Sailing the Seas of BDS

CCSS Math K-5

Our Objectives:

• To increase K-5 teachers understanding of the Mathematical Practice Standards and the Mathematical Content Standards specific to their grade band.

You would never hear someone say:

I am not very good at reading.

I can’t read.

You do hear people say:

I am not good at math.I can’t do math.

When I was a kid math was my worst subject

What is the difference?

The difference between the USA and other higher performing nations is that a culture of learning math is established from the beginning of a students career in school.

Students are informed and taught everyone can do math.

Essential Question #1

• How do the Mathematical Practice Standards help me to improve my math instruction?

Common Core Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively3. Construct viable arguments and critique the

reasoning of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated

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

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

• Students: make sense of the meaning of the task find an entry point or a way to start the task focus on concrete manipulatives before moving

to pictorial representations develop a foundation for problem solving

strategies reexamine the task when they are stuck ask, “Does my answer make sense?”

CCSS Practice #1Practice: #1 Make Sense of problems and persevere in solving them

Three Main Points: Make a Plan Self Monitor and Explain Demonstrate understanding by

corresponding

Example: Student Friendly Definition:I can make a plan, explain my answers, and show how I did it; then I can try another way

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What does SMP1 look like in a K-2 classroom?

–“Old” problem (little or no rigor) Tina had 10 balloons. She gave 7 of them away. How many balloons did Tina have then?

–“New” problem (with rigor)Burger Barn has 1 small table that can seat 4 people. They also have 1 large table that can seat double that amount. 15 people came in at lunch time. How many people did not get a seat?

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SMP 2 - Reason abstractly and quantitatively.

• Students: make sense of quantities and the

relationships decontextualize contextualize use abstract reasoning:

when they measure and compare lengths of objects

when they partition 2-D geometric figures into halves and fourths

as they begin to use standard measurement units

CCSS Practice #2Practice: #2: Reason Abstractly and Quantitatively

Three Main Points: Makes sense of quantities and

their relationship to the problem Bring complementary abilities

together Use reasoning that entails creating

a coherent representation

Example: Student Friendly Definition:I can think about the math problem in my head first

Math StringsMental Math

• The number of fingers on two human hands• Subtract the number of toes on one human

foot• Multiply it by the number doughnuts in a half

dozen• Divide by the number of eyes on a human

face• Add to it the number of hearts in a human

body• The answer is? 16

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What does SMP2 look like in a K-2 classroom?

–I had two pencils. My mom gave me some more. Now I have five pencils. How many pencils did my mom give me?

–Decontextualize:

2 + □ = 5

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

• Students: use mathematical terms to construct

arguments use definitions and previously

established solutions in their arguments engage in discussions about problem

solving strategies recognize and discuss reasonableness of

strategies recognize and discuss similarities and

differences between strategies

CCSS Practice #3Practice: #3: Construct viable arguments and critique the reasoning of others

Three Main Points: Understand and use stated

assumptions and definitions Construct arguments using

objects, drawings, and actions Listen, read, and critique to find

out what makes sense

Example: Student Friendly Definition:I can make a plan.I can tell my partner how I did it and listen to how they did it too!Talk about it!

Find the Fiction

• My number is 100.1.I can be broken into 4 parts equally2.I represent a millennium3.My quantity in pennies is equal to a

dollar

Find the Fiction

• On your board write the number of the statement that is fiction and write the word fiction next to that number (DO NOT SHOW ANYONE)

• Example: 4 Fiction • When you hear the signal word discuss

with your group one at a time your answer. Come to a consensus

• Answer: 2 is the Fiction • Praise: Expert Thinking

Farmer Fred looks in his field, but can only see feet. He knows that he has four animals. He knows that he has cows and chickens. Farmer Fred sees 14 feet. What are his four animals?

A model of mathematical modeling (pg. 72, CCSSM)

•Identifying important quantities in a practical situation•Making assumptions and approximations to simplify a complicated situation•Mapping relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas•Analyzing relationships mathematically to draw conclusions•Interpreting mathematical results in the context of the situation•Reflecting on whether the results make sense•Improving the model if it has not served its purpose.

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SMP 4 - Model with mathematics.

• Students: Apply math to solve problems in

everyday life. Represent math by using symbols,

pictures, concrete representation, graphs or equation writing.

Analyze relationships and draw conclusions

Interpret results in context Reflect if results make sense

CCSS Practice #4Practice: #4 Model with mathematics

Three Main Points: Reflect on whether the results

make sense Apply the math they know to

solve problems in everyday life Map relationships using tools

Example: Student Friendly Definition:I can use symbols and numbers to solve problems

What does this Array tell us?

Why Arrays?

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SMP 5 - Use appropriate tools strategically.

• Students: have access to a variety of tools

counters, place value blocks, hundred boards, number lines, geometric shapes, paper/pencil, etc.

determine which tools are most appropriate to use explain why they used a specific tool use tools appropriately have experiences with educational technology

Calculators, virtual manipulatives, games that support conceptual understanding or higher order thinking skills

CCSS Practice #5Practice: #5 Use appropriate tools strategically

Three Main Points: Be familiar with and consider all

available tools Use Technology tools to deepen

understanding Identify and Use other math

resources

Example: Student Friendly Definition:I can use all available tools and technology appropriately when solving math problems

Number Lines

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What does SMP5 look like in a K-2 classroom?

–Students have easy access to math tools.

–Students select their own tools.

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SMP 6 – Attend to precision.

• Students: are precise in their communication are precise in their calculations are precise in their measurements (no gaps

or overlaps) describe their actions and strategies clearly use grade-level appropriate mathematical

vocabulary accurately give precise explanations and reasoning check their work for accuracy and

reasonableness of solutions

CCSS Practice #6Practice: #6 Attend to Precision

Three Main Points: Communicate precisely and use

clear definitions State the meaning of the symbol Specify units of measure

Example: Student Friendly Definition:I can carefully explain to my partner how I came across my answer and why I think it is correct

Equality

Insert link from cpalms

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What does SMP6 look like in a K-2 classroom?

–How does a student (or teacher) talk through a subtraction with regrouping model?

413-168

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SMP 7 - Look for and make use of structure.

• Students: look for patterns and structures in the number

system and other areas of mathematics begin to recognize the commutative property begin to recognize that numbers can be

decomposed into tens and leftovers (ones) work with subtraction as missing addend

problems (How much more do I need to get to ___?)

skip count by tens off the decade to solve addition and subtraction problems

recognize that ten ones equals a ten, and ten tens equals a hundred

CCSS Practice #7Practice: #7 Look for and make use of structures

Three Main Points: Find a pattern Use what you know Solve the problem!!!

Example: Student Friendly Definition:I can use what I already know to solve a problem using patterns

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1 2 543 6 7 10981112 151413 1617 2019182122 252423 2627 3029283132 353433 3637 4039384142 454443 4647 5049485152 555453 5657 6059586162 656463 6667 7069687172 757473 7677 8079788182 858483 8687 9089889192 959493 9697 10

09998

53 + 23 = ?53 + 23 = 7682 - 14 = ?82 - 14 = 68

What does SMP7 look like in a K-2 classroom?

Word ProblemTake a Deep Breath!

• Mr. Centeno had a fruit fly problem.• On day 1 there were two fruit flies.• On day 2 there were four fruit flies.• On day 3 there were six fruit flies.• How many fruit flies would Mr. Centeno have

on day 5?• Answer: 10• What was the pattern? +2

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SMP 8 - Look for and express regularity in repeated reasoning.

• Students: look for regularity in problem structures when

solving mathematical tasks compose and decompose numbers in different ways use composition to make a ten and add the extras use decomposition and composition to break apart

numbers by place value to add and subtract larger numbers

look for the most efficient strategies for computations (including doubles, doubles 1 or 2, make a ten, counting on, etc.)

use repeated reasoning when solving a task with multiple correct solutions

check for reasonableness during and after task

What does SMP7 look like in a K-2 classroom?

• Equivalent Representations 4 + 6 = 10

3 + 7 = 10

9 + 1 = 10 0 + 10 = 10

6 + 4 = 10

8 + 2 = 10

7 + 3 = 10

10 + 0 = 104 + 6 = 103 + 7 = 10

9 + 1 = 10

0 + 10 = 10

6 + 4 = 10

8 + 2 = 107 + 3 = 10

10 + 0 = 10

• Addition facts that equal ten• How does a student know they have found them all?

CCSS Practice #8Practice: #8 Look for and express regularity in repeated reasoning

Three Main Points: Notice if calculations are repeated

by looking for methods Attend to detail Often evaluate results (check

understanding)

Example: Student Friendly Definition:I can look for shortcuts, use steps to solve problems, and see if it makes sense

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What is ten more than …… 3?

13?23?33?43?53?

What does SMP8 look like in a K-2 classroom?

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What does SMP8 look like in a K-2 classroom?

6 + 7

4 3

10+3=13

=13

• Make a Ten (Add the Extra)

Repeated Pattern Activity

Directions. Multiply the middle number by itself. Multiply the outer numbers to each other. Compare the products

5,6,73,4,56,7,8What conjecture can you come up with?What is 29x31 and why?What would the Algebraic formula look like?

Mathematics – Grades K - 5• 5 Domains

–Counting and Cardinality–Operations and Algebraic Thinking

–Number and Operations in Base Ten

–Measurement and Data–Geometry

Mathematics – Grades 1st – 5th

• 4 Domains–Operations and Algebraic Thinking(OA)

–Number and Operations in Base Ten(NBT)

–Measurement and Data(MD)–Geometry(G)

Let’s examine coherence in Operations and algebraic thinking

Kindergarten: Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

1st Grade: Represent and solve problems involving addition and subtraction.

2nd grade: Represent and solve problems involving addition and subtraction.

2nd grade: Work with equal groups of objects to gain foundations for multiplication.

3rd grade: Represent and solve problems involving multiplication and division.

3rd grade: Solve problems involving the four operations, and identify and explain patterns in arithmetic.

4th grade: Use the 4 operations with whole numbers to solve problems.

4th grade: Generate and analyze patterns.

Two critical areas in Kindergarten

• In Kindergarten, instructional time should focus on two critical areas:

• (1) representing, relating, and operating on whole numbers, initially with sets of objects (2) describing shapes and space.

• More learning time in Kindergarten should be devoted to number than to other topics.

http://www.corestandards.org/the-standards/mathematics/kindergarten/introduction/

Four critical areas in 1st Grade

• In Grade 1, instructional time should focus on four critical areas:– (1) developing understanding of addition, subtraction,

and strategies for addition and subtraction within 20; – (2) developing understanding of whole number

relationships and place value, including grouping in tens and ones;

– (3) developing understanding of linear measurement and measuring lengths as iterating length units; and

– (4) reasoning about attributes of, and composing and decomposing geometric shapes.

http://www.corestandards.org/the-standards/mathematics/grade-1/introduction/

Four critical areas in 2nd Grade

• In Grade 2, instructional time should focus on four critical areas: – (1) extending understanding of

base-ten notation– (2) building fluency with addition

and subtraction; – (3) using standard units of

measure; and– (4) describing and analyzing

shapes.http; ://www.corestandards.org/the-standards/mathematics/grade-2/introduction/

Four critical areas in 3rd Grade

• In 3rd Grade, instructional time should focus on four critical areas: – Developing understanding of multiplication

and division strategies for multiplication and division within 100

– Developing understanding of fractions, especially unit fractions (fractions with numerator 1)

– Developing understanding of the structure of rectangular arrays and of area

– Describing and analyzing two-dimensional shapes.

http://www.corestandards.org/the-standards/mathematics/grade-3/introduction/

Three critical areas in 4th Grade

• In 4th Grade, instructional time should focus on three critical areas: – Developing understanding and fluency with multi-digit

multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends.

– Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers.

– Understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.

http://www.corestandards.org/the-standards/mathematics/grade-4/introduction/

Three critical areas in 5th Grade

• In 5th Grade, instructional time should focus on three critical areas: – Developing fluency with addition and subtraction

of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases.

– Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations.

– Developing understanding of volume.http://www.corestandards.org/the-standards/mathematics/grade-5/introduction/

Essential Question #2

• How will unpacking and chunking the Mathematical Content Standards improve my math instruction?

Key Instructional Shifts

• Focus strongly where the Standards focus

• Coherence: Think across grades, and link to major topics within grades

• Rigor - In major topics, pursue conceptual understanding, procedural skill and fluency, and application with EQUAL intensity

Instructional shift #1

FOCUS strongly where the Standards focus

“… standards must address the problem of a curriculum that is ‘a mile wide and an inch deep.’ These Standards are a substantial answer to that challenge” (CCSS, 2010, p. 3).

Critical AreasThere are 2-4 Critical Areas for

instruction in the Introduction for each grade level

1st Grade Critical Areas – Introduction

They bring focus to the standards at each grade by grouping and summarizing the big ideas that educators can use to build their curriculum and to guide instruction.

–The goal of this activity is to help teachers:Become familiar with the Critical Areas and Content Standards

Understand chunking standards into Critical Ideas (Big Ideas)

Directions

In Grade-level groups, read through the Critical Areas and their descriptions on the Introduction page.

Then, read each of the content standards and mark on the recording sheet with a: X - when a standard strongly matches a Critical Area or

? - when you are not sure

1.OA.1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.1

3.OA.1.Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

Questions to Consider

• Did every standard fall within a Critical Area?

• Are there standards that fall within more than one Critical Area?

• Do all standards within a cluster fall within the same Critical Area?

A Process for Unpacking Standards

Developed by Steven M. Weber (2008)Adapted from Ainsworth (2003)

Step 1 Select a standard or a set of standards.

Step 2 Circle the verbs (skills).

Step 3 Underline the nouns and noun phrases (concepts).

Step 4 Create a chart or graphic organizer of the Unpacked Standards.

Step 5 Analyze and discuss the Unpacked Standards with co-workers.

Step 6 Write Generalizations/Big Ideas.

Step 7 Write Essential Questions. - http://www.authenticeducation.org/bigideas/article.lasso?artId=53What is an Essential Question? by Grant Wiggins

Step 8 Create Common Formative Assessments.

Step 9 Discuss and determine appropriate pacing.

Step 10 Develop Units of Study which will teach the identified Declarative and Procedural Knowledge, Key Concepts and Skills.

Step 11 Instruction

Step 12 Meet as a Professional Learning Community - http://www.mesd.k12.or.us/si/dufour_PLCs.pdfWhat is a Professional Learning Community? by Richard DuFour

Step 13 Review/Reteach the Generalizations/Big Ideas and the Unpacked Standards.

Step 14 Administer Common Formative Assessment.

Step 15 Meet as a Professional Learning Community.

Step 16 Begin the next unit of instruction, focusing on the Unpacked Standards.

http://www.k12curriculumdevelopment.com/1/post/2009/8/unpacking-standards.html

Step 1 Select a Standard

• Identify the Common Core State Standard for Mathematics

• Remember to start with the specific course description.

COMMON CORESTATE STANDARDS

Step 2 and Step 3Identify Skills and Concepts

Unpacking a standard is the process of identifying what the students will know (the nouns) and be able to do (the verbs) when they have mastered the standard.

Step 4 Create a Chart or Graphic

Organizer of the Unpacked Standard

Graphic Organizer

Standard Componen

ts

MACC.K.CC.2.5(Count to tell the number of objects)

Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.

Count to answer “how many?”

questions about as many as 20 things

arranged in a rectangular array.

Step 4

Count to answer “how many?”

questions about as many as 20 things

arranged in a circle.

Count to answer “how many?”

questions about as many as 20 things arranged in a line.

Count to answer “how many?”

questions about as many as 10 things

in a scattered configuration.

Count to answer “how many?”

questions about as many as 20 things

arranged in a rectangular array.

Step 4 (continued)

Count to answer “how many?”

questions about as many as 20 things

arranged in a circle.

Count to answer “how many?”

questions about as many as 20 things arranged in a line.

Count to answer “how many?”

questions about as many as 10 things

in a scattered configuration.

5 things10 things

15 things

10 things

15 things

10 things

15 things

Step 5 Analyze and Discuss

Analyze and discuss the Unpacked Standards with co-workers.

Mathematics – Grade 1– Domain: MACC.1.OA

• Operation and Algebraic Thinking– Cluster: MACC.1.OA.1

• Represent and solve problems involving addition and subtraction.

– Standard: MACC.1.OA.1.1• Use addition and subtraction within 20 to solve

word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Note: See Table 1

Unpack the Standard

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Unpack the Standard• Solve addition word

problems– Unknowns in all

positions– Adding to– Taking from– Putting together– Taking apart– Comparing

• Using– Objects– Drawings– Equations

• Solve subtraction word problems– Unknowns in all

positions– Adding to– Taking from– Putting together– Taking apart– Comparing

• Using– Objects– Drawings– Equations

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Unpacking Grade 4

• MACC.4.NBT.1: Generalize place value understanding for multi-digit whole numbers.

1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that by applying concepts of place value and division.

2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meaning of the digits in each place, using >, =, and < symbols to record the results of comparisons.

3. Use place value understanding to round multi-digit whole numbers to any place.

Marking Skills and Concepts

Exit Survey

Tell us about the questions that are still circling in your mind.

Describe three things that you found useful today.