This document provides an overview of scaffolding instruction for addition and subtraction. It discusses using graphic organizers and manipulatives to build conceptual understanding from concrete to abstract levels. Specific lessons are outlined for pre-kindergarten through second grade that model addition and subtraction using objects, pictures, and number sentences. Representational tools like linking cubes, part-part-whole mats, and base ten blocks are employed to scaffold learning.

1. Addition and Subtraction Scaffolding Instruction

2. 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

3. TEKS Focus Use organizers to build understanding of addition and subtraction of two-digit numbers. Use organizers to bridge the development of addition and subtraction to the abstract of writing number sentences. Use story mats and manipulatives to develop joining and separating up to 20 objects. (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. (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. (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. Modeling Addition and Subtraction Modeling Addition and Subtraction Modeling Addition and Subtraction Second Grade First Grade Kindergarten

4. TEKS Focus Use organizers and games to select and solve addition and subtraction situations. Use organizers to develop, bridge, and build basic addition and subtraction facts. (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 (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) (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) Selecting Addition or Subtraction to Solve Problems Basic Addition and Subtraction Facts Basic Addition and Subtraction Facts Second Grade Second Grade First Grade

5. Scaffolding Instructional Focus: Scaffolding

6. 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)

7. 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

8. Video Reflection How does the instructor scaffold Tomika’s learning? [ Tomika Video]

9. Reading Academies

10. 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.

11. What is Scaffolding in Mathematics Instruction? What Is Scaffolding in Mathematics Instruction?

12. Scaffolding Focus Bridging conceptual understanding using: Graphic organizers Representational Tools

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

14. 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).

15. Video Reflection How do the teacher’s questions help Shania become successful with the problem 15 minus 1? [Shania Video]

16. Pre Kindergarten and Kindergarten Lesson Modeling Addition and Subtraction

17. 1 st Series of Lessons: Modeling Addition & Subtraction (3.3) The student is expected to model addition and subtraction using pictures, words, and numbers. (2.3B) The student is expected to model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers. (1.3A) The student is expected to model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences. (K.4) The student is expected to model and create addition and subtraction problems in real situations with concrete objects. PK Guidelines Combines separates and names “how many” concrete objects. (3.3) The student adds and subtracts to solve meaningful problems involving whole numbers. (2.3) The student adds and subtracts whole numbers to solve problems. (1.3) The student recognizes and solves problems in addition and subtraction. (K.4) The student models addition (joining) and subtraction) separating. PK Guidelines Number and Operation

18. Create a shell story. Trace the shells to retell at home. Extend Students choose beach mat or zoo mat and are given 3 scenarios to solve with manipulatives ; students will create a scenario. Evaluate Students will again experience joining and separating of sets with a beach mat and sea shells. Elaborate 2 Students will create their own animal cards and add to the story using blank cards. Elaborate 1 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). Explore & Explain There are questions eliciting prior knowledge about experiences going to zoos and being in plays. Read the story. Engage Model Addition & Subtraction for PK-K

19. One Gorilla

20. 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

22. One gorilla is out of his cage, Oh my, what a rage!

24. Two elephants followed in line, Parading, strutting, looking fine!

26. Three tigers let out a roar, As they join the fun galore.

28. Four parrots flew the coop, What a crazy looking group.

30. Time to Reflect Which manipulative is easier to use and to understand the concepts for this activity?

31. Part-Part-Whole Mat for Addition Explore

33. Part-Part-Whole Mat 10 IIII IIII

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

36. Five monkeys scream and shout. It doesn’t take long for them to get out! 10 5 + = 15

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

39. Cling! Clang! It’s dinner time! Five monkeys swing home in rhyme. 15 5 - = 10

40. Four parrots hungry for seed Quickly fly back to feed.

42. Three tigers smell red meat, As they swiftly spring to their feet.

44. Two elephants run for food. They must eat not to be rude!

46. One gorilla, sad and blue, What do you think he should do?

47. Elaborate 2

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

49. Beach Mat Filled Filled Filled Creating Animal Stories Zoo story – Zoo mat with cards 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. Act It Out Animal Card Placards Students model the poem wearing pictures of animals around their neck. 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. Zoo story – Acting it out as in a play Abstract/ Generalization Pictorial Concrete Activity

50. Create a shell story. Trace the shells to retell at home. Extend Students choose beach mat or zoo mat and are given 3 scenarios to solve with manipulatives ; students will create a scenario. Evaluate Students will again experience joining and separating of sets with a beach mat and sea shells. Elaborate 2 Students will create their own animal cards and add to the story using blank cards. Elaborate 1 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). Explore & Explain There are questions eliciting prior knowledge about experiences with zoos and being in plays. Read the story. Engage Model Addition & Subtraction for PK-K

51. First Grade Lesson Modeling Addition and Subtraction

52. 1st Series of Lessons: Modeling Addition & Subtraction (3.3) The student is expected to model addition and subtraction using pictures, words, and numbers. (2.3B) The student is expected to model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers. (1.3A) The student is expected to model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences. (K.4) The student is expected to model and create addition and subtraction problems in real situations with concrete objects. PK Guidelines Combines separates and names “how many” concrete objects. (3.3) The student adds and subtracts to solve meaningful problems involving whole numbers. (2.3) The student adds and subtracts whole numbers to solve problems. (1.3) The student recognizes and solves problems in addition and subtraction. (K.4) The student models addition (joining) and subtraction) separating. PK Guidelines Number and Operation

53. Uses the Linking Cubes on a String and accompanying recording sheets. Optional Tool There is a pizza game with ten frames or five frames. Extend The students will create and write a new scenario , then give the story to a partner and let the partner write the number sentence. Evaluate Students will embellish the zoo story by drawing the new story and writing the matching number sentence. Elaborate Finish the story and experience using the part/part whole mat with the subtraction portion of the story. Explain 2 Subtraction will be modeled with the visual of the part/part whole mat and zoo mat with the poem. Explore 2 Read the story and experience using the part/part whole mat with the addition portion of the story. Explain 1 Addition will be modeled with the visual of the part/part whole mat and zoo mat with the poem. Explore 1 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. Engage Model Addition & Subtraction for Grade 1

54. Linking Cubes on a String Engage Optional Tool

55. Recording Sheet for Linking Cubes on a String Up to 15 Optional Recording Sheet 6 + 4 = 10

56. Recording Sheet for Linking Cubes on a String Up to 20 6 + 4 = 10 Optional Recording Sheet

57. Time to Reflect Which representational tools have we used so far? How have we used them? Fill the Reflection Sheet

58. Second Grade Lesson Modeling Addition and Subtraction

59. Progression of Knowledge and Skill Statements (3.3) The student is expected to model addition and subtraction using pictures, words, and numbers. (2.3B) The student is expected to model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers. (1.3A) The student is expected to model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences. (K.4) The student is expected to model and create addition and subtraction problems in real situations with concrete objects. PK Guidelines Number and Operations (3.3) The student adds and subtracts to solve meaningful problems involving whole numbers. (2.3) The student adds and subtracts whole numbers to solve problems. (1.3) The student recognizes and solves problems in addition and subtraction. (K.4) The student models addition (joining) and subtraction) separating. PK Guidelines Number and Operations

60. Show transparencies of questions for students to solve . Evaluate Partners will compare their two pinecones and formulate questions and answers. Elaborate 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. Explain 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. Explore Estimate the scales on a pinecone. Engage Model Addition & Subtraction for Grades 2-3

61. 9 8 7 6 5 4 3 2 1 Units Tens 10 9 8 7 6 5 4 3 2 1 100 90 80 70 60 50 40 30 20 10

62. Linking Cube Organizer 9 8 7 6 5 4 3 2 1 Ones Tens 10 9 8 7 6 5 4 3 2 1 100 90 80 70 60 50 40 30 20 10 Elaborate + Ones Tens Hundred

63. 15 + 23 9 8 7 6 5 4 3 2 1 Ones Tens 10 9 8 7 6 5 4 3 2 1 100 90 80 70 60 50 40 30 20 10 Elaborate + 5 1 Ones Tens Hundred

64. 15 + 23 9 8 7 6 5 4 3 2 1 Ones Tens 10 9 8 7 6 5 4 3 2 1 100 90 80 70 60 50 40 30 20 10 Elaborate 3 8 3 2 + 5 1 Ones Tens Hundred

65. 15 + 23 Describe the Results in Words and Numbers 9 8 7 6 5 4 3 2 1 Ones Tens 10 9 8 7 6 5 4 3 2 1 100 90 80 70 60 50 40 30 20 10 Elaborate = 38 3 tens and 8 ones 30 + 8 38

66. 24 + 36 9 8 7 6 5 4 3 2 1 Ones Tens 10 9 8 7 6 5 4 3 2 1 100 90 80 70 60 50 40 30 20 10 Elaborate + 4 2 Ones Tens Hundred

67. 24 + 36 9 8 7 6 5 4 3 2 1 Ones Tens 10 9 8 7 6 5 4 3 2 1 100 90 80 70 60 50 40 30 20 10 Elaborate 0 1 6 6 3 + 4 2 Ones Tens Hundred

68. 24 + 36 Describe Results in Words and Numbers 9 8 7 6 5 4 3 2 1 Ones Tens 10 9 8 7 6 5 4 3 2 1 100 90 80 70 60 50 40 30 20 10 Elaborate 6 tens and 0 ones 60 + 0 60

69. 24 + 38 9 8 7 6 5 4 3 2 1 Ones Tens 10 9 8 7 6 5 4 3 2 1 100 90 80 70 60 50 40 30 20 10 Elaborate 2 1 6 8 3 + 4 2 Ones Tens Hundred

70. 24 + 38 Describe Results in Words and Numbers 9 8 7 6 5 4 3 2 1 Ones Tens 10 9 8 7 6 5 4 3 2 1 100 90 80 70 60 50 40 30 20 10 Elaborate 6 tens and 2 ones 60 + 2 62

71. Base Ten Blocks Proportional Relationship between the pieces Unit Rod Flat Cube Represent whole numbers

72. Base Ten Blocks Organizer 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 67 + 45 Explain

73. Base Ten Blocks Organizer 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 67 + 45 Explain

74. Base Ten Blocks Organizer 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 67 + 45 Explain

75. Describe the Results in Words and 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 1 hundred 1 ten 2 ones 100 + 10 + 2 67 + 45 = 112

76. 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)

77. Video Reflection How does the teacher in the video demonstrate how to find the sum of four fives? [Allen Video]

78. 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? Reflection on the Second Grade Lesson How did we scaffold the lessons in the second and third grade lesson by bridging from the concrete to the pictorial to the abstract?

79. First and Second Grade Lesson Basic Addition and Subtraction Facts

80. TEKS Focus Use organizers and games to select and solve addition and subtraction situations. Use organizers to develop, bridge and build basic addition and subtraction facts. (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 (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) (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) Selecting Addition or Subtraction to Solve Problems Basic Addition and Subtraction Facts Basic Addition and Subtraction Facts Second Grade Second Grade First Grade

81. Play 3 in a row! with cards which have the quantity of the numerals in pictorial form or numerical form. Evaluate Have students solve + =13 using a variety of manipulatives and organizers. Elaborate Students will identify all possible strategies to solve five facts. Explain 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. Explore Solve a problem 9 + 8. Engage Basic Facts for Addition & Subtraction for Grades 1-2

82. Addition and Subtraction Strategies Counting On Counting Back Doubles Near Doubles Make Ten Splitting Related Facts Compensation

83. Doubles and Near Doubles

84. Doubles 5 + 5 = 10 6 + 6 = 7 + 7 = 12 14

85. 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.

86. 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

87. 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.

88. Video Reflection How does the teacher scaffold the students’ learning of the concept for adding various numbers? [Mrs. MacDonald Video]

89. 6 + 8 = How can the students use the Make Ten Strategy to solve this problem?

90. 6 + 8 =

91. 6 + 8 =

92. 6 + 8 =

93. Video Reflection How does Harrison use his mental figuration of make ten to describe how to solve 8 + 5? [Harrison Video]

94. 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

95. Splitting Strategy from Young Mathematicians at Work 28 + 44 = 72 10 + + 2 + 2 70 + 8 20 + 40 4 60 12

96. Splitting Strategy from Young Mathematicians at Work 28 + 44 20 + 8 + 40 + 4 60 + 12 60 + 10 + 2 70 + 2 = 72

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

98. Second and Third Grade Lesson Selecting Addition or Subtraction to Solve Problems

99. Give the students several manipulatives and a variety of organizers to use when solving problems. Evaluate Play Race to Win! solving the problem. Elaborate 2 Play Race to Win! giving only the number sentence. Elaborate 1 Have the students solve a problem using a problem solving organizer. Explain Give a problem and demonstrate the on the problem solving mat. Explore/ Explain Play 99 with organizers. How can you play 99 with the least amount of rolls? Engage Select Addition & Subtraction for Problem Solving for Grades 2-3

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