Big Picture of Session

• TEKS Focus: Addition and Subtraction

• Instructional Focus: Scaffolding

• Scaffolding Focus: Effective use of graphic organizers and representational tools to develop, bridge, and build conceptual understanding

Kindergarten First Grade Second Grade

Modeling Addition and Subtraction

Modeling Addition and Subtraction

Modeling Addition and Subtraction

(K.4) The student models addition (joining) and subtraction) separating(K.4) The student is expected to model and create addition and subtraction problems in real situations with concrete objects.

(1.3) The student recognizes and solves problems in addition and subtraction (1.3A) The student is expected to model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences.

(2.3) The student adds and subtracts whole numbers to solve problems(2.3B) The student is expected to model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers.

Use story mats and manipulatives to develop joining and separating up to 20 objects.

Use organizers to bridge the development of addition and subtraction to the abstract of writing number sentences.

Use organizers to build understanding of addition and subtraction of two-digit numbers.

TEKS Focus

First Grade Second Grade Second Grade

Basic Addition and Subtraction Facts

Basic Addition and Subtraction Facts

Selecting Addition or Subtraction to Solve Problems

(1.3) The student recognizes and solves problems in addition and subtraction (1.3) (B) use concrete and pictorial models to apply basic addition and subtraction facts (up to 9 + 9 = 18 and 18 – 9 = 9)

(2.3) The student adds and subtracts whole numbers to solve problems(2.3) (A) recall and apply basic addition and subtraction facts (to 18)

(2.3) The student adds and subtracts whole numbers to solve problems(2.3) (C) select addition or subtraction to solve problems using two-digit numbers, whether or not regrouping is necessary

Use organizers to develop, bridge, and build basic addition and subtraction facts.

Use organizers and games to select and solve addition and subtraction situations.

TEKS Focus

ScaffoldingInstructional Focus: Instructional Focus: ScaffoldingScaffolding

According to Lev Vygotsky . . .

“The zone of proximal development is the

distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers.”

L.S. Vygotsky, (1978)

Scaffolding

“Through scaffolding, the teacher was able to build bridges from the unknown and not understood to the known and understood.”

Henderson, Many, Wellborn, and Ward. (2002, Summer). Reading Research and Instruction, 41(4), p. 310.

Known

Unknown

Video Reflection

• How does the instructor scaffold Tomika’s learning?

[Tomika Video]

Reading Academies

First Turn/Last Turn

• Read individually. Highlight 2-3 items.• In turn, share one of your items but do not comment on it.• Group members comment in round-robin fashion* – about

the item (without cross-talk).• The initial person who named the item then shares his or

her thinking about the item and takes the last turn, making the final comments.

• Repeat the pattern around the table.

*Round-robin is a highly structured participation strategy. Group members speak in turns, moving around the table in one direction.

What is Scaffolding in Mathematics Instruction?

What Is Scaffolding in Mathematics Instruction?

Scaffolding FocusScaffolding Focus

Bridging conceptual understanding using:

• Graphic organizers

• Representational Tools

Video Reflection

What are some examples of videos shown yesterday that use a graphic organizer to enhance understanding?

Scaffolding Instruction Through Questioning

“Teachers can use questions as a kind of scaffolding to help students reach higher

levels of thinking and learning. . . In the asking of questions, teachers are thinking actively

and helping students be active thinkers” (Walsh and Sattes, 2005, p 23).

Video Reflection

How do the teacher’s questions help Shania become successful with the problem 15 minus 1?

[Shania Video]

Pre Kindergarten and Pre Kindergarten and Kindergarten LessonKindergarten Lesson

1st Series of Lessons: Modeling Addition & Subtraction

PK Guidelines

Number and Operation

(K.4) The student models addition (joining) and subtraction) separating.

(1.3) The student recognizes and solves problems in addition and subtraction.

(2.3) The student adds and subtracts whole numbers to solve problems.

(3.3) The student adds and subtracts to solve meaningful problems involving whole numbers.

PK Guidelines

Combines separates and names “how many” concrete objects.

(K.4) The student is expected to model and create addition and subtraction problems in real situations with concrete objects.

(1.3A) The student is expected to model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences.

(2.3B) The student is expected to model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers.

(3.3) The student is expected to model addition and subtraction using pictures, words, and numbers.

Model Addition & Subtraction for PK-K

Engage There are questions eliciting prior knowledge about experiences going to zoos and being in plays. Read the story.

Explore & Explain

Reread the story with students acting it out physically and with manipulatives of a zoo mat and animal cards pointing out separating, joining, and relating to the size of the group (zoo mat and linking cubes are also available).

Elaborate 1 Students will create their own animal cards and add to the story using blank cards.

Elaborate 2 Students will again experience joining and separating of sets with a beach mat and sea shells.

Evaluate Students choose beach mat or zoo mat and are given 3 scenarios to solve with manipulatives; students will create a scenario.

Extend Create a shell story. Trace the shells to retell at home.

Linking Cubes to Represent Characters

1 gorilla – 1 blue cube

2 elephants – 2 white cubes

3 tigers – 3 orange cubes

4 parrots – 4 red cubes

5 monkeys – 5 brown cubes

One gorilla is out of his cage,

Oh my, what a rage!

Two elephants followed in line,

Parading, strutting, looking fine!

Three tigers let out a roar,

As they join the fun galore.

Four parrots flew the coop,

What a crazy looking group.

Which manipulative is easier to use and to understand the concepts for this activity?

Part-Part-Whole Mat for Addition

Explore

Part-Part-Whole Mat

10IIII

IIII

Five monkeys

scream and shout.

It doesn’t take long for them to get out!

Five monkeys scream and shout. It doesn’t take long for them to get out!

10 5

+ =

15

Cling! Clang! It’s dinner time!

Five monkeys swing home in

rhyme.

Cling! Clang! It’s dinner time!Five monkeys swing home in rhyme.

15

5

- = 10

Four parrots hungry for seed

Quickly fly back to feed.

Three tigers smell red meat,

As they swiftly spring to their feet.

Two elephants run for food.

They must eat not to be rude!

One gorilla, sad and blue,

What do you think he should do?

Elaborate 2

How did we use visuals to move the students from the concrete to the abstract level of learning?

Activity Concrete PictorialAbstract/

Generalization

Zoo story – Acting it out as in a play

•Kids are experiencing a tactile and kinesthetic activity.

•Kids see a physical representation and interaction with the story.

•The purpose of acting it out is to allow all students to create a common experience on which they can draw.

•Acting it out provides students an opportunity to connect language with action.

Act It Out Animal Card Placards

• Students model the poem wearing pictures of animals around their neck.

• Students begin to generalize that joining means that the group is getting larger.

• Students begin to generalize that separates means the group is getting smaller.

• Students can also generalize that if no animals join, then the group stays the same.

• Students begin to generalize what happens to the group size and the reasonableness to their answers.

Zoo story – Zoo mat with cards

Creating Animal Stories Filled Filled Filled

Beach Mat

Model Addition & Subtraction for PK-K

Engage There are questions eliciting prior knowledge about experiences with zoos and being in plays. Read the story.

Explore & Explain

Reread the story with students acting it out physically and using manipulatives of a zoo mat and animal cards pointing out separating, joining, and the relationship to the size of the group (zoo mat and linking cubes are also available).

Elaborate 1 Students will create their own animal cards and add to the story using blank cards.

Elaborate 2 Students will again experience joining and separating of sets with a beach mat and sea shells.

Evaluate Students choose beach mat or zoo mat and are given 3 scenarios to solve with manipulatives; students will create a scenario.

Extend Create a shell story. Trace the shells to retell at home.

First Grade LessonFirst Grade Lesson

PK Guidelines

Number and Operation

(K.4) The student models addition (joining) and subtraction) separating.

(1.3) The student recognizes and solves problems in addition and subtraction.

(2.3) The student adds and subtracts whole numbers to solve problems.

(3.3) The student adds and subtracts to solve meaningful problems involving whole numbers.

PK Guidelines

Combines separates and names “how many” concrete objects.

(K.4) The student is expected to model and create addition and subtraction problems in real situations with concrete objects.

(1.3A) The student is expected to model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences.

(2.3B) The student is expected to model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers.

(3.3) The student is expected to model addition and subtraction using pictures, words, and numbers.

1st Series of Lessons: Modeling Addition & Subtraction

Model Addition & Subtraction for Grade 1

Engage There are questions eliciting prior knowledge about experiences with zoos and being in plays. Read the story while students physically act it out with manipulatives.

Explore 1 Addition will be modeled with the visual of the part/part whole mat and zoo mat with the poem.

Explain 1 Read the story and experience using the part/part whole mat with the addition portion of the story.

Explore 2 Subtraction will be modeled with the visual of the part/part whole mat and zoo mat with the poem.

Explain 2 Finish the story and experience using the part/part whole mat with the subtraction portion of the story.

Elaborate Students will embellish the zoo story by drawing the new story and writing the matching number sentence.

Evaluate The students will create and write a new scenario, then give the story to a partner and let the partner write the number sentence.

Extend There is a pizza game with ten frames or five frames.

Optional Tool

Uses the Linking Cubes on a String and accompanying recording sheets.

Linking Cubes on a String

Engage Optional Tool

Recording Sheet for Linking Cubes on a String Up to 15

Optional Recording Sheet

6 + 4 = 10

Recording Sheet for Linking Cubes on a String Up to 20

6 + 4 = 10

Optional Recording Sheet

•Which representational tools have we used so far?•How have we used them?•Fill the Reflection Sheet

Second Grade LessonSecond Grade Lesson

Progression of Knowledge and Skill Statements

PK Guidelines

Number and Operations

(K.4) The student models addition (joining) and subtraction) separating.

(1.3) The student recognizes and solves problems in addition and subtraction.

(2.3) The student adds and subtracts whole numbers to solve problems.

(3.3) The student adds and subtracts to solve meaningful problems involving whole numbers.

PK Guidelines

Number and Operations

(K.4) The student is expected to model and create addition and subtraction problems in real situations with concrete objects.

(1.3A) The student is expected to model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences.

(2.3B) The student is expected to model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers.

(3.3) The student is expected to model addition and subtraction using pictures, words, and numbers.

Model Addition & Subtraction for Grades 2-3

Engage Estimate the scales on a pinecone.

Explore Students will discuss the strategies for finding the amount of scales on a pinecone and explore how to use a base ten or linking cubes organizer.

Explain Students share and explain how they found the number of scales on their pinecone and explain how to use linking cubes or a base ten organizer.

Elaborate Partners will compare their two pinecones and formulate questions and answers.

Evaluate Show transparencies of questions for students to solve.

9

8

7

6

5

4

3

2

1

UnitsTens

10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

9

8

7

6

5

4

3

2

1

OnesTens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Elaborate

Linking Cube Organizer

Hundred Tens Ones

+

9

8

7

6

5

4

3

2

1

OnesTens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Elaborate

15 + 23

Hundred Tens Ones

1 5

+

9

8

7

6

5

4

3

2

1

OnesTens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Elaborate

15 + 23

Hundred Tens Ones

1 5

+ 2 3

3 8

9

8

7

6

5

4

3

2

1

OnesTens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Elaborate

15 + 23 Describe the Results in Words and Numbers

= 38

3 tens and 8 ones

30 + 8

38

9

8

7

6

5

4

3

2

1

OnesTens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Elaborate

24 + 36

Hundred Tens Ones

2 4

+

9

8

7

6

5

4

3

2

1

OnesTens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Elaborate

24 + 36

Hundred Tens Ones

2 4

+ 3 6

0

1

6

9

8

7

6

5

4

3

2

1

OnesTens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Elaborate

24 + 36Describe Results in Words and Numbers

6 tens and 0 ones

60 + 0

60

9

8

7

6

5

4

3

2

1

OnesTens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Elaborate

24 + 38

Hundred Tens Ones

2 4

+ 3 8

2

1

6

9

8

7

6

5

4

3

2

1

OnesTens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Elaborate

24 + 38Describe Results in Words and Numbers

6 tens and 2 ones

60 + 2

62

Base Ten Blocks

• Proportional Relationship between the pieces– Unit– Rod– Flat– Cube

• Represent whole numbers

Ones

9

8

7

6

5

4

3

2

1

Tens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Base Ten Blocks Organizer

67 + 45Explain

Ones

9

8

7

6

5

4

3

2

1

Tens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Base Ten Blocks Organizer

67 + 45Explain

Ones

9

8

7

6

5

4

3

2

1

Tens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Base Ten Blocks Organizer

67 + 45Explain

Ones

9

8

7

6

5

4

3

2

1

Tens 10 9 8 7 6 5 4 3 2 1

100 90 80 70 60 50 40 30 20 10

Describe the Results in Words and Numbers

1 hundred 1 ten 2 ones

100 + 10 + 2

67 + 45 = 112

Which term do we use: Carry? Borrow? Regroup? Rename?

• “Carry” and “borrow” are misleading mathematically - They may promote mechanical manipulation of symbols.

• The term “regroup” is appropriate when manipulatives for a quantity are grouped differently.

• The term “rename” is mathematically correct; the quantity is actually given a different name. – For example, when computing 273-186, 2 hundreds + 7 tens + 3

ones is renamed as 2 hundreds + 6 tens + 13 ones” . (Ashlock, 2002, p. 63)

Video Reflection

How does the teacher in the video demonstrate how to find the sum of four fives?

[Allen Video]

Reflection on the First Series of Lessons

How do you scaffold lessons for students to be able to model addition and subtraction?

What tools could help students to better understand addition and subtraction?

First and Second Grade LessonFirst and Second Grade Lesson

First Grade Second Grade Second Grade

Basic Addition and Subtraction Facts

Basic Addition and Subtraction Facts

Selecting Addition or Subtraction to Solve Problems

(1.3) The student recognizes and solves problems in addition and subtraction (1.3) (B) use concrete and pictorial models to apply basic addition and subtraction facts (up to 9 + 9 = 18 and 18 – 9 = 9)

(2.3) The student adds and subtracts whole numbers to solve problems(2.3) (A) recall and apply basic addition and subtraction facts (to 18)

(2.3) The student adds and subtracts whole numbers to solve problems(2.3) (C) select addition or subtraction to solve problems using two-digit numbers, whether or not regrouping is necessary

Use organizers to develop, bridge and build basic addition and subtraction facts.

Use organizers and games to select and solve addition and subtraction situations.

TEKS Focus

Basic Facts for Addition & Subtraction for Grades 1-2Engage Solve a problem 9 + 8.

Explore Discuss various strategies and share with the students the make ten mat, doubles and near doubles mat, and part/part whole mat used for related facts, and splitting strategies, compensation.

Explain Students will identify all possible strategies to solve five facts.

Elaborate Have students solve + =13 using a variety of manipulatives and organizers.

Evaluate Play 3 in a row! with cards which have the quantity of the numerals in pictorial form or numerical form.

Addition and Subtraction Strategies

• Counting On

• Counting Back

• Doubles

• Near Doubles

• Make Ten

• Splitting

• Related Facts

• Compensation

Doubles and Near Doubles

                              

Doubles

5 + 5 =

10

6 + 6 = 7 + 7 =

12 14

Video Reflection

How is the organizer used to teach the students doubles from the pictorial to the concrete level?

[Zachary Video]

Fill this section on the Reflection sheet.

Doubles and Near Doubles:

5 + 5 =

10

6 + 6 = 7 + 7 =

12 14

5 + 6 = 6 + 7 =

11 13

Explore thinking strategies like these or realizing that 7 + 8 is the same as 7+7+1 will help students see the meaning of the operations. Such explorations also help teachers learn what students are thinking. NCTM

Video Reflection

How is organizer used to teach the students near doubles from the concrete level to the abstract level?

[Avery Video]

Fill this section on the Reflection sheet.

Video Reflection

How does the teacher scaffold the students’ learning of the concept for adding various numbers?

[Mrs. MacDonald Video]

6 + 8 =

How can the students use the Make Ten Strategy to solve this problem?

6 + 8 =

6 + 8 =

6 + 8 =

Video Reflection

How does Harrison use his mental figuration of make ten to describe how to solve 8 + 5?

[Harrison Video]

Splitting Strategy

“Splitting strategy is a strategy that children develop almost on their own, as soon as they begin to understand place value. They split the numbers up into friendly pieces, usually into hundreds, tens, and ones.”

Young Mathematicians at Work, pp. 134-135

Splitting Strategy from Young Mathematicians at Work

28 + 44

60 12

= 72

10

+

+ 2

+ 270

+ 820 +40 4

Splitting Strategy from Young Mathematicians at Work

28 + 44

20 + 8 + 40 + 4

60 + 12

60 + 10 + 2

70 + 2 = 72

Reflect:

What modes of learning were used to help students learn their basic facts? Fill the reflection sheet.

Second and Third Grade Second and Third Grade LessonLesson

Select Addition & Subtraction for Problem Solving for Grades 2-3

Engage Play 99 with organizers. How can you play 99 with the least amount of rolls?

Explore/Explain

Give a problem and demonstrate the on the problem solving mat.

Explain Have the students solve a problem using a problem solving organizer.

Elaborate 1 Play Race to Win! giving only the number sentence.

Elaborate 2 Play Race to Win! solving the problem.

Evaluate Give the students several manipulatives and a variety of organizers to use when solving problems.

Reflect:

Look at your scaffolding handout and highlight a scaffolding technique you would like to start using immediately.