does it have? Find all solutions of the equation. Problem 5. The opera theater contains 1,200 seats, with three different prices. The seats cost $45 dollars per seat, $50 per seat, and $60 per seat. The opera needs to gross $63,750 on seat sales. There are twice as many $60 seats as $45 seats. How many seats in each level need to be sold? Problem 6. A publishing company publishes a total of no more than 100 magazines every year. At least 30 of these are women’s magazines, but the company always publishes at least as many women’s magazines as men’s magazines. Find a system of inequalities that describes the possible number of men’s and women’s magazines that the company can produce each year consistent with these policies. Graph the solution set.

From http://www.azed.gov/standards-practices/mathematics-standards/

http://www.azed.gov/standards-practices/mathematics-standards/

Exploring the Depth of the CCSS Sample Lesson Plan

Understanding Algebraic Factoring Students sometimes think algebra and geometry are two static, unrelated subjects that were "invented" by some historical figure and appear in books but have no redeeming value or purpose. One way to make these topics more meaningful and less mysterious is to look at the words themselves. While working through algebraic factoring it is helpful to remember that all you are doing is working with breaking up and putting back together squares and rectangles!

Objective: To show the geometric basis of algebraic factoring. Materials: One set of algebra tiles

Procedure: Introduce (or review) the notations used for multiplication.

3 X 3 3 · 3 (3)(3) and show these ideas using an array of the 1 x 1 unit squares.

Use the notation (3)(3) so that when the transition is made from numerals to algebraic expressions the notation will be the same. If you think it is necessary, repeat this idea with several whole numbers until you think the students have the concept of how multiplication can be shown geometrically (using squares) as well as numerically.

Main Activity:

1. Introduce the algebra tiles.

What shape is this? Answer: (square) What shape is this? Answer: (rectangle)

From http://mathforum.org/alejandre/algfac.html Page 1


How long is this side? Answer: (x units)

How long is this side? Answer: (1 unit) Before continuing, make certain that it is clear that you are working with pieces that are one of the following:

one unit square one rectangle that is one unit wide and "x" units long. one square that is "x" units long by "x" units wide.

2. Using algebra tiles, show that (x + 1)(x + 3)

can be shown using the tiles:

Rearrange the tiles so that they look like this:

What is that? It is

one square that is "x" units long by "x" units wide four rectangles that are one unit wide and "x" units long, and 3 unit squares

which can also be written as:

3. Now have students try: (x+4)(x+2) (x+5)(x+1)

etc. Once the students have an understanding of what algebraic factoring really is, introducing the FOIL method will have more meaning.



Procedure 1:

Ask the students to form a square or rectangle using the algebra tiles that shows

One possibility would look like this:

Questions To Ask:

1. What figure does this make? A square or a rectangle? Why? 2. What is the length of each side? 3. If you rearrange the squares and rectangles making up the larger square, what do you

have? Answer:



Procedure 2:

Ask the students to form a square or rectangle using the algebra tiles that shows


Procedure 3: Ask the students to form a square or rectangle using the algebra tiles that shows



Exploring the Depth of the Common Core State Standards Critical Thinking Problems – Selected Response

Source: Lo, J. & Tsai, F. (2011). Taiwanese Arithmetic and Algebra. Mathematics Teaching in the Middle School, 16(7), pp. 422-429.


Capture Sheet for STEM Conversation Consider your personal experiences in light of your content and practices explorations today. Can you think of connections between mathematics and science, technology, and/or engineering? It is important that the connections be natural and appropriate, seamless, rigorous, and based in logic and inquiry. Consider the science inquiry model of the 5 E’s: Engage Explore Explain Elaborate Evaluate

time activity who (high effect strategy)education.ucf.edu/rtp3/docs/ccss_mathematics_july.pdf ·...

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