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Transitioning to the Common Core State Standards – Mathematics. Pam Hutchison p [email protected] AGENDA. Party Flags Overview of CCSS-M Standards for Mathematical Practice Standards for Mathematical Content Word Problems and Model Drawing Math Facts

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Transitioning to the common core state standards mathematics

Transitioning to the Common Core State Standards – Mathematics

Pam Hutchison

[email protected]


Agenda
AGENDA

  • Party Flags

  • Overview of CCSS-M

    • Standards for Mathematical Practice

    • Standards for Mathematical Content

    • Word Problems and Model Drawing

  • Math Facts

    • Quick review – Addition and Subtraction Facts

    • Strategies – Multiplication and Division Facts

  • Area Models and Multiplication


Expectations
Expectations

  • We are each responsible for our own learning and for the learning of the group.

  • We respect each others learning styles and work together to make this time successful for everyone.

  • We value the opinions and knowledge of all participants.


Erica is putting up lines of colored flags for a party.

The flags are all the same size and are spaced equally along the line.

1. Calculate the length of the sides of each flag, and the space between flags.

Show all your work clearly.

2. How long will a line of n flags be?

Write down a formula to show how long a line of n flags would be.


Caccss m
CaCCSS-M

  • Find a partner

  • Decide who is “A” and who is “B”

  • At the signal, “A” takes 30 seconds to talk

  • Then at the signal, switch, “B” takes 30 seconds to talk.

    “What do you know about the CaCCSS-M?”


Caccss m1
CaCCSS-M

“What do you know about the CaCCSS-M?”

Using the fingers on one hand, please show me how much you know about the CaCCSS-M


National math advisory panel final report
National Math Advisory PanelFinal Report

“This Panel, diverse in experience, expertise, and philosophy, agrees broadly that the delivery system in mathematics education—the system that translates mathematical knowledge into value and ability for the next generation — is broken and must be fixed.”

(2008, p. xiii)


Common Core State Standards

Developed through

Council of Chief State School Officers

and

National Governors Association


Common Core State Standards


How are the ccss different
How are the CCSS different?

The CCSS are reverse engineered from an analysis of what students need to be college and career ready.

The design principals were focusand coherence. (No more mile-wide inch deep laundry lists of standards)


How are the ccss different1
How are the CCSS different?

Real life applicationsand mathematical modelingare essential.


How are the ccss different2
How are the CCSS different?

  • The CCSS in Mathematics have two sections:

  • Standards for Mathematical CONTENT

  • and

  • Standards for Mathematical PRACTICE

  • The Standards for Mathematical Content are what students should know.

  • The Standards for Mathematical Practice are what students should do.

    • Mathematical “Habits of Mind”


Standards for mathematical practice

Standards for Mathematical Practice


Mathematical practice
Mathematical Practice

  • 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.

  • Use appropriate tools strategically.

  • Attend to precision.

  • Look for and make use of structure.

  • Look for and express regularity in repeated reasoning.


Ccss mathematical practices
CCSS Mathematical Practices

REASONING AND EXPLAINING

Reason abstractly and quantitatively

Construct viable arguments and critique the reasoning of others

MODELING AND USING TOOLS

Model with mathematics

Use appropriate tools strategically

OVERARCHING HABITS OF MIND

Make sense of problems and persevere in solving them

Attend to precision

SEEING STRUCTURE AND GENERALIZING

Look for and make use of structure

Look for and express regularity in repeated reasoning


Ccss mathematical practices1
CCSS Mathematical Practices

  • Cut apart the Eight Standards for Mathematical Practice (SMPs)

  • Look over each Taxedo image and decide which image goes with which practice

    • The more frequently a word is used, the larger the image

  • Using the Standards for Mathematical Practice handout…did you get them right?

    • Glue the Practice title to the appropriate image.

  • What did you notice about the SMPs?


Reflection
Reflection

  • How are these practices similar to what you are already doing when you teach?

  • How are they different?

  • What do you need to do to make these a daily part of your classroom practice?


Supporting the smp s
Supporting the SMP’s

  • Summary

  • Questions to Develop Mathematical Thinking

    Common Core State Standards Flip Book

  • Compiled from a variety of resources, including CCSS, Arizona DOE, Ohio DOE and North Carolina DOE

  • http://katm.org/wp/wp-content/uploads/ flipbooks


Standards for mathematical content

Standards for Mathematical Content


Content standards
Content Standards

  • Are a balanced combination of procedure and understanding.

  • Stressing conceptual understanding of key concepts and ideas


Content standards1
Content Standards

  • Continually returning to organizing structures to structure ideas

    • place value

    • properties of operations

  • These supply the basis for procedures and algorithms for base 10 and lead into procedures for fractions and algebra


Understand
“Understand”

means that students can…

  • Explain the concept with mathematical reasoning, including

    • Concrete illustrations

    • Mathematical representations

    • Example applications


Organization k 8
Organization K-8

  • Domains

    • Larger groups of related standards. Standards from different domains may be closely related.


Domains k 5
Domains K-5

  • Counting and Cardinality (Kindergarten only)

  • Operations and Algebraic Thinking

  • Number and Operations in Base Ten

  • Number and Operations-Fractions (Starts in 3rd Grade)

  • Measurement and Data

  • Geometry


Organization k 81
Organization K-8

  • Clusters

    • Groups of related standards. Standards from different clusters may be closely related.

  • Standards

    • Defines what students should understand and be able to do.

    • Numbered


Word problems and model drawing

Word Problems and Model Drawing


Model drawing
Model Drawing

  • A strategy used to help students understand and solve word problems

  • Pictorial stage in the learning sequence of

    concrete – pictorial – abstract

  • Develops visual-thinking capabilities and algebraic thinking.


Steps to model drawing
Steps to Model Drawing

  • Read the entire problem, “visualizing” the problem conceptually

  • Decide and write down (label) who and/or what the problem is about

  • Rewrite the question in sentence form leaving a space for the answer.

  • Draw the unit bars that you’ll eventually adjust as you construct the visual image of the problem

H


Steps to model drawing1
Steps to Model Drawing

  • Chunk the problem, adjust the unit bars to reflect the information in the problem, and fill in the question mark.

  • Correctly compute and solve the problem.

  • Write the answer in the sentence and make sure the answer makes sense.


Missing numbers 1
Missing Numbers 1

Mutt and Jeff both have money. Mutt has $34 more than Jeff. If Jeff has $72, how much money do they have altogether?

H


Missing numbers 2
Missing Numbers 2

Mary has 94 crayons. Ernie has 28 crayons less than Mary but 16 crayons more than Shauna. How many crayons does Shauna have?


Missing numbers 3
Missing Numbers 3

Bill has 12 more than three times the number of baseball cards Chris has. Bill has 42 more cards than Chris. How many baseball cards does Chris have? How many baseball cards does Bill have?


Missing numbers 4
Missing Numbers 4

Amy, Betty, and Carla have a total of 67 marbles. Amy has 4 more than Betty. Betty has three times as many as Carla. How many marbles does each person have?


Representation
Representation

  • Getting students to focus on the relationships and NOT the numbers!



Teaching for understanding
Teaching for Understanding

Telling students a procedure for solving computation problems and having them practice repeatedly

rarely results in fluency

Because we rarely talk about how and why the procedure works.


Teaching for understanding1
Teaching for Understanding

  • Students do need to learn procedures for solving computation problems

  • But emphasis (at earliest possible age) should be on why they are performing certain procedure


Learning progression
Learning Progression

Concrete

 Representational

 Abstract


Research
Research

  • Students who learn rules before they learn concepts tend to score significantly lower than do students who learn concepts first

  • Initial rote learning of a concept can create interference to later meaningful learning


Fact fluency
Fact Fluency

  • Institute of Educational Sciences Practice Guide “Assisting Students Struggling with Mathematics: Response to Intervention for Elementary and Middle Schools”

  • Recommends approximately 10 minutes per day building fact fluency


Fact fluency1
Fact Fluency

  • The intent IS NOT to administer basic fact tests!

  • Teachers need to build basic fact strategy lessons for conceptual development, which builds fluency.


Fact fluency2
Fact Fluency

  • Fact fluency must be based on an understanding of operations and thinking strategies.

  • Students must

    • Construct visual representations to develop conceptual understanding.

    • Connect facts to those they know

    • Use mathematics properties and relationships to make associations



Concept learning
Concept Learning

  • Goal

    • Understand the meaning of addition, subtraction, multiplication, and division

  • Techniques

    • Use concrete objects, pictures, and symbols to develop


Fact strategies
Fact Strategies

  • Goal

    • Recognize clusters of facts

    • Understanding relationships between facts

  • Techniques

    • Addition and Subtraction – counting on, counting back, making 5’s, making 10’s, doubles, doubles plus 1, compensation, derived facts

    • Multiplication and Division – skip counting, repeated addition, repeated subtraction, derived facts, distributive property


Automaticity
Automaticity

  • Goal

    • Know facts so that they can be recalled quickly and accurately, and be retained over time

  • Techniques

    • Schedule short frequent practices

    • Reinforce facts already known

    • Use concepts and strategies to develop missing facts


Math fact strategies
Math Fact Strategies

  • Direct modeling / Counting all

  • Counting on / Counting back / Skip Counting

  • Derived Fact Strategies

    • Composing / Decomposing

    • Mental strategies

  • Automaticity



4 + 3


Addition 4 3
Addition – 4 + 3

  • Make fives

    4 + 3

1

2

7


8 + 6


8 + 6


Addition 8 6
Addition – 8 + 6

  • Make fives

    8 + 6

3

1

14


8 + 6


Addition 8 61
Addition – 8 + 6

  • Make ten

    8 + 6

2

4

14


7 + 5


7 + 5

7 + 5

3

2

  • 12



7 – 3


13 – 6


Subtraction 13 6
Subtraction: 13 – 6

  • Decompose with tens

    13 – 6 =

3

3


Subtraction 15 7
Subtraction: 15 – 7

  • Decompose with tens

    15 – 7 =

5

2


Commutative property
Commutative Property

  • 4 + 9

  • 9 + 4

  • 5 + 7

  • 7 + 5


Fact families
Fact Families

  • 7 + 5 = 12

  • 5 + 7 = 12

  • 12 – 5 = 7

  • 12 – 7 = 5



Reasoning about Multiplication and Division

http://fw.to/sQh6P7I



Multiplication1
Multiplication

  • 3 x 2

    • 3 groups of 2

    • Repeated Addition

2 + 2 + 2


Multiplication2
Multiplication

  • 3 rows of 2

  • This is called an “array” or an “area model”


Advantages of arrays as a model
Advantages of Arraysas a Model

  • Models the language of multiplication

4 groups of 6

or

4 rows of 6

or

6 + 6 + 6 + 6


Advantages of arrays as a model1
Advantages of Arraysas a Model

  • Students can clearly see the difference between (the sides of the array) and the (the area of the array)

factors

product

7 units

4 units

28 squares


Advantages of arrays
Advantages of Arrays

  • Commutative Property of Multiplication

    4x 6 = 6 x 4


Advantages of arrays1
Advantages of Arrays

  • Associative Property of Multiplication

    (4 x 3) x 2 = 4 x (3 x 2)


Advantages of arrays2
Advantages of Arrays

  • Distributive Property

    3(5 + 2) = 3 x 5 + 3 x 2


Advantages of arrays as a model2
Advantages of Arraysas a Model

  • They can be used to support students in learning facts by breaking problem into smaller, known problems

    • For example, 7 x 8

8

8

4

5

3

4

7

7

+

= 56

35

21

28

= 56

28

+



Group 1
Group 1

  • Repeated addition

  • Skip counting

  • Drawing arrays and counting

  • Connect to prior knowledge

    Build to automaticity


Multiplication3
Multiplication

  • 3 x 2

    • 3 groups of 2

1

3

5

6

4

2


Multiplication4
Multiplication

  • 3 x 2

    • 3 groups of 2

6

2

4


Multiplication5
Multiplication

  • 3 x 2

    • 3 groups of 2

2 + 2 + 2


Multiplying by 2
Multiplying by 2

Doubles Facts

  • 3 + 3

  • 2 x 3

  • 5 + 5

  • 2 x 5


Multiplying by 4
Multiplying by 4

Doubling

  • 2 x 3 (2 groups of 3)

  • 4 x 3 (4 groups of 3)

  • 2 x 5 (2 groups of 5)

  • 4 x 5 (4 groups of 5)


Multiplying by 3
Multiplying by 3

Doubles, then add on

  • 2 x 3 (2 groups of 3)

  • 3 x 3 (3 groups of 3)

  • 2 x 5 (2 groups of 5)

  • 3 x 5 (3 groups of 5)



Group 2
Group 2

  • Building on what they already know

    • Breaking apart areas into smaller known areas

  • Distributive property

    Build to automaticity



Teaching multiplication facts2
Teaching Multiplication Facts

Group 1

Group 2

Group 3


Group 3
Group 3

  • Commutative property

    Build to automaticity


Teaching multiplication facts3
Teaching Multiplication Facts

Group 1

Group 2

Group 4

Group 3


Group 4
Group 4

  • Building on what they already know

    • Breaking apart areas into smaller known areas

  • Distributive property

    Build to automaticity


Connecting multiplication and division
Connecting Multiplication and Division


Division
Division

  • What does 6  2 mean?

    • Repeated subtraction

1 group

2 groups

3 groups

6

-2

4

-2

2

-2

0

3 groups


Measurement division
Measurement Division

  • I have 21¢ to buy candies with. If each gumdrop costs 3¢, how many gumdrops can I buy?


Fair share division
Fair Share Division

  • Mr. Gomez has 12 cupcakes. He wants to put the cupcakes into 4 boxes so that there’s the same number in each box. How many cupcakes can go in each box?


Difference in counting
Difference in counting?

  • Measurement

    • 4 for you, 4 for you, 4 for you

      • And so on

    • Like measuring out an amount

  • Fair Share

    • 1 for you, 1 for you, 1 for you, 1 for you

    • 2 for you, 2 for you, 2 for you, 2 for you

      • And so on

    • Like dealing cards


Measurement division1
Measurement Division

  • What does 6  2 mean?

    • 6 split into groups of 2


Measurement division quotative
Measurement Division(Quotative)

  • You know

    • The total amount of objects

    • The number of objects in each group

  • You’re trying to find

    • The number of groups

  • Counting

    • 1, 2, 3, 4, ….

    • 1, 2, 3, 4, …..


Fair share division1
Fair Share Division

  • What does 6  2 mean?

    • 6 split evenly into 2 groups


Fair share division partitive
Fair Share Division(Partitive)

  • You know

    • The total amount of objects

    • The number of groups

  • You’re trying to find

    • The number of objects in each group

  • Counting

    • 1 for you, 1 for you, 1 for you, 1 for you,….

    • 2 for you, 2 for you, 2 for you, 2 for you, etc.


Models for division
Models for Division

  • Repeated subtraction

  • Groups

    • Finding the number in each group

    • Finding the number of groups

  • Arrays – finding the missing side


Repeated subtraction
Repeated Subtraction

  • 21 ÷ 3

    • 1 group for you 21 – 3 = 18 left

    • 1 group for you 18 – 3 = 15 left

    • 1 group for you 15 – 3 = 12 left

    • 1 group for you 12 – 3 = 9 left

    • 1 group for you 9 – 3 = 6 left

    • 1 group for you 6 – 3 = 3 left

    • 1 group for you 3 – 3 = 0 left

      7 groups of 3


Groups and subtraction
Groups and Subtraction

  • 36 ÷ 4

    • Should I put 1, 2, or 3 in each group?

    • How many cubes did I give away?

    • How many cubes are left?


Skip counting
Skip Counting

30 ÷ 5

  • 5

  • 10

  • 15

  • 20

  • 25

  • 30

1

2

3

4

5

6


Using arrays
Using Arrays

8

4

?

12

3

?

40

5


Using arrays1
Using Arrays

5

15

3

?

6

?

24

4


Using arrays2
Using Arrays

4

?

8

6)48

28

7


Connection to multiplication
Connection to Multiplication

  • 42 ÷ 7 =

    think “7 x _?_ = 42”

  • Fact Families




Flash card practice
Flash Card Practice

Facts I Am Still Learning

Facts I Know Quickly

Facts I Can Figure Out Quickly

Create 2 representations for each fact

Create 1 representation for each fact



Fluency assessments
Fluency Assessments

  • 20-25 facts

  • 2 colors of pencils (or pens)

  • After 60 seconds, call switch. Students change the color of the pencil they are using.

  • Give students another 60-90 seconds

  • If students finish before time to stop, continue to write and solve your own fact problems


Advantages
Advantages

  • All students get to finish!

  • Let’s you assess both fluency and accuracy.



Using arrays to multiply
Using Arrays to Multiply

23

x 4

80

4 rows of 20

= 80

12

4 rows of 3

= 12

92


Using arrays to multiply1
Using Arrays to Multiply

23

x 4

12

4 rows of 3

= 12

80

4 rows of 20

= 80

92


Using arrays to multiply2
Using Arrays to Multiply

  • Use Base 10 blocks and an area model to solve the following:

21

x 13




Partial products
Partial Products

31

x 14

300

(10  30)

10

(10  1)

120

(4  30)

4

(4  1)

434


Partial products1
Partial Products

31

x 14

4

(4  1)

120

(4  30)

10

(10  1)

300

(10  30)

434


Pictorial representation
Pictorial Representation

80 + 4

84

x 57

50

+

7

50  4

50  80

4,000

200

7  80

7  4

560

28


Pictorial representation1
Pictorial Representation

30 + 7

37

x 94

90

+

4

90  7

90  30

2,700

630

4  30

4  7

120

28


Pictorial representation2
Pictorial Representation

300 + 40 + 7

347

x 68

60

+

8

2,400

18,000

420

320

2,400

56


Decimals
Decimals

0.4

0.4 x 0.6

0.6


Fractions
Fractions

3

2

3

5


Patterns in multiplication
Patterns in Multiplication

Groups of 6

groups

1’s

10’s

100’s

1

2

3

.

.

.

5

8

60

120

180

.

.

.

300

480

600

1200

1800

.

.

.

3000

4800

6

12

18

.

.

.

30

48


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