ADAPT, by Tim Harford In Adapt, Harford credits Peter Palchinsky for three principles that pave the road for success: (1) Try new things, (2) try them on a scale where failure is survivable, and (3) seek feedback and learn from mistakes. Each day, our students try new things. They make mistakes and correct them, with critical learning along the way.

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What’s Math Got to Do With It? Jo Boaler

“The very best way to teach children helpful strategies is to provide interesting settings, problems, and puzzles that require the strategies and then to share and discuss successful methods and strategies at frequent intervals.”

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How Children Succeed Paul Tough

“One of the most important executive functions is cognitive flexibility. Cognitive flexibility is the ability to see alternative solutions to problems, to think outside the box, and to negotiate unfamiliar situations.”

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The World is Flat Thomas Friedman Friedman believes that future employment will focus not on lifetime employment, but on lifetime employability -- guaranteed opportunities to remain current enough to stay employed. Knowing how to ‘learn how to learn’ will be one of the most important assets any worker can have, because job churn will come faster, because innovation will happen faster.

4

This

shift in job growth

requires a

shift in learning,

which can be

accomplished only if

we make a

corresponding

shift in instruction.

Today, we will…

• Identify and analyze the key shifts in the Common Core State Standards.

• Explore the Common Core Standards Progressions and Illustrative Mathematics documents.

• Connect the Key Shifts with Progressions and Illustrative Mathematics.

• Create a plan to disseminate the information about the Progressions and Illustrative Mathematics documents to colleagues.

Today, We Will…

Common Core State Standards

http://corestandards.org/

Introduction: UNDERSTANDING

MATHEMATICS

• The Standards define what students should

understand and be able to do in their study of

mathematics.

• Understanding is the ability to explain why a

particular mathematical statement is true or

where a mathematical rule comes from. The

student who can explain a rule understands the

mathematics and can apply it.

Introduction: UNDERSTANDING

MATHEMATICS

• Mathematical understanding and procedural skill

are equally important, and both are assessable

using mathematical tasks of sufficient richness.

Standards for Mathematical Practice

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

2. Reason abstractly and quantitatively.

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

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Key Shifts

The Common Core State Standards for

Mathematics build on the best of existing standards

and reflect the skills and knowledge students will

need to succeed in college, career, and life.

Understanding how the standards differ from

previous standards—and the necessary shifts they

call for—is essential to implementing them.

There are three key shifts called for by the

Common Core.

Common Core Resources

The Progressions Documents describe teaching methods, strategies, and algorithms.

Progressions http://ime.math.arizona.edu/progressions/

The structure of mathematics One aspect of the structure of mathematics is reliance on a small collection of general properties rather than a large collection of specialized properties. (Focus)

For example, as number systems expand from whole numbers to fractions in Grades 3–5, to rational numbers in Grades 6–8, to real numbers in high school, the same key ideas are used to define operations within each system. (Coherence)

From the Progressions Documents

http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_frontmatter_2013_07_30.pdf

From the Progressions Documents

K-5 Number and Operations in Base Ten Standard algorithms for base-ten computations with the four operations rely on decomposing numbers written in base ten notation into base-ten units. The properties of operations then allow any multi-digit computation to be reduced to a collection of single-digit computations, sometimes requiring the composition or decomposition of a base-ten unit (regrouping).

From the Progressions Documents

K-5 Operations and Algebra The Progression in Operations and Algebraic Thinking deals with the basic operations: (a) the kinds of quantitative relationships they model, (b) the kinds of problems they can be used to solve, and (c) their mathematical properties and relationships. Most of the standards organized under the OA heading involve whole numbers, but the importance of the Progression is that it describes concepts, properties, and representations that extend to other number systems, to measures, and to algebra.

From the Progressions Documents

K-5 Operations and Algebra Over time, students build their understanding of the properties of arithmetic: commutativity and associativity of addition and multiplication, and distributivity of multiplication over addition. Initially, they build intuitive understandings of these properties, using intuitive understandings in strategies to solve real-world and mathematical problems. Later, these understandings become more explicit and allow students to extend operations into the system of rational numbers.

From the Progressions Documents

4.NF.1

Two Grades 6–8 domains are important in preparing students for Algebra in high school.

• The Number System prepares students to (a) see all numbers as part of a unified system, and (b) become fluent in finding and using the properties of operations to find the values of numerical expressions that include those numbers.

• The Expressions and Equations domain ask students to extend their use of these properties to linear equations and expressions with letters. These extend uses of the properties of operations in earlier grades: in Grades 3–5 Number and Operations—Fractions, in K–5 Operations and Algebraic Thinking, and K–5 Number and Operations in Base Ten.

From the Progressions Documents

Grade 6-8, Number Systems Students build on two important conceptions which have developed throughout K–5, in order to understand the rational numbers as a number system. The first is the representation of whole numbers and fractions as points on the number line, and the second is a firm understanding of the properties of operations on whole numbers and fractions.

From the Progressions Documents

6-8 Expressions and Equations An expression is a phrase in a sentence about a mathematical or real-world situation. You can read a lot from an algebraic expression, and it is a goal of this progression for students to (a) see expressions as objects in their own right, and to (b) read the general appearance and fine details of algebraic expressions. An equation is a statement that two expressions are equal. It is an important aspect of equations that the two expressions on either side of the equal sign might not actually always be equal; that is, the equation might be a true statement for some values of the variable(s) and a false statement for others. For example, 3x + 1 = 4 is only true when x = 1.

From the Progressions Documents

NS and EE

36 x 94

36 x 94 = (30 + 6) x (90 + 4)

[(30 + 6) x 90] + [(30 + 6) x 4]

(30 x 90) + (6 x 90) + (30 x 4) + (6 x 4) (FOIL)

ab x cd = (a + b) x (c + d)

[(a + b) x c)] + (a + b) x d]

ac + bc + ad + bd

High School Algebra The study of algebra occupies a large part of a student’s high school career. • The Progressions Documents give some general

guidance about ways to treat the material and ways to tie it together.

• It notes key connections among standards, points out cognitive difficulties and pedagogical solutions, and gives more detail on particularly knotty areas of the Mathematics.

From the Progressions Documents

In Grade Level Groups

Go to the Progressions documents. Find your grade level and/or course. Peruse and find something useful.

Progressions http://ime.math.arizona.edu/progressions/

The Progressions Documents

Common Core Resources

Illustrative Mathematics documents provide “tasks” with supportive commentary that illustrate how the Standards might be applied or assessed.

Illustrative Mathematics https://www.illustrativemathematics.org

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Illustrative Mathematics, Algebra I/Geometry

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Illustrative Mathematics, Algebra I/Geometry

35

Jessica and some friends have ordered two pizzas. One is a medium sized pizza while the other one is a large.

Jessica eats two slices of the medium pizza. Has she eaten 2/16 of the two pizzas? Explain your reasoning and draw a picture to illustrate your explanation.

Illustrative Mathematics, Grade 4

36

Illustrative Mathematics, Grade 4 Commentary The focus of this task is on understanding that fractions, in an explicit context, are fractions of a specific whole. In this problem, there are three different wholes: the medium pizza, the large pizza, and the two pizzas taken together. Because the medium and large pizza are not the same size and Jessica has taken two slices of the medium pizza, it is not the case that she has eaten 2/16 or 1/8 of the two pizzas. The two pizzas together can be naturally divided into 8 equal portions, but this would mean taking one piece of the medium and one piece of the large pizza. This task is best suited for instruction. Students can practice explaining their reasoning to each other in pairs or pas part of a whole group discussion.

Illustrative Mathematics, Grade 3

37

3.NBT.A.1

When rounding to the nearest ten:

• What is the smallest whole number that will round to 50?

• What is the largest whole number that will round to 50?

• How many different whole numbers will round to 50?

When rounding to the nearest hundred:

• What is the smallest whole number that will round to 500?

• What is the largest whole number that will round to 500?

• How many different whole numbers will round to 500?

Illustrative Mathematics, Grade 3

3.OA.A.2 • There are 4 tanks and 3 fish in each tank. • The total number of fish can be expressed

as 4 x 3 = 12. • Describe what is meant in this situation by (a) 12 ÷ 3 = 4 and (b) 12 ÷ 4 = 3.

Illustrative Mathematics, Grade 4

39

4.NBT.A a. Find a number greater than 0 and less than 1,000

that:

– Is closer to 500 than 0,

and – Is closer to 200 than 500.

b. There are many correct answers to this problem. Describe all of the numbers that are correct.

Illustrative Mathematics, Grade 4

4.OA.A There are two snakes at the zoo, Jewel and Clyde. Jewel was six feet and Clyde was eight feet. A year later Jewel was eight feet and Clyde was 10 feet. Which one grew more?

Illustrative Mathematics, Grade 5

5.NBT.A.2

Marta made an error while finding 84.15 x 10.

In your own words, explain Marta’s misunderstanding. Explain what she should do to get the correct answer. Include the correct answer in your response.

Illustrative Mathematics, Grade 5

42

5.OA.A.2 Eric is playing a video game. At a certain point in the game, he has 31500 points. Then the following events happen, in order: • He earns 2450 additional points. • He loses 3310 points. • The game ends, and his score doubles. • Write an expression for the number of points Eric has at the

end of the game. Do not evaluate the expression. The expression should keep track of what happens in each step listed above.

Eric's sister Leila plays the same game. When she is finished playing, her score is given by the expression • 3(24500+3610)−6780. • Describe a sequence of events that might have led to Leila

earning this score.

Illustrative Mathematics, Grade 6 6.NS.A.1

Alice, Raul, and Maria are baking cookies. They had all of the ingredients except the flour and butter, so they each brought what they had at home. They need ¾ cup of flour and 1/3 cup of butter to make a batch of cookies.

Alice had 2 cups of flour and ¼ cup of butter. Raul had 1 cup of flour and ½ cup of butter. Maria had 1 ¼ cups of flour and ¾ cups of butter. How many batches of cookies can they make?

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Illustrative Mathematics, Grade 6

meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $27.50 for dinner. What is the cost of her dinner without tax or tip?

6.EE Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her

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Illustrative Mathematics, Grade 7

7.NS.A.3 The three seventh grade classes at Sunview Middle School collected the most boxtops for a school fundraiser, and so they won a $600 prize to share among them. Mr. Aceves’ class collected 3,760 box tops, Mrs. Baca’s class collected 2,301, and Mr. Canyon’s class collected 1,855. How should they divide the money so that each class gets the same fraction of the prize money as the fraction of the box tops that they collected?

Illustrative Mathematics, Grade 7

46

7.EE.A

If we multiply x/2 + 3/4 by 4, we get 2x + 3.

Is 2x + 3 an equivalent expression to x/2 + 3/4?

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8.NS.A.2 For each pair of numbers, decide which is larger without using a calculator. Explain your choices. 1. π2 or 9 2. √50 or √51 3. √50 or 8 4. −2π or −6

Illustrative Mathematics, Grade 8

Illustrative Mathematics, Grade 8

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8.EE.A.4 This headline appeared in a newspaper: Every day 7% of Americans eat at B-burger! Decide whether this headline is true using the following information: • There are about 8 x 103 B-burger restaurants in

America. • Each restaurant serves on average 2.5 x 103 people

every day. • There are about 3 x 108 Americans. Explain your reasons and show clearly how you figured it out.

A-SSE.A

In the equations (a)–(d), the solution x to the equation depends on the constant a. Assuming a is positive, what is the effect of increasing a on the solution? Does it increase, decrease, or remain unchanged? Give a reason for your answer that can be understood without solving the equation.

a) x − a = 0

b) ax = 1

c) ax = a

d) xa = 1

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Illustrative Mathematics, Algebra

Illustrative Mathematics, Algebra

50

A-SSE.B Judy is working at a retail store over summer break. A customer buys a $50 shirt that is on sale for 20% off. Judy computes the discount, then adds sales tax of 10%, and tells the customer how much he owes. The customer insists that Judy first add the sales tax and then apply the discount. He is convinced that this way he will save more money because the discount amount will be larger. a. Is the customer right? b. Does your answer to part (a) depend on the numbers

used or would it work for any percentage discount and any sales tax percentage? Find a convincing argument using algebraic expressions and/or diagrams for this more general scenario.

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The Common Core Standards: What to Teach and How to Teach Them

Robyn Silbey www.robynsilbey.com ◊ robyn@robynsilbey.com